Monotone operators and "bigger conjugate" functions
Heinz H. Bauschke, Jonathan M. Borwein, Xianfu Wang, Liangjin Yao
aa r X i v : . [ m a t h . F A ] A ug Monotone operators and “bigger conjugate” functions
Heinz H. Bauschke ∗ , Jonathan M. Borwein † , Xianfu Wang ‡ , and Liangjin Yao § August 12, 2011
Abstract
We study a question posed by Stephen Simons in his 2008 monograph involving “big-ger conjugate” (BC) functions and the partial infimal convolution. As Simons demon-strated in his monograph, these function have been crucial to the understanding andadvancement of the state-of-the-art of harder problems in monotone operator theory,especially the sum problem.In this paper, we provide some tools for further analysis of BC–functions whichallow us to answer Simons’ problem in the negative. We are also able to refute asimilar but much harder conjecture which would have generalized a classical result ofBr´ezis, Crandall and Pazy. Our work also reinforces the importance of understandingunbounded skew linear relations to construct monotone operators with unexpectedproperties.
Primary 47A06, 47H05; Secondary 47B65, 47N10, 90C25
Keywords:
Adjoint, BC–function, Fenchel conjugate, Fitzpatrick function, linear relation,maximally monotone operator, monotone operator, multifunction, normal cone operator,partial infimal convolution. ∗ Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada.E-mail: [email protected] . † CARMA, University of Newcastle, Newcastle, New South Wales 2308, Australia. E-mail: [email protected] . Distinguished Professor King Abdulaziz University, Jeddah. ‡ Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada.E-mail: [email protected] . § Mathematics, Irving K. Barber School, University of British Columbia, Kelowna, B.C. V1V 1V7, Canada.E-mail: [email protected] . Introduction
Throughout this paper, we assume that X is a real reflexive Banach space with norm k · k ,that X ∗ is the continuous dual of X , and that X and X ∗ are paired by h· , ·i .Let A : X ⇒ X ∗ be a set-valued operator (also known as a multifunction) from X to X ∗ ,i.e., for every x ∈ X , Ax ⊆ X ∗ , and let gra A := (cid:8) ( x, x ∗ ) ∈ X × X ∗ | x ∗ ∈ Ax (cid:9) be the graph of A . The domain of A is dom A := (cid:8) x ∈ X | Ax = ∅ (cid:9) , and ran A := A ( X ) for the range of A . Recall that A is monotone if(1) h x − y, x ∗ − y ∗ i ≥ , ∀ ( x, x ∗ ) ∈ gra A ∀ ( y, y ∗ ) ∈ gra A, and maximally monotone if A is monotone and A has no proper monotone extension (in thesense of graph inclusion). Let S ⊆ X × X ∗ . We say S is a monotone set if there exists amonotone operator A : X ⇒ X ∗ such that gra A = S , and S is a maximally monotone set if there exists a maximally monotone operator A such that gra A = S . Let A : X ⇒ X ∗ bemonotone and ( x, x ∗ ) ∈ X × X ∗ . We say ( x, x ∗ ) is monotonically related to gra A if h x − y, x ∗ − y ∗ i ≥ , ∀ ( y, y ∗ ) ∈ gra A. Maximally monotone operators have proven to be a potent class of objects in modernOptimization and Analysis; see, e.g., [6, 7, 8], the books [2, 9, 10, 13, 16, 17, 15, 19] and thereferences therein.We adopt standard notation used in these books especially [9, Chapter 2] and [6, 16, 17]:Given a subset C of X , int C is the interior of C , C is the norm closure of C . The supportfunction of C , written as σ C , is defined by σ C ( x ∗ ) := sup c ∈ C h c, x ∗ i . The indicator function of C , written as ι C , is defined at x ∈ X by ι C ( x ) := ( , if x ∈ C ;+ ∞ , otherwise . (2)For every x ∈ X , the normal cone operator of C at x is defined by N C ( x ) = (cid:8) x ∗ ∈ X ∗ | sup c ∈ C h c − x, x ∗ i ≤ (cid:9) , if x ∈ C ; and N C ( x ) = ∅ , if x / ∈ C . For x, y ∈ X , we set [ x, y ] = { tx + (1 − t ) y | ≤ t ≤ } . The closed unit ball is B X := (cid:8) x ∈ X | k x k ≤ (cid:9) , and N := { , , , . . . } .If Z is a real Banach space with dual Z ∗ and a set S ⊆ Z , we denote S ⊥ by S ⊥ := { z ∗ ∈ Z ∗ | h z ∗ , s i = 0 , ∀ s ∈ S } . The adjoint of an operator A , written A ∗ , is defined bygra A ∗ := (cid:8) ( x, x ∗ ) ∈ X × X ∗ | ( x ∗ , − x ) ∈ (gra A ) ⊥ (cid:9) .
2e say A is a linear relation if gra A is a linear subspace. We say that A is skew if gra A ⊆ gra( − A ∗ ); equivalently, if h x, x ∗ i = 0 , ∀ ( x, x ∗ ) ∈ gra A . Furthermore, A is symmetric ifgra A ⊆ gra A ∗ ; equivalently, if h x, y ∗ i = h y, x ∗ i , ∀ ( x, x ∗ ) , ( y, y ∗ ) ∈ gra A .Let f : X → ] −∞ , + ∞ ]. Then dom f := f − ( R ) is the domain of f , and f ∗ : X ∗ → [ −∞ , + ∞ ] : x ∗ sup x ∈ X ( h x, x ∗ i − f ( x )) is the Fenchel conjugate of f . We say f is properif dom f = ∅ . Let f be proper. The subdifferential of f is defined by ∂f : X ⇒ X ∗ : x
7→ { x ∗ ∈ X ∗ | ( ∀ y ∈ X ) h y − x, x ∗ i + f ( x ) ≤ f ( y ) } . We now turn to the objects of the present paper: representative and
BC-functions . Let F : X × X ∗ → ] −∞ , + ∞ ], and define pos F [17] bypos F := (cid:8) ( x, x ∗ ) ∈ X × X ∗ | F ( x, x ∗ ) = h x, x ∗ i (cid:9) . We say F is a BC–function (BC stands for “bigger conjugate”) [17] if F is proper and convexwith F ∗ ( x ∗ , x ) ≥ F ( x, x ∗ ) ≥ h x, x ∗ i , ∀ ( x, x ∗ ) ∈ X × X ∗ . (3)The prototype for a BC function is the Fitzpatrick function [11, 17, 9].Let now Y be another real Banach space. We set P X : X × Y → X : ( x, y ) x . Let F , F : X × Y → ] −∞ , + ∞ ]. Then the partial inf-convolution F (cid:3) F is the function definedon X × Y by F (cid:3) F : ( x, y ) inf v ∈ Y F ( x, y − v ) + F ( x, v ) . The importance of BC-functions associated with monotone operators is that along withappropriate partial convolutions, they provide the most powerful current method to establishthe maximality of the sum of two maximally monotone operators [17, 9]. The two problemsconsidered below are closely related to constructions of maximally monotone operators assums (see also Remark 5.4).The following question was posed by S. Simons [17, Problem 34.7]:
Problem 2.1 (Simons)
Let F , F : X × X ∗ → ] −∞ , + ∞ ] be proper lower semicontinuousand convex functions with P X dom F ∩ P X dom F = ∅ . Assume that F , F are BC–functions and that there exists an increasing function j : [0 , + ∞ [ → [0 , + ∞ [ such that the3mplication( x, x ∗ ) ∈ pos F , ( y, y ∗ ) ∈ pos F , x = y and h x − y, y ∗ i = k x − y k · k y ∗ k⇒ k y ∗ k ≤ j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) holds. Then, is it true that, for all ( z, z ∗ ) ∈ X × X ∗ , there exists x ∗ ∈ X ∗ such that F ∗ ( x ∗ , z ) + F ∗ ( z ∗ − x ∗ , z ) ≤ ( F (cid:3) F ) ∗ ( x ∗ , z )?In Example 4.4 of this paper, we construct a comprehensive negative answer to Prob-lem 2.1. This in turn prompts another question: Problem 2.2
Let F , F : X × X ∗ → ] −∞ , + ∞ ] be proper lower semicontinuous and convexfunctions with P X dom F ∩ P X dom F = ∅ . Assume that F , F are BC–functions and thatthere exists an increasing function j : [0 , + ∞ [ → [0 , + ∞ [ such that the implication( x, x ∗ ) ∈ pos F , ( y, y ∗ ) ∈ pos F , x = y and h x − y, y ∗ i = k x − y k · k y ∗ k⇒ k y ∗ k ≤ j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) holds. Then, is it true that, for all ( z, z ∗ ) ∈ X × X ∗ , there exists v ∗ ∈ X ∗ such that F ∗ ( v ∗ , z ) + F ∗ ( z ∗ − v ∗ , z ) ≤ ( F (cid:3) F ) ∗ ( z ∗ , z )?(4)This is a quite reasonable question and somewhat harder to answer. An affirmative re-sponse to Problem 2.2 would rederive Simons’ theorem (Fact 3.4). Precisely, when the latterconjecture holds, we can deduce that F := F (cid:3) F is a BC-function. It follows that pos F (i.e., M in Fact 3.4) is a maximally monotone set; by Simons’ result [17, Theorem 21.4].However, Example 5.2 shows that the conjecture fails in general.We are now ready to set to work. The remainder of the paper is organized as follows. InSection 3, we collect auxiliary results for future reference and for the reader’s convenience.Our main result (Theorem 4.3) is established in Section 4. In Example 4.4, we provide thepromised negative answer to Problem 2.1. In Section 5, we provide a negative answer toProblem 2.2. Fact 3.1 (Rockafellar) (See [14, Theorem A], [19, Theorem 3.2.8], [17, Theorem 18.7]or [12, Theorem 2.1])
Let f : X → ] −∞ , + ∞ ] be a proper lower semicontinuous convexfunction. Then ∂f is maximally monotone.
4e now turn to prerequisite results on Fitzpatrick functions, monotone operators, andlinear relations.
Fact 3.2 (Fitzpatrick) (See [11, Corollary 3.9 and Proposition 4.2] and [6, 9].)
Let A : X ⇒ X ∗ be maximally monotone, and set (5) F A : X × X ∗ → ] −∞ , + ∞ ] : ( x, x ∗ ) sup ( a,a ∗ ) ∈ gra A (cid:0) h x, a ∗ i + h a, x ∗ i − h a, a ∗ i (cid:1) , which is the Fitzpatrick function associated with A . Then F A is a BC–function and pos F A =gra A . Fact 3.3 (Simons and Z˘alinescu) (See [18, Theorem 4.2] or [17, Theorem 16.4(a)].)
Let Y be a real Banach space and F , F : X × Y → ] −∞ , + ∞ ] be proper, lower semicontinuous,and convex. Assume that for every ( x, y ) ∈ X × Y , ( F (cid:3) F )( x, y ) > −∞ and that S λ> λ [ P X dom F − P X dom F ] is a closed subspace of X . Then for every ( x ∗ , y ∗ ) ∈ X ∗ × Y ∗ , ( F (cid:3) F ) ∗ ( x ∗ , y ∗ ) = min u ∗ ∈ X ∗ [ F ∗ ( x ∗ − u ∗ , y ∗ ) + F ∗ ( u ∗ , y ∗ )] . The following Simons’ result generalizes the result of Br´ezis, Crandall and Pazy [5].
Fact 3.4 (Simons) (See [17, Theorem 34.3].)
Let F , F : X × X ∗ → ] −∞ , + ∞ ] be properlower semicontinuous and convex functions with P X dom F ∩ P X dom F = ∅ . Assume that F , F are BC–functions and that there exists an increasing function j : [0 , + ∞ [ → [0 , + ∞ [ such that the implication ( x, x ∗ ) ∈ pos F , ( y, y ∗ ) ∈ pos F , x = y and h x − y, y ∗ i = k x − y k · k y ∗ k⇒ k y ∗ k ≤ j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) holds. Then M := (cid:8) ( x, x ∗ + y ∗ ) | ( x, x ∗ ) ∈ pos F , ( x, y ∗ ) ∈ pos F (cid:9) is a maximally monotoneset. We start with two technical tools which relate Fitzpatrick functions and skew operators. Wefirst give a direct proof of the following result.5 act 4.1 (See [1, Corollary 5.9].)
Let C be a nonempty closed convex subset of X . Then F N C = ι C ⊕ ι ∗ C .Proof . Let ( x, x ∗ ) ∈ X × X ∗ . Then we have F N C ( x, x ∗ ) = sup ( c,c ∗ ) ∈ gra N C [ h x, c ∗ i + h c, x ∗ i − h c, c ∗ i ]= sup ( c,c ∗ ) ∈ gra N C ,k ≥ [ h x, kc ∗ i + h c, x ∗ i − h c, kc ∗ i ]= sup ( c,c ∗ ) ∈ gra N C ,k ≥ [ k ( h x, c ∗ i − h c, c ∗ i ) + h c, x ∗ i ](6)By (6), ( x, x ∗ ) ∈ dom F N C ⇒ sup ( c,c ∗ ) ∈ gra N C [ h x, c ∗ i − h c, c ∗ i ] ≤ ⇔ inf ( c,c ∗ ) ∈ gra N C [ −h x, c ∗ i + h c, c ∗ i ] ≥ ⇔ inf ( c,c ∗ ) ∈ gra N C [ h c − x, c ∗ − i ] ≥ ⇔ ( x, ∈ gra N C (by Fact 3.1) ⇔ x ∈ C. (7)Now assume x ∈ C . By (6), F N C ( x, x ∗ ) = ι ∗ C ( x ∗ ) . (8)Combine (7) and (8), F N C = ι C ⊕ ι ∗ C . (cid:4) Fact 4.2 (See [3, Proposition 5.5].)
Let A : X ⇒ X ∗ be a monotone linear relation suchthat gra A = ∅ and gra A is closed. Then F ∗ A ( x ∗ , x ) = ι gra A ( x, x ∗ ) + h x, x ∗ i , ∀ ( x, x ∗ ) ∈ X × X ∗ . (9)We are now ready to establish our main result. Theorem 4.3
Let A : X ⇒ X ∗ be a maximally monotone linear relation that is at mostsingle-valued, and let C = { } be a bounded closed and convex subset of X such that S λ> λ [dom A − C ] is a closed subspace of X . Let j : [0 , + ∞ [ → [0 , + ∞ [ be an increas-ing function such that j ( γ ) ≥ γ for every γ ∈ [0 , + ∞ [ . Then the following hold. (i) F A and F N C = ι C ⊕ σ C are BC-functions. F ∗ A ( x ∗ , x ) + F ∗ N C ( y ∗ − x ∗ , x ) = ι gra A ∩ C × X ∗ ( x, x ∗ ) + h x, x ∗ i + σ C ( y ∗ − x ∗ ) , ∀ ( x, x ∗ , y ∗ ) ∈ X × X ∗ × X ∗ . (iii) For every ( x, x ∗ ) ∈ X × X ∗ , (10) ( F A (cid:3) F N C ) ∗ ( x ∗ , x ) = ( h x, Ax i + σ C ( x ∗ − Ax ) , if x ∈ C ∩ dom A ; + ∞ , otherwise. (iv) There exists ( z, z ∗ ) ∈ X × X ∗ such that z ∈ dom A ∩ C and σ C ( z ∗ − Az ) > . (v) Assume that ( z, z ∗ ) ∈ X × X ∗ satisfies z ∈ dom A ∩ C and σ C ( z ∗ − Az ) > . Then F ∗ A ( x ∗ , z ) + F ∗ N C ( z ∗ − x ∗ , z ) > ( F A (cid:3) F N C ) ∗ ( x ∗ , z ) , ∀ x ∗ ∈ X ∗ . (11)(vi) Moreover, assume that X is a Hilbert space and C = B X . Then the implication ( x, x ∗ ) ∈ pos F A , ( y, y ∗ ) ∈ pos F N C , x = y and h x − y, y ∗ i = k x − y k · k y ∗ k⇒ k y ∗ k ≤ k x ∗ + y ∗ k ≤ j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) (12) holds.Proof . (i): Combine Fact 4.1 and Fact 3.2.(ii): Let ( x, x ∗ , y ∗ ) ∈ X × X ∗ × X ∗ . Then by Fact 4.2 and (i), we have F ∗ A ( x ∗ , x ) + F ∗ N C ( y ∗ − x ∗ , x ) = ι gra A ( x, x ∗ ) + h x, x ∗ i + ( ι ∗ C ⊕ σ ∗ C )( y ∗ − x ∗ , x )= ι gra A ( x, x ∗ ) + h x, x ∗ i + ι C ( x ) + σ C ( y ∗ − x ∗ )= ι gra A ∩ C × X ∗ ( x, x ∗ ) + h x, x ∗ i + σ C ( y ∗ − x ∗ ) . (iii): By [3, Lemma 5.8], we have [ λ> λ (cid:0) P X (dom F A ) − P X (dom F N C ) (cid:1) is a closed subspace of X. (13)Then for every ( x, x ∗ ) ∈ X × X ∗ and u ∗ ∈ X ∗ , by (i), F A ( x, u ∗ ) + F N C ( x, x ∗ − u ∗ ) ≥ h x, u ∗ i + h x, x ∗ − u ∗ i = h x, x ∗ i . Hence ( F A (cid:3) F N C )( x, x ∗ ) ≥ h x, x ∗ i > −∞ . (14) 7y (13), (14), Fact 3.3, and (ii), for every ( x, x ∗ ) ∈ X × X ∗ , there exists z ∗ ∈ X ∗ suchthat ( F A (cid:3) F N C ) ∗ ( x ∗ , x ) = min y ∗ ∈ X ∗ F ∗ A ( y ∗ , x ) + F ∗ N C ( x ∗ − y ∗ , x )= ι gra A ∩ C × X ∗ ( x, z ∗ ) + h x, z ∗ i + σ C ( x ∗ − z ∗ ) . (15)This implies (10).(iv): By the assumption, there exists z ∈ dom A ∩ C . Since C = { } , there exists z ∗ ∈ X ∗ such that σ C ( z ∗ − Az ) > x ∗ ∈ X ∗ . By the assumptions, (iii) and the boundedness of C , we have( F A (cid:3) F N C ) ∗ ( x ∗ , z ) = h z, Az i + σ C ( x ∗ − Az ) < + ∞ . (16)We consider two cases. Case 1 : x ∗ = Az .Then ( z, x ∗ ) / ∈ gra A and so ι gra A ∩ C × X ∗ ( z, x ∗ ) = + ∞ . In view of (ii) and (16), (11) holds. Case 2 : x ∗ = Az .By (ii) and (16), we have F ∗ A ( x ∗ , z ) + F ∗ N C ( z ∗ − x ∗ , z ) = h z, Az i + σ C ( z ∗ − Az ) > h z, Az i + 0 = h z, Az i + σ C (0)= ( F A (cid:3) F N C ) ∗ ( x ∗ , z ) . Hence (11) holds as well.(vi): We start with a well known formula whose short proof we include for completeness.Let x ∈ X . Then N B X ( x ) = , if k x k < , ∞ [ · x, if k x k = 1; ∅ , otherwise . (17)Clearly, N B X ( x ) = 0 if k x k <
1, and N B X ( x ) = ∅ if x / ∈ B X . Assume k x k = 1. Then x ∗ ∈ N B X ( x ) ⇔ k x ∗ k = k x ∗ k · k x k ≥ h x ∗ , x i ≥ sup h x ∗ , B X i = k x ∗ k⇔ h x ∗ , x i = k x ∗ k · k x k⇔ x ∗ = γx, γ ≥ . Hence (17) holds. 8ow let ( x, x ∗ ) ∈ pos F A , ( y, y ∗ ) ∈ pos F N C and x = y be such that h x − y, y ∗ i = k x − y k ·k y ∗ k . By Fact 3.2, x ∗ = Ax and y ∗ ∈ N B X ( y ) . (18)Now we show that k x ∗ + y ∗ k ≥ k y ∗ k . (19)Clearly, (19) holds if y ∗ = 0. Thus, we assume that y ∗ = 0. By (18) and (17), there exists γ > y ∗ = γ y, (20)where k y k = 1 . (21)Since h x − y, y ∗ i = k x − y k · k y ∗ k , we have y ∗ = k y ∗ kk x − y k ( x − y ) . (22)We claim that x = 0 . (23)Suppose to the contrary that x = 0. Then by (22) and (21), we have y ∗ = − k y ∗ kk y k y = −k y ∗ k y ,which contradicts (20). Hence (23) holds.By (20), (22) and (23), we have x k x k = y ∗ k y ∗ k . (24)Then (18) and the monotonicity of A imply k x ∗ + y ∗ k ≥ h x ∗ + y ∗ , x k x k i ≥ h y ∗ , y ∗ k y ∗ k i = k y ∗ k . Therefore, (19) holds.Then by the assumption, we have j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) ≥ j (cid:0) k x ∗ + y ∗ k (cid:1) ≥ k x ∗ + y ∗ k≥ k y ∗ k . Hence (12) holds, (cid:4)
We are now ready to exploit Theorem 4.3 to resolve Problem 2.1.9 xample 4.4
Suppose that X is a Hilbert space, and let A : X ⇒ X ∗ be a maximallymonotone linear relation that is at most single-valued, and set C = B X . Let j : [0 , + ∞ [ → [0 , + ∞ [ be an increasing function such that j ( γ ) ≥ γ for every γ ∈ [0 , + ∞ [. Then thefollowing hold.(i) Let z ∗ = 0. Then F ∗ A ( x ∗ ,
0) + F ∗ N C ( z ∗ − x ∗ , > ( F A (cid:3) F N C ) ∗ ( x ∗ , , ∀ x ∗ ∈ X. (ii) The implication( x, x ∗ ) ∈ pos F A , ( y, y ∗ ) ∈ pos F N C , x = y and h x − y, y ∗ i = k x − y k · k y ∗ k⇒ k y ∗ k ≤ k x ∗ + y ∗ k ≤ j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) holds. Proof . Set z = 0. Then Az = 0 ⇒ z ∗ − Az = z ∗ = 0 ⇒ σ C ( z ∗ − Az ) = σ C ( z ∗ ) = k z ∗ k > (cid:4) Remark 4.5
Example 4.4 yields a negative answer to Simons’ Problem 2.1 ([17, Prob-lem 34.7]) for many linear relations — including the rotation by 90 degrees in the plane.
We now move to the second problem. Its resolution depends on the following fact concerninga maximally monotone operator on ℓ , the real Hilbert space of square-summable sequences. Fact 5.1 (See [4, Propositions 3.5, 3.6 and 3.7 and Lemma 3.18].)
Suppose that X = ℓ ,and that A : ℓ ⇒ ℓ is given by Ax := (cid:18) P i
Example 5.2
Suppose that X and A are as in Fact 5.1. Set e := (1 , , . . . , , . . . ), i.e., thereis a 1 in the first place and all others entries are 0, and C := [0 , e ]. Let j : [0 , + ∞ [ → [0 , + ∞ [be an increasing function such that j ( γ ) ≥ γ for every γ ∈ [0 , + ∞ [. Then the following hold.(i) F A ∗ and F N C = ι C ⊕ σ C are BC–functions.(ii) ( F A ∗ (cid:3) F N C )( x, x ∗ ) = ( h x, A ∗ x i + σ C ( x ∗ − A ∗ x ) , if x ∈ C ;+ ∞ , otherwise, ∀ ( x, x ∗ ) ∈ X × X ∗ .(iii) Then F ∗ A ∗ ( x ∗ ,
0) + F ∗ N C ( A ∗ e − x ∗ , > ( F A ∗ (cid:3) F N C ) ∗ ( A ∗ e , , ∀ x ∗ ∈ X. (iv) The implication( x, x ∗ ) ∈ pos F N C , ( y, y ∗ ) ∈ pos F A ∗ , x = y and h x − y, y ∗ i = k x − y k · k y ∗ k⇒ k y ∗ k ≤ k y k ≤ j (cid:0) k x k + k x ∗ + y ∗ k + k y k + k x − y k · k y ∗ k (cid:1) holds.(v) A ∗ + N C is maximally monotone. Proof . (i): Combine Fact 5.1(ii), Fact 3.2 and Fact 4.1.11ii): Using Fact 5.1(iii), we see that for every ( x, x ∗ ) ∈ X × X ∗ ,( F A ∗ (cid:3) F N C )( x, x ∗ ) = inf y ∗ ∈ X ∗ ι gra A ∗ ( x, y ∗ ) + h x, y ∗ i + ι C ( x ) + σ C ( x ∗ − y ∗ )= ( h x, A ∗ x i + σ C ( x ∗ − A ∗ x ) , if x ∈ dom A ∗ ∩ C ;+ ∞ , otherwise, . The identity now follows since C ⊆ dom A ∗ .(iii): Let x ∗ ∈ X . Then by Fact 5.1(iii) we have F ∗ A ∗ ( x ∗ ,
0) + F ∗ N C ( A ∗ e − x ∗ ,
0) = ι { } ( x ∗ ) + σ C ( A ∗ e − x ∗ )= σ C ( A ∗ e ) + ι { } ( x ∗ )= sup t ∈ [0 , (cid:8) t h e , A ∗ e i (cid:9) + ι { } ( x ∗ )= h e , A ∗ e i + ι { } ( x ∗ )= + ι { } ( x ∗ ) (by Fact 5.1(iv)) . (27)On the other hand, by (ii) and C ⊆ dom A ∗ by Fact 5.1, we have( F A ∗ (cid:3) F N C ) ∗ ( A ∗ e ,
0) = sup x ∈ C,x ∗ ∈ X (cid:8) h A ∗ e , x i − h x, A ∗ x i − σ C ( x ∗ − A ∗ x ) (cid:9) ≤ sup x ∈ C,x ∗ ∈ X (cid:8) h A ∗ e , x i − h x, A ∗ x i (cid:9) (by 0 ∈ C )= sup t ∈ [0 , (cid:8) t h A ∗ e , e i − t h e , A ∗ e i (cid:9) = h A ∗ e , e i = (by Fact 5.1(iv)) < F ∗ A ∗ ( x ∗ ,
0) + F ∗ N C ( A ∗ e − x ∗ ,
0) (by (27)) . Hence (iii) holds.(iv): Let ( x, x ∗ ) ∈ pos F N C , ( y, y ∗ ) ∈ pos F A ∗ , and x = y be such that h x − y, y ∗ i = k x − y k · k y ∗ k . By Fact 3.2, x ∗ ∈ N C ( x ) and y ∗ = A ∗ y. (28)Now we show k y k ≥ k y ∗ k . (29)Clearly, (29) holds if y ∗ = 0. Now assume that y ∗ = 0. Then by h x − y, y ∗ i = k x − y k · k y ∗ k and x ∈ C , there exist t ≥ γ > x = t e and y ∗ = γ ( t e − y ) . (30) 12rite y = ( y n ) n ∈ N . By (26) and (30), we have X i>n y i = − γ y n − y n , ∀ n ≥ . (31)Thus X i>n +1 y i = − γ y n +1 − y n +1 , ∀ n ≥ . (32)Subtracting (32) from (31), we obtain y n +1 = ( − γ − )( y n − y n +1 ) , ∀ n ≥ . (33)Since γ >
0, by (33), we have y n +1 γ − γ + = y n , ∀ n ≥ . (34)Now we claim that y n = 0 , ∀ n ≥ . (35)Suppose to the contrary that there exists i ≥ y i = 0 . (36)Then by (34), we have y i = y i +1 γ − γ + 12 . Thus, γ = 12 . (37)Then by (34), we have y n +1 = γ + γ − y n , ∀ n ≥ . (38)Set α := γ + 12 γ − . Then by γ > | α | > . (39)By (38) and Fact 5.1, we have P i> y i = y P i ≥ α i and the former series is convergent.Thus (39) implies that y = 0 and then y n = 0 , ∀ n > y ∗ = ( y , , , . . . , , . . . ) . (40) 13ence k y ∗ k ≤ k y k and thus (29) holds. Then by the assumption, we have k y ∗ k ≤ k y k ≤ (cid:0) k x k + k y k + k x ∗ + y ∗ k + k x − y k · k y ∗ k (cid:1) ≤ j ( k x k + k y k + k x ∗ + y ∗ k + k x − y k · k y ∗ k ) . Hence the implication (iv) holds.(v): By Fact 3.2 and Fact 5.1(ii), pos F A ∗ = gra A ∗ and pos F N C = gra N C . Then directlyapply (i)&(iv) and Fact 3.4. (cid:4) Remark 5.3
Example 5.2 provides a negative answer to Problem 2.2 as asserted.
Remark 5.4
It is not as easy to find a counterexample to Problem 2.2 as it is for Prob-lem 2.1. Indeed, Fact 3.4 and Fact 3.3 imply that, to find a counterexample, we need tostart with two maximally monotone operators
A, B : X ⇒ X ∗ such that A + B is maxi-mally monotone but it does not satisfy the well known sufficient transversality condition forthe maximal monotonicity of the sum operator in a reflexive space [18, Lemma 5.1] and [4,Lemma 5.8], that is: [ λ> λ [dom A − dom B ] is a closed subspace of X. (41)Otherwise, (41) ensures that (4) in Problem 2.2 holds by Fact 3.3 and [4, Lemma 5.8].Finally, as we mentioned in Section 2 an affirmative answer to Problem 2.2 would rederiveSimons’ theorem (Fact 3.4). Indeed, Simons [17, Corollary 34.5] shows in detail how todeduce the classic result of Br´ezis, Crandall and Pazy [5] from his result. Acknowledgments.
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