On Characterizations of Metric Regularity of Multi-valued Maps
aa r X i v : . [ m a t h . F A ] N ov On Characterizations of Metric Regularityof Multi-valued Maps ∗ M. Ivanov and N. ZlatevaNovember 21, 2018
Dedicated to Professor Alexander D. Ioffe
Abstract
We provide a new proof along the lines of the recent book of A. Ioffeof a 1990’s result of H. Frankowska showing that metric regularity of amulti-valued map can be characterized by regularity of its contingentvariation – a notion extending contingent derivative.
Keywords: surjectivity, metric regularity, multi-valued map.
AMS Subject Classification : 49J53, 47H04, 54H25.
Metric regularity, as well as, the equivalent to it linear openness and pseudo-Lipschitz property of the inverse, are very important concepts in VariationalAnalysis. They have been intensively studied as it can be seen in a numberof recent monographs, e.g. [1, 8, 3, 7] and the references therein. A very richand instructive survey on metric regularity is the book of A. Ioffe [6].It may be noted in Chapter V of [6] that the modulus of regularity ofa multi-valued map between Banach spaces is estimated in terms of the ∗ Research supported by the Scientific Fund of Sofia University under grant 80-10-133/25.04.2018.
X, d ) and (
Y, d ) be metric spaces and let F : X ⇒ Y be a multi-valued map. If V ⊂ Y the restriction F V is defined by F V ( x ) := F ( x ) ∩ V, ∀ x ∈ X, see [6, p.54]. The properties related to the so restricted map are called restricted .For example, the multi-valued map F : X ⇒ Y is called restrictedlyMilyutin regular on ( U, V ), where U ⊂ X and V ⊂ Y , if there exists anumber r > B ( v, rt ) ∩ V ⊂ F ( B ( x, t ))whenever ( x, v ) ∈ Gr F ∩ ( U × V ) and B ( x, t ) ⊂ U , where B ( x, t ) is theclosed ball with center x and radius t : B ( x, t ) := { u ∈ X : d ( u, x ) ≤ t } , andGr F = { ( x, v ) : v ∈ F ( x ) } .The supremum of all such r is called modulus of surjection , denoted bysur m F V ( U | V ) . By convention, sur m F V ( U | V ) = 0 means that F is not restrictedly Milyutinregular on ( U, V ).This notion taken from [6] is explained in great detail in Section 2 below.In the literature, e.g. [6, Section 5.2], there are various estimates ofsur m F V ( U | V ) and related moduli in terms of derivative-like objects. Unlikethe so called co-derivative criterion , see [6, Section 5.2.3], most of the primal estimates are not characteristic in general. Here we re-establish one primalcriterion which complements [6, Section 5.2.2] and is, moreover, characteris-tic. It is essentially done by H. Frankowska in [4], see also [5]. There a newderivative-like object is defined as follows.2et ( X, d ) be a metric space, ( Y, k · k ) be a Banach space, F : X ⇒ Y be a multi-valued map. For ( x, y ) ∈ Gr F the contingent variation of F at( x, y ) is the closed set F (1) ( x, y ) := lim sup t → + F ( B ( x, t )) − yt , where lim sup stands for the Kuratowski limit superior of sets.Equivalently, v ∈ F (1) ( x, y ) exactly when there exist a sequence of reals t n ↓ x n , y n ) ∈ Gr F such that d ( x, x n ) ≤ t n and (cid:13)(cid:13)(cid:13)(cid:13) v − y n − yt n (cid:13)(cid:13)(cid:13)(cid:13) → , when n → ∞ . This notion extends the so-called contingent, or graphical, derivative usu-ally denoted by DF ( x, y ), e.g. [6, pp.163, 202].Our main result can now be stated. As usual, B Y denotes the closed unitball of the Banach space ( Y, k · k ). Theorem 1.
Let ( X, d ) be a metric space and ( Y, k · k ) be a Banach space,let U ⊂ X and V ⊂ Y be non-empty open sets. Let F : X ⇒ Y be a multi-valued map with complete graph. F is restrictedly Milytin regular on ( U, V ) with sur m F V ( U | V ) ≥ r > ifand only if F (1) ( x, v ) ⊃ rB Y for all ( x, v ) ∈ Gr F ∩ ( U × V ) . (1)This result is essentially established by H. Frankowska in [4, Theorem 6.1and Corollary 6.2]. However, there it is presented as a characterization oflocal modulus of regularity in terms of the local variant of the condition (1).Here we render the characterization global. The technique in [4] is different,but it again depends on Ekeland Variational Principle.The rest of the article is organized as follows. In Section 2 we providefor reader’s convenience the relevant material from [6]. We also present inanother form the first criterion for Milyutin regularity from [6]. In Section 3we prove Theorem 1. 3 Milyutin regularity
Let (
X, d ) and (
Y, d ) be metric spaces. Let U ⊂ X and V ⊂ Y , let F : X ⇒ Y be a multi-valued map and let γ ( · ) be extended real-valued function on X assuming positive values (possibly infinite) on U . Definition 2. ( linear openness , [6, Definition 2.21]) F is said to be γ -open at linear rate on ( U, V ) if there is an r > such that B ( F ( x ) , rt ) ∩ V ⊂ F ( B ( x, t )) , if x ∈ U and t < γ ( x ) , i.e. B ( v, rt ) ∩ V ⊂ F ( B ( x, t )) , whenever ( x, v ) ∈ Gr F , x ∈ U and t < γ ( x ) . Denote by sur γ F ( U | V ) the upper bound of all such r > mod-ulus of γ -surjection of F on ( U, V ). If no such r exists, set sur γ F ( U | V )=0. Definition 3. ( metric regularity , [6, Definition 2.22]) F is said to be γ -metrically regular on ( U, V ) if there is κ > such that d ( x, F − ( y )) ≤ κd ( y, F ( x )) , provided x ∈ U , y ∈ V and κd ( y, F ( x )) < γ ( x ) . Denote by reg γ F ( U | V ) the lower bound of all such κ > modulus of γ -metric regularity of F on ( U, V ). If no such κ exists, setreg γ F ( U | V ) = ∞ . Theorem 4. ( equivalence theorem , [6, Theorem 2.25]) The following areequivalent for any metric spaces X , Y , any F : X ⇒ Y , any U ⊂ X , V ⊂ Y and any extended real-valued function γ ( · ) which is positive on U :a) F is γ -open at linear rate un ( U, V ) ;b) F is γ -metrically regular on ( U, V ) .Moreover (under the convention . ∞ = 1 ), sur γ F ( U | V ) . reg γ F ( U | V ) = 1 . Definition 5. ( regularity , [6, Definition 2.26]) We say that F : X ⇒ Y is γ -regular on ( U, V ) if the equivalent properties of Theorem 4 are satisfied. efinition 6. ( Miluytin regularity , [6, Definition 2.28]) Set m U ( x ) := d ( x, X \ U ) . We shall say that F is Milyutin regular on ( U, V ) if it is γ -regular on ( U, V ) with γ ( x ) = m U ( x ) . We will need also
Ekeland Variational Principle (see [9, p.45]): Let(
M, d ) be a complete metric space, and f : M → R ∪ { + ∞} be a proper,lower semicontinuous and bounded from below function. Assume that f ( x ) ≤ inf f + λε for some x ∈ M and λε >
0. Then there is y ∈ M such that(i) f ( y ) ≤ f ( x ) − λd ( x, y );(ii) d ( x, y ) ≤ ε ;(iii) f ( x ) + λd ( x, y ) ≥ f ( y ), for all x ∈ M .The following characterization of Milyutin regularity is very similar inform (in fact equivalent) to the so called first criterion for Milyutin reg-ularity , see [6, Theorem 2.47]. It is also similar to [2, Proposition 2.2],but there it is stated in local form. We present here a proof for reader’sconvenience.Following [6, p.35] for ξ > d ξ the product metric d ξ (( x , y ) , ( x , y )) := max { d ( x , x ) , ξd ( y , y ) } , (2)where x i ∈ X , y i ∈ Y , i = 1 ,
2, and (
X, d ) and (
Y, d ) are metric spaces.
Theorem 7.
Let ( X, d ) , ( Y, d ) be metric spaces. Let F : X ⇒ Y be amulti-valued map with complete graph. Let U ⊂ X and V ⊂ Y . Then sur m F ( U | V ) = sup { r ≥ ∃ ξ > such that ∀ ( x, v ) ∈ Gr F, x ∈ U, y ∈ V satisfying < d ( y, v ) < rm U ( x ) ∃ ( u, w ) ∈ Gr F such that d ( y, w ) < d ( y, v ) − rd ξ (( x, v ) , ( u, w )) } . (3) Proof.
Let us denote by s the left hand side of the above equation, i.e. s := sur m F ( U | V ). In other words, s = sup { r ≥ B ( v, rt ) ∩ V ⊂ F ( B ( x, t )) , ∀ ( x, v ) ∈ Gr F, x ∈ U, t < m U ( x ) } . Denote by s the right hand side of the equation.We need to show that s = s . 5irst, we will show that s ≤ s .If s = 0 we have nothing to prove.Let s >
0. Take 0 < r < r ′ < s . Let x ∈ U , v ∈ F ( x ) be fixed. Let y ∈ V be such that 0 < d ( y, v ) < rm U ( x ). In particular 0 < d ( y, v ) < r ′ m U ( x ).Set t := d ( y, v ) r ′ . Then t < m U ( x ). By r ′ < s = sur m F ( U | V ) and by thedefinition of sur m F ( U | V ) it holds that y ∈ B ( v, r ′ t ) ∩ V ⊂ F ( B ( x, t )), i.e. y ∈ F ( B ( x, t )). So, there exists u ∈ B ( x, t ) such that y ∈ F ( u ).Fix ξ such that 0 < ξr ′ <
1. Then d ξ (( x, v ) , ( u, y )) = max { d ( x, u ) , ξd ( v, y ) } ≤ max { t, ξr ′ t } = t max { , ξr ′ } = t, so r ′ d ξ (( x, v ) , ( u, y )) ≤ r ′ t = d ( y, v ) . Observe that d ξ (( x, v ) , ( u, y )) > d ( v, y ) >
0. The latter and r ′ > r yield rd ξ (( x, v ) , ( u, y )) < r ′ t < d ( y, v ) , or 0 < d ( y, v ) − rd ξ (( x, v ) , ( u, y )) . Since 0 = d ( y, y ) we get that d ( y, y ) < d ( y, v ) − rd ξ (( x, v ) , ( u, y ))and (3) holds with w = y as ( u, y ) ∈ Gr F .This means that r ≤ s . Finally, s ≤ s .Second, we will prove that s ≤ s .If s = 0 we have nothing to prove.Let now s >
0. Let 0 < r < s . Let us fix x ∈ U , v ∈ F ( x ) and0 < t < m U ( x ).Fix y ∈ V such that d ( y, v ) ≤ rt , i.e. y ∈ B ( v , rt ) ∩ V . Let M := Gr F ,and let ξ > r in the definition of s . It is clear that ( M, d ξ )is a complete metric space.Consider the function f : M → R defined as f ( u, w ) := d ( w, y ).Then f ≥ M . Since f ( x , v ) = d ( v , y ) ≤ rt ,by Ekeland Variational Principle there exists ( x , v ) ∈ M such that(i) f ( x , v ) ≤ f ( x , v ) − rd ξ (( x , v ) , ( x , v ));(ii) d ξ (( x , v ) , ( x , v )) ≤ t ; 6iii) f ( u, w ) + rd ξ (( u, w ) , ( x , v )) ≥ f ( x , v ), for all ( u, w ) ∈ M .Or, equivalently(i) d ( v , y ) ≤ d ( v , y ) − rd ξ (( x , v ) , ( x , v )) ≤ rt − rd ξ (( x , v ) , ( x , v ));(ii) d ( x , x ) ≤ t, ξd ( v , v ) ≤ t ;(iii) d ( w, y ) + rd ξ (( u, w ) , ( x , v )) ≥ d ( v , y ), for all ( u, w ) ∈ M .Set p := d ( v , y ).Assume that p >
0. Take t ′ such that t < t ′ < m U ( x ). For x ∈ B (cid:16) x , pr + t ′ − t (cid:17) we have that d ( x, x ) ≤ d ( x, x ) + d ( x , x ) ≤ pr + t ′ − t + d ( x , x )(using (i)) ≤ rt − rd ( x , x ) r + t ′ − t + d ( x , x )= t − d ( x , x ) + t ′ − t + d ( x , x )= t ′ . Hence B (cid:16) x , pr + t ′ − t (cid:17) ⊂ B ( x , t ′ ) ⊂ U . Then pr + t ′ − t ≤ m U ( x ), and pr < m U ( x ) because t ′ − t >
0. Hence, 0 < d ( v , y ) < rm U ( x ). But now (3)contradicts (iii).Therefore, p = 0 and then y = v ∈ F ( x ). Since by (ii) x ∈ B ( x , t ), wehave y ∈ F ( B ( x , t )) ∩ V .Since x ∈ U , v ∈ F ( x ), y ∈ B ( v , rt ) ∩ V and 0 < t < m U ( x ) werearbitrary, this means that r ≤ s . Since 0 < r < s was arbitrary, s ≤ s ,and the proof is completed.In the definitions of regularity properties it is not required that F ( x ) ⊂ V .Such requirements can be included in the definitions as follows. Definition 8. ( restricted regularity , [6, Definition 2.35]) Set F V ( x ) := F ( x ) ∩ V . We define restricted γ -openness at linear rate and restricted γ -metric regularity on ( U, V ) by replacing F by F V . The equivalence Theorem 4 also holds for the restricted versions of theproperties. The case is the same with Theorem 7, where the proof needs onlysmall adjustments when working with F V instead of F .7 Proof of the main result
The proof of our main result relies on the following Lemma.
Lemma 9.
Let ( X, d ) be a metric space and ( Y, k · k ) be a Banach space, let U ⊂ X and V ⊂ Y be non-empty sets and let F : X ⇒ Y be a multi-valued map.If for some r > it holds that F (1) ( x, v ) ⊃ rB Y for all ( x, v ) ∈ Gr F ∩ ( U × V ) , then for any < r ′ < r and any ξ ∈ ( r − , ( r ′ ) − ) it holds that for any x ∈ U and any v ∈ F V ( x ) and y ∈ V \ { v } there is ( u, w ) ∈ Gr F such that k y − w k < k y − v k − r ′ d ξ (( x, v ) , ( u, w )) . Proof.
Let r ′ ∈ (0 , r ) be fixed.Fix ξ > r ′ ) − > ξ > r − .Take ( x, v ) such that ( x, v ) ∈ Gr F ∩ ( U × V ).Fix y ∈ V such that 0 < k y − v k .Set ¯ v := r y − v k y − v k . Obviously k ¯ v k = r . By assumption, F (1) ( x, v ) ∋ ¯ v .By definition of the contingent variation there exist t n ↓ u n ∈ X as wellas w n ∈ Y and z n ∈ Y such that w n ∈ F ( u n ), d ( x, u n ) ≤ t n , k z n k → v + t n ¯ v = w n + t n z n . (4)Note first that for n large enough ξ k w n − v k > t n ≥ d ( x, u n ) ⇒ d ξ (( x, v ) , ( u n , w n )) = ξ k w n − v k . (5)Indeed, k w n − v k = t n k ¯ v − z n k ≥ t n ( r − k z n k ) and, since ξ ( r − k z n k ) → ξr > n → ∞ , we have ξ k w n − v k > t n for n large enough.From (4) we have y − w n = y − v − t n v + t n z n . (6)Since y − v − t n v = (cid:0) − t n r k y − v k − (cid:1) ( y − v ) , − t n r k y − v k − > n large enough, we have for such n that k y − v − t n v k = (cid:0) − t n r k y − v k − (cid:1) k y − v k = k y − v k − t n r. Combining the latter with (6) we get for n large enough k y − w n k = k y − v − t n ¯ v + t n z n k≤ k y − v − t n ¯ v k + t n k z n k = k y − v k − t n ( r − k z n k ) . (7)On the other hand, (4) can be rewritten as w n − v = t n ¯ v − t n z n , hence k w n − v k = t n k v − z n k ≤ t n ( r + k z n k ) , and using this estimate we obtain thatlim inf n →∞ t n ( r − k z n k ) r ′ ξ k v − w n k ≥ lim inf n →∞ t n ( r − k z n k ) r ′ ξt n ( r + k z n k ) = 1 r ′ ξ > . From this and (7) we have that for large n k y − w n k < k y − v k − r ′ ξ k v − w n k . Using (5) we finally obtain that for all n large enough k y − w n k < k y − v k − r ′ d ξ (( x, v ) , ( u n , w n ))and the claim follows.Proving our main result is now straightforward. Theorem 1.
Let ( X, d ) be a metric space and ( Y, k · k ) be a Banach space,let U ⊂ X and V ⊂ Y be non-empty open sets. Let F : X ⇒ Y be a multi-valued map with complete graph. F is restrictedly Milytin regular on ( U, V ) with sur m F V ( U | V ) ≥ r > ifand only if F (1) ( x, v ) ⊃ rB Y for all ( x, v ) ∈ Gr F ∩ ( U × V ) . roof. Let F (1) ( x, v ) ⊃ rB Y for all ( x, v ) ∈ Gr F ∩ ( U × V ) . From Lemma 9 and Theorem 7 it follows that sur m F V ( U | V ) ≥ r .Conversely, let sur m F V ( U | V ) ≥ r >
0. This means that B ( v, rt ) ∩ V ⊂ F ( B ( x, t ))whenever ( x, v ) ∈ Gr F , x ∈ U , v ∈ V and t < m U ( x ).Take arbitrary ( x, v ) ∈ Gr F V , x ∈ U and note that m U ( x ) > U is open. Take positive t such that t < m U ( x ).For any y ∈ rB Y it holds that v + ty ∈ B ( v, rt ). Moreover, v + ty ∈ V will be true for small t because V is open. Then, by assumption, v + ty ∈ F ( B ( x, t )), so y ∈ F ( B ( x, t )) − vt which means that y ∈ F (1) ( x, v ). Hence, F (1) ( x, v ) ⊃ rB Y . Acknowledgements.
We wish to express our gratitude for the interestingdiscussions we have had with Professor Ioffe in the summer of 2018 on sometopics in his recent monograph [6], and for his kind attention.10 eferences [1] J. M. Borwein and Q. J. Zhu,
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