Computational Problems in Metric Fixed Point Theory and their Weihrauch Degrees
COMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORYAND THEIR WEIHRAUCH DEGREES ∗ EIKE NEUMANNTechnische Universit¨at Darmstadt, Germany e-mail address : [email protected]
Abstract.
We study the computational difficulty of the problem of finding fixed pointsof nonexpansive mappings in uniformly convex Banach spaces. We show that the fixedpoint sets of computable nonexpansive self-maps of a nonempty, computably weakly closed,convex and bounded subset of a computable real Hilbert space are precisely the nonempty,co-r.e. weakly closed, convex subsets of the domain. A uniform version of this result allowsus to determine the Weihrauch degree of the Browder-G¨ohde-Kirk theorem in computablereal Hilbert space: it is equivalent to a closed choice principle, which receives as input aclosed, convex and bounded set via negative information in the weak topology and outputsa point in the set, represented in the strong topology. While in finite dimensional uniformlyconvex Banach spaces, computable nonexpansive mappings always have computable fixedpoints, on the unit ball in infinite-dimensional separable Hilbert space the Browder-G¨ohde-Kirk theorem becomes Weihrauch-equivalent to the limit operator, and on the Hilbertcube it is equivalent to Weak K˝onig’s Lemma. In particular, computable nonexpansivemappings may not have any computable fixed points in infinite dimension. We also studythe computational difficulty of the problem of finding rates of convergence for a large classof fixed point iterations, which generalise both Halpern- and Mann-iterations, and provethat the problem of finding rates of convergence already on the unit interval is equivalentto the limit operator. Introduction
Metric fixed point theory is the study of fixed point properties of mappings that arise fromthe geometric structure of the underlying space or the geometric properties of the mappingsthemselves. An important classical framework for metric fixed point theory is the study ofnonexpansive mappings in uniformly convex Banach spaces. A Banach space E is called [ Mathematics of computing ]: Mathematical analysis; Continuous mathematics;[
Theory of computation ]: Logic—Constructive mathematics.
Key words and phrases: computable analysis, functional analysis, nonexpansive mappings, fixed pointtheory, Weihrauch degrees. ∗ This paper is essentially a condensed version of the author’s master’s thesis [61], written under thesupervision of Ulrich Kohlenbach at Technische Universit¨at Darmstadt.The author was partly supported by the German Research Foundation (DFG) with Project Zi 1009/4-1and by the Royal Society International Exchange Grant IE111233.
LOGICAL METHODS l IN COMPUTER SCIENCE DOI:10.2168/LMCS-11(4:20)2015 c (cid:13)
E. Neumann CC (cid:13) Creative Commons
E. NEUMANN strictly convex , if for all x, y ∈ E with x = y and || x || = || y || = 1, we have || x + y || <
1. It iscalled uniformly convex , if ∀ ε ∈ (0 , . ∃ δ ∈ (0 , . ∀ x, y ∈ B E . (cid:18) || x − y || ≥ ε → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ − δ (cid:19) . Here, B E denotes the closed unit ball of E . Clearly, every uniformly convex Banach space isstrictly convex. A function η E : (0 , → (0 ,
1] witnessing the existential quantifier is calleda modulus of convexity for E . By the parallelogram law, every Hilbert space H is uniformlyconvex with computable modulus of convexity η H ( ε ) = 1 − q − ε . More generally, all L p -spaces with 1 < p < ∞ are uniformly convex with a computable modulus of uniformconvexity. A mapping f : ⊆ E → E is called nonexpansive if it is Lipschitz-continuous withLipschitz-constant one, i.e. if || f ( x ) − f ( y ) || ≤ || x − y || for all x, y ∈ dom f. We have the following existence result:
Theorem 1.1 (Browder-G¨ohde-Kirk) . Let E be a uniformly convex Banach space, let K ⊆ E be nonempty, bounded, closed, and convex, and let f : K → K be nonexpansive.Then f has a fixed point. Theorem 1.1 was proved independently by Browder [24], G¨ohde [34], and Kirk [40] in1965 (Kirk’s version is even more general than the version stated here). Throughout thispaper we denote the fixed point set of a mapping f by Fix( f ). A considerable amountof attention is dedicated to the study of so-called fixed point iterations, which start withan initial guess x for a fixed point of f and successively improve the guess by applying acomputable operation, which yields a sequence ( x n ) n of points in K that is then shown toconverge (weakly or strongly) to a fixed point. Many of these results are modifications ofeither of two classical theorems. Theorem 1.2 (Wittmann, [81]) . Let H be a Hilbert space, let K ⊆ H be nonempty, bounded,closed, and convex, and let f : K → K be nonexpansive. Choose a starting point x ∈ K and an “anchor point” y ∈ K and consider the sequence ( x n ) n , where x = x and x n +1 = n +2 y + (1 − n +2 ) f ( x n ) . Then the sequence ( x n ) n converges to the uniquely defined fixedpoint of f which is closest to the anchor point y . Theorem 1.3 (Krasnoselski, [51]) . Let E be a uniformly convex Banach space, let K ⊆ E be nonempty, bounded, closed, and convex, and let f : K → K be nonexpansive and f ( K ) becompact. Then for any x ∈ K the sequence ( x n ) n , where x = x and x n +1 = ( f ( x n ) + x n ) / ,converges to a fixed point of f . The iteration employed in Theorem 1.2 is a special case of a general iteration scheme,typically referred to as
Halpern iteration , as it was first introduced by Halpern [35]. Ithas the general form x n +1 = (1 − α n ) y + α n f ( x n ), where α n ∈ (0 , α n ) n . The iteration usedin Theorem 1.3 can be similarly generalised to the scheme x n +1 = (1 − α n ) x n + α n f ( x n ),and again there are certain conditions that guarantee convergence. This iteration schemeis typically called Krasnoselski-Mann iteration or simply
Mann iteration . In Hilbert space,Krasnoselski’s iteration converges weakly to a fixed point, even in the absence of compact-ness (cf. [62]). While these iterations do allow us to compute a sequence of approximations
OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 3 which is guaranteed to eventually converge to a fixed point, it is well known that the re-quirement of mere convergence is too weak to constitute a satisfactory notion of effectiveapproximation, as there exist for instance computable sequences of rational numbers whoselimit encodes the special halting problem (cf. [72]). It is hence important to understand thequantitative convergence behaviour of the approximation sequence. Quantitative aspectsof metric fixed point theory have been very successfully studied within the programme of proof mining (the standard reference is [46], see also e.g. [43, 45, 44, 57, 47, 50, 69]), whichis concerned with the extraction of hidden effective data from non-effective proofs. Mostof the applications of proof mining in fixed point theory focus on the extraction of eitherof two types of effective data. Firstly, one considers rates of asymptotic regularity of theiteration, which in this context mean rates of convergence of the sequence ( || f ( x n ) − x n || ) n towards zero. These allow us to compute arbitrarily good ε -fixed points, i.e. points x ε satisfying || f ( x ε ) − x ε || < ε , up to arbitrary precision with an a-priori running time esti-mate. Secondly, one considers so-called rates of metastability (see also [53, 54, 75, 76]),which constitute a more refined quantitative measure of approximation quality. A functionΦ : N N × N → N is called a rate of metastability for the sequence ( x n ) n if it satisfies ∀ n ∈ N . ∀ g : N → N . ∃ k ≤ Φ( g, n ) . ∀ i, j ∈ [ k ; k + g ( k )] (cid:0) || x i − x j || < − n (cid:1) . (1.1)Note that (1.1) is classically (but not constructively) equivalent to the statement that ( x n ) n is a Cauchy sequence, so that metastability can be viewed as a finitary version of convergence.Also note that in the case of Krasnoselski’s iteration, asymptotic regularity is the specialcase of metastability where g ( k ) = 1 for all k ∈ N . Of course, both types of informationare strictly weaker than actual rates of convergence. In fact, effective uniform rates ofconvergence cannot exist, as the existence result fails to be computably realisable alreadyin the case where K = [0 , Theorem 1.4 ([43]) . The multi-valued operator which receives as input a nonexpansive selfmap f of the compact unit interval [0 , and returns some fixed point of f is not computable. While Theorem 1.4 already shows that there exists no algorithm for computing a rateof convergence for Krasnoselski’s or Halpern’s iteration uniformly in the input function andthe starting point, it leaves several questions open: whether every computable nonexpan-sive mapping has a computable fixed point, whether there exist non-uniformly computablerates of convergence for the Mann- or Halpern-iteration for every computable nonexpansivemapping, at least for certain suitable starting points, whether fixed points are uniformlycomputable relative to discrete advice, what the exact relation between the computationalcontent of the three theorems is, and how their computational content relates to the com-putational content of other mathematical theorems, such as Brouwer’s fixed point theorem.In this paper, we study the computational content of Theorems 1.1, 1.3, and 1.2, aswell as related computational problems in terms of
Weihrauch degrees , which have beenproposed by Gherardi and Brattka [17] as a framework for classifying mathematical theoremsaccording to their computational content. Many classical mathematical theorems have beenclassified over the recent years. Recently, Brattka, Le Roux and Pauly [19] have shown thatBrouwer’s fixed point theorem in dimension n is equivalent to the closed choice principleon the closed unit ball in R n restricted to connected sets, and that it is equivalent to WeakK˝onig’s Lemma from dimension three upwards. Their work is based on a characterisationof the fixed point sets of computable self-maps of the unit ball in R n , due to Miller [59]. We E. NEUMANN provide a similar characterisation for the fixed point sets of computable nonexpansive self-maps of nonempty, convex, closed, and bounded subsets of computable Hilbert space, whichwe can use to determine the Weihrauch degree of the Browder-G¨ohde-Kirk theorem. Thiswill in particular allow us to compare the computational content of the Browder-G¨ohde-Kirk theorem and the problem of finding rates of convergence for fixed point iterations toBrouwer’s classic result. 2.
Preliminaries
Here we review some basic notions from computable and functional analysis and the theoryof Weihrauch reducibility. Most of the results in this section are more or less folklore,and none of them are original, except maybe Proposition 2.15. Standard references incomputable analysis are [66] and [79]. A more general treatment of the theory of computablemetric spaces can be found in [15] and [20]. The results in functional analysis reviewed herecan for instance be found in [58] or [80]. We will closely follow the approach to computableanalysis taken by Matthias Schr¨oder [70], and more recently by Arno Pauly [64], particularlyconcerning the canonical constructions of hyperspaces. Also, we adopt most of the notationand terminology from [64], which differs from standard terminology at certain points (seeCaveat 2.2).A numbering of a nonempty countable set S is a surjective partial mapping ν : ⊆ N → S .A representation of a nonempty set X is a surjective partial mapping δ : ⊆ N N → X . If δ is a representation of X , we call the tuple ( X, δ ) a represented space . If the underlying rep-resentation is clear from context, we will often simply write X for ( X, δ ) and by conventiondenote the underlying representation δ of X by δ X . If δ and ε are representations of thesame set X , we denote continuous reduction by δ ≤ t ε and computable reduction by δ ≤ ε .A representation δ : ⊆ N N → X is admissible if it is continuous and maximal with respectto continuous reduction. A represented topological space is a tuple ( X, δ ), where X is atopological space and δ is an admissible representation for X . If the representation is clearfrom the context, we will simply write X for ( X, δ ) and by convention denote the underlyingadmissible representation δ by δ X . We say that a partial mapping F : ⊆ N N → N N is a realiser for a partial multi-valued mapping (or “multimapping”) f : ⊆ X ⇒ Y betweenrepresented spaces and write F ⊢ f if δ Y ( F ( p )) ∈ f ( x ), whenever δ X ( p ) = x . We call f computable if it has a computable realiser, and realiser-continuous if it has a continuousrealiser. If we want to emphasise the underlying representations, we will write that f is( δ X , δ Y )-computable or ( δ X , δ Y )-continuous respectively. We denote by ρ the standard rep-resentation of real numbers. If δ : ⊆ N N → X and ε : ⊆ N N → Y are representations(or numberings), we denote by [ δ → ε ] the canonical representation of the space [ X → Y ]of functions with continuous realiser. If X and Y are represented topological spaces, then[ X → Y ] coincides with the space C seq ( X, Y ) of sequentially continuous functions from X to Y . If X is first-countable, [ X → Y ] furthermore coincides with the space C ( X, Y ) ofcontinuous functions from X to Y . Moreover, we denote by δ × ε the canonical representa-tion of the product space X × Y , by δ ω the canonical representation of the space X N andby δ ∗ the canonical representation of X ∗ = S n ∈ N X n . If δ and ε are representations of the Like Schr¨oder, and as opposed to Pauly, we will mostly work with a specific topology for X in mind.To emphasize this, we call the spaces of interest represented topological spaces, rather than “representedspaces”. In general, the topology on X will not be the final topology of its representation, and topologicalcontinuity may differ from realiser-continuity if the topology of X is not sequential. OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 5 same space X , we let δ ⊓ ε = π ( δ × ε ) | ∆( X ) , where ∆( X ) = { ( x, y ) ∈ X | x = y } and π is the projection onto the first coordinate. If K ⊂ X is a subset of X , we sometimes write δ K for δ X | K .As already mentioned, the following constructions are essentially due to [70] and [64].For any represented space X , the canonical function-space construction gives rise to canon-ical representations of the hyperspaces of “open” and “closed” subsets of X , by postulatingthat openness corresponds to semi-decidability. Let S = { , } denote Sierpi´nski space withtopology {∅ , { , } , { }} and representation σ ( p ) = 0 : ⇔ p = 0 . The characteristic function χ U : X → { , } of a set U is defined as χ U ( x ) = 1 : ⇔ x ∈ U . Definition 2.1.
Let X be a represented space. We call a set U ⊆ X open , if its characteristicfunction χ U : X → S is realiser-continuous, i.e. χ U ∈ [ X → S ]. A θ X -name of an open set U ⊆ X is a [ δ X → σ ]-name of its characteristic function χ U . The set of all open subsetsof X with representation θ X defines the represented space O ( X ). Dually, we define therepresented space A ( X ) of closed subsets of X by identifying a closed set A ⊆ X with itscomplement in O ( X ) and call the underlying representation ψ X .We will often just write ψ for ψ X if the underlying space is clear from context. Wecall the computable points of O ( X ) semi-decidable and the computable points in A ( X ) co-semi-decidable . Note that, just like the notion of realiser continuity may differ fromtopological continuity, the notions of closedness and openness for subsets of representedspaces are a-priori different from the notions of topological openness and closedness. If X isan admissibly represented topological space, then the set O ( X ) coincides with the set of allsequentially open subsets of X and A ( X ) coincides with the set of all sequentially closedsubsets of X . If in addition X is second-countable then O ( X ) is the set of open subsets of X and A ( X ) is the set of all closed subsets of X . Caveat . Note that the terminology introduced here, which is mainly due to [64], isdifferent from the usual terminology used in computable analysis, which is for instanceused in Weihrauch’s book [79]. In [79], the space A ( X ) is denoted by A > ( X ) and itscomputable elements are called co-r.e. closed, rather than co-semi-decidable. Although wehave introduced our A ( X ) as “the space of closed subsets” of X , we deliberately refrainfrom referring to its computable points as “computably closed”, so as to avoid confusionwith topological closedness on one hand, and with Weihrauch’s terminology on the other.The symbol A ( X ) is used in [79] to denote the space of closed and overt subsets of X , tobe introduced below. Also note that in the abstract we used Weihrauch’s terminology.The space A ( X ) can be thought of as “the space of closed sets encoded via negativeinformation”. The following definition provides in a certain sense a notion of “closed setsencoded via positive information”. Definition 2.3.
Let X be a represented space. We define the represented space V ( X ) ofovert closed subsets of X to be the represented space of closed subsets of X , where a closedset A ⊆ X is represented by a [ θ X → S ]-name of the functionintersects? A : O ( X ) → S , U ( U ∩ A = ∅ ,0 otherwise. E. NEUMANN
We denote the standard representation of V ( X ) by υ X or simply υ and call the com-putable points of V ( X ) computably overt . Computably overt closed sets are those, for whichintersection with an open set can be effectively verified. The space V ( X ) hence correspondsto the space A < ( X ) in [79], and hence is sometimes called the “space of closed sets, rep-resented with positive information”. In [64] it is argued that from an intrinsic perspective,the word “closed” is rather misleading, because the closure properties of the space V ( X )differ significantly from the closure properties of closed sets (e.g. union is computable butintersection is not, the image of a computably overt set under a computable function iscomputably overt, but the preimage is not), and we agree with this position. Overtness isrelated to effective separability, which yields a convenient criterion for computable overtness(see e.g. [20, Theorem 3.8 (1)]). Proposition 2.4.
Let X be a separable represented topological space. Define a representa-tion δ enum of the set of nonempty closed subsets of X as follows: δ enum ( p ) = A : ⇔ δ ωX ( p ) is dense in A. Then δ enum ≤ υ (cid:12)(cid:12) V ( X ) \{∅} .Proof. Suppose we are given a dense sequence ( x n ) n in a closed set A , and an open set U ∈ O ( X ). In order to verify if A ∩ U , check if there exists n ∈ N such that x n ∈ U . Thisproves the claim.Our representation δ enum is called δ range in [15] and [20]. Next we define the canonicalrepresentation of the hyperspace of compact subsets of a Hausdorff represented topologicalspace X . In a countably based T space, a compact set can be represented as a list of allits finite open covers by basic neighbourhoods. It is easy to see that this representation ischaracterised by the property that containment in an open set is semi-decidable. This canbe used to generalise the definition to arbitrary represented topological spaces, and in factto arbitrary represented spaces. For the sake of simplicity we restrict ourselves to the caseof Hausdorff represented topological spaces. Definition 2.5.
Let X be a Hausdorff represented topological space. The represented space K ( X ) of compact subsets of X is the set of all compact subsets of X , where a compact set K ∈ K ( X ) is represented as a [ θ X → σ ]-name of the function contained? K : O ( X ) → S ,contained? K ( U ) = 1 ⇔ K ⊆ U. We denote the canonical representation of K ( X ) by κ and call the computable pointsof K ( X ) computably compact . Note that, like A ( X ), our space K ( X ) only encodes “neg-ative” information on compact sets. Weihrauch [79] hence uses the notation “ κ > ” for our κ .Similarly as in the case of A ( X ), computable points in our K ( X ) are called “co-r.e. com-pact” by some authors. Definition 2.5 can be generalised to arbitrary represented spaces,essentially by using the same approach as in Definition 2.1, and calling a subset K of arepresented space X compact if the function contained? K is an element of [ O ( X ) → S ]. Ingeneral this will only yield a representation of the space of saturated compact sets (cf. [64]),or a multi-valued representation of the space of compact sets (cf. [70]). If X is a T rep-resented topological space, then the thus obtained space K ( X ) coincides with the set ofall compact subsets of the sequentialisation seq( X ) of X , whose open sets are the sequen-tially open sets of X . For details see [70]. By Proposition 3.3.2 (3) in [70], the notions ofcompactness, sequential compactness, and compactness in the sequentialisation coincide forHausdorff represented topological spaces, so we obtain Definition 2.5. OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 7
It will sometimes be convenient to work with an intrinsic notion of computable com-pactness for represented spaces, which we introduce next.
Definition 2.6.
A Hausdorff represented topological space X is called computably compact ,if the mapping empty? : A ( X ) → S , A ( A = ∅ , X is a computably compact space if and only if X is a computable point in K ( X ). Thenext proposition is a converse to this in some sense. Proposition 2.7.
Let X be a Hausdorff represented topological space and K ⊆ X be anonempty co-semi-decidable subset, such that the represented space ( K, δ X (cid:12)(cid:12) K ) is computablycompact. Then K is a computably compact subset of X , i.e. a computable point in K ( X ) .Proof. It follows immediately from the definition of O ( X ) that the mapping ∩ K : O ( X ) → O ( K ) , U U ∩ K is computable. Now, U ⊇ K if and only if K \ ( U ∩ K ) = ∅ . It again follows from thedefinition, that the mapping O ( K ) → A ( K ) , U K \ U is computable. Since K is a computably compact represented space, the mapping A ( K ) → S , A ( A = ∅ , A = ∅ is computable. It follows that the set of open subsets of X containing K is computablyopen, i.e. K is computably compact.A closed subset of a compact space is compact, and in a Hausdorff space, every compactset is closed. We have an effective counterpart of this in the theory of represented spaces.A represented space X is called effectively Hausdorff if the mapping X → A ( X ) , x
7→ { x } is computable. Proposition 2.8.
Let X be a Hausdorff computably compact represented topological space. (i) The mapping id : A ( X ) → K ( X ) is well-defined and computable. (ii) If X is effectively Hausdorff, then the mapping id : K ( X ) → A ( X ) is well-definedand computable.Proof. (i) We are given a closed set A ∈ A ( X ) which we want to compute as a compact set A ∈ K ( X ). Given an open set U ∈ O ( X ) we want to verify if U ⊇ A . In order to doso, check if U ∪ A C covers X , using that X is computably compact.(ii) We are given a compact set K ∈ K ( X ) which we want to compute as a closed set K ∈ A ( X ). Given a point x ∈ X we want to verify if x / ∈ K . In order to do so,compute { x } ∈ A ( X ), using that X is effectively Hausdorff, and verify if { x } C ⊇ K ,using the compactness information on K . E. NEUMANN
It is easy to see that the computability and well-definedness of the mappingid : A ( X ) → K ( X ) characterises computably compact represented spaces (cf. also [64]). Theorem 2.9.
Let K be a computably compact represented topological space, containing acomputable dense sequence. Then the mapping max : [ K → R ] → R , f max { f ( x ) (cid:12)(cid:12) x ∈ K } is well-defined and computable.Proof. Since K is adequately represented, we have [ K → R ] = C seq ( K, R ), and K is sequen-tially compact thanks to Proposition 3.3.2 (3) in [70]. It follows that for any f ∈ [ K → R ],the set f ( K ) is sequentially compact in R and thus compact. This shows that max iswell-defined. It remains to show that max( f ) is computable relative to f . Let ( x n ) n bea computable dense sequence in K . Then the sequence ( f ( x n )) n is computable relativeto f , with sup n ∈ N f ( x n ) = max( f ). On the other hand, for every computable b ∈ R , theset U b = { x ∈ K (cid:12)(cid:12) f ( x ) < b } is semi-decidable relative to f , so that by the computablecompactness of K , the predicate ∀ x ∈ K. ( f ( x ) < b ) is semi-decidable relative to f for all b ∈ Q . We can use this to construct a sequence ( b n ) n of real numbers which is computablerelative to f and satisfies max( f ) = inf n ∈ N b n . Since max( f ) can hence be approximatedarbitrarily well “from above” as well as “from below”, it is computable relative to f .It follows from Theorem 2.9 that every finite dimensional uniformly convex computableBanach space E has a computable modulus of uniform convexity η E , since we may put η E ( ε ) = inf n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) x, y ∈ B E , || x − y || ≥ ε o , and the set { ( x, y ) ∈ B E × B E (cid:12)(cid:12) || x − y || ≥ ε } is computably compact relative to ε andcontains a computable, dense sequence relative to ε . Since the proof of Theorem 2.9 isuniform in K , the claim follows. Theorem 2.10 (Kreinovich’s theorem, [52]) . Let K be a computably compact representedtopological space. Then the mapping UC K : ⊆ A ( K ) → K, { x } 7→ x is computable. Definition 2.11. A computable metric space is a triple ( M, d, ν M ), where ( M, d ) is a metricspace and ν M : N → A is a numbering of a dense subset A ⊆ M , such that d : A × A → R is ( ν M × ν M , ρ )-computable. With a computable metric space we associate the representedspace ( M, δ M ), where δ M ( p ) = x : ⇔ d ( ν M ( p ( n )) , x ) ≤ − n for all n ∈ N . One can show that the above defined canonical representation of a computable metricspace M is admissible and that d : M × M → R is computable. This canonical representation δ M is also called the Cauchy representation induced by ν M . We will refer to the points inim ν M as the rational points of the represented space M . Note that any computable metricspace is separable, and hence Hausdorff, by definition. In fact, every computable metricspace is effectively Hausdorff, since the predicate d ( x, y ) > x and y . In any metric space M we denote by B ( x, r ) = { y ∈ M (cid:12)(cid:12) d ( x, y ) < r } the open OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 9 ball of radius r centred at x , and by B ( x, r ) = { y ∈ M (cid:12)(cid:12) d ( x, y ) ≤ r } the closed ball ofradius r centred at x .The following result is more or less folklore, and justifies the more abstract Definition2.5 of computable compactness. Proposition 2.12.
A computable metric space is computably compact if and only if itis complete and computably totally bounded , i.e. if and only if there exists a function α : N → M ∗ such that ∀ n ∈ N . ∀ x ∈ M. ∃ k ≤ lth( α ( n )) . (cid:0) d ( x, α ( n ) k ) < − n (cid:1) . We call α ( n ) a − n -net in M .Proof. Suppose that M is computably compact. Then M is compact and thus complete.Let ( a k ) k be a dense computable sequence in M . Since M is computably compact, we canverify for all n, m ∈ N if the open set B (˜ a , − n − ) ∪ · · · ∪ B (˜ a m , − n − ) is equal to all of M , where ˜ a k is a rational approximation to a k to up error 2 − n − . In that case, a , . . . , a m is a 2 − n -net in M . On the other hand, since ( a k ) k is dense in M and M is compact, thisprocess has to finish after a finite number of steps for each n ∈ N . It follows that M iscomputably totally bounded.Suppose now that M is complete and computably totally bounded. Let S = { B ( c , r ) , . . . , B ( c k , r k ) } be a collection of rational balls, i.e. balls whose radii are rational numbers and whosecentres are rational points in M . We show that we can verify if S is a cover of A . It is astandard argument that this suffices in order to establish that M is computably compact.Let ( h a n , . . . , a nl ( n ) i ) n ∈ N be a computable sequence of 2 − n -nets in M . Then S covers M if andonly if there exists n ∈ N such that for all i ∈ { , . . . , l ( n ) } there exists j ∈ { , . . . , k } suchthat d ( a ni , c j ) < r j − − n . This property is semi-decidable, so M is computably compact.In complete computable metric spaces, overtness is characterised by separability, in thesense that Proposition 2.4 admits a converse (cf. [20, Theorem 3.8 (2)]). Proposition 2.13.
Let M be a complete computable metric space. Define the represen-tation δ enum of the set of nonempty closed subsets of M as in Proposition 2.4. Then δ enum ≡ (cid:16) υ (cid:12)(cid:12) V ( M ) \{∅} (cid:17) .Proof. The direction δ enum ≤ υ was already proved in Proposition 2.4, so it remains to prove υ ≤ δ enum . Suppose we are given a closed set A ∈ V ( M ). We can compute an enumeration( B m ) m of all open rational balls (i.e. balls with rational centre and radius) with radius atmost 1 intersecting A . We use this to construct a dense sequence ( x m ) m in A . The m th element in the sequence is computed as follows: the first approximation x (0) m to x m is thecentre of B m . Let 1 ≥ ε > B m . We claim that we can find an openrational ball B (1) m with radius at most ε which is contained in B m and intersects A . Let a ∈ B m ∩ A . Then there exists rational ε > δ > d ( a, x (0) m ) < ε − δ . Let ˜ a bea rational approximation of a up to error δ/
2. Then d (˜ a, x (0) m ) < ε − δ/
2. In particular, B (˜ a, δ/ ∩ A = ∅ and B (˜ a, δ/ ⊆ B ( x (0) m , ε ). On the other hand, we can verify for a givenrational a and δ < ε that d ( a, x (0) m ) < ε − δ and that B ( a, δ ) ∩ A = ∅ . We may hence searchfor such a and δ , and put B (1) m = B ( a, δ ) and x (1) m = a . Continuing in this manner, we obtain a Cauchy sequence x ( n ) m with d ( x ( n ) m , x ( n + k ) m ) < − n for all k, n ∈ N . Since M is complete,the sequence x ( n ) m converges to some element x m ∈ A with d ( x ( n ) m , x m ) ≤ − n . Applyingthis to all ( B m ) m in parallel, we obtain a computable sequence ( x m ) m . It remains to showthat ( x m ) m is dense in A . Let a ∈ A , and let ε be a rational number satisfying 1 > ε > x satisfying d ( a, x ) < ε . In particular, B ( x, ε ) ∩ A = ∅ , so B ( x, ε ) = B k for some k ∈ N . It follows from the construction of x k that d ( x k , a ) < ε .If M is a computable metric space, we have another natural notion of computabilityfor closed sets, by identifying a closed set A ⊆ M with its distance function d A : M → R , d A ( x ) = inf { d ( x, y ) | y ∈ A } . (2.1)Define represented spaces R < = ( R , ρ < ) and R > = ( R , ρ > ) via ρ < ( p ) = x : ⇔ sup n ∈ N ν ω Q ( p )( n ) = x and ρ > ( p ) = x : ⇔ inf n ∈ N ν ω Q ( p )( n ) = x. Computable elements of R < are called left-r.e. numbers , and computable elements of R > are called right-r.e. numbers . Obviously, a number is computable if and only if it is bothright- and left-r.e., whereas a classic result due to Specker [72] asserts the existence of bothuncomputable left-r.e.- and uncomputable right-r.e. numbers. Definition 2.14.
Let M be a computable metric space.(i) The represented space A dist ( M ) is the space of nonempty closed subsets of M , wherea closed subset A ⊆ M is represented via a [ δ M → ρ ]-name of its distance function(2.1).(ii) The represented space A dist < ( M ) is the space of nonempty closed subsets of M , wherea closed subset A ⊆ M is represented via a [ δ M → ρ < ]-name of its distance function(2.1).(iii) The represented space A dist > ( M ) is the space of nonempty closed subsets of M , wherea closed subset A ⊆ M is represented via a [ δ M → ρ > ]-name of its distance function(2.1).Computable points of A dist ( M ) are called located , computable points of A dist < ( M ) are called lower semi-located , and computable points of A dist > ( M ) are called upper semi-located . UsingProposition 2.13, it is easy to see that for any complete computable metric space M , thecanonical representations of the spaces A dist > ( M ) and V ( M ) \ {∅} are equivalent (see also[20, Theorem 3.7]). It is also easy to see that id : A dist < ( M ) → A ( M ) is computable (see[20, Theorem 3.11 (1)]). In [79, Lemma 5.1.7] it is proved, that for M = R d , the canonicalrepresentations of A dist < ( M ) and A ( M ) \ {∅} are equivalent, and the argument readilygeneralises to any complete computable metric space with (effectively) compact closed balls(see [20, Theorem 3.11 (3)]). Any such space is locally compact. Local compactness is infact necessary for the reduction to hold: Proposition 2.15.
Let M be a complete computable metric space. If the identity mapping id : A ( M ) \ {∅} → A dist < ( M ) is computable, then M is locally compact.Proof. If M is a singleton, the claim is trivial, so we may assume that M consists of at leasttwo points. Given x ∈ M we show that we can compute B ( x, r ) as a compact subset of M , for r sufficiently small. We search for a rational y ∈ M and r ∈ Q + with d ( y, x ) > r .By an argument similar to Proposition 2.12, it suffices to compute for every k ∈ N acover of B ( x, r ) by balls of radius 2 − k , whose centres are rational points in M . Since y is OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 11 computable, the singleton { y } is co-semi-decidable, so we can compute an enumeration ofballs with rational centres and radii exhausting the complement of { y } such that every ballhas radius at most 2 − k . We feed this enumeration into the machine computing the identityid : A ( M ) \ {∅} → A dist < ( M ) and use the [ δ M → ρ < ]-name provided by the machine tocompute d( x, { y } ) from below. After having processed finitely many balls, the machine willoutput the lower bound r on d( x, { y } ) = d ( x, y ). This means that B ( x, r ) is covered bythese finitely many balls, since otherwise we could force the machine computing the identityto err.The proof of Proposition 2.15 shows that we can even compute a witness for the localcompactness of M , namely a function f : M → K ( M ) which maps a point x to a compactclosed ball containing x . Definition 2.16. A (real) computable normed space is a normed real vector space E to-gether with a numbering e : N → E such that span { e ( n ) | n ∈ N } is dense in E and ( E, d, ν E )is a computable metric space, where d ( x, y ) = || x − y || and ν E is a canonical notation of all(finite) Q -linear combinations of im e .A complete computable normed space is called a computable Banach space. A com-putable normed space which is also a Hilbert space is called a computable Hilbert space.The inner product in a computable Hilbert space is computable by the polarisation identity.A computable normed space becomes a represented space when endowed with the Cauchyrepresentation induced by the numbering ν E . In this representation the vector space op-erations and the norm are computable functions and 0 ∈ E is a computable point of therepresented space E . An important feature of (infinite dimensional) computable normedspaces is that without loss of generality the fundamental sequence is linearly independent(cf. [66, p. 142]). Lemma 2.17 (Effective independence lemma) . Let ( E, e ) be an infinite dimensional com-putable normed space. Then there exists a computable function f : N → N such that e ◦ f : N → E has dense span in E and consists of linearly independent vectors. Corollary 2.18. (i)
Every computable real Hilbert space H has a computable orthonormal basis, i.e. anorthonormal basis which is a (potentially finite) computable sequence in H . (ii) Every finite dimensional computable real Hilbert space is computably isometricallyisomorphic to R d for some d ∈ N . Every infinite dimensional real Hilbert space iscomputably isometrically isomorphic to ℓ . Let us now introduce some basic notions from the theory of Weihrauch degrees. We willtreat this paragraph somewhat informally, as we will not need to develop the theory very far.A formal and comprehensive treatment of everything stated here can be found in [17, 13, 16],and in [19], where the Weihrauch degree of Brouwer’s fixed point theorem is determined.
Definition 2.19.
Let h· , ·i : N N × N N → N N denote some computable pairing function onBaire Space. A multimapping g : X ⇒ Y between represented spaces X and Y is said to Weihrauch reduce to h : Z ⇒ W , in symbols g ≤ W h , if there exist computable functions K, N : ⊆ N N → N N such that K h HN, id i is a realiser of g , whenever H is a realiser of h . If f ≤ W g and g ≤ W f we say that g and f are Weihrauch equivalent and write f ≡ W g . Theequivalence classes with respect to ≡ W are called Weihrauch degrees . The Weihrauch degrees together with the ≤ W -relation are known to form a boundedlattice. A very important and useful tool for studying Weihrauch degrees are so-called closed choice principles on represented spaces. Definition 2.20.
Let X be a represented space.(i) The closed choice principle on X is the multimappingC X : A ( X ) \ {∅} ⇒ X, A A. The unique choice principle UC X on X is C X restricted to singleton sets and the connected choice principle CC X is C X restricted to connected sets.(ii) Let X additionally be a closed subset of a computable Banach space E . We definethe convex choice principle ConvC X as the restriction of C X to the space A co ( X ) ofconvex closed subsets of X .Let us now introduce some concrete Weihrauch degrees that will be useful in our furtherstudies. The limit operator lim : ⊆ N N → N N takes as input a (suitably encoded) convergentsequence ( p n ) n ∈ ( N N ) N ≃ N N and outputs its limit. Weak K˝onig’s Lemma, WKL, takes asinput an infinite binary tree and outputs an infinite path. The intermediate value theorem,IVT, takes as input a continuous function f : [0 , → R with f (0) · f (1) < x ∈ [0 ,
1] such that f ( x ) = 0. Brouwer’s fixed point theorem in n -dimensionalspace, BFT n , takes as input a continuous function f : [0 , n → [0 , n and outputs somefixed point of f . Their relation is summarised in the following Fact 2.21. (i) WKL ≡ W C { , } ω . (ii) CC [0 , n ≡ W BFT n ≤ W WKL for all n . (iii) IVT ≡ W ConvC [0 , ≡ W CC [0 , ≡ W BFT < W BFT ≤ W BFT ≡ W WKL . (iv) WKL < W lim < W C N N . An important property of computably compact spaces is that their closed choice principleis of low degree.
Theorem 2.22.
Let K be a computably compact represented topological space. Then themultimapping C K : A ( K ) \ {∅} ⇒ K, A A satisfies C K ≤ W WKL . Finally, we need a few observations from (computable) functional analysis. The firsttheorem is the so-called projection theorem , which can be found in virtually any functionalanalysis textbook (cf. e.g. [80, Satz V.3.2 & Lemma V.3.3]).
Theorem 2.23. (i)
Let E be a uniformly convex real Banach space and let K ⊆ E be nonempty, closed,and convex. For every x ∈ E there exists a unique y ∈ K such that d ( x, y ) = d ( x, K ) = inf { d ( x, z ) | z ∈ K } . We denote this element by P K ( x ) . The mapping P K is a continuous retraction onto K , called the metric projection . OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 13 (ii)
Let H be a real Hilbert space and K ⊆ H be nonempty, closed, and convex. Then P K is a nonexpansive mapping, and for all x ∈ H the element P K ( x ) is characterised bythe variational inequality( x − P K ( x ) , y − P K ( x )) ≤ for all y ∈ K. Proof. (i) We may assume that x / ∈ K and x = 0. Put r = inf {|| z || (cid:12)(cid:12) z ∈ K } > y n ) n be a sequence in K with lim n →∞ || y n || = r . We show that( y n ) n is a Cauchy sequence. Let ε >
0. There exists n ∈ N such that || y n + k || < r + ε for all k ≥
0. We have y n + y n + k ∈ K for all k ≥ || y n + y n + k || ≥ r . If δ ∈ (0 , η ( δ ) > εr + ε , then (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n + y n + k (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ r > ( r + ε ) (1 − η ( δ )) . Applying the contraposition of uniform convexity to y n + k r + ε and y n r + ε thus yields || y n − y n + k || < ( r + ε ) δ. Since εr + ε → ε →
0, and η ( δ ) > δ ∈ (0 , || y n − y n + k || → n → ∞ , i.e. the sequence ( y n ) n is a Cauchy sequence. Since K is closed and E iscomplete, it converges to some element y ∈ K , which satisfies || y || = lim n →∞ || y n || = r . Uniqueness: Suppose that the points y , y ∈ K with y = y satisfy || y i || = r > y , y ∈ B (0 , r ). Since K is convex, y + y ∈ K and since E is strictlyconvex and y = y , we have || y + y || < r . Contradiction.(ii) On one hand we have for all α ∈ [0 ,
1] and z ∈ K : || x − P K ( x ) || ≤ || x − ( αz + (1 − α ) P K ( x )) || = ( x − P K ( x ) − α ( z − P K ( x )) , x − P K ( x ) − α ( z − P K ( x )))= || x − P K ( x ) || − α ( x − P K ( x ) , z − P K ( x )) + α || z − P K ( x ) || and thus ( x − P K ( x ) , z − P K ( x )) ≤ α || z − P K ( x ) || for all α ∈ (0 , P K satisfies the variational inequality. On the other hand, if p ∈ K satisfies( x − p, z − p ) ≤ z ∈ K , then for all z ∈ K we have || x − z || = || ( x − p ) + ( p − z ) || = || x − p || + 2( x − p, p − z ) + || p − z || ≥ || x − p || and thus p = P K ( x ) by Theorem 2.23. It remains to show that P K is nonexpansive.Let x, y ∈ H . We may assume that x = y and P K ( x ) = P K ( y ). Since P K ( x ) , P K ( y ) ∈ K , we may use the variational inequality to obtain( P K ( y ) − P K ( x ) , x − P K ( x )) ≤ P K ( x ) − P K ( y ) , y − P K ( y )) ≤ . Adding both inequalities yields( P K ( y ) − P K ( x ) , x − y + P K ( y ) − P K ( x )) ≤ . And hence || P K ( y ) − P K ( x ) || ≤ ( P K ( x ) − P K ( y ) , x − y ) ≤ || P K ( x ) − P K ( y ) || · || x − y || , where the last inequality is the Cauchy-Schwarz inequality. We thus obtain || P K ( x ) − P K ( y ) || ≤ || x − y || . The next important result is that projections onto located convex sets in uniformly convexcomputable Banach spaces are computable relative to a modulus of convexity. This willfollow from a highly uniform proof mining result due to Kohlenbach:
Theorem 2.24 ([46, Proposition 17.4]) . There exists a computable functional
Φ : N N × N × N → N such that if E is a uniformly convex normed space with modulus of uniform convexity η E , µ is any functional satisfying − µ ( n ) ≤ η E (2 − n ) , K ⊆ E is nonempty, closed, and convex,and x ∈ E with d ( x, K ) ≤ d then φ ( n ) = Φ( µ, d, n ) is a modulus of uniqueness for the projection onto K . This means that if p, q ∈ K satisfy || p − x || ≤ d ( x, K ) + 2 − φ ( n ) and || q − x || ≤ d ( x, K ) + 2 − φ ( n ) , then || p − q || < − n . Corollary 2.25.
Let E be a uniformly convex computable Banach space, let C ⊆ E benonempty, convex, and computably overt. Let η E be a modulus of uniform convexity for E . Let A codist ( C ) denote the represented space of nonempty convex closed subsets of C ,represented via their distance function. Then the mapping P : A codist ( C ) → C ( C, C ) , K P K , is computable relative to η E . In fact it is computable relative to any µ : N → N satisfying − µ ( n ) ≤ η E (2 − n ) .Proof. Let µ : N → N be such that 2 − µ ( n ) ≤ η E (2 − n ) and let Φ be the functional fromTheorem 2.24. We are given a set K ∈ A codist ( C ), a point x ∈ C and a number n ∈ N and want to compute an approximation to P K ( x ) up to error 2 − n . Since we are giventhe distance function to K , we can compute an integer upper bound d to d ( x, K ). Againusing the distance function, we can compute a dense sequence in K . This allows us to finda point p ∈ K with || p − x || ≤ d ( x, K ) + 2 − Φ( µ,d,n ) . It follows from Theorem 2.24 that || p − P K ( x ) || < − n .The special case where E has a computable modulus of convexity and C = E yields: Corollary 2.26 ([14]) . Let E be a uniformly convex computable Banach space with com-putable modulus of uniform convexity. Then the mapping P : A codist ( E ) → C ( E, E ) , K P K , is computable. In particular, if K ⊆ E is a nonempty located and convex set, then P K is ( δ E , δ E ) -computable. OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 15
The (for our purpose) most important structural feature of fixed point sets of nonex-pansive mappings in strictly convex Banach spaces is that they are always convex. This isa standard exercise in functional analysis. We will prove it here anyway, to give a simpleexample of a proof exploiting the convexity of the underlying space.
Proposition 2.27.
Let E be a strictly convex Banach space, let K ⊆ E be nonempty, closed,bounded, and convex. Let f : K → K be nonexpansive. Then the set Fix( f ) is convex.Proof. Since f is continuous, it suffices to show that for each x, y ∈ Fix( f ), the convexcombination x + y is again contained in Fix( f ) (since then it follows that the set of dyadicconvex combinations of x and y , which is dense in the line segment joining x and y , consistsentirely of fixed points). Since f is nonexpansive, we have (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) x + y (cid:19) − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) f (cid:18) x + y (cid:19) − f ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Similarly, || f ( x + y ) − y || ≤ || x − y || and obviously the same inequality holds if we replace f ( x + y ) by x + y . Now, suppose that a, b ∈ E satisfy || a − y || ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , || a − x || ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , || b − x || ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , || b − y || ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Then, if a = b , strict convexity yields (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a + b − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) a + b − x (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) and hence || x − y || ≤ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y − a + b (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < || x − y || , contradiction. It follows, that f ( x + y ) = x + y .3. Computability of Fixed points and Rates of Convergence
In this section we study the computability-theoretic complexity of the problems of find-ing fixed points of nonexpansive mappings on compact domains, and of obtaining ratesof convergence of certain fixed point iterations. Let us first state some natural computa-tional problems associated with the fixed point properties of nonexpansive mappings anddetermine their rough relation.
Definition 3.1.
Let E be a uniformly convex real Banach space, let K ⊆ E be a nonemptysubset of K , and let N ( K ) denote the set of nonexpansive self-maps of K . A fixed pointiteration on K is a mapping I : N ( K ) × K → K N such that for all f ∈ N ( K ), x ∈ K , wehave lim n →∞ I ( f, x )( n ) ∈ Fix( f ). Definition 3.2.
Let E be a uniformly convex computable Banach space, let K ⊆ E benonempty, co-semi-decidable, computably overt, bounded, and convex. Let N ( K ) be therepresented space of nonexpansive self-maps of K with representation [ δ K → δ K ] (cid:12)(cid:12)(cid:12) N ( K ) .Let I : N ( K ) × K → K N be a computable fixed point iteration. Consider the followingcomputational problems: (i) The realiser problem for the Browder-G¨ohde-Kirk theorem BGK K : Given a nonex-pansive function f : K → K , output a fixed point for f . More formally:BGK K : N ( K ) ⇒ K, f Fix( f ) . (ii) The projection problem Proj K : Given a nonexpansive function f : K → K , and apoint x ∈ K output the metric projection of x onto Fix( f ). More formally:Proj K : N ( K ) × K → K, ( f, x ) P Fix( f ) ( x ) . (iii) The limit problem lim( I ) for I : given a nonexpansive function f : K → K and astarting point x ∈ K , output lim n ∈ N I ( f, x )( n ). More formally:lim( I ) : N ( K ) × K → K, ( f, x ) lim n →∞ I ( f, x )( n ) . (iv) The rate of convergence problem Conv I for I : given a nonexpansive function f : K → K and a starting point x ∈ K , output a rate of convergence of the sequence ( I ( f, x )( n )) n .More formally:Conv I : N ( K ) × K ⇒ N N , ( f, x )
7→ { ϕ ∈ N N | ∀ n ∈ N . ∀ l ≥ ϕ ( n ) . || I ( f, x )( l ) − lim k →∞ I ( f, x )( k ) || < − n } . Most fixed point iterations considered in the literature are of a far more particular form thanjust computable mappings. This can be exploited to obtain stronger uncomputability resultsfor particular classes of fixed point iterations. We summarise some common properties.
Definition 3.3.
Let E be a uniformly convex real Banach space, and let K ⊆ E benonempty, convex, closed, and bounded. Let I : N ( K ) × K → K N be a fixed point iteration.(i) I is called projective if for all f ∈ N ( K ) and x ∈ K , the limit lim n →∞ I ( f, x )( n ) isthe unique fixed point of f which is closest to x .(ii) I is called retractive if for all f ∈ N ( K ) and x ∈ Fix( f ), we have lim n →∞ I ( f, x )( n ) = x .(iii) I is called avoidant if for all f ∈ N ( K ) and x ∈ K , we have the implication( ∃ n.f ( I ( f, x )( n )) = I ( f, x )( n )) ⇒ f ( x ) = x. (iv) I is called simple if it is of the form I ( f, x )(0) = xI ( f, x )( n + 1) = n X k =0 α nk I ( f, x )( k ) + n X j,k =0 β nj,k f ( j ) ( I ( f, x )( k )) , with α nk ≥ β nk,j ≥ k, n, j .The notion of projectiveness is well-defined thanks to Theorem 2.23 and Proposition 2.27.Any projective fixed point iteration is clearly retractive. Note that Halpern’s iteration(where by convention we always choose the anchor point to be equal to the starting point)is projective and simple and that the Krasnoselski-Mann iteration is simple, retractive, andavoidant. Proposition 3.4. (i)
Let I : N ([0 , × [0 , → [0 , N be a simple and retractive fixed point iteration. Then I is projective when restricted to the set of all monotonically increasing functions. OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 17 (ii)
Let E be a uniformly convex real Banach space, and let K ⊆ E be nonempty, convex,closed, and bounded. Let I : N ( K ) × K → K N be an avoidant fixed point iteration.Then for all nonexpansive f : K → K and x / ∈ Fix( f ) , lim n →∞ I ( f, x )( n ) is a pointon the boundary of Fix( f ) .Proof. (i) By induction one easily verifies that if x ≤ y , then I ( f, x )( n ) ≤ I ( f, y )( n ) for allmonotonically increasing f : [0 , → [0 , n →∞ I ( f, x )( n ) ≤ lim n →∞ I ( f, y )( n ) . By Proposition 2.27, the set Fix( f ) is an interval of the form [ a, b ], possibly with a = b .If x ≤ a , then, since the iteration is retractive,lim n →∞ I ( f, x )( n ) ≤ lim n →∞ I ( f, a )( n ) = a, so lim n →∞ I ( f, x )( n ) = a . An analogous argument applies if x ≥ b . It follows that themapping λx. lim n →∞ I ( f, x )( n ) is the metric projection onto [ a, b ], i.e. the iterationis projective.(ii) Is trivial.The following proposition establishes the more obvious relationships between the problemsintroduced in Definition 3.2. Proposition 3.5.
Let E be a uniformly convex computable Banach space, let K ⊆ E benonempty, co-semi-decidable, computably overt, bounded, and convex. Then BGK K ≤ W Proj K . If I : N ( K ) × K → K N is a computable fixed point iteration, then BGK K ≤ W lim( I ) ≤ W Conv I ≤ W lim . If I is projective, then Proj K ≤ W lim( I ) . Next we prove a general upper bound on the Weihrauch degree of Proj K (and thus ofBGK K ). We need two lemmas which constitute the main steps in Goebel’s proof [32] of theBrowder-G¨ohde-Kirk theorem (see also the proof of [80, Theorem IV.7.13]). Lemma 3.6.
Let E be a uniformly convex Banach space, let K ⊆ E be nonempty, convex,closed, and bounded. Then there exists a function ϕ : (0 , → (0 , with lim ε → ϕ ( ε ) = 0 ,such that for every nonexpansive mapping f : K → K and all x, y ∈ K we have the impli-cation ( || x − f ( x ) || < ε ∧ || y − f ( y ) || < ε ) → (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y − f (cid:18) x + y (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) < ϕ ( ε ) . Actually, ϕ ( ε ) is given by a very simple term involving ε and the modulus of uniformconvexity of E , but we do not need this fact here. Lemma 3.7.
Let E be a uniformly convex Banach space, let K ⊆ E be nonempty, convex,closed, and bounded, and let f : K → K be nonexpansive. Let A ⊆ K be nonempty, closed,and convex. Then A intersects the fixed point set of f if and only if inf {|| f ( x ) − x || (cid:12)(cid:12) x ∈ A } = 0 . Proof.
Clearly, if A intersects the fixed point set of f , then inf {|| f ( x ) − x || (cid:12)(cid:12) x ∈ A } = 0.On the other hand, suppose that inf {|| f ( x ) − x || (cid:12)(cid:12) x ∈ A } = 0. Let µ ( s ) = inf {|| f ( x ) − x || (cid:12)(cid:12) x ∈ A, || x || ≤ s } and r = inf { s > (cid:12)(cid:12) µ ( s ) = 0 } . Since K is bounded and inf {|| f ( x ) − x || (cid:12)(cid:12) x ∈ A } = 0, r is a well-defined real number. Let( x n ) n be a sequence in A with lim n →∞ || f ( x n ) − x n || = 0and lim n →∞ || x n || = r. We will show that ( x n ) n is a Cauchy sequence. It then follows that ( x n ) n converges to afixed point, which proves the claim. Suppose that ( x n ) n is not a Cauchy sequence. Then r > ε > y n ) n of ( x n ) n such that || y n +1 − y n || ≥ ε for all sufficiently large n . Let 2 r ≥ s > r be such that (cid:16) − η E (cid:16) ε r (cid:17)(cid:17) s < r. For n sufficiently large we have || y n || ≤ s , so that we have the inequalities (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n +1 s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n s − y n +1 s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≥ εs . Applying uniform convexity, we obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n + y n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s (cid:16) − η E (cid:16) εs (cid:17)(cid:17) , which yields (using that without loss of generality, η E is monotonically increasing) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n + y n +1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ s (cid:16) − η E (cid:16) ε r (cid:17)(cid:17) < r. Now, by Lemma 3.6 we havelim n →∞ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) y n + y n +1 − f (cid:18) y n + y n +1 (cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 . This contradicts the minimality of r . Hence, ( x n ) n is a Cauchy sequence. Proposition 3.8.
Let E be a uniformly convex computable Banach space, let K ⊆ E benonempty, co-semi-decidable, computably overt, bounded, and convex. Then we have Proj K ≤ W lim ◦ lim . If K is computably compact or E is a Hilbert space, then Proj K ≤ W lim . Proof.
We are given as input a nonexpansive function f : K → K and a point x ∈ K andwant to obtain the point p = P Fix( f ) ( x ) ∈ K . In the case where E is a Hilbert space, we canuse Halpern’s iteration (1.2) to obtain a computable sequence converging to p and applylim to obtain p itself.In the case where K is computably compact, we can compute Fix( f ) as an element of A dist < ( K ) (see the discussion after Definition 2.14). In particular we can compute a sequence( d ( x n , Fix( f )) n ∈ N , where ( x n ) n is a computable dense sequence in K , as an element of R N < . OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 19
Using the standard identification of N N with (cid:0) N N (cid:1) N , it is easy to see that we can use asingle instance of lim to obtain countably many instances of lim in parallel (cf. also e.g. [13]or [17]). Using lim, we can hence compute the sequence ( d ( x n , Fix( f )) n ∈ N as an element of R N , which allows us to compute λy.d ( y, Fix( f )) as an element of C ( K, R ), since we have | d ( y, Fix( f )) − d ( z, Fix( f )) | ≤ d ( y, z )for all y, z ∈ K .Independently, we can use the same instance of lim to obtain a modulus of uniformconvexity η E for E : we may put η E ( ε ) = inf n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x + y (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) x, y ∈ B E , || x − y || ≥ ε o , and if ( x n ) n is a computable dense sequence in B E (we may choose e.g. the sequence ofrational points contained in the open unit ball) we have η E ( ε ) = inf n − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x i + x j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:12)(cid:12) i, j ∈ N , || x i − x j || > ε o , which is clearly limit-computable in ε . This allows us to compute the restriction of η E tothe rational numbers using countably many applications of lim. In particular we can limit-compute a function µ : N → N satisfying 2 − µ ( n ) ≤ η E (2 − n ). Using the distance functionof Fix( f ) and the function µ , we apply Corollary 2.25 to obtain the projection of x ontoFix( f ).In the general case, we cannot a-priori compute Fix( f ) as an element of A dist < ( K ),because of Proposition 2.15. We can however use lim to obtain Fix( f ) as an element of A dist < ( K ): since K is computably overt, we can list all rational closed balls B ( a, r ) forwhich the open ball B ( a, r ) intersects K . Given such a rational closed ball B ( a, r ) in E , wecan compute a dense sequence in B ( a, r ) ∩ K : choose a computable dense sequence ( x n ) n in K and filter out those points x n which satisfy d ( x n , a ) < r . Using the convexity of K it is easy to see that the resulting sequence is dense in B ( a, r ) ∩ K . This allows us tolimit-compute inf (cid:8) || f ( x ) − x || (cid:12)(cid:12) x ∈ B ( a, r ) ∩ K (cid:9) . Again, we can do this for all suitable closed rational balls in parallel. We can then enumeratethose balls B ( a, r ) satisfying inf {|| f ( x ) − x || (cid:12)(cid:12) x ∈ B ( a, r ) ∩ K } >
0, which by Lemma 3.7is equivalent to B ( a, r ) ∩ Fix( f ) = ∅ . This allows us to compute the distance function toFix( f ) from below (cf. [20, Theorem 3.9 (1)] or the proof of [79, Lemma 5.1.7]). Now weapply the limit-computable method used in the compact case above to obtain the projection.Since the Weihrauch degree of the composition of two limit-computable mappings is belowlim ◦ lim (see e.g. Fact 8.2 in [18]), the result follows.Note that if E is finite dimensional, then K is always computably compact, so thatthe stronger upper bound of Proposition 3.8 applies. We will show in Theorem 6.10 thatlim is an upper bound on Proj K in all uniformly convex and uniformly smooth computableBanach spaces, and thus for instance in all L p -spaces with 1 < p < ∞ .We now begin a discussion on the computability of fixed points and the Weihrauchdegree of the Browder-G¨ohde-Kirk theorem. Proposition 2.27 yields an immediate upperbound for the Weihrauch degree of BGK K . We call a multimapping limit-computable if its Weihrauch degree is below lim.
Proposition 3.9.
Let E be a uniformly convex computable Banach space. Let K ⊆ E be nonempty, co-semi-decidable, computably overt, bounded, and convex. Then we have BGK K ≤ W ConvC K . In finite dimension, this already implies that BGK is always strictly weaker than WKL.In fact it is non-uniformly computable thanks to the following result due to Le Roux andZiegler.
Theorem 3.10 ([56]) . Let E be a finite dimensional computable Banach space. Let K ⊆ E be nonempty, co-semi-decidable, and convex. Then K contains a computable point.Proof sketch. We may assume that K is compact, since we can always intersect K with asufficiently large closed ball. We may also assume that E is represented by ρ d for some d ∈ N .We proceed by induction on dim E . If dim E = 1, then K is either a singleton and hencecomputable, or it is an interval and hence contains a rational point. If dim E = d , thenthe projection of K onto the x -axis is still nonempty, co-semi-decidable, and convex, andcontains a computable point x by induction hypothesis. Now, the intersection of { x } × R d − with K is nonempty, convex, and co-semi-decidable, of dimension strictly smaller than d .Again by induction hypothesis, the intersection, and in particular K , contains a computablepoint.The above theorem even shows that K has a dense subset of computable points, sincethe intersection of K with a small rational ball is again co-semi-decidable and compact. Amore uniform version, using Weihrauch degrees, has been given in [55]. Corollary 3.11.
Let E be a finite dimensional, strictly convex computable Banach space.Let K ⊆ E be nonempty, computably overt, co-semi-decidable, bounded, and convex and let f : K → K be computable and nonexpansive. Then f has a computable fixed point. Corollary 3.11 in particular shows that the Browder-G¨ohde-Kirk theorem in finite di-mension is strictly more effective than the (in this case more general) Brouwer fixed pointtheorem: a construction due to Orevkov [63] and Baigger [4] shows that there exists a com-putable function on the unit square in R without computable fixed points, while in everyfinite dimension some fixed points whose existence is guaranteed by the Browder-G¨ohde-Kirk theorem are computable. Note that Corollary 3.11 really only uses the fact that f iscomputable and its fixed point set is convex. Thus, Theorem 3.10 presents a fairly generalnon-uniform computability result: if a computable equation on a finite dimensional spacehas a convex set of solutions, then it has a computable solution. A nontrivial applicationis based on the following result . A self-map f : K → K of a nonempty subset K of a realHilbert space H is called pseudocontractive if it satisfies( f ( x ) − f ( y ) , x − y ) ≤ || x − y || for all x, y ∈ K . Theorem 3.12 ([82]) . Let H be a real Hilbert space, let K ⊆ H be nonempty, closed andconvex, and let f : K → K be pseudocontractive. Then Fix( f ) is closed and convex. Corollary 3.13.
Let H be a finite dimensional computable Hilbert space. Let K ⊆ H benonempty, computably overt, co-semi-decidable, bounded, and convex, and let f : K → K be computable and pseudocontractive. Then f has a computable fixed point. This application was pointed out to the author by Ulrich Kohlenbach.
OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 21
Corollary 3.13 is strictly more general than the Euclidean version of Corollary 3.11, asthe following example, due to [28], shows.
Proposition 3.14.
There exists a computable Lipschitz-continuous pseudocontractive map-ping on the unit ball of Euclidean R , which is not nonexpansive.Proof. For x = ( x , x ), we put x ⊥ = ( − x , x ). Let f ( x ) = ( x + x ⊥ if || x || ≤ , x || x || − x + x ⊥ if || x || ≥ . It is easy to see that f is computable and Lipschitz-continuous with Lipschitz constant 5.The proof of pseudocontractiveness is somewhat technical and can be found in [28].The function from Proposition 3.14 actually provides an example of a pseudocontractivemapping with a unique fixed point, for which the Krasnoselski-Mann iteration, which isguaranteed to converge for nonexpansive mappings, fails to converge. This is proved in [28].In infinite dimension, there exist computable firmly nonexpansive mappings withoutcomputable fixed points, already on compact sets. A mapping f : K → K defined on anonempty subset K of a real Hilbert space H is called firmly nonexpansive if it satisfies || f ( x ) − f ( y ) || ≤ ( x − y, f ( x ) − f ( y ))for all x, y ∈ K . Clearly, every firmly nonexpansive mapping is nonexpansive. It is notdifficult to see that a mapping is firmly nonexpansive if and only if it is of the form f ( x ) = ( x + g ( x )), where g is a nonexpansive mapping. Let H = { x ∈ ℓ | ≤ x ( n ) ≤ − n for all n ∈ N } denote the Hilbert cube in ℓ (represented by δ ℓ (cid:12)(cid:12) H ). Theorem 3.15.
There exists a computable firmly nonexpansive mapping f : H → H without computable fixed points.Proof. Put g n ( x ) = (1 − − n ) x and h n ( x ) = 2 − n + (1 − − n ) x . Then ( g n ) n and ( h n ) n are computable sequences of nonexpansive self-maps of [0 , | g n ( x ) − x | ≤ − n , | h n ( x ) − x | ≤ − n for all x ∈ [0 , g n ) = { } , and Fix( h n ) = { } .Let A, B ⊆ N be two disjoint, recursively enumerable, and recursively inseparable sets.Let α be the G¨odel number of an algorithm with halting set A and β be the G¨odel numberof an algorithm with halting set B . Consider the sequence of functions ( f n ) n with f n ( x ) = g i ( x ) if α halts on input n within i steps, h i ( x ) if β halts on input n within i steps, x if both α and β diverge on input n. Note that since A and B are disjoint, both α and β cannot halt on the same input, so f n is well-defined. The sequence ( f n ) n is a computable sequence: in order to compute f n ( x ) up to error 2 − m , we simulate α and β simultaneously on input n for m steps. If α (respectively β ) halts within k ≤ m steps, we output g k ( x ) (respectively h k ( x )) up to error2 − m . If neither α nor β halt after m steps, we may output x as an approximation, since | g m + k ( x ) − x | ≤ − m − k and | h m + k ( x − x ) | ≤ − m − k for all k ≥
0. Now, suppose there existsa computable sequence ( x n ) n with f n ( x n ) = x n . In order to arrive at a contradiction, weuse ( x n ) n to construct a computable set S ⊆ N separating A and B . Membership for S is decided as follows: for a given n ∈ N , run the tests x n > x n < x n >
0, we decidethat n / ∈ S . If the first test to succeed is x n <
1, we decide that n ∈ S . Note that in thecase where both x n > x n <
1, the outcome of the decision procedure may dependon the Cauchy sequence of dyadic rational numbers representing x n . We claim that S ⊇ A and S C ⊇ B . If n ∈ A , then α halts on input n after i ∈ N steps. So f n = g i , and thus x n = 0. The test x n > x n < n ∈ S .If n ∈ B , then β halts on input n after i ∈ N steps, so f n = h i and thus x n = 1. It followsthat n / ∈ S . So S separates A and B . Contradiction. Now define a nonexpansive mapping g : H → H via g ( x )( n ) = 2 − n f n (2 n x ( n )). Then g is computable, for in order to computean approximation to g ( x ) in ℓ up to error 2 − n , it suffices to compute the real numbers g ( x )(0) , . . . , g ( x )( n + 1) up to error 2 − n − / ( n + 2). Any fixed point for g can be usedto compute a sequence of fixed points for ( f n ) n . In particular, g has no computable fixedpoints. In order to obtain a firmly nonexpansive mapping f we put f = (id + g ).Theorem 3.15 in particular shows that fixed points of (firmly) nonexpansive mappingsare not computable relative to any discrete advice. Let us now consider the computabil-ity of rates of convergence of certain fixed point iterations. While in infinite dimension,the non-uniform uncomputability of fixed points in particular implies that there exist com-putable mappings such that no computable fixed point iteration has a computable rate ofconvergence for any computable starting point, and Theorem 1.4 tells us that there is nogeneral algorithm for obtaining rates of convergence uniformly in the input function and inthe starting point, it might still be the case (at least in finite dimension) that there existsa computable fixed point iteration I such that for every computable nonexpansive function f there exists a computable starting point x / ∈ Fix( f ) such that the sequence I ( f, x ) hasa computable rate of convergence. This could still be practically relevant, since in a givenpractical scenario one might be able to exploit additional information on the input functionin order to choose the starting point of the iteration in such a way that the rate of conver-gence becomes computable. We will however see that this fails to be the case for a largeclass of fixed point iterations, already on the compact unit interval.We prove a special case of our main result (Theorem 5.1) where the underlying set is thethe compact unit interval [0 , , Proposition 3.16.
Let I = { ( a, b ) ∈ ( R < × R > ) | a ≤ b } . Then the mapping A co ( R ) \ {∅} → I , [ a, b ] ( a, b ) and its inverse I → A co ( R ) \ {∅} , ( a, b ) [ a, b ] are computable. Theorem 3.17. (i)
The mapping
Fix : N ([0 , → A co ([0 , \ {∅} , f Fix( f ) is computable. (ii) And so is its multivalued inverse
Fix − : A co ([0 , \ {∅} ⇒ N ([0 , . OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 23
Proof.
The first claim immediately follows from Proposition 2.27 and the well-known resultthat the set of zeroes of a continuous mapping f is co-semi-decidable in f .Let us now prove the second claim. Suppose we are given a nonempty, closed interval[ a, b ] ∈ A co ([0 , a n ) n of rational numbers converging from below to a , and a monotonically decreasing list( b n ) n of rational numbers converging from above to b . We may assume without loss ofgenerality that a n ≥ b n ≤ n ∈ N .From ( a n ) and ( b n ) we can compute a sequence ( f n ) n of nonexpansive functions via f n ( x ) = a n if x ≤ a n , x if a n ≤ x ≤ b n , b n if x ≥ b n .Finally, we compute f ( x ) = X n ∈ N − n − f n ( x ) . Then f is nonexpansive, and maps [0 ,
1] into [0 , f ) = [ a, b ]. If x ∈ [ a, b ] then f n ( x ) = x for all n ∈ N , so f ( x ) = P n ∈ N − n − x = x . Suppose now withoutloss of generality that x < a . Then f n ( x ) ≥ x for all n ∈ N , and there exists m ∈ N suchthat x < a m and hence f m ( x ) > x . It follows that f ( x ) = X n ∈ N − n − f n ( x ) > x, and hence x / ∈ Fix( f ). An analogous argument applies if x > b . We may hence putFix − ([ a, b ]) = f .From the proof we obtain the following non-uniform corollary, which is slightly strongerthan the non-uniform version of Theorem 3.17. Corollary 3.18.
Let [ a, b ] ⊆ [0 , be a co-semi-decidable interval. Then there exists amonotonically increasing, firmly nonexpansive, computable function f : [0 , → [0 , with Fix( f ) = [ a, b ] .Proof. The algorithm we use in the proof of Theorem 3.17 to compute Fix − maps anynonempty co-semi-decidable interval to a monotonically increasing, nonexpansive function g : [0 , → [0 ,
1] with Fix( g ) = [ a, b ]. In order to obtain a firmly nonexpansive function f ,we put f ( x ) = ( x + g ( x )).Since by Proposition 3.16 any left-r.e. number can be the left endpoint of a co-semi-decidable interval, we obtain the announced result together with Proposition 3.4. Corollary 3.19.
Let I : N ([0 , × [0 , → [0 , N be an either projective, or simple andretractive, or avoidant computable fixed point iteration. Let ε > . There exists a com-putable, firmly nonexpansive function f : [0 , → [0 , with diam(Fix( f )) < ε such that forno computable x / ∈ Fix( f ) , the sequence I ( f, x ) has a computable rate of convergence.Proof. Let a ∈ (0 ,
1) be an uncomputable left-r.e. number and b ∈ (0 ,
1) be an uncomputableright-r.e. number with | a − b | < ε . Then the closed interval [ a, b ] ⊆ [0 ,
1] is co-semi-decidableby Proposition 3.16. Using Corollary 3.18 we obtain a monotonically increasing firmly non-expansive function f : [0 , → [0 ,
1] with Fix( f ) = [ a, b ]. If x / ∈ Fix( f ), then by Proposition3.4, we have lim n →∞ I ( f, x )( n ) ∈ { a, b } . In particular, lim n →∞ I ( f, x )( n ) is uncomputable. Since x is computable, the sequence ( I ( f, x )( n )) n is a computable sequence of real numbers,so if it had a computable rate of convergence, its limit would be computable. This provesthe claim.Both the Halpern iteration and the Krasnoselski-Mann iteration are simple and retrac-tive, so Corollary 3.19 applies to them. Using Weihrauch degrees, we can state our presentresults more uniformly. Proposition 3.20.
We have
Proj [0 , ≡ W lim . If I : N ([0 , × [0 , → [0 , N is aneither projective, or simple and retractive, or avoidant computable fixed point iteration,then lim( I ) ≡ W Conv I ≡ W lim .Proof. It is well known that the identity id : R < → R is Weihrauch-equivalent to lim, evenwhen restricted to the unit interval. Given a ∈ R < ∩ [0 , a, ∈ A co ([0 , f : [0 , → [0 ,
1] with Fix( f ) = [ a, [0 , ( f,
0) = a , so lim ≤ W Proj [0 , .Together with Proposition 3.5 we obtain Proj [0 , ≡ W lim. If I is a projective, sim-ple and retractive, or avoidant, computable fixed point iteration, then by Proposition3.4 we obtain Proj [0 , ≤ W lim( I ), and thus lim ≤ W lim( I ) ≤ W Conv I ≤ W lim, i.e.lim( I ) ≡ W Conv I ≡ W lim. Proposition 3.21.
BGK [0 , ≡ w ConvC [0 , ≡ W IVT ≡ W BFT . The equivalence BGK [0 , ≡ W ConvC [0 , follows immediately from Theorem 3.17, theequivalence ConvC [0 , ≡ W IVT ≡ W BFT was already stated in Fact 2.21. So far, thereseems to be a significant discrepancy between the computational content of the existenceresult BGK and the “constructive” theorems by Wittmann and Krasnoselski. We will see inSection 5 that this discrepancy disappears on non-compact domains in infinite-dimensionalHilbert space, where the Browder-G¨ohde-Kirk theorem is Weihrauch equivalent to lim, andhence to Wittmann’s theorem. 4. Weak Topologies
In order to be able to prove our main result in full generality, we have to introduce anadmissible representation for the weak topology on a reflexive Banach space E . Such arepresentation has first been introduced by Brattka and Schr¨oder [21]. We denote thecontinuous dual of a normed space E by E ′ and define the mapping( · , · ) : E × E ′ → R , ( x, x ′ ) x ′ ( x ) . Definition 4.1.
Let E be a computable Banach space. The represented space E ′ w is thespace E ′ , represented via the co-restriction of [ δ E → ρ ] to all continuous linear functionals. Theorem 4.2 ([21]) . The representation [ δ E → ρ ] (cid:12)(cid:12)(cid:12) E ′ is admissible with respect to the weak*topology on E ′ . OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 25
Since the points of E , viewed as functionals on E ′ , separate the points of E ′ , the weak*topology on E ′ is Hausdorff. By Theorem 4.2, the space E ′ w is then a Hausdorff representedtopological space, so that the space K ( E ′ w ) is well-defined and coincides extensionallywith the set of all weak* compact subsets of E ′ . Note that this crucially relies on theseparability of E , since since weak* sequential compactness and weak* compactness neednot coincide on duals of inseparable spaces. The fact that they do coincide in the separablecase also follows from the well-known fact that the weak* topology on the dual space ofa separable Banach space is metrisable on the unit ball (cf. Theorem 4.4 below). Alsonote that the weak* topology on E ′ is in general not sequential (i.e. there exist sequentiallyweak* closed sets which are not weak* closed). Consequently, the spaces O ( E ′ w ) and A ( E ′ w )do not coincide with the hyperspaces of weak* open and weak* closed sets respectively, butwith the hyperspaces of weak* sequentially open and weak* sequentially closed sets. If A ∈ A ( E ′ w ), we write A w for the represented space ( A, δ E ′ w (cid:12)(cid:12) A ) to emphasize the underlyingrepresentation.If E is a reflexive real Banach space with computable dual E ′ , we obtain a canonicalrepresentation for E with respect to the weak topology, by identifying E with E ′′ and putting E w = ( E ′ ) ′ w , i.e. E w is the represented space E ′′ with representation [ δ E ′ → ρ ] (cid:12)(cid:12) E ′′ (usingthat in this case the weak* topology on E ′′ coincides with the weak topology on E ). Again,the space K ( E w ) and the space of weakly compact subsets of E coincide extensionally ,but the caveat on O ( E ′ W ) and A ( E ′ w ) also applies to O ( E w ) and A ( E w ). As in the caseof E ′ w , if A ∈ A ( E w ), we write A w for the represented space ( A, δ E w ). We will often usethe adjective “weak” when referring to elements in hyperspaces constructed from E w . Forinstance, we may call the computable points of A ( E w ) “weakly co-semi-decidable” and thecomputable points of A ( E ′ w ) “weak* co-semi-decidable” etc. Remark . Note that in the definition of E w we only require E ′ , but not E itself, to bea computable Banach space. By definition, the mapping ( · , · ) is ( δ E w × δ E ′ , ρ )-computable.If both E and E ′ are computable Banach spaces, it is natural to require that the mapping( · , · ) be ( δ E × δ E ′ , ρ )-computable, so that id : E w → E becomes computable (see for instanceTheorem 4.4 (iii), Corollary 4.5 and Propositions 4.8, 4.13 and 4.15 below). This is forinstance the case for the spaces L p ([0 , < p < ∞ , since we have ( L p ([0 , ′ = L q ([0 , p + q = 1, and if f ∈ L p ([0 , g ∈ L q ([0 , f, g ) is given bythe effective formula ( f, g ) = Z f ( x ) g ( x ) dx . Next, we prove some basic properties of the space K ( E ′ w ). We will need a few effectivecounterparts to classical results from functional analysis. The first is an effective version ofthe separable Banach-Alaoglou theorem, which was proved by Brattka [12]. Theorem 4.4 (Computable separable Banach-Alaoglou theorem, [12]) . Let E be a com-putable Banach space. Let B E ′ w denote the unit ball in E ′ , viewed as a subset of the repre-sented space E ′ w (thus bearing the weak* topology). Then (i) B E ′ w ∈ K ( E ′ w ) . (ii) More generally, let K ⊆ E ′ w be a co-semi-decidable subset of E ′ w . If K is bounded,then K is computably weak* compact. Thus, Theorem 4.2 together with Proposition 3.3.2 (3) in [70] provide an interesting proof of the separable
Eberlein-ˇSmulian theorem as well as its analogue for the weak* topology in duals of separable spaces. (iii) If E ′ is a computable Banach space and ( · , · ) is ( δ E × δ E ′ , ρ ) -computable, then B E ′ w admits the structure of a computably compact computable metric space.Proof. It is proved in [12] that there exists a computable embedding i : B E ′ w → X into acomputably compact computable metric space X , such that i ( B E ′ w ) is computably compactas a subset of X and the partial inverse i − : i ( B E ′ w ) → B E ′ w is computable. It follows that i ( B E ′ w ) is a computably compact represented space, and i induces a computable isomorphismbetween B E ′ w and i ( B E ′ w ), so that B E ′ w is computably compact as a represented space. Itfollows from Proposition 2.7 that B E ′ w is a computably compact subset of E ′ w . This provesthe first claim.For the second claim, observe that the mappingsmult : (0 , ∞ ) × A ( E ′ w ) → A ( E ′ w ) , ( α, A ) αA = { αx (cid:12)(cid:12) x ∈ A } and mult : (0 , ∞ ) × K ( E ′ w ) → K ( E ′ w ) , ( α, K ) αK = { αx (cid:12)(cid:12) x ∈ K } are computable. Proposition 2.8 (i) asserts that f : A ( B E ′ w ) → K ( B E ′ w ) , A A is computable. Trivially, given A ∈ A ( E ′ w ), such that A ⊆ B E ′ w , we can compute A as aset in A ( B E ′ w ). Given a bounded set A ∈ A ( E ′ w ) with bound b , we hence obtain A as anelement of K ( E ′ w ) by computing mult( b, f (mult( b , A ))).The third claim follows immediately from the proof of the first. We may pull backthe metric d X on X via i to obtain a metric on B E ′ w , i.e. put d ( x, y ) = d X ( i ( x ) , i ( y )) for x, y ∈ B E ′ w . As the set of rational points in B E ′ w we may choose those rational points ofthe computable Banach space E ′ whose norm is strictly smaller than one. One now easilyverifies that B E ′ w is computably compact as a computable metric space, and that the Cauchyrepresentation on B E ′ w is computably equivalent to [ δ E → ρ ] (cid:12)(cid:12) B E ′ w .As a corollary we get an effective version of a classical result in functional analysis(cf. [80, Korollar VIII.3.13]) in the reflexive case. Corollary 4.5.
Let E be a reflexive computable Banach space with computable dual E ′ ,such that the mapping ( · , · ) is ( δ E × δ E ′ , ρ ) -computable. Then E is computably isometricallyisomorphic to a co-semi-decidable and computably overt subspace of a function space C ( M ) over a computably compact metric space M .Proof. By Theorem 4.4 (iii), M = B E ′ w (with the weak* topology) is a computably compactcomputable metric space. We show that the mapping i : E → C ( M ) , x λx ′ . ( x, x ′ )is a ( δ E , [ δ B E ′ w → ρ ])-computable isometric embedding with co-semi-decidable and com-putably overt image. The ( δ E , [ δ B E ′ w → ρ ])-computability is obvious, and the fact that it isan isometry follows from || x || = sup x ′ ∈ B E ′ | ( x, x ′ ) | , which in turn is an easy corollary of theHahn-Banach theorem (cf. e.g. [80, Korollar III.1.7]). Clearly, i ( E ) is computably overt. Itis also co-semi-decidable, for if we are given a continuous function f on B E ′ w , we can verifyif it is nonlinear. It remains to show that its inverse i − : i ( E ) → E is computable. Given a OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 27 [ δ B E ′ w → ρ ]-name of i ( x ) ∈ i ( E ) and a ν E -name of a rational point y ∈ E we can computea [ δ B E ′ w → ρ ]-name of i ( x ) − i ( y ), and hence computemax {| ( x − y, x ′ ) | (cid:12)(cid:12) x ′ ∈ B E ′ w } = || i ( x ) − i ( y ) || = || x − y || , using Theorem 2.9. For every n ∈ N we may hence search for a rational point y n in E satisfying || y n − x || < − n , which allows us to compute a δ E -name of x . Remark . In Corollary 5 in [21], it is claimed that the representation δ wE of E defined by δ wE ( p ) = x : ⇔ [ δ ′ E → ρ ]( p ) = ι ( x ) , where ι : E → E ′′ is the canonical embedding and δ ′ E = [ δ E → ρ ] (cid:12)(cid:12)(cid:12) E ′ , is admissible for theweak topology on E . This contradicts the proof of our Corollary 4.5, which suggests that δ wE is admissible with respect to the norm topology. To convince ourselves that the claimis false, we consider the simple example of E = ℓ . For the scope of this remark we willadopt the notation used in [21]. The representation δ ≥ ℓ , where a δ ≥ ℓ -name of ( x n ) n ∈ ℓ is a ρ ω -name of a sequence ( b, x , x , . . . ) with b ≥ || ( x n ) n || , is equivalent to δ ℓ w (which isdenoted by δ ′ ℓ in [21]). It follows that given a δ wE ( p )-name of x ∈ ℓ we can compute ( x, x ′ )for any x ′ ∈ ℓ , provided that we know x ′ ( n ) for all n ∈ N and some bound on || x ′ || . Inparticular we can compute x ( n ) = ( x, e n ) for every n ∈ N . Since δ ℓ ≤ δ ℓ w , a δ wℓ -name of x allows us to compute a [ δ ℓ → ρ ]-name of x . Theorem 5.1. in [11] then asserts that wecan compute some bound b on || x || . It follows that we can compute || x || = ( x, x ), and so δ wℓ ≡ δ = ℓ , which entails that in fact δ wℓ is admissible with respect to the (strictly stronger)norm topology.The flaw in the argument seems to be the claim that for every compatible representation δ of a separable Banach space X , the dual representation δ ′ = [ δ → δ F ] (cid:12)(cid:12) X ′ is admissible withrespect to the weak* topology on X ′ , the reasoning being that for represented topologicalspaces A and B , the canonical function space representation [ δ A → δ B ] is admissible withrespect to the sequentially-compact-open topology on [ A → B ], and that weak*-convergenceon X ′ coincides with compact-open-convergence on C ( X, F ) (cf. Theorem 4.2 and Proposi-tion 1 in [21]). However, if δ is admissible with respect to the weak topology on X , then δ ′ = [ δ → ρ ] is admissible with respect to the weakly-compact-open topology and notnecessarily with respect to the (norm-)compact-open topology. Thus, if X is a reflexivecomputable Banach space, and we start with the standard representation δ X of X , whichis compatible and admissible with respect to the strong topology, then δ ′ X = [ δ X → ρ ]is compatible and admissible with respect to the compact-open topology on X ′ , which isjust the weak* topology. Applying the construction again, we see that ( δ ′ X ) ′ = [ δ ′ X → ρ ]is compatible and admissible with respect to the weak*-compact-open topology, which ingeneral is strictly stronger than the (norm-)compact-open-topology.Let us now turn to some special properties of convex sets. Mazur’s lemma asserts thata convex set is weakly sequentially closed if and only if it is strongly closed.
Theorem 4.7 (Mazur’s lemma) . Let E be a Banach space and K ⊆ E be convex. If ( x n ) n is a sequence in K which converges weakly to x ∈ E , then there exists a sequence of finiteconvex combinations of the x n ’s, converging strongly to x . It follows that strongly overt convex sets are weakly overt.
Proposition 4.8.
Let E be a reflexive computable Banach space with computable dual E ′ ,such that the mapping ( · , · ) is ( δ E × δ E ′ , ρ ) -computable. Then the identity id : V co ( E ) \ {∅} → V co ( E w ) \ {∅} , where V co ( E ) denotes the hyperspace of convex overt closed subsets of E , is well-definedand computable.Proof. The mapping id is well-defined by Theorem 4.7. Since E is a computable metric spaceand E w is separable, we may use the characterisation of overtness given in Proposition 2.13and the sufficient condition given in Proposition 2.4. If K ∈ V co ( E ), then by Proposition2.13, we can compute a δ ωE -name of a norm-dense sequence ( x n ) n in K . Since the weaktopology is coarser than the norm topology, the weak sequential closure of ( x n ) n contains K , and by Theorem 4.7, any weak limit of ( x n ) n is already contained in K , so that K isthe closure of ( x n ) n with respect to the sequentialisation of the weak topology. Since ( · , · )is computable, we have δ E ≤ δ E w , so that we can compute a δ ωE w -name of ( x n ) n . Thus wecan compute K as an element of V co ( E w ) using Proposition 2.4.Propositions 4.8 and 2.4 imply that a convex subset of a reflexive computable Banachspace E is weakly overt if and only if it has a computable norm-dense sequence.Finally, we prove a useful uniform characterisation of computably weakly compact con-vex sets in a reflexive Banach space E with computable dual E ′ , which will be an importantingredient for the proof of our main result. Note that by Mazur’s lemma, a convex subsetof E is weakly compact if and only if it is closed and bounded (cf. also [58, Proposition2.8.1]). Definition 4.9.
Let E be a reflexive Banach space with computable dual E ′ .(i) A rational half space is a nonempty set of the form h = { x ∈ E | ( x, x ′ h ) + a h ≤ } where x ′ h is a rational point in E ′ and a h ∈ Q . A ν HB -name of a rational half spaceis a ν E ′ × ν Q -name of ( x ′ h , a h ) ∈ E ′ × Q .(ii) Let K ⊆ E be closed, convex and bounded. A κ HB -name of K is a ν ω HB × κ E w -nameof all rational half spaces containing K in their interior and a weakly compact set L ∈ K ( E w ) containing K .Note that by Theorem 4.4 (ii), a κ HB -name of a closed, convex and bounded set can becomputed from a list of all rational half spaces containing K in their interior and a rationalbound on sup {|| x || (cid:12)(cid:12) x ∈ K } . It may not be immediately obvious that κ HB is a well-defined representation. This follows however from the following easy consequence of theHahn-Banach separation theorem (cf. e.g. [80, Theorem III.2.5]). Lemma 4.10.
Let E be a reflexive Banach space with computable dual E ′ . Let K ⊆ E beclosed, bounded and convex, let x / ∈ K . Then there exists a rational half space h such that K ⊆ h ◦ and x ∈ h C . Obviously, the boundedness condition on K cannot be dropped, as the example of astraight line with irrational slope in R shows. Theorem 4.11.
Let E be a reflexive Banach space with computable dual E ′ . Then we have κ HB ≡ (cid:0) κ E w (cid:1)(cid:12)(cid:12) K co ( E w ) , where K co ( E w ) denotes the space of convex weakly compact subsetsof E w . OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 29
Proof. h κ HB ≤ (cid:0) κ E w (cid:1)(cid:12)(cid:12) K co ( E w ) i : Suppose we are given a κ HB -name of K ∈ K co ( E w ). Sincethe name provides us with a weakly compact set L ∈ K ( E w ) containing K , and sinceid : A ( L w ) → K ( L w ) is computable by Proposition 2.8 (i), it suffices to show that we cancompute a ψ E w -name of K . Given x ∈ E w and a ν HB -name of a functional f = ( · , x ′ h ) + a h we can compute f ( x ) ∈ R . Now, if the sequence of half spaces containing K given by the κ HB -name is defined by the sequence of affine linear functionals ( f n ) n , we can compute thecharacteristic function of K C into Sierpi´nski-space as follows: given x ∈ E w , if there existsan n ∈ N such that f n ( x ) > ψ E w -name of K , which proves the claim. h(cid:0) κ E w (cid:1)(cid:12)(cid:12) K co ( E w ) ≤ κ HB i : Suppose we are given a κ E w -name of K . We need to compute aweakly compact set L ∈ K ( E w ) with L ⊇ K and a list of all rational half spaces containing K in their interior. Since K contains itself and is given as a κ E w -name, we may put L = K ,so that it suffices to show that we can enumerate all rational half spaces containing K intheir interior. We show that given a ν HB -name of an affine linear functional (technically,of the half space defining the functional) f : E → R of the form f ( x ) = ( x, x ′ ) + a , we canverify if f ( x ) < x ∈ K . We can computably translate the ν HB -name of f intoa δ E ′ -name of x ′ . Then by definition of E w , the mapping f : E w → R , x ( x, x ′ ) + a iscomputable. It follows that U f = { x ∈ E w (cid:12)(cid:12) f ( x ) < } is semi-decidable relative to f . Bydefinition of κ , the relation K ⊆ U f is semi-decidable relative to f as well, which proves theclaim.The proof of Theorem 4.11 shows that the definition of κ HB can be slightly relaxed. Lemma 4.12.
Let E be a reflexive Banach space with computable dual E ′ . Define a newrepresentation ˜ κ HB of K co ( E w ) as follows: a ˜ κ HB -name of K ∈ K co ( E w ) is a ν ω HB × κ E w -name of a sequence ( h n ) n of rational half spaces such that K = T n ∈ N h ◦ n and a weaklycompact set L ∈ K ( E w ) containing K . Then ˜ κ HB ≡ κ HB .Proof. Clearly, κ HB ≤ ˜ κ HB . For the converse direction, note that the proof of the reduc-tion κ HB ≤ (cid:0) κ E w (cid:1)(cid:12)(cid:12) K co ( E w ) in Theorem 4.11 actually establishes the stronger reduction˜ κ HB ≤ (cid:0) κ E w (cid:1)(cid:12)(cid:12) K co ( E w ) , so that we obtain the reduction chain˜ κ HB ≤ (cid:0) κ E w (cid:1)(cid:12)(cid:12) K co ( E w ) ≤ κ HB and thus ˜ κ HB ≡ κ HB .On compact subsets of a Banach space E , the weak topology and the norm topologycoincide. This is effectively witnessed by our representation. Proposition 4.13.
Let E be a reflexive computable Banach space with computable dual,such that the mapping ( · , · ) is ( δ E × δ E ′ , ρ ) -computable. Let K ⊆ E be a computably compactand computably overt subset of E . Then we have δ E w (cid:12)(cid:12) K ≡ δ E (cid:12)(cid:12) K .Proof. Since ( · , · ) is computable, we have δ E ≤ δ E w , so we only have to show the conversereduction δ E w | K ≤ δ E | K . Define the represented spaces K = (cid:16) K, δ E (cid:12)(cid:12) K (cid:17) and K w = (cid:16) K, δ E w (cid:12)(cid:12) K (cid:17) . Firstly, observe that E w is effectively Hausdorff, i.e. the mapping E w → A ( E w ) , x
7→ { x } is computable: we can verify if two given elements in E w are different by comparing theirvalues on the rational points of E ′ . It follows that the identity id : O ( K ) → O ( K w ) iscomputable via the following chain of maps: O ( K ) −−−−→ A ( K ) (1) −−−−→ K ( K ) (2) −−−−→ K ( K w ) (3) −−−−→ A ( K w ) −−−−→ O ( K w ) U −−−−→ K \ U −−−−→ K \ U −−−−→ K w \ U −−−−→ K w \ U −−−−→ U. The computability of (1) follows from the computable compactness of K together withProposition 2.8 (i). The computability of (2) can be derived from the computability ofid : O ( E w ) → O ( E ), which in turn follows from the computability of id : E → E w , and thecomputability of (3) follows from the fact that E w is effectively Hausdorff, together withProposition 2.8 (ii).We then obtain the mapping id : K w → K , i.e. the reduction δ E w | K ≤ δ E | K , via thefollowing chain of maps: K w (4) −−−−→ A ( K w ) (5) −−−−→ K ( K w ) (6) −−−−→ K ( K ) (7) −−−−→ Kx −−−−→ { x } −−−−→ { x } −−−−→ { x } −−−−→ x. Mapping (4) is computable since E w is effectively Hausdorff. To establish the computabilityof (5), observe that the computability of (2) and the computable compactness of K implythat K ∈ K ( K w ) and apply Proposition 2.8 (i). The computability of (6) follows fromthe computability of id : O ( K ) → O ( K w ), which we have established above. For the com-putability of (7), observe that we can verify if a rational ball of the form B ( a, − n ) contains { x } , which yields a Cauchy sequence effectively converging to x by exhaustive search overall rational balls.Finally, we observe that computably overt, co-semi-decidable subsets of E ′ w are (uni-formly) located. The following proposition guarantees that this actually makes sense. Proposition 4.14.
Let E be a Banach space. Let A ⊆ E be weakly sequentially closed.Then A is closed with respect to the norm topology.Proof. Since E is a metric space, A is closed if and only if it is sequentially closed. Let( x n ) n be a sequence in A with limit x ∈ E . Then x is a weak limit of ( x n ) n , so x ∈ A , since A is weakly sequentially closed. It follows that A is sequentially closed, and thus closed. Proposition 4.15.
Let E be a reflexive computable Banach space with computable dual,such that the mapping ( · , · ) is ( δ E × δ E ′ , ρ ) -computable. Then the canonical embedding i : A ( E w ) \ {∅} → A dist < ( E ) , A A is computable.Proof. Given a sequentially weakly closed set A ∈ A ( E w ), it suffices to show that we canuniformly computably enumerate all closed balls with rational centres and radii containedin the complement A C of A . The result then follows from [20, Theorem 3.9 (1)]. The proofof Theorem 4.4 (ii) allows us to uniformly translate a computable number r ∈ R into aname of B (0 , r ) as a weakly compact subset of E w , i.e. the mapping(0 , ∞ ) → K ( E w ) , r B (0 , r )is computable. It is easy to see that the mapping E w × E w → E w , ( x, c ) x + c OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 31 is computable. Hence, the mapping E w × (0 , ∞ ) → K ( E w ) , ( c, r ) B ( c, r )is computable. It follows that the mapping A ( E w ) × E w × (0 , ∞ ) → S , ( A, c, r ) ( B ( c, r ) ⊆ A C ,0 otherwiseis computable. Since ( · , · ) is computable, we have δ E ≤ δ E w , so that in particular themapping A ( E w ) × E × (0 , ∞ ) → S , ( A, c, r ) ( B ( c, r ) ⊆ A C ,0 otherwiseis computable. Using this mapping we can enumerate all rational closed balls contained inthe complement of A .Proposition 4.15 in particular implies that any nonempty weakly co-semi-decidablesubset of E is lower semi-located, and hence every nonempty weakly co-semi-decidable andcomputably overt subset of E is located (which by Proposition 2.15 is at least not uniformlytrue for co-semi-decidable subsets of E , if E is infinite dimensional).Let us introduce some further Weihrauch degrees. Let E be a computable Banachspace with computable dual, such that the mapping ( · , · ) is ( δ E × δ E ′ , ρ )-computable. Let A ⊆ E be nonempty and weakly closed. The weak closed choice principle C w → w A on A isthe closed choice principle C A w on the represented space A w = ( A, δ E w ). The weak-strongclosed choice principle is the multimappingC w → n A : A ( A w ) ⇒ A, S S, where the image is represented by δ E | A . Similarly, we define ConvC w → w A , UC w → w A , ConvC w → n A and UC w → n A . We may also define a (computationally) weaker version of the Browder-G¨ohde-Kirk theorem. Let K be nonempty, computably overt, weakly co-semi-decidable, boundedand convex. The weak Browder-G¨ohde-Kirk theorem is the mappingWBGK K : N ( K ) ⇒ K w , f Fix( f ) , where we are given a nonexpansive mapping like in the case of the Browder-G¨ohde-Kirktheorem, but are only required to compute a fixed point with respect to the weak topology.Note that in ℓ this amounts to computing a fixed point with respect to an orthonormalbasis, but not necessarily computing its ℓ -norm (cf. also [9]).5. Characterisation of the Fixed Point Sets of Computable NonexpansiveMappings in Computable Hilbert Space
We may now prove our main result. Throughout this section we will work on a computableHilbert space H . Note that in this case H ′ ≃ H is again a computable Hilbert space, andthat the mapping ( · , · ) : H × H ′ → R is the usual inner product on H , which is computableby the polarisation identity. In particular, we can use Definition 4.1 to construct the space H w , whose representation is admissible for the weak topology on H . Theorem 5.1.
Let H be a computable Hilbert space, let K ⊆ H be weakly co-semi-decidable,computably overt, bounded, and convex. Then (i) The mapping
Fix : N ( K ) → K co ( K w ) \ {∅} , f Fix( f ) is computable. (ii) And so is its multivalued inverse
Fix − : K co ( K w ) \ {∅} ⇒ N ( K ) . Let us sketch the proof of the second claim. Given a nonempty, weakly closed, boundedand convex subset A of K , by Theorem 4.11 we can enumerate a sequence of half spaceswhose intersection is equal to A . Now, the projections onto these half spaces are nonexpan-sive, thanks to Theorem 2.23 (ii), and computable: Lemma 5.2.
Let H be a computable Hilbert space. There exists a computable functionwhich takes as input a rational half space h ⊆ H , encoded as a ν HB -name, and returns asoutput the metric projection onto h as an element of C ( H, H ) .Proof. Let h = { x ∈ H | ( x, x h ) + a h ≤ } , where x h is a rational point in H and a h ∈ Q .It follows from Lemma 2.17 that we can without loss of generality assume that the set { n ∈ N | ν H ( n ) = 0 } is decidable (cf. also [12, Lemma 3]). Thus, we can decide if x h = 0,and if this is the case we necessarily have a h = 0 (since h is nonempty), and the projectiononto h is the identity on H . If x h = 0, put ˜ x h = x h || x h || , ˜ a h = a h || x h || , and p = x − α ˜ x h , where α = max { , ( x, ˜ x h ) + ˜ a h } . One easily verifies that p ∈ h and that p satisfies the variationalinequality (Theorem 2.23 (ii)). It follows that P h ( x ) = p . This proves the claim.We can hence compute a sequence of nonexpansive mappings such that A is the inter-section of the fixed point sets of these mappings. The following theorem due to Bruck allowsus to construct a single nonexpansive mapping whose fixed point set is the intersection ofthe fixed point sets of our sequence of mappings. Theorem 5.3 ([27]) . Let E be a strictly convex real normed space, let K ⊆ E be nonempty,closed, bounded, and convex. Let ( λ n ) n be any sequence in (0 , satisfying P n λ n = 1 .Let ( f n ) n be a family of nonexpansive mappings on K with T n ∈ N Fix( f n ) = ∅ . Then themapping f = X n λ n f n is well-defined, nonexpansive and satisfies Fix( f ) = \ n ∈ N Fix( f n ) . In the final step, we project back onto K in order to construct a self-map of K . Lemma 5.4.
Let H be a real Hilbert space, K ⊆ H be closed and convex and f : K → H be nonexpansive and suppose that Fix( f ) = ∅ . Let P K denote the metric projection onto K .Then P K ◦ f is nonexpansive as well with Fix( P K ◦ f ) = Fix( f ) .Proof. It is clear that P K ◦ f is nonexpansive and that Fix( f ) ⊆ Fix( P K ◦ f ). Suppose thereexists x ∈ K with f ( x ) = x and P K ( f ( x )) = x . Let y be some fixed point of f . Then || f ( x ) − f ( y ) || = || y − x || + || x − f ( x ) || − f ( x ) − x, y − x ) . By assumption, x = P K ( f ( x )), so by the variational inequality (Theorem 2.23)( f ( x ) − x, y − x ) ≤ y ∈ K. OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 33
We also assumed that f ( x ) = x , i.e. || f ( x ) − x || >
0, hence || f ( x ) − f ( y ) || > || x − y || . Contradicting the assumption that f is nonexpansive. Proof of Theorem 5.1 ( ii ) . We prove that given a κ HB -name φ of a nonempty, closed, convexsubset A ⊆ K we can compute the name of a nonexpansive function f : K → K withFix( f ) = A . The name φ encodes a sequence of rational half spaces ( h k ) k containing A in their interior. Using Lemma 5.2, given φ we can compute a [ δ H → δ H ] ω -name of somesequence ( P k ) k of projections, where P k is the projection onto the rational half space h k .By Theorem 5.3, the mapping g = P k ∈ N − k − P k will satisfy Fix( g ) = T k ∈ N Fix( P k ) = A .By Lemma 5.4, the mapping P K ◦ g : K → K will have the same set of fixed points. Notethat P K is computable by Corollary 2.26, since K is located by Proposition 4.15.In order to prove item (i) of Theorem 5.1, we need to inspect Lemma 5.4 a little closer.We first need another simple lemma. Lemma 5.5.
Let E be real normed space, let C ⊆ E be closed and convex, let S ⊆ C bea dense subset of C and let h be a half space in E . If C ∩ h ◦ is nonempty, then S ∩ h ◦ isdense in C ∩ h .Proof. Let x ∈ C ∩ h , and let c ∈ C ∩ h ◦ . Then the line segment joining c and x is containedin C ∩ h and contains an element b ∈ B ( x, ε/ ∩ h ◦ . Now we choose a ∈ S in a sufficientlysmall ball around b , so that d ( x, a ) < ε .Lemma 5.5 guarantees that intersections of weakly closed and overt sets and rationalhalf spaces are (uniformly) overt. This is a special property, as in general the intersectionoperator on closed sets is (( ψ ⊓ υ ) × ( ψ ⊓ υ ) , υ )-discontinuous (cf. [79, Theorem 5.1.13]). Corollary 5.6.
Let E be a computable Banach space with computable dual E ′ , such thatthe mapping ( · , · ) is ( δ E × δ E ′ , ρ ) -computable. Then intersection of weakly closed convex sets C and closed rational half spaces h with C ∩ h ◦ = ∅ is (cid:0) ( ψ E w ⊓ υ E w ) × ν HB , ( ψ E w ⊓ υ E w ) (cid:1) -computable, and hence (cid:0) ( ψ E w ⊓ υ E w ) × ν HB , ψ dist (cid:1) -computable.Proof. Since ν HB ≤ ψ , we can always uniformly compute a ψ -name of the intersection. Inorder to compute an υ -name, enumerate all elements given by the υ -name of C which arealso contained in h ◦ . The above lemma guarantees that this yields an υ -name of C ∩ h .The second claim follows from ( ψ E w ⊓ υ E w ) ≡ ψ dist , which in turn follows from Proposition4.15. Lemma 5.7.
Let H be a real Hilbert space, K ⊆ H be closed, bounded and convex, let f : K → K be nonexpansive, h be a half space such that h ◦ ∩ K = ∅ and let S ⊆ K be densein K . Let A = h ∩ K . Then Fix( f ) ∩ h = ∅ if and only if ∃ x ∈ h ◦ ∩ S. ∃ n ∈ N . (cid:18) || f ( x ) − x || > − n ∧ || P A ( f ( x )) − x || < − n − B (cid:19) , (5.1) where B ≥ sup {|| x || + 1 (cid:12)(cid:12) x ∈ K } .Proof. Let us first prove the forward direction. By the Browder-G¨ohde-Kirk Theorem, P A ◦ f | A : A → A has a fixed point ˜ x ∈ h ∩ K . Since, by assumption, ˜ x is not a fixed point of f , there exists an m ∈ N with || f (˜ x ) − ˜ x || > − m . By Lemma 5.5, S ∩ h ◦ is dense in h ∩ K , so that we may choose x ∈ S ∩ h ◦ with || x − ˜ x || < − m − B . Then || P A ( f ( x )) − x || ≤ || P A ( f (˜ x )) − ˜ x || + || P A ( f ( x )) − P A ( f (˜ x )) || + || x − ˜ x ||≤ || x − ˜ x || < − m − B = 2 − m +1) − B .
And (using B ≥ || f ( x ) − x || ≥ || f (˜ x ) − ˜ x || − || x − ˜ x || > − m − − m − B ≥ − m − − m − > − m − . For the converse direction, we proceed by contrapositive. We suppose that there exists y ∈ Fix( f ) ∩ h (and hence y ∈ Fix( P A ◦ f | A )) and show ∀ x ∈ h ◦ ∩ S. ∀ n ∈ N . (cid:18) || f ( x ) − x || > − n → || P A ( f ( x )) − x || ≥ − n − B (cid:19) . Let x ∈ h ◦ ∩ S with || f ( x ) − x || > − n . Since f is nonexpansive, we have || y − x || ≥ || f ( y ) − f ( x ) || = || f ( y ) − P A ( f ( x )) || + || P A ( f ( x )) − f ( x ) || + 2( y − P A ( f ( x )) , P A ( f ( x )) − f ( x )) . Now, by the variational inequality, 2( y − P A ( f ( x )) , P A ( f ( x )) − f ( x )) ≥
0, so that || y − x || ≥ || f ( y ) − P A ( f ( x )) || + || P A ( f ( x )) − f ( x ) || = || y − x || + || x − P A ( f ( x )) || + 2( y − x, x − P A ( f ( x )))+ || P A ( f ( x )) − x || + || x − f ( x ) || + 2( P A ( f ( x )) − x, x − f ( x )) , which entails that0 ≥ || f ( x ) − x || + 2( P A ( f ( x )) − x, x − f ( x ) − y + x ) ≥ || f ( x ) − x || − || P A ( f ( x )) − x || · || x − f ( x ) − y + x ||≥ || f ( x ) − x || − || P A ( f ( x )) − x || B, and hence || P A ( f ( x )) − x || ≥ − n − B .
Note that it follows from Corollary 5.6 that the projection onto A in Lemma 5.7 is com-putable. Proof of Theorem 5.1 ( i ) . Given a nonexpansive mapping f : K → K , we want to computea κ HB -name of Fix( f ). We need to compute a weakly compact set L ∈ K ( E w ) with L ⊇ Fix( f ), and a list of all rational half spaces containing f . Since K contains Fix( f ) andis computably weakly compact by Theorem 4.4 (ii) we may put L = K , so it suffices to listall rational half spaces containing Fix( f ). In fact, by Lemma 4.12 it suffices to computea list of rational half spaces ( h n ) n satisfying T n ∈ N h ◦ n = Fix( f ). In order to do so, weenumerate two different lists L and L of half spaces and interleave them. The first list L consists of all rational half spaces containing K in their interior. This list is computablesince K is computably weakly compact, and hence κ HB -computable by Theorem 4.11. Inorder to compute the second list L , we first enumerate all rational half spaces h such that h ◦ ∩ K = ∅ and h C ∩ K = ∅ . This is possible because K is computably overt. Out of these OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 35 half spaces we only enumerate those which satisfy Fix( f ) ∩ ( h ◦ ) C = ∅ . In order to verifythis property we apply Lemma 5.7 to the half space ( h ◦ ) C . Note that we can compute theprojection onto K ∩ ( h ◦ ) C by Corollary 5.6, so that the property (5.1) in Lemma 5.7 becomessemi-decidable. Now, Fix( f ) ∩ ( h ◦ ) C = ∅ is equivalent to h ◦ ⊇ Fix( f ), and it is easy to seethat the list ( h n ) n we obtain by interleaving L and L satisfies T n ∈ N h ◦ n = Fix( f ).Theorem 5.1 now allows us to determine the Weihrauch degree of the weak and strongBrowder-G¨ohde-Kirk theorem. Theorem 5.8.
Let H be a computable Hilbert space and K ⊆ H be nonempty, bounded,convex, computably weakly closed, and computably overt. Then BGK K ≡ W ConvC w → n K , and WBGK ≡ W ConvC w → w K . If K is computably compact, then BGK K ≡ W WBGK K ≡ W ConvC K . Proof.
The equivalences BGK K ≡ W ConvC w → n K and WBGK ≡ W ConvC w → w K follow fromTheorem 5.1, together with the fact that by Theorem 4.4 K is computably weak* compact,so that id : A co ( K ) → K co ( K ) is computable. If K is computably compact, then byProposition 4.13 we have δ E (cid:12)(cid:12) K ≡ δ E w (cid:12)(cid:12) K , which yields BGK K ≡ W WBGK K ≡ W ConvC K (note that if two representations of the same space are equivalent, then the induced canonicalrepresentations of closed sets are - by construction - equivalent as well).Theorem 5.8 also shows that on a non-compact domain, negative information on theweak closedness of a set is much stronger than negative information on its norm-closedness.We have for instance BGK B ℓ ≤ W Proj B ℓ ≡ W lim, and so ConvC w → n B ℓ ≤ W lim, while al-ready UC n → n B ℓ is equivalent to the extremely non-effective principle C N N . In finite dimension,the degree of BGK K is always strictly below WKL because of Corollary 3.11. On the unitball in ℓ this is no longer the case. Theorem 5.9.
There exists a computable mapping T : ⊆ [0 , N → K co ( B ℓ w ) with dom T = { x ∈ [0 , N | x ( n ) ≤ x ( n +1) } such that for all x ∈ dom T we have T ( x ) = { a } with || a || = lim x ( n ) .Proof. Let x ∈ dom T . Put a (0) = x (0) and a ( n + 1) = p x ( n + 1) − x ( n ) . Then we have a ( n ) + · · · + a (0) = x ( n ) . Now, put T ( x ) = { a } . Note that a is δ ℓ w -computable relativeto x , so we can compute { a } in K co ( B ℓ w ): in order to compute the characteristic functionof { a } C into Sierpi´nski space we simply check for inequality with a component-wise. Thisallows us to compute { a } as a point in A co ( B ℓ w ), and thus as a point of K co ( B ℓ w ), usingthat the identity id : A co ( B ℓ w ) → K co ( B ℓ w ) is computable, since B ℓ w is computably weaklycompact by Theorem 4.4. Choosing from a set in K co ( B ℓ ) hence allows us computably translate a ρ < -name toa ρ -name of a given real number x ∈ [0 , Corollary 5.10.
BGK B ℓ ≡ W UC w → n B ℓ ≡ W lim . In particular BGK B ℓ ≡ W Proj B ℓ .Proof. We have UC w → n B ℓ ≤ W ConvC w → n B ℓ ≡ W BGK B ℓ , the latter by Theorem 5.8. Also,BGK B ℓ ≤ W lim by Proposition 3.5 and Proposition 3.8. It follows from Theorem 5.9that UC w → n B ℓ allows us to determine the limit of any computable monotonically increasingsequence x ∈ [0 , N , since lim n →∞ x ( n ) = || UC w → n B ℓ ( T ( x )) || and ||·|| is ( δ ℓ , ρ )-computable. We hence have lim ≤ W UC w → n B ℓ , which finishes the proof.In particular we have the following non-uniform corollary: Corollary 5.11.
There exists a computable nonexpansive self-map of the closed unit ballin ℓ with a unique fixed point, which is uncomputable. Compare Corollary 5.11 to Theorem 3.15: on a compact domain, any computablefunction without computable fixed points necessarily has uncountably many fixed points,since otherwise it has at least one isolated fixed point which is then computable by Theorem2.10. If we drop compactness, even unique solutions may be uncomputable. Note however,that since the unit ball in ℓ is still computably weakly compact, unique fixed points on B ℓ are still “weakly computable”, in the sense that they are computable as elements in therepresented space ℓ w . In particular, their coordinates with respect to an orthonormal basisare still computable.On a computably compact domain, the Weihrauch degree of the theorem is still at mostWKL. We can now show that it is in fact equivalent to WKL on the Hilbert cube. In orderto do so, we will first have to define the parallelisation of Weihrauch degrees, which wasintroduced in [17]. Definition 5.12.
Let f : ⊆ X ⇒ Y be a partial multimapping. The parallelisation ˆ f of f is the multimapping ˆ f : ⊆ X N ⇒ Y N , ˆ f ( λn.x ( n )) = λn.f ( x ( n )).It is not hard to see that f ≤ W g implies ˆ f ≤ W ˆ g (cf. also Proposition 4.2 in [17]).The following theorem is essentially due to [17] (cf. also Theorem 6.2 and the subsequentcomment in [16]). Theorem 5.13. d IVT ≡ W WKL . Theorem 5.14.
Let H = { P i ∈ N α i e i | α i ∈ [0 , − i ] } be the Hilbert cube in ℓ . Then BGK H ≡ W WKL .Proof.
Clearly, H is computably compact, so BGK H ≡ W ConvC H ≤ W C H ≡ W WKL. Inorder to prove the converse direction, we show that d IVT ≤ W ConvC H . Since d IVT ≡ W WKLand ConvC H ≡ W BGK H , it follows that WKL ≤ W BGK H . By Proposition 3.21 we haveIVT ≡ W ConvC [0 , and so d IVT ≡ W \ ConvC [0 , . Let ([ a n , b n ]) n be a sequence of closedintervals in ( A co ([0 , N . Consider the set A = { P i ∈ N α i e i | α i ∈ [ a i − i , b i − i ] } ⊆ H .Then A is computable as a point in K co ( H ) relative to ([ a n , b n ]) n . Clearly, choosing apoint in A allows us to choose a point in ([ a n , b n ]) n , so \ ConvC [0 , ≤ W ConvC H ≡ W BGK H . OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 37
Theorem 5.14 can (essentially) be viewed as a uniform strengthening of Theorem3.15. Notice that the proof of Theorem 3.15 can be utilized to establish the reduction \ LLPO ≤ W BGK H , which yields a slightly different proof of Theorem 5.14, since \ LLPO ≡ W WKL (again, cf. [17]). The proof of Theorem 5.14 can also be used to show WBGK B ℓ ≡ W WKL. We now have a fairly good idea of the computational content of the Browder-G¨ohde-Kirk theorem. It follows from [55] that (BGK [0 , n ) n ∈ N is a strictly increasing sequenceof Weihrauch degrees, all strictly below WKL. On the compact but infinite dimensionalHilbert cube H the theorem becomes equivalent to WKL. If we drop compactness andconsider the theorem on the unit ball in ℓ , it becomes even more non-effective, and inparticular equivalent to computing rates of convergence for fixed point iterations, but isstill much more effective than full choice on B ℓ .In finite dimension, a computable nonexpansive self-map of a computably compactdomain always has computable fixed points by Theorem 3.10, and this relies solely on thefact that the fixed point set is convex. This is reminiscent of the fact that unique zeroes ofcomputable functions are always (in this case even uniformly) computable. A typical featureof such results is that they assert the existence of computable objects, but the computationalcomplexity of these objects is unbounded. This is also the case here: using similar techniquesas in [41], we can strengthen Theorem 5.1 ( ii ) to assert for every nonempty co-semi-decidableand convex A ⊆ K the existence of a polynomial-time computable nonexpansive f : K → K such that Fix( f ) = A , at least in the case where K is computably compact, and so inparticular in the finite-dimensional case (if K is not computably compact there is no uniformmajorant on the names of the points in K , so one would have to work in the framework of second-order complexity [39]). This allows us to characterise the computational complexityof fixed points of Lipschitz-continuous polynomial-time computable functions according totheir Lipschitz constant. Theorem 5.15.
Let [0 , be the unit square in Euclidean space R . Let f : [0 , → [0 , be polynomial-time computable and Lipschitz-continuous with Lipschitz constant L . Then: − If L < , f has a unique polynomial time computable fixed point, which is uniformlycomputable relative to the promise that L < and uniformly polynomial time computablerelative to the promise that L < − ε for some fixed ε > . − If L = 1 , the fixed point set of f can be any nonempty co-semi-decidable convex subset of [0 , . The multi-valued operator mapping f to some fixed point is realiser-discontinuousand hence uncomputable, but f still has computable fixed points. However, there is nocomputable bound on the computational complexity of the fixed points of f . − If L > , f may not have any computable fixed points. The third claim in Theorem 5.15 follows from a strengthening of the results in [19], whichthe authors of that paper have recently obtained, but which seems to be unpublished as ofyet. 6.
Further Results and Possible Generalisations
The special case of Theorem 5.1 where the underlying Hilbert space is two-dimensionalseems to generalise to uniformly convex and smooth real Banach spaces of dimension two.Note that the first item of the theorem becomes trivial in finite dimension. A Banach spaceis called smooth , if its dual space is strictly convex and uniformly smooth if its dual spaceis uniformly convex. For instance, all L p -spaces with 1 < p < ∞ are uniformly convex and uniformly smooth. The two notions of smoothness and uniform smoothness coincide infinite dimension. Conjecture 6.1.
Let E be a uniformly convex, smooth, computable Banach space of di-mension two, and let K ⊆ E be bounded, convex, and located . Then the multi-valuedmapping Fix − : K co ( K ) \ {∅} ⇒ N ( K ) is computable. The proof of this result would be almost identical to that of Theorem 5.1. The onlyplaces where we used that the underlying space is a Hilbert space were Theorem 2.23, whichasserts that the projection onto each convex, closed set is nonexpansive, and Lemma 5.4.In general the projection onto a closed and convex subset of a Banach space will not benonexpansive. In fact, this property characterises Hilbert spaces (cf. [65]). However, weonly need the existence of a computable nonexpansive retraction onto each located convexsubset. A retraction Q : E → K of E onto a nonempty subset K ⊆ E is called sunny , if Q ( αx + (1 − α ) Q ( x )) = Q ( x ) for all x ∈ E, α ∈ [0 , . Geometrically, this means that for all x / ∈ K , all points on the ray defined by x and Q ( x )with initial point Q ( x ) are mapped onto the same point Q ( x ). It is well known that in asmooth Banach space of dimension two, sunny nonexpansive retractions onto closed convexsubsets exist and are unique. Consequently, they are computable. Theorem 6.2 ([38]) . Let E be a smooth real Banach space of dimension two. Then forevery nonempty closed convex subset C of E , there exists a nonexpansive sunny retractionof E onto C . Theorem 6.3 ([26]) . Let E be a smooth real Banach space. Let K ⊆ C be two nonempty,closed, and convex subsets of E . Then there exists at most one sunny nonexpansive retrac-tion of C onto K . Theorem 6.4.
Let E be a smooth computable Banach space of dimension two. Let C ⊆ E be nonempty, convex, bounded, and located. Then the mapping SRet : A codist ( C ) \ {∅} 7→ N ( C ) that maps K to the unique sunny nonexpansive retraction of C onto K , is computable.Proof (sketch). The set of nonexpansive self-maps of C is computably compact, since itis equicontinuous and C is compact. We can verify if a given map f : C → C doesnot leave all points of K fixed, if it maps a point of C to a point outside of K , and if f ( αx + (1 − α ) f ( x )) = f ( x ) for some x ∈ C , α ∈ [0 , C onto K is co-semi-decidable relative to (a ψ dist -name of) K .Theorems 6.2 and 6.3 assert that it is a singleton. It follows that the operator is uniformlycomputable. Recall that in finite dimension a nonempty closed set is located if and only if it is co-semi-decidable andcomputably overt (cf. also [79]).
OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 39
The other result that relies on Hilbert space techniques is Lemma 5.4, which uses thenonexpansiveness of the projection and the variational inequality. In principle we couldreplace the projection by the sunny nonexpansive retraction onto the domain, but thequestion remains whether this will always leave the fixed point set unchanged.
Conjecture 6.5.
Let E be a smooth and uniformly convex Banach space of dimension two,let K ⊆ E be nonempty, closed, bounded, and convex and let f : K → E be nonexpansivewith Fix( f ) = ∅ . Let P : E → K be the sunny nonexpansive retraction onto K . Then wehave Fix( P ◦ f ) = Fix( f ) .Proof of to Conjecture 6.1 up to Conjecture 6.5. We could now prove Conjecture 6.1 anal-ogously to Theorem 5.1: we are given a convex, closed subset A of K as a κ HB -name andwant to construct a nonexpansive mapping f : K → K with Fix( f ) = A . Let ( h n ) n be thesequence of half spaces given by the κ HB -name. Since h n ∩ K = ∅ for all n , we can computea ψ ω dist -name of the sequence ( h n ∩ K ) n ∈ N thanks to Corollary 5.6. Now, Theorem 6.4 allowsus to compute a [ δ K → δ K ] ω -name of the sequence ( f n ) n of sunny nonexpansive retractionsof K onto h n ∩ K . Applying Theorem 5.3, we obtain a nonexpansive mapping g : K → E with Fix( g ) = T n ∈ N h n = A . Finally, we use the computable nonexpansive sunny retractiononto K and Conjecture 6.5 to obtain a self-map f of K with Fix( f ) = A .The only “missing piece” in this proof is Conjecture 6.5. By replacing this conjectureby a weaker statement that we can prove, we obtain a weaker version of Conjecture 6.1,which is almost as good. Lemma 6.6.
Let E be a uniformly convex Banach space of dimension two, let K ⊆ E benonempty, closed, bounded, and convex. Suppose that K ◦ is nonempty and that ∂K doesnot contain any line segments, and let f : K → E be nonexpansive with Fix( f ) = ∅ . Let P : E → K be the sunny nonexpansive retraction onto K . Then P ◦ f is nonexpansive aswell with Fix( P ◦ f ) = Fix( f ) .Proof. Clearly, P ◦ f is nonexpansive with Fix( f ) ⊆ Fix( P ◦ f ). Suppose that there exists x ∈ Fix( P ◦ f ), which is not a fixed point of f . Since P is sunny, x ∈ ∂K . Let y ∈ Fix( f ).Since Fix( f ) is closed, there exists ε > B ( x, ε ) ⊆ Fix( f ) C . Since P ◦ f isnonexpansive, the line segment L joining y and x is contained in Fix( P ◦ f ). By hypothesis,the line segment without its endpoints has to lie in K ◦ (it is easy to see that if a convex setcontains three points of a line segment in its boundary, it contains the whole line segmentin its boundary). Hence, there exists z ∈ K ◦ ∩ L ∩ B ( x, ε ). Contradiction. Remark . A similar proof shows that we may replace the condition that ∂K contains noline segments, by the condition that Fix( f ) ∩ K ◦ = ∅ . In this case we do not even requirethe retraction to be sunny. Theorem 6.8.
Let E be a uniformly convex, smooth computable Banach space of dimen-sion two, and let K ⊆ E be nonempty, bounded, convex, and located. Suppose that either dim K = 2 and ∂K contains no line segments or dim K = 1 . then the multi-valued mapping Fix − : K co ( K ) \ {∅} ⇒ N ( K ) is computable. In a uniformly convex space the unit ball contains no line segments, so B E is an exampleof an admissible domain K . In particular, every co-semi-decidable, convex subset of B E isthe fixed point set of some computable, nonexpansive self-map of B E . Proof of Theorem 6.8.
If dim K = 1, we introduce suitable coordinates in which K is con-tained in the x -axis and use the construction of Theorem 3.17. If dim K = 2, we use theproof of Conjecture 6.1. Note that here we may replace Conjecture 6.5 by Lemma 6.6, sothe proof is complete.The obvious question at this point is whether Conjecture 6.1 might generalise to higherdimensional Banach spaces. While most of the results we used in the proof at least generaliseto finite-dimensional smooth and uniformly convex computable Banach spaces the mainobstruction appears to be the existence of nonexpansive retractions. Our proof uses thefact that there exist nonexpansive retractions onto every rational half space, but if E isa Banach space of dimension at least three and there exist nonexpansive retractions ontoeach two-dimensional subspace, then E is a Hilbert space. Similarly, the unit ball of an atleast three-dimensional Banach space E is a nonexpansive retract of E if and only if E is aHilbert space (cf. [25]). On the other hand, every fixed point set of a nonexpansive mapping f : K → K is a nonexpansive retraction of K . In view of these results it seems likely thatTheorem 5.1 characterises computable Hilbert space of dimension three or higher.Finally, we extend the stronger upper bound obtained in Proposition 3.8 for compactsets and Hilbert space to the noncompact case in uniformly convex and uniformly smoothspaces. For this we need a generalisation of Theorem 1.2 due to Reich [67]. We will onlystate a special case. Theorem 6.9 ([67]) . Let E be a uniformly smooth, uniformly convex Banach space, let K ⊆ E be nonempty, closed, bounded and convex, let f : K → K be nonexpansive, and let x ∈ K . Put α n = 1 − ( n + 2) − . Then the sequence ( x n ) n defined by the iteration scheme x = x and x n +1 = (1 − α n ) x + α n f ( x n ) converges to a fixed point of f . Note that the iteration defined in Theorem 6.9 converges to a retraction onto the fixedpoint set of f . In fact, one can show that the sequence ( x n ) n converges to Q ( x ), where Q is the unique sunny nonexpansive retraction of K onto Fix( f ). Theorem 6.10.
Let E be a uniformly convex, uniformly smooth computable Banach space.Let K ⊆ E be nonempty, bounded, convex, co-semi-decidable, and computably overt. Then Proj K ≤ W lim . Proof.
We use similar ideas as in the proof of Proposition 3.8. Again we exploit the factthat we can actually compute countably many instances of lim in parallel. As in the proofof the general upper bound in Proposition 3.8, we use countably many instances of lim toobtain a function µ : N → N satisfying 2 − µ ( n ) ≤ η E (2 − n ), where η E is a modulus of uniformconvexity for E , and another batch of countably many instances to obtain an approximationto the distance function to Fix( f ) from below. Since K is computably overt, it contains acomputable dense sequence ( x n ) n . Let x n = x n and x k +1 n = (1 − α k ) x n + α k f ( x kn ) with α k asin Theorem 6.9. Using another countable batch of instances of lim, we obtain the sequence(lim k →∞ x kn ) n , which is dense in Fix( f ), since the iteration defines a retraction of K ontoFix( f ). Using this sequence we can compute the distance function to Fix( f ) from above,so that we obtain the distance function to Fix( f ) as an element of C ( K, R ). Together withCorollary 2.25 this establishes the reduction. OMPUTATIONAL PROBLEMS IN METRIC FIXED POINT THEORY 41
Acknowledgements.
The present work was motivated by a question by Ulrich Kohlenbach, whether the Krasnosel-ski-Mann iteration has nonuniformly computable rates of convergence. He has also providedmany valuable insights both concerning fixed point theory and computability theory. Thiswork has greatly benefited from discussions with Vasco Brattka, Arno Pauly, Guido Gher-ardi, and Martin Ziegler. I would also like to thank the anonymous referees for pointingout many shortcomings in the original version of this paper.
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