On the generalized parallel sum of two maximal monotone operators of Gossez type (D)
aa r X i v : . [ m a t h . F A ] J un On the generalized parallel sum of two maximal monotone operators ofGossez type (D)
Radu Ioan Bot¸ ∗ Szil´ard L´aszl´o † August 21, 2018
Abstract.
The generalized parallel sum of two monotone operators via a linear continuous mapping is definedas the inverse of the sum of the inverse of one of the operators and with inverse of the composition of the secondone with the linear continuous mapping. In this article, by assuming that the operators are maximal monotoneof Gossez type (D), we provide sufficient conditions of both interiority- and closedness-type for guaranteeing thattheir generalized sum via a linear continuous mapping is maximal monotone of Gossez type (D), too. This resultwill follow as a particular instance of a more general one concerning the maximal monotonicity of Gossez type (D)of an extended parallel sum defined for the maximal monotone extensions of the two operators to the correspondingbiduals.
Key Words. monotone operator, maximal monotone operator of Gossez type (D), representative function, convexconjugate duality
AMS subject classification.
Having two nonempty sets A and B and a multivalued operator M : A ⇒ B , we denote by G ( M ) = { ( a, b ) ∈ A × B : b ∈ M ( a ) } its graph and by M − : B ⇒ A the inverse operator of M , which is the multivalued operatorhaving as graph the set G ( M − ) := { ( b, a ) ∈ B × A : ( a, b ) ∈ G ( M ) } . When X is a real nonzero Banach space and X ∗ its topological dual space, the parallel sum of two multivalued monotone operators S, T : X ⇒ X ∗ is defined as S || T : X ⇒ X ∗ , S || T ( x ) := ( S − + T − ) − ( x ) ∀ x ∈ X. This notion has been first considered in Hilbert spaces by Passty in [22], where the interested reader can find somepractical interpretations of this notion including some preliminary investigations on the maximal monotonicity ofthe parallel sum of two maximal monotone operators. The latter problem was also addressed in Hilbert spacesin [21] and in reflexive Banach spaces in [1, 28], the weakest condition for the maximal monotonicity of the parallelsum available in the latter setting in the literature being recently introduced in [26]. Since S and T are maximalmonotone if and only if their inverse S − and, respectively, T − are maximal monotone, the sufficient conditionsfor the maximal monotonicity of S || T in reflexive Banach spaces can be gathered from the ones formulated for themaximal monotonicity of the sum of two maximal monotone operators, applied to S − + T − .When Y is another real nonzero Banach space with Y ∗ its topological dual space, S : X ⇒ X ∗ and T : Y ⇒ Y ∗ are two monotone operators and A : X → Y is a linear continuous mapping with adjoint mapping A ∗ : Y ∗ → X ∗ ,Penot and Z˘alinescu proposed in [26] the following generalized parallel sums of S and T defined via AS || A T : Y ⇒ Y ∗ , S || A T ( y ) := ( AS − A ∗ + T − ) − ( y ) ∀ y ∈ Y and S || A T : X ⇒ X ∗ , S || A T ( x ) := ( S − + ( A ∗ T A ) − ) − ( x ) ∀ x ∈ X, ∗ Faculty of Mathematics, Chemnitz University of Technology, D-09107 Chemnitz, Germany, e-mail: [email protected]. Research partially supported by DFG (German Research Foundation), project WA 922/1-3. † Faculty of Mathematics and Computer Science, Babe¸s-Bolyai University, Cluj-Napoca, Romania, e-mail: [email protected] done during the stay of the author in the academic year 2010/2011 at Chemnitz University of Technology as a guest of theChair of Applied Mathematics (Approximation Theory). The author wishes to thank for the financial support provided from programsco-financed by The Sectoral Operational Programme Human Resources Development, Contract POSDRU 6/1.5/S/3 - “Doctoral studies:through science towards society”. X = Y and A is the identity mapping on X , then they both collapseinto S || T . As the monotonicity of S and T gives rise to the same property for S || A T and S || A T , the question of howto guarantee for these maximal monotonicity, provided that S and T are maximal monotone, comes automatically.This question was already addressed by Stephen Simons in [33] in general Banach spaces in what concerns thegeneralized parallel sum S || A T . Under the assumption that S and T are maximal monotone operators of Gosseztype (D), in the mentioned paper, interiority-type regularity conditions for ensuring that S || A T is a maximalmonotone operator of Gossez type (D), too, have been formulated. Due to its nature, at least in reflexive spaces,statements on the maximal monotonicity of the parallel sum S || A T and corresponding interiority- and closedness-type regularity conditions can be derived from the statements given in the literature for the sum of a monotoneoperator with the composition of a second one with a linear continuous mapping (see [5, 6, 27]).On the other hand, the approach suggested above for S || A T , regarding the direct derivation of sufficient condi-tions for maximal monotonicity from the already existent ones, cannot be applied to S || A T accordingly. This factrepresented the starting point of the investigations made in this paper, where we want to provide interiority- and closedness-type regularity conditions for the maximal monotonicity of Gossez type (D) of S || A T , whenever S and T are maximal monotone operators of Gossez type (D).The outline of the paper is the following. In the remaining of this section we recall some elements of convexanalysis and introduce the necessary apparatus of notions and results referring to monotone operators in generalBanach spaces. In Section 2 we investigate the fulfilment in an exact sense of a generalized bivariate infimalconvolution formula for which we provide, by making use of a special conjugate formula, equivalent closedness-typeconditions, but also sufficient interiority-type ones. This formula represents the premise for ensuring in Section3 maximal monotonicity of Gossez type (D) of a generalized parallel sum of the maximal monotone operators ofGossez type (D) S and T , defined by making use of their extensions to the corresponding biduals. The maximalmonotonicity of Gossez type (D) of S || A T will follow as a particular instance of this general result. A specialattention will be also given to the formulation of further sufficient conditions for the interiority-type regularitycondition and to the situation when these became equivalent. Finally, in Section 4, some particular instances, towhich the general results on the maximal monotonicity of S || A T give rise, are considered. Let X a real separated locally convex space and X ∗ its topological dual space. We denote by w ( X, X ∗ ) (or, forshort, w ) the weak topology on X induced by X ∗ and by w ( X ∗ , X ) (or, for short, w ∗ ) the weak ∗ topology on X ∗ induced by X . We denote by h x ∗ , x i the value of the continuous linear functional x ∗ ∈ X ∗ at x ∈ X . For a givenset D ⊆ X , we denote by co D, aff D, int D and cl D , its convex hull, affine hull, interior and closure , respectively.When Z ⊆ X is a given set we say that D is closed regarding the set Z if cl D ∩ Z = D ∩ Z . The conic hull of theset D will be denoted by cone D = ∪ λ> λD , while its relative interior is defined as (see [40])ri D = (cid:26) rint D, if aff D is a closed set , ∅ , otherwise , where rint D := int aff D D . The algebraic interior (or core ) of D is the set (see [15, 29, 40])core D = { u ∈ X | ∀ x ∈ X ∃ δ > ∀ λ ∈ [0 , δ ] : u + λx ∈ D } , while its relative algebraic interior (or intrinsic core ) is the set (see [15, 40])icr D = { u ∈ X | ∀ x ∈ aff( D − D ) ∃ δ > ∀ λ ∈ [0 , δ ] : u + λx ∈ D } . One always has that rint D ⊆ icr D . The intrinsic relative algebraic interior of D (see [40, 41]) is defined as ic D = (cid:26) icr D, if aff D is a closed set , ∅ , otherwise . Thus we have, in general, that ri D ⊆ ic D. (1)In the case when D is a convex set, the above generalized interiority notions can be characterized as follows: • core D = { x ∈ D : cone( D − x ) = X } (see [29, 40]); • icr D = { x ∈ D : cone( D − x ) is a linear subspace of X } (see [15, 40]); • ic D = { x ∈ D : cone( D − x ) is a closed linear subspace of X } (see [40, 41]);2 x ∈ ic D if and only if x ∈ icr D and aff( D − x ) is a closed linear subspace of X (see [40, 41])and we have the following inclusions int D ⊆ core D ⊆ ic D ⊆ icr D ⊆ D, (2)they being in general strict.When Y is another real separated locally convex space and A : X −→ Y a linear continuous mapping we considerthe following notation ∆ AX := { ( x, Ax ) : x ∈ X } , which becomes when A = id X : X −→ X with id X ( x ) = x for all x ∈ X (the identity mapping on X ) the diagonal subspace ∆ X := { ( x, x ) : x ∈ X } of X × X . The following result,which is of interest independently of the purposes of this article, will play an important role in the sequel. Lemma 1.1.
Let X and Y be separated locally convex spaces, U ⊆ X and V ⊆ Y two given convex sets and A : X −→ Y a linear continuous mapping. Then it holds (0 , ∈ ic (cid:0) U × V − ∆ AX (cid:1) ⇔ ∈ ic ( V − A ( U )) . Proof.
In the proof we use the following two characterizations:(0 , ∈ ic (cid:0) U × V − ∆ AX (cid:1) ⇔ C := cone (cid:0) U × V − ∆ AX (cid:1) is a closed linear subspace of X × Y and 0 ∈ ic ( V − A ( U )) ⇔ D := cone( V − A ( U )) is a closed linear subspace of Y. ” ⇒ ” Suppose that C is a closed linear subspace. Since U and V are convex sets, one has that D is a convexcone. In order to proof that D is a linear subspace, we show that − D ⊆ D . Take an arbitrary d ∈ D . Thus d = α ( v − Au ) for α > , u ∈ U and v ∈ V, hence (0 , d ) = ( α ( u − u ) , α ( v − Au )) = α (( u, v ) − ( u, Au )) ∈ C. But C is a linear space, hence (0 , − d ) ∈ C, that is (0 , − d ) = β (( u , v ) − ( x, Ax )) , with β > , u ∈ U, v ∈ V and x ∈ X. It results that u − x = 0 , hence x = u ∈ U . Thus − d = β ( v − Au ) with β > , u ∈ U, v ∈ V, hence − d ∈ D. We prove next that D is closed and consider therefore an arbitrary element d ∈ cl D. Thus there exist ( λ α ) α ∈ I ⊆ R + , ( u α ) α ∈ I ⊆ U and ( v α ) α ∈ I ⊆ V such that d α = λ α ( v α − Au α ) −→ d. But (0 , d α ) ∈ C for all α ∈ I and C isclosed, thus (0 , d ) = β ( u − x, v − Ax ) , with β > , u ∈ U, v ∈ V and x ∈ X. Hence, x = u ∈ U and, consequently, d = β ( v − Au ) ∈ D. ” ⇐ ” Suppose now that D is a closed linear subspace. The convexity of the sets U and V guarantees that C is aconvex cone. Next we prove that − C ⊆ C and consider to this end an arbitrary c ∈ C . Thus c = α ( u − x, v − Ax ) , with α > , u ∈ U, v ∈ V, x ∈ X. Hence, c = α (0 , v − Au ) + α ( u − x, A ( u − x )) . Obviously, α ( v − Au ) ∈ D and since D is a linear space, we have − α ( v − Au ) = β ( v − Au ) , with β > , u ∈ U and v ∈ V. Thus − c = β (0 , v − Au ) − α ( u − x, A ( u − x )) = β (cid:16) u − (cid:0) u + α/β ( u − x ) (cid:1) , v − A (cid:0) u + α/β ( u − x ) (cid:1)(cid:17) ∈ C. In order to show that C is closed we consider an element c := ( c , c ) ∈ cl C and show that c ∈ C. Thus there exist( λ α ) α ∈ I ⊆ R + , ( u α ) α ∈ I ⊆ U, ( v α ) α ∈ I ⊆ V and ( x α ) α ∈ I ⊆ X such that c α = λ α ( u α − x α , v α − Ax α ) −→ c = ( c , c ) . Obviously, λ α ( u α − x α ) −→ c , hence λ α A ( u α − x α ) −→ Ac and from here we obtain that λ α ( v α − Au α ) −→ c − Ac . But λ α ( v α − Au α ) ∈ D for all α ∈ I and D is closed, hence c − Ac = β ( v − Au ) , with β > , u ∈ U and v ∈ V. Thus ( c , c ) = (cid:16) u − (cid:0) u − /βc (cid:1) , v − A (cid:0) u − /βc (cid:1)(cid:17) ∈ C and this concludes the proof.The indicator function of a set D ⊆ X is defined as δ D : X −→ R := R ∪ {±∞} , δ D ( x ) = (cid:26) , if x ∈ D, + ∞ , otherwise . For E and F two nonempty sets we consider the projection operator pr E : E × F → E , pr E ( e, f ) = e for all( e, f ) ∈ E × F . For G and H two further nonempty sets and k : E → G and l : F → H two given functions wedenote by k × l : E × F → G × H the function defined as k × l ( e, f ) = ( k ( e ) , l ( f )) for all ( e, f ) ∈ E × F . Throughoutthe paper, when an infimum is attained we write min instead of inf.Having a function f : X −→ R we denote its domain by dom f = { x ∈ X : f ( x ) < + ∞} and its epigraph byepi f = { ( x, r ) ∈ X × R : f ( x ) ≤ r } . We call f proper if dom f = ∅ and f ( x ) > −∞ for all x ∈ X . By cl f : X → R we denote the lower semicontinuous hull of f , namely the function whose epigraph is the closure of epi f , that isepi(cl f ) = cl(epi f ) . We consider also co f : X → R , the convex hull of f , which is the greatest convex functionmajorized by f . For x ∈ X such that f ( x ) ∈ R we define the subdifferential of f at x by ∂f ( x ) = { x ∗ ∈ X ∗ : f ( y ) − f ( x ) ≥ h x ∗ , y − x i ∀ y ∈ X } . f ( x ) ∈ {±∞} we take by convention ∂f ( x ) = ∅ .The Fenchel-Moreau conjugate of f is the function f ∗ : X ∗ −→ R defined by f ∗ ( x ∗ ) = sup x ∈ X {h x ∗ , x i − f ( x ) } ∀ x ∗ ∈ X ∗ . One always has the
Young-Fenchel inequality f ∗ ( x ∗ ) + f ( x ) ≥ h x ∗ , x i ∀ x ∈ X ∀ x ∗ ∈ X ∗ . Consider Y another separated locally convex space and a mapping h : X −→ Y . We denote by h ( D ) = { h ( x ) : x ∈ D } the image of a set D ⊆ X through h and by h − ( E ) = { x ∈ X : h ( x ) ∈ E } the inverse of a set E ⊆ Y through h .For A : X −→ Y a linear continuous mapping, Im A := A ( X ) denotes the image space of A , while its adjointoperator A ∗ : Y ∗ −→ X ∗ is defined by h A ∗ y ∗ , x i = h y ∗ , Ax i for all y ∗ ∈ Y ∗ and x ∈ X . When X and Y are normedspaces, the biadjoint operator of A , A ∗∗ : X ∗∗ −→ Y ∗∗ , is defined as being the adjoint operator of A ∗ . Consider further X a nonzero real Banach space, X ∗ its topological dual space and X ∗∗ its topological bidual space.Throughout the paper we identify X with its image under the canonical injection of X into X ∗∗ . A multivaluedoperator S : X ⇒ X ∗ is said to be monotone if h y ∗ − x ∗ , y − x i ≥ , whenever y ∗ ∈ S ( y ) and x ∗ ∈ S ( x ) . A monotone operator S is called maximal monotone if its graph G ( S ) is not properly contained in the graph ofany other monotone operator S ′ : X ⇒ X ∗ . For the operator S we consider also its domain D ( S ) := { x ∈ X : S ( x ) = ∅} = pr X ( G ( S )) and its range R ( S ) := ∪ x ∈ X S ( x ) = pr X ∗ ( G ( S )). The most prominent example of amaximal monotone operator is the subdifferential of a proper, convex and lower semicontinuous function (see [30]).However, there exist maximal monotone operators which are not subdifferentials (see [31, 32]).To an arbitrary monotone operator S : X ⇒ X ∗ we associate the Fitzpatrick function ϕ S : X × X ∗ −→ R ,defined by ϕ S ( x, x ∗ ) = sup {h y ∗ , x i + h x ∗ , y i − h y ∗ , y i : y ∗ ∈ S ( y ) } , which is obviously convex and weak × weak ∗ lower semicontinuous. Introduced by Fitzpatrick in 1988 (see [11]) andrediscovered after some years in [10,20], it proved to be very important in the theory of maximal monotone operators,revealing important connections between convex analysis and monotone operators (see [2–10,19,23–26,31,35,36,39]and the references therein).Denoting by c : X × X ∗ → R , c ( x, x ∗ ) = h x ∗ , x i for all ( x, x ∗ ) ∈ X × X ∗ the coupling function of X × X ∗ ,one can easily show that ϕ S ( x, x ∗ ) = c ∗ S ( x ∗ , x ) for all ( x, x ∗ ) ∈ X × X ∗ , where c S : X × X ∗ → R , c S = c + δ G ( S ) .Well-linked to the Fitzpatrick function is the function ψ S : X × X ∗ → R , ψ S = cl k·k×k·k ∗ (co c S ), where the closureis taken in the strong topology of X × X ∗ . For all ( x, x ∗ ) ∈ X × X ∗ we have ψ ∗ S ( x ∗ , x ) = ϕ S ( x, x ∗ ), while when X is a reflexive Banach space the equality ϕ ∗ S ( x ∗ , x ) = ψ S ( x, x ∗ ) holds (see [10, Remark 5.4]). The most importantproperties of the Fitzpatrick function of a maximal monotone operator follow. Lemma 1.2. (see [11]) Let S : X ⇒ X ∗ be a maximal monotone operator. Then(i) ϕ S ( x, x ∗ ) ≥ h x ∗ , x i for all ( x, x ∗ ) ∈ X × X ∗ ,(ii) G ( S ) = { ( x, x ∗ ) ∈ X × X ∗ : ϕ S ( x, x ∗ ) = h x ∗ , x i} . They gave rise to the following notion introduced in connection to a monotone operator.
Definition 1.1.
For S : X ⇒ X ∗ a monotone operator, we call representative function of S a convex and lowersemicontinuous (in the strong topology of X × X ∗ ) function h S : X × X ∗ −→ R fulfilling h S ≥ c and G ( S ) ⊆ { ( x, x ∗ ) ∈ X × X ∗ : h S ( x, x ∗ ) = h x ∗ , x i} . If G ( S ) = ∅ (which is the case when S is maximal monotone), then every representative function of S is proper.Obviously, the Fitzpatrick function associated to a maximal monotone operator is a representative function ofthe operator. From [10] we have the following properties for the representative function of a maximal monotoneoperator. 4 roposition 1.1. Let S : X ⇒ X ∗ be a maximal monotone operator and h S be a representative function of S .Then the following statements are true:(i) ϕ S ≤ h S ≤ ψ S ;(ii) the function ( x, x ∗ ) h ∗ S ( x ∗ , x ) is also a representative function of S ;(iii) { ( x, x ∗ ) ∈ X × X ∗ : h S ( x, x ∗ ) = h x ∗ , x i} = { ( x, x ∗ ) ∈ X × X ∗ : h ∗ S ( x ∗ , x ) = h x ∗ , x i} = G ( S ) . By Proposition 1.1 it follows that a convex and lower semicontinuous function f : X × X ∗ −→ R is a repre-sentative function of the maximal monotone operator S if and only if ϕ S ≤ f ≤ ψ S , in particular, ϕ S and ψ S are representative functions of S . Let us also notice that if f : X −→ R is a proper, convex and lower semicon-tinuous function, then a representative function of the maximal monotone operator ∂f : X ⇒ X ∗ is the function( x, x ∗ ) f ( x ) + f ∗ ( x ∗ ). Moreover, according to [8, Theorem 3.1] (see also [23, Example 3]), if f is a sublinear andlower semicontinuous function, then the operator ∂f : X ⇒ X ∗ has a unique representative function, namely thefunction ( x, x ∗ ) f ( x ) + f ∗ ( x ∗ ). For more on the properties of representative functions we refer to [3, 10, 19, 26]and the references therein.Next we give a maximality criteria for a monotone operator valid in reflexive Banach spaces (cf. [9, Theorem3.1] and [26, Proposition 2.1]; see also [32] for other maximality criteria in reflexive spaces). Theorem 1.1.
Let X be a reflexive Banach space and f : X × X ∗ −→ R a proper, convex and lower semicontinuousfunction such that f ≥ c . Then the operator whose graph is the set { ( x, x ∗ ) ∈ X × X ∗ : f ( x, x ∗ ) = h x ∗ , x i} ismaximal monotone if and only if f ∗ ( x ∗ , x ) ≥ h x ∗ , x i for all ( x, x ∗ ) ∈ X × X ∗ . For the following generalization of this result to general Banach spaces we refer to [17, Theorem 4.2].
Theorem 1.2.
Let X be a nonzero Banach space and f : X × X ∗ −→ R a proper, convex and lower semicontinuousfunction such that f ≥ c and f ∗ ( x ∗ , x ∗∗ ) ≥ h x ∗∗ , x ∗ i for all ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ . Then the operator whose graphis the set { ( x, x ∗ ) ∈ X × X ∗ : f ( x, x ∗ ) = h x ∗ , x i} is maximal monotone and it holds { ( x, x ∗ ) ∈ X × X ∗ : f ( x, x ∗ ) = h x ∗ , x i} = { ( x, x ∗ ) ∈ X × X ∗ : f ∗ ( x ∗ , x ) = h x ∗ , x i} . In the last part of this section we turn our attention to a particular class of maximal monotone operators ongeneral Banach spaces.
Definition 1.2. (see [14]) Let S : X ⇒ X ∗ be a maximal monotone operator.(a) Gossez’s monotone closure of S is the operator S : X ∗∗ ⇒ X ∗ whose graph is G ( S ) = { ( x ∗∗ , x ∗ ) ∈ X ∗∗ × X ∗ : h x ∗∗ − y, x ∗ − y ∗ i ≥ ∀ ( y, y ∗ ) ∈ G ( S ) } . (b) The operator S : X ⇒ X ∗ is said to be of Gossez type (D) if for any ( x ∗∗ , x ∗ ) ∈ G ( S ) there exists a boundednet { ( x α , x ∗ α ) } α ∈ I ⊆ G ( S ) which converges to ( x ∗∗ , x ∗ ) in the w ∗ × k · k ∗ -topology of X ∗∗ × X ∗ . Gossez proved in [13] that a maximal monotone operator S : X ⇒ X ∗ of Gossez type (D) has a unique maximal monotone extension to the bidual, namely, its Gossez’s monotone closure S : X ∗∗ ⇒ X ∗ . The followingcharacterization of the maximal monotone operators of Gossez type (D) was recently provided in [18] (see also [16]). Theorem 1.3.
Let X be a nonzero real Banach space and S : X ⇒ X ∗ a maximal monotone operator. Thefollowing statements are equivalent:(a) S is of Gossez type (D);(b) S is of Simons negative infimum type (NI) (see [34]), namely inf ( y,y ∗ ) ∈ G ( S ) h y − x ∗∗ , y ∗ − x ∗ i ≤ ∀ ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ ; (c) there exists a representative function h S of S such that h ∗ S ( x ∗ , x ∗∗ ) ≥ h x ∗∗ , x ∗ i ∀ ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ ; (d) for every representative function h S of S one has h ∗ S ( x ∗ , x ∗∗ ) ≥ h x ∗∗ , x ∗ i ∀ ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ . A representative function h S of a maximal monotone operator S : X ⇒ X ∗ fulfilling the inequality in the item(c) (or (d)) of the above theorem is called strong representative function of S (see [37]). The Fitzpatrick function ϕ S of a maximal monotone operator S : X ⇒ X ∗ of Gossez type (D) is a strong representative function andone has ϕ S | X × X ∗ = ϕ S . When h S : X × X ∗ → R is a representative function of a maximal monotone operatorof Gossez type (D) S : X ⇒ X ∗ , then h ∗ S : X ∗ × X ∗∗ → R is a representative function of the inverse operator S − : X ∗ → X ∗∗ of Gossez’s monotone closure S of S (for these statements we refer the reader to [18]).5 A generalized bivariate infimal convolution formula
In this section we provide, by making use of an appropriate conjugate formula , sufficient conditions for an extendedbivariate infimal convolution formula , which we use in the sequel.
Let
X, Y, Z be real separated locally convex spaces with topological duals X ∗ , Y ∗ and Z ∗ , respectively. Theorem 2.1.
Let f : X −→ R and g : Y −→ R be proper, convex and lower semicontinuous functions and A : Z −→ X and B : Z −→ Y be linear continuous mappings such that A − (dom f ) ∩ B − (dom g ) = ∅ .(a) For every set U ⊆ Z ∗ the following statements are equivalent:(i) The set { ( A ∗ x ∗ + B ∗ y ∗ , r ) : r ∈ R , f ∗ ( x ∗ ) + g ∗ ( y ∗ ) ≤ r } is closed regarding U × R in ( Z ∗ , w ∗ ) × R ;(ii) ( f ◦ A + g ◦ B ) ∗ ( z ∗ ) = min { f ∗ ( x ∗ ) + g ∗ ( y ∗ ) : ( x ∗ , y ∗ ) ∈ X ∗ × Y ∗ , A ∗ x ∗ + B ∗ y ∗ = z ∗ } for all z ∗ ∈ U. (b) If X, Y and Z are Fr´echet spaces and (0 , ∈ ic (dom f × dom g − ( A × B )(∆ Z )) , then the statements (i) and (ii) are valid for every U ⊆ Z ∗ .Proof. (a) Consider an arbitrary set U ⊆ Z ∗ and the perturbation functionΦ : Z × X × Y −→ R , Φ( z, x, y ) = f ( Az + x ) + g ( Bz + y ) , which is proper, convex and lower semicontinuous and fulfills(0 , ∈ pr X × Y (dom Φ) = dom f × dom g − ( A × B )(∆ Z ) . Its conjugate function looks for all ( z ∗ , x ∗ , y ∗ ) ∈ Z ∗ × X ∗ × Y ∗ likeΦ ∗ ( z ∗ , x ∗ , y ∗ ) = δ { } ( z ∗ − A ∗ x ∗ − B ∗ y ∗ ) + f ∗ ( x ∗ ) + g ∗ ( y ∗ ) . Thus (ii) is nothing else than (Φ( · , , ∗ ( z ∗ ) = min ( x ∗ ,y ∗ ) ∈ X ∗ × Y ∗ Φ ∗ ( z ∗ , x ∗ , y ∗ ) ∀ z ∗ ∈ U. According to [5, Theorem 2], this is further equivalent topr Z ∗ × R (epi Φ ∗ ) is closed regarding U × R in ( Z ∗ , w ∗ ) × R . (3)As one can easily see, it holdspr Z ∗ × R (epi Φ ∗ ) = { ( A ∗ x ∗ + B ∗ y ∗ , r ) : f ∗ ( x ∗ ) + g ∗ ( y ∗ ) ≤ r } and in this way the equivalence (i) ⇔ (ii) is proven.(b) Since X, Y and Z are Fr´echet spaces and (0 , ∈ ic (cid:0) pr X × Y (dom Φ) (cid:1) , by [40, Corollary 2.7.3] it follows thatfor all z ∗ ∈ Z ∗ (Φ( · , , ∗ ( z ∗ ) = min ( x ∗ ,y ∗ ) ∈ X ∗ × Y ∗ Φ ∗ ( z ∗ , x ∗ , y ∗ )or, equivalently,( f ◦ A + g ◦ B ) ∗ ( z ∗ ) = min { f ∗ ( x ∗ ) + g ∗ ( y ∗ ) : ( x ∗ , y ∗ ) ∈ X ∗ × Y ∗ , A ∗ x ∗ + B ∗ y ∗ = z ∗ } , which concludes the proof. Remark 2.1.
In the hypotheses of Theorem 2.1, when X , Y and Z are Fr´echet spaces, then, according to [40,Proposition 2.7.2], ic (dom f × dom g − ( A × B )(∆ Z )) = ri(dom f × dom g − ( A × B )(∆ Z )) . emark 2.2. We refer the reader to [4] for examples where, even X , Y and Z are finite dimensional spaces, thestatements (i) and (ii) in Theorem 2.1(a) are fulfilled, while the interiority-type condition in Theorem 2.1(b) fails. Remark 2.3.
According to the previous theorem, one obtains when Z = X , A = id X and X and Y are Fr´echetspaces as a sufficient condition for the exact conjugate formula( f + g ◦ B ) ∗ ( z ∗ ) = min { f ∗ ( z ∗ − B ∗ y ∗ ) + g ∗ ( y ∗ ) : y ∗ ∈ Y ∗ } ∀ z ∗ ∈ X ∗ (4)the interiority-type condition (0 , ∈ ic (dom f × dom g − ∆ BZ ) . Via Lemma 1.1 it follows that this is nothing else than0 ∈ ic (dom g − B (dom f )) , which is a regularity condition for (4) that has been already considered in literature (see, for instance, [40]). Let X and Y be two Banach spaces with X ∗ and Y ∗ their topological dual spaces and X ∗∗ and Y ∗∗ their topologicalbidual spaces, respectively. Further, let f : X × X ∗ −→ R and g : Y × Y ∗ −→ R be two given functions and A : X −→ Y a linear continuous mapping. In this subsection we deal with the following extended bivariate infimalconvolutions f (cid:13) A g : X × X ∗ −→ R ,( f (cid:13) A g )( x, x ∗ ) = inf { f ( u, x ∗ ) + g ( Aw, v ∗ ) : u, w ∈ X, v ∗ ∈ Y ∗ , u + w = x, A ∗ v ∗ = x ∗ } , and f ∗ (cid:13) A g ∗ : X ∗ × X ∗∗ −→ R ,( f ∗ (cid:13) A g ∗ )( x ∗ , x ∗∗ ) = inf { f ∗ ( x ∗ , u ∗∗ ) + g ∗ ( v ∗ , A ∗∗ w ∗∗ ) : u ∗∗ , w ∗∗ ∈ X ∗∗ , v ∗ ∈ Y ∗ , u ∗∗ + w ∗∗ = x ∗∗ , A ∗ v ∗ = x ∗ } , respectively. By making use of Theorem 2.1, we can prove the following result. Theorem 2.2.
Assume that f : X × X ∗ −→ R and g : Y × Y ∗ −→ R are proper, convex and lower semicontinuousfunctions such that dom g × pr X ∗ (dom f ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ .(a) The following statements are equivalent:(i) The set { ( u ∗ , A ∗ v ∗ , A ∗∗ u ∗∗ + v ∗∗ , r ) : r ∈ R , f ∗ ( u ∗ , u ∗∗ ) + g ∗ ( v ∗ , v ∗∗ ) ≤ r } is closed regarding ∆ X ∗ × Im A ∗∗ × R in ( X ∗ , w ∗ ) × ( X ∗ , w ∗ ) × ( Y ∗∗ , w ∗ ) × R ;(ii) ( f (cid:13) A g ) ∗ ( x ∗ , x ∗∗ ) = ( f ∗ (cid:13) A g ∗ )( x ∗ , x ∗∗ ) and f ∗ (cid:13) A g ∗ is exact (that is, the infimum in the definitionof ( f ∗ (cid:13) A g ∗ )( x ∗ , x ∗∗ ) is attained) for every ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ . (b) If (0 , , ∈ ic (cid:16) dom g × pr X ∗ (dom f ) − Im A × ∆ A ∗ Y ∗ (cid:17) , then the statements (i) and (ii) are true.Proof. Consider the proper, convex and lower semicontinuous functions F : X × X × X ∗ −→ R , F ( u, w, u ∗ ) = f ( u, u ∗ ) and G : X × Y × Y ∗ −→ R , G ( u, v, v ∗ ) = g ( v, v ∗ ) and the linear continuous mappings M : X × X × Y ∗ → X × X × X ∗ , M = id X × id X × A ∗ , and N : X × X × Y ∗ → X × Y × Y ∗ , N = id X × A × id Y ∗ . Sincedom g × pr X ∗ (dom f ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ , we obtain that M − (dom F ) ∩ N − (dom G ) = ∅ . (a) According to Theorem 2.1(a), applied for U := ∆ X ∗ × Im A ∗∗ ⊆ X ∗ × X ∗ × Y ∗∗ , we have that { ( M ∗ ( u ∗ , w ∗ , u ∗∗ ) + N ∗ ( u ∗ , v ∗ , v ∗∗ ) , r ) : r ∈ R , F ∗ ( u ∗ , w ∗ , u ∗∗ ) + G ∗ ( u ∗ , v ∗ , v ∗∗ ) ≤ r } is closed regarding∆ X ∗ × Im A ∗∗ × R in ( X ∗ , w ∗ ) × ( X ∗ , w ∗ ) × ( Y ∗∗ , w ∗ ) × R (5)if and only if ( F ◦ M + G ◦ N ) ∗ ( x ∗ , x ∗ , A ∗∗ x ∗∗ ) =min ( u ∗ ,w ∗ ,u ∗∗ ) ∈ X ∗× X ∗× X ∗∗ ( u ∗ ,v ∗ ,v ∗∗ ) ∈ X ∗ × Y ∗ × Y ∗∗ { F ∗ ( u ∗ , w ∗ , u ∗∗ ) + G ∗ ( u ∗ , v ∗ , v ∗∗ ) : M ∗ ( u ∗ , w ∗ , u ∗∗ ) + N ∗ ( u ∗ , v ∗ , v ∗∗ ) = ( x ∗ , x ∗ , A ∗∗ x ∗∗ ) } for all ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ . (6)7ince F ∗ ( u ∗ , w ∗ , u ∗∗ ) = δ { } ( w ∗ )+ f ∗ ( u ∗ , u ∗∗ ) for all ( u ∗ , w ∗ , u ∗∗ ) ∈ X ∗ × X ∗ × X ∗∗ and G ∗ ( u ∗ , v ∗ , v ∗∗ ) = δ { } ( u ∗ )+ g ∗ ( v ∗ , v ∗∗ ) for all ( u ∗ , v ∗ , v ∗∗ ) ∈ X ∗ × Y ∗ × Y ∗∗ , one can easily see that { ( M ∗ ( u ∗ , w ∗ , u ∗∗ ) + N ∗ ( u ∗ , v ∗ , v ∗∗ ) , r ) : r ∈ R , F ∗ ( u ∗ , w ∗ , u ∗∗ ) + G ∗ ( u ∗ , v ∗ , v ∗∗ ) ≤ r } = { ( u ∗ , A ∗ v ∗ , A ∗∗ u ∗∗ + v ∗∗ , r ) : f ∗ ( u ∗ , u ∗∗ ) + g ∗ ( v ∗ , v ∗∗ ) ≤ r } , which means that the statement in (5) is nothing else than (i).On the other hand, for all ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ it holds( F ◦ M + G ◦ N ) ∗ ( x ∗ , x ∗ , A ∗∗ x ∗∗ ) =sup ( u,w,v ∗ ) ∈ X × X × Y ∗ {h ( x ∗ , x ∗ , A ∗∗ x ∗∗ ) , ( u, w, v ∗ ) i − ( F ◦ M )( u, w, v ∗ ) − ( G ◦ N )( u, w, v ∗ ) } =sup ( u,w,v ∗ ) ∈ X × X × Y ∗ {h ( x ∗ , x ∗∗ ) , ( u + w, A ∗ v ∗ ) i − f ( u, A ∗ v ∗ ) − g ( Aw, v ∗ ) } =sup ( s,s ∗ ) ∈ X × X ∗ (cid:26) h ( x ∗ , x ∗∗ ) , ( s, s ∗ ) i − inf ( u,w,v ∗ ) ∈ X × X × Y ∗ { f ( u, s ∗ ) + g ( Aw, v ∗ ) : u + w = s, A ∗ v ∗ = s ∗ } (cid:27) =( f (cid:13) A g ) ∗ ( x ∗ , x ∗∗ )and min ( u ∗ ,w ∗ ,u ∗∗ ) ∈ X ∗× X ∗× X ∗∗ ( u ∗ ,v ∗ ,v ∗∗ ) ∈ X ∗ × Y ∗ × Y ∗∗ { F ∗ ( u ∗ , w ∗ , u ∗∗ ) + G ∗ ( u ∗ , v ∗ , v ∗∗ ) : M ∗ ( u ∗ , w ∗ , u ∗∗ ) + N ∗ ( u ∗ , v ∗ , v ∗∗ ) = ( x ∗ , x ∗ , A ∗∗ x ∗∗ ) } =min ( u ∗ , ,u ∗∗ ) ∈ X ∗× X ∗× X ∗∗ (0 ,v ∗ ,v ∗∗ ) ∈ X ∗ × Y ∗ × Y ∗∗ { f ∗ ( u ∗ , u ∗∗ ) + g ∗ ( v ∗ , v ∗∗ ) : M ∗ ( u ∗ , , u ∗∗ ) + N ∗ (0 , v ∗ , v ∗∗ ) = ( x ∗ , x ∗ , A ∗∗ x ∗∗ ) } =min ( u ∗∗ ,w ∗∗ ,v ∗ ) ∈ X ∗∗ × X ∗∗ × Y ∗ { f ∗ ( x ∗ , u ∗∗ ) + g ∗ ( v ∗ , A ∗∗ w ∗∗ ) : A ∗ v ∗ = x ∗ , u ∗∗ + w ∗∗ = x ∗∗ } =( f ∗ (cid:13) A g ∗ )( x ∗ , x ∗∗ ) , which means that the the statement in (6) says actually that ( f (cid:13) A g ) ∗ ( x ∗ , x ∗∗ ) = ( f ∗ (cid:13) A g ∗ )( x ∗ , x ∗∗ ) and f ∗ (cid:13) A g ∗ is exact for every ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ . This leads to the desired conclusion.(b) The assertion is a direct consequence of Theorem 2.1(b), as, obviously,(0 , , , , , ∈ ic (dom F × dom G − ( M × N )(∆ X × X × Y ∗ )) ⇔ (0 , , , , , ∈ ic (cid:16) X × X × X × (cid:16) dom g × pr X ∗ (dom f ) − Im A × ∆ A ∗ Y ∗ (cid:17)(cid:17) ⇔ (0 , , ∈ ic (cid:16) dom g × pr X ∗ (dom f ) − Im A × ∆ A ∗ Y ∗ (cid:17) . Remark 2.4.
In the hypotheses of Theorem 2.2 and by keeping the notations used in its proof, according toRemark 2.1, we have ic (dom F × dom G − ( M × N )(∆ X × X × Y ∗ )) = ri (dom F × dom G − ( M × N )(∆ X × X × Y ∗ )) , which is equivalent to ic (cid:16) dom g × pr X ∗ (dom f ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ri (cid:16) dom g × pr X ∗ (dom f ) − Im A × ∆ A ∗ Y ∗ (cid:17) . In reflexive Banach spaces the equivalence in Theorem 2.2(a) gives rise to the following result.
Corollary 2.1.
Let X and Y be reflexive Banach spaces and f : X × X ∗ −→ R and g : Y × Y ∗ −→ R proper,convex and lower semicontinuous functions such that pr X ∗ (dom f ) ∩ A ∗ (pr Y ∗ (dom g )) = ∅ . Then the followingstatements are equivalent:(i) the set { ( u ∗ , A ∗ v ∗ , Au + v, r ) : r ∈ R , f ∗ ( u ∗ , u ) + g ∗ ( v ∗ , v ) ≤ r } is closed regarding ∆ X ∗ × Im A × R in ( X ∗ , k · k ∗ ) × ( X ∗ , k · k ∗ ) × ( Y, k · k ) × R ;(ii) ( f (cid:13) A g ) ∗ ( x ∗ , x ) = ( f ∗ (cid:13) A g ∗ )( x ∗ , x ) and f ∗ (cid:13) A g ∗ is exact for every ( x ∗ , x ) ∈ X ∗ × X. The maximal monotonicity of Gossez type (D) of S || A T In what follows we assume that X and Y are real nonzero Banach spaces, that S : X ⇒ X ∗ and T : Y ⇒ Y ∗ are two monotone operators and that A : X −→ Y is a linear continuous mapping. For S : X ∗∗ ⇒ X ∗ and T : Y ∗∗ ⇒ Y ∗ , Gossez’s monotone closures of S and T , respectively, we consider their extended generalized parallelsum defined via A , which is the multivalued operator defined as S || A T : X ⇒ X ∗ , S || A T ( x ) := ( S − + ( A ∗ T A ∗∗ ) − ) − ( x ) ∀ x ∈ X. The following result proposes two sufficient conditions ensuring the maximal monotonicity of Gossez type (D)of S || A T , provided that both operators are maximal monotone of Gossez type (D), and it will give rise to acharacterization of the maximal monotonicity of the generalized parallel sum of S and T defined via A , S || A T : X ⇒ X ∗ , S || A T ( x ) := ( S − + ( A ∗ T A ) − ) − ( x ) ∀ x ∈ X. Theorem 3.1.
Let S : X ⇒ X ∗ and T : Y ⇒ Y ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, and A : X −→ Y a linear continuous mapping such that dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ . Assume that one of the following conditions is fulfilled:(a) (0 , , ∈ ic (dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ ) ;(b) the set { ( u ∗ , A ∗ v ∗ , A ∗∗ u ∗∗ + v ∗∗ , r ) : r ∈ R , h ∗ S ( u ∗ , u ∗∗ )+ h ∗ T ( v ∗ , v ∗∗ ) ≤ r } is closed regarding ∆ X ∗ × Im A ∗∗ × R in ( X ∗ , w ∗ ) × ( X ∗ , w ∗ ) × ( Y ∗∗ , w ∗ ) × R .Then the function h : X × X ∗ −→ R , h ( x, x ∗ ) = cl k·k×k·k ∗ ( h S (cid:13) A h T )( x, x ∗ ) , is a strong representative function of S || A T and the extended generalized parallel sum S || A T is a maximal monotone operator of Gossez type (D).Proof. Obviously, h : X × X ∗ −→ R is convex and (strong) lower semicontinuous and, due to the feasibilitycondition dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ , h is not identical to + ∞ . Since one of the conditions (a) and(b) is fulfilled, then one has, via Theorem 2.2, that h ∗ ( x ∗ , x ∗∗ ) = ( h S (cid:13) A h T ) ∗ ( x ∗ , x ∗∗ ) = ( h ∗ S (cid:13) A h ∗ T )( x ∗ , x ∗∗ ) and h ∗ S (cid:13) A h ∗ T is exact for every ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ .Take an arbitrary ( x, x ∗ ) ∈ X × X ∗ . Then we have( h S (cid:13) A h T )( x, x ∗ ) = inf { h S ( u, x ∗ ) + h T ( Aw, v ∗ ) : u, w ∈ X, v ∗ ∈ Y ∗ , u + w = x, A ∗ v ∗ = x ∗ }≥ inf {h x ∗ , u i + h x ∗ , w i : u, w ∈ X, u + w = x } = h x ∗ , x i . Hence, h ( x, x ∗ ) = cl k·k×k·k ∗ ( h S (cid:13) A h T )( x, x ∗ ) ≥ h x ∗ , x i , which implies that h ≥ c , concomitantly ensuring that h is proper.Take an arbitrary ( x ∗ , x ∗∗ ) ∈ X ∗ × X ∗∗ . Then we have h ∗ ( x ∗ , x ∗∗ ) = ( h ∗ S (cid:13) A h ∗ T )( x ∗ , x ∗∗ )= inf { h ∗ S ( x ∗ , u ∗∗ ) + h ∗ T ( v ∗ , A ∗∗ w ∗∗ ) : u ∗∗ , w ∗∗ ∈ X ∗∗ , v ∗ ∈ Y ∗ , u ∗∗ + w ∗∗ = x ∗∗ , A ∗ v ∗ = x ∗ }≥ inf {h u ∗∗ , x ∗ i + h w ∗∗ , x ∗ i : u ∗∗ , w ∗∗ ∈ X ∗∗ , u ∗∗ + w ∗∗ = x ∗∗ } = h x ∗∗ , x ∗ i . Thus, according to Theorem 1.2 and Theorem 1.3, the operator with the graph { ( x, x ∗ ) ∈ X × X ∗ : h ( x, x ∗ ) = h x ∗ , x i} is maximal monotone of Gossez type (D) and one has { ( x, x ∗ ) ∈ X × X ∗ : h ( x, x ∗ ) = h x ∗ , x i} = { ( x, x ∗ ) ∈ X × X ∗ : h ∗ ( x ∗ , x ) = h x ∗ , x i} . In order to conclude the proof, we show that G ( S || A T ) = { ( x, x ∗ ) ∈ X × X ∗ : h ∗ ( x ∗ , x ) = h x ∗ , x i} and this will mean that h is a strong representative function of S || A T .Let ( x, x ∗ ) ∈ G ( S || A T ) . Then x ∈ S − ( x ∗ ) + ( A ∗ T A ∗∗ ) − ( x ∗ ) , hence there exists u ∗∗ ∈ S − ( x ∗ ) and w ∗∗ ∈ ( A ∗ T A ∗∗ ) − ( x ∗ ) such that x = u ∗∗ + w ∗∗ . Thus ( u ∗∗ , x ∗ ) ∈ G ( S ) and, as x ∗ ∈ A ∗ T A ∗∗ ( w ∗∗ ), there exists9 ∗ ∈ T ( A ∗∗ w ∗∗ ) such that A ∗ v ∗ = x ∗ . Consequently, h ∗ S ( x ∗ , u ∗∗ ) = h u ∗∗ , x ∗ i and h ∗ T ( v ∗ , A ∗∗ w ∗∗ ) = h A ∗∗ w ∗∗ , v ∗ i and, so, h ∗ ( x ∗ , x ) = ( h ∗ S (cid:13) A h ∗ T )( x ∗ , x ) ≤ h ∗ S ( x ∗ , u ∗∗ ) + h ∗ T ( v ∗ , A ∗∗ w ∗∗ ) = h u ∗∗ , x ∗ i + h w ∗∗ , x ∗ i = h x ∗ , x i . On the other hand, as shown above, h ∗ ( x ∗ , x ) ≥ h x ∗ , x i for all ( x, x ∗ ) ∈ X × X ∗ , hence h ∗ ( x ∗ , x ) = h x ∗ , x i , implyingthat G ( S || A T ) ⊆ { ( x, x ∗ ) ∈ X × X ∗ : h ( x, x ∗ ) = h x ∗ , x i} .Conversely, let be ( x, x ∗ ) ∈ X × X ∗ such that h ∗ ( x ∗ , x ) = h x ∗ , x i . Using that h ∗ S (cid:13) A h ∗ T is exact at ( x ∗ , x ), thereexists ( u ∗∗ , w ∗∗ , v ∗ ) ∈ X ∗∗ × X ∗∗ × Y ∗ such that u ∗∗ + w ∗∗ = x, A ∗ v ∗ = x ∗ and h x ∗ , x i = h ( x, x ∗ ) = h ∗ S ( x ∗ , u ∗∗ ) + h ∗ T ( v ∗ , A ∗∗ w ∗∗ ). Since, on the other hand, h ∗ S ( x ∗ , u ∗∗ ) + h ∗ T ( v ∗ , A ∗∗ w ∗∗ ) ≥ h u ∗∗ , x ∗ i + h A ∗∗ w ∗∗ , v ∗ i = h x ∗ , x i , itfollows that h ∗ S ( x ∗ , u ∗∗ ) = h u ∗∗ , x ∗ i and h ∗ T ( v ∗ , A ∗∗ w ∗∗ ) = h A ∗∗ w ∗∗ , v ∗ i .But h ∗ S and h ∗ T are representative functions of S − and T − , respectively, which means that ( u ∗∗ , x ∗ ) ∈ G ( S )and ( A ∗∗ w ∗∗ , v ∗ ) ∈ G ( T ). We have u ∗∗ ∈ S − ( x ∗ ) and, since w ∗∗ = x − u ∗∗ , we obtain v ∗ ∈ T A ∗∗ ( x − u ∗∗ ),hence x ∗ = A ∗ v ∗ ∈ A ∗ T A ∗∗ ( x − u ∗∗ ) or, equivalently, x − u ∗∗ ∈ ( A ∗ T A ∗∗ ) − ( x ∗ ) . Thus x = u ∗∗ + ( x − u ∗∗ ) ∈ ( S − + ( A ∗ T A ∗∗ ) − )( x ∗ ) and so ( x, x ∗ ) ∈ G ( S || A T ).Hence, G ( S || A T ) = { ( x, x ∗ ) ∈ X × X ∗ : h ∗ ( x ∗ , x ) = h x ∗ , x i} and this concludes the proof.Under the additional assumption that the domain of Gossez’s closure of S is a subset of X , the conditions (a) and(b) of the previous theorem become sufficient for the maximal monotonicity of Gossez type (D) of the generalizedparallel sum S || A T . One can notice that D ( S ) ⊆ X is particulary fulfilled when X is a reflexive Banach space . Theorem 3.2.
Let S : X ⇒ X ∗ and T : Y ⇒ Y ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, and A : X −→ Y a linear continuous mapping such that dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ and D ( S ) ⊆ X . Assume that one of the following conditions is fulfilled:(a) (0 , , ∈ ic (dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ ) ;(b) the set { ( u ∗ , A ∗ v ∗ , A ∗∗ u ∗∗ + v ∗∗ , r ) : r ∈ R , h ∗ S ( u ∗ , u ∗∗ )+ h ∗ T ( v ∗ , v ∗∗ ) ≤ r } is closed regarding ∆ X ∗ × Im A ∗∗ × R in ( X ∗ , w ∗ ) × ( X ∗ , w ∗ ) × ( Y ∗∗ , w ∗ ) × R .Then the function h : X × X ∗ −→ R , h ( x, x ∗ ) = cl k·k×k·k ∗ ( h S (cid:13) A h T )( x, x ∗ ) , is a strong representative function of S || A T and the generalized parallel sum S || A T is a maximal monotone operator of Gossez type (D).Proof. We need only to show that S || A T = S || A T , whenever D ( S ) ⊆ X . Indeed, ( x, x ∗ ) ∈ G ( S || A T ) if and only ifthere exist u ∗∗ ∈ S − ( x ∗ ) ⊆ X and w ∗∗ ∈ ( A ∗ T A ∗∗ ) − ( x ∗ ) such that x = u ∗∗ + w ∗∗ . This is further equivalent tothe existence of u ∗∗ and w ∗∗ in X such that ( u ∗∗ , x ∗ ) ∈ G ( S ), x ∗ ∈ A ∗ T A ∗∗ ( w ∗∗ ) = A ∗ T ( Aw ∗∗ ) = A ∗ T ( Aw ∗∗ ) and x = u ∗∗ + w ∗∗ . But this is the same with x ∈ S − ( x ∗ ) + ( A ∗ T A ) − ( x ∗ ) or, equivalently, ( x, x ∗ ) ∈ G ( S || A T ). Remark 3.1.
Concerning the two sufficient conditions for maximal monotonicity considered in Theorem 3.1 andTheorem 3.2, one can notice, according to Theorem 2.2, that condition (b) is fulfilled whenever condition (a) isfulfilled. In the last section of the paper we provide a situation where the latter fails, while condition (b) is valid(see Example 4.1).In the last part of this section we turn our attention to the formulation of further interiority-type regularityconditions for the maximal monotonicity of Gossez type (D) of the generalized parallel sums S || A T , respectively, S || A T , this time expressed by means of the graph of T and of the range of S . We start with the following result. Theorem 3.3.
Let S : X ⇒ X ∗ and T : Y ⇒ Y ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, and A : X −→ Y a linear continuous mapping such that dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ . Then it holds: ic (cid:16) G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⊆ ic (cid:16) co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⊆ ic (cid:16) dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ri (cid:16) dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ (cid:17) . Proof.
Let us denote by C := dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ and by D := G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ .Then co D = co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ and, obviously, ic D ⊆ ic (co D ). On the other hand, as pointed outin Remark 2.4, we have ic C = ri C . Thus, it remains to show that ic (co D ) ⊆ ic C .10ince, co D ⊆ C , one has aff(co D ) = aff D ⊆ aff C . Thus, in order to prove that ic (co D ) ⊆ ic C , it is enough toshow that that aff C ⊆ cl(aff D ). The proof will rely on [31, Lemma 20.4(b)] (for another result, where this lemmafound application we refer to [38]). What we will actually prove, is thatdom ϕ T × pr X ∗ (dom ϕ S ) ⊆ cl(aff D ) , (7)where ϕ S and ϕ T denote the Fitzpatrick functions of the operators S and T , respectively. If (7) is true, then onegets C ⊆ dom ϕ T × pr X ∗ (dom ϕ S ) − Im A × ∆ A ∗ Y ∗ ⊆ cl (cid:16) aff D − Im A × ∆ A ∗ Y ∗ (cid:17) = cl(aff D ) , which leads to the desired conclusion.In order to show (7), we assume without loss of generality that (0 , ∈ G ( S ) and (0 , ∈ G ( T ). Suppose thatthere exists ( v, v ∗ , u ∗ ) ∈ dom ϕ T × pr X ∗ (dom ϕ S ) such that ( v, v ∗ , u ∗ ) cl(aff D ). Then, according to a strongseparation theorem, there exist δ ∈ R and ( q ∗ , q ∗∗ , p ∗∗ ) ∈ Y ∗ × Y ∗∗ × X ∗∗ such that h ( q ∗ , q ∗∗ , p ∗∗ ) , ( v, v ∗ , u ∗ ) i > δ > sup {h ( q ∗ , q ∗∗ , p ∗∗ ) , ( y, y ∗ , x ∗ ) i : ( y, y ∗ , x ∗ ) ∈ cl(aff D ) } . As 0 ∈ D , aff D is a linear subspace. Thus h ( q ∗ , q ∗∗ , p ∗∗ ) , ( y, y ∗ , x ∗ ) i = 0 for all ( y, y ∗ , x ∗ ) ∈ aff D and,consequently, δ >
0. In other words, h ( q ∗ , q ∗∗ , p ∗∗ ) , ( y − Au, y ∗ − v ∗ , x ∗ − A ∗ v ∗ ) i = 0 ∀ ( y, y ∗ ) ∈ G ( T ) ∀ x ∗ ∈ R ( S ) ∀ u ∈ X ∀ v ∗ ∈ Y ∗ . (8)By taking ( y, y ∗ , x ∗ ) := (0 , , ∈ G ( T ) × R ( S ), we obtain q ∗∗ = − A ∗∗ p ∗∗ and from here it results that h q ∗ , Au i = h A ∗ q ∗ , u i = 0 ∀ u ∈ X, which means that A ∗ q ∗ = 0. Hence, h q ∗∗ , q ∗ i = h− A ∗∗ p ∗∗ , q ∗ i = h− p ∗∗ , A ∗ q ∗ i = 0 . On the other hand, from (8), we have h ( q ∗ , q ∗∗ , p ∗∗ ) , ( y, y ∗ , x ∗ ) i = 0 for all ( y, y ∗ ) ∈ G ( T ) and all x ∗ ∈ R ( S ), hence h ( q ∗ , q ∗∗ ) , ( y, y ∗ ) i = 0 ∀ ( y, y ∗ ) ∈ G ( T )and h p ∗∗ , x ∗ i = 0 ∀ x ∗ ∈ R ( S ) . Take now an arbitrary ( y ∗∗ , y ∗ ) ∈ G ( T ) . Then there exists ( y α , y ∗ α ) α ∈ I ∈ G ( T ) such that ( y α ) α ∈ I converges to y ∗∗ in the weak ∗ topology of Y ∗∗ and ( y ∗ α ) α ∈ I converges to y ∗ in the strong topology of Y ∗ . Since ( y α , y ∗ α ) ∈ G ( T ) , we have h ( q ∗ , q ∗∗ ) , ( y α , y ∗ α ) i = 0 for every α ∈ I , hence h ( q ∗ , q ∗∗ ) , ( y ∗∗ , y ∗ ) i = 0. Consequently, h ( q ∗ , q ∗∗ ) , ( y ∗∗ , y ∗ ) i = 0 ∀ ( y ∗∗ , y ∗ ) ∈ G ( T )and one can prove in a similar way that h p ∗∗ , x ∗ i = 0 ∀ x ∗ ∈ R ( S ) . From here, according to [31, Lemma 20.4(b)], one has (as h q ∗∗ , q ∗ i = 0) h ( q ∗ , q ∗∗ ) , ( y ∗∗ , y ∗ ) i = 0 ∀ ( y ∗∗ , y ∗ ) ∈ dom ϕ T and (as, obviously, h p ∗∗ , i = 0) h (0 , p ∗∗ ) , ( x ∗∗ , x ∗ ) i = 0 ∀ ( x ∗ , x ∗∗ ) ∈ dom ϕ S . But ( v, v ∗ , u ∗ ) ∈ dom ϕ T × pr X ∗ (dom ϕ S ) and, as ϕ S | X × X ∗ = ϕ S and ϕ T | Y × Y ∗ = ϕ T , it follows that h ( q ∗ , q ∗∗ , p ∗∗ ) , ( v, v ∗ , u ∗ ) i = 0 , which is a contradiction to δ >
0. Consequently, (7) is valid and, so, ic (co D ) ⊆ ic ( C ) . S || A T . Corollary 3.1.
Let S : X ⇒ X ∗ and T : Y ⇒ Y ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, and A : X −→ Y a linear continuous mapping such that dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ . If (0 , , ∈ ic (cid:16) G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) or (0 , , ∈ ic (cid:16) co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) , then the extended generalized parallel sum S || A T is a maximal monotone operator of Gossez type (D). As follows from the following result, under the supplementary assumption that D ( S ) ⊆ X , the inclusion relationsin Theorem 3.3 become equalities. Theorem 3.4.
Let S : X ⇒ X ∗ and T : Y ⇒ Y ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, and A : X −→ Y a linear continuous mapping such that dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ and D ( S ) ⊆ X . Then it holds: ic (cid:16) G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ri (cid:16) G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ic (cid:16) co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ri (cid:16) co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ic (cid:16) dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ (cid:17) = ri (cid:16) dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ (cid:17) . Proof.
By keeping the notations introduced in the proof of Theorem 3.3, let us prove first that ic C ⊆ D . Take anarbitrary ( v, v ∗ , u ∗ ) ∈ ic C , hence (0 , , ∈ ic ( C − ( v, v ∗ , u ∗ )). Consider the functions˜ f : X × X ∗ −→ R , ˜ f ( x, x ∗ ) = h S ( x, x ∗ + u ∗ ) − h u ∗ , x i and ˜ g : Y × Y ∗ −→ R , ˜ g ( y, y ∗ ) = h T ( y + v, y ∗ + v ∗ ) − ( h v ∗ , y i + h y ∗ , v i + h v ∗ , v i )and the operators e S : X ⇒ X ∗ defined by G ( e S ) = { ( x, x ∗ ) ∈ X × X ∗ : ˜ f ( x, x ∗ ) = h x ∗ , x i} and e T : Y ⇒ Y ∗ defined by G ( e T ) = { ( y, y ∗ ) ∈ Y × Y ∗ : ˜ g ( y, y ∗ ) = h y ∗ , y i} . It can be easily observed, that G ( e S ) = G ( S ) − (0 , u ∗ )and G ( e T ) = G ( T ) − ( v, v ∗ ). Consequently, e S and e T are maximal monotone operators of Gossez type (D) and ˜ f , respectively, ˜ g are strong representative functions for them. Since D ( S ) ⊆ X , the domain of Gossez’s closure of e S is a subset of X , too. Hence, according to Theorem 3.2, the condition(0 , , ∈ ic ( C − ( v, v ∗ , u ∗ )) = ic (cid:16) dom ˜ g × pr X ∗ (dom ˜ f ) − Im A × ∆ A ∗ Y ∗ (cid:17) ensures the maximal monotonicity of e S || A e T . Hence, G ( e S || A e T ) = ∅ , thus there exists x ∗ ∈ ( e S − + ( A ∗ e T A ) − ) − ( x )for some x ∈ X. This means that there exist u, w ∈ X such that ( u, x ∗ ) ∈ G ( e S ) and ( w, x ∗ ) ∈ G ( A ∗ e T A ) and u + w = x . As G ( e S ) = G ( S ) − (0 , u ∗ ), we have(0 , u ∗ ) ∈ G ( S ) − ( u, x ∗ ) . On the other hand, as x ∗ ∈ A ∗ e T A ( w ) , there exists y ∗ ∈ Y ∗ , such that y ∗ ∈ e T ( Aw ) and x ∗ = A ∗ y ∗ . Thus, for y := Aw , we have ( y, y ∗ ) ∈ G ( e T ) = G ( T ) − ( v, v ∗ ) , hence( v, v ∗ ) ∈ G ( T ) − ( y, y ∗ ) . In conclusion, ( v, v ∗ , u ∗ ) ∈ G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ = D and, so, ic C ⊆ D .If ic C = ri C is empty, then by Theorem 3.3 it holds ic D = ic (co D ) = ic C = ri C = ∅ . Consequently,ri D = ri(co D ) = ∅ .Assume now that ic C is nonempty. Since ic C ⊆ D ⊆ co D ⊆ C , one gets that ic D = ic (co D ) = ic C = ri C .Moreover, it holds aff( ic C ) = aff C and, as ri C = ic C ⊆ D ⊆ co D ⊆ C , we have aff C = aff D , these sets beingclosed. Thus ri C = ri D = ri(co D ) and this provides the desired conclusion.12e close the section by the following characterization of the maximal monotonicity of Gossez type (D) of S || A T ,which follows from Theorem 3.2 and Theorem 3.4. Corollary 3.2.
Let S : X ⇒ X ∗ and T : Y ⇒ Y ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, and A : X −→ Y a linear continuous mapping such that dom h T × pr X ∗ (dom h S ) ∩ Im A × ∆ A ∗ Y ∗ = ∅ and D ( S ) ⊆ X . Then one has the following sequence of equivalencies (0 , , ∈ ic (cid:16) G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⇔ (0 , , ∈ ri (cid:16) G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⇔ (0 , , ∈ ic (cid:16) co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⇔ (0 , , ∈ ri (cid:16) co G ( T ) × co R ( S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⇔ (0 , , ∈ ic (cid:16) dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ (cid:17) ⇔ (0 , , ∈ ri (cid:16) dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ (cid:17) and each of these conditions guarantees that the generalized parallel sum S || A T is a maximal monotone operatorof Gossez type (D). In this section we will consider two particular instances of the generalized parallel sum defined via a linear continuousmapping and show what the results provided in Section 3 become in these special settings. S || T Assume that X is a real nonzero Banach space and S : X ⇒ X ∗ and T : X ⇒ X ∗ are two monotone operators.By taking A = id X : X → X , their extended generalized parallel sum defined via A and their generalized parallelsum defined via A become the extended parallel sum of S and TS || T : X ⇒ X ∗ , S || T ( x ) := ( S − + T − ) − ( x ) ∀ x ∈ X and the classical parallel sum of S and T , S || T : X ⇒ X ∗ , S || T ( x ) := ( S − + T − ) − ( x ) ∀ x ∈ X, respectively.Having h S : X × X ∗ −→ R and h T : X × X ∗ −→ R representative functions of S and T , respectively, theextended infimal convolutions of them, namely h S (cid:13) A h T and h ∗ S (cid:13) A h ∗ T , turn out to be the following classicalbivariate infimal convolutions (see, for instance, [5, 31, 35, 37]) h S (cid:3) h T : X × X ∗ → R , ( h S (cid:3) h T )( x, x ∗ ) = inf { h S ( u, x ∗ ) + h T ( w, x ∗ ) : u, w ∈ X, u + w = x } and h ∗ S (cid:3) h ∗ T : X ∗ × X ∗∗ → R , ( h ∗ S (cid:3) h ∗ T )( x ∗ , x ∗∗ ) = inf { h ∗ S ( x ∗ , u ∗∗ ) + h ∗ T ( x ∗ , w ∗∗ ) : u ∗∗ , w ∗∗ ∈ X ∗∗ , u ∗∗ + w ∗∗ = x ∗∗ } , respectively. Theorem 4.1.
Let S : X ⇒ X ∗ and T : X ⇒ X ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, such that pr X ∗ (dom h S ) ∩ pr X ∗ (dom h T ) = ∅ and assumethat one of the following conditions is fulfilled:(a) ∈ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) ;(b) the set { ( u ∗ , v ∗ , u ∗∗ + v ∗∗ , r ) : r ∈ R , h ∗ S ( u ∗ , u ∗∗ ) + h ∗ T ( v ∗ , v ∗∗ ) ≤ r } is closed regarding ∆ X ∗ × X ∗∗ × R in ( X ∗ , w ∗ ) × ( X ∗ , w ∗ ) × ( X ∗∗ , w ∗ ) × R .Then the following statements are true:(i) The function h : X × X ∗ −→ R , h ( x, x ∗ ) = cl k·k×k·k ∗ ( h S (cid:3) h T )( x, x ∗ ) , is a strong representative function of S || T and the extended parallel sum S || T is a maximal monotone operator of Gossez type (D). ii) If D ( S ) ⊆ X (or, if D ( T ) ⊆ X ), then the function h : X × X ∗ −→ R , h ( x, x ∗ ) = cl k·k×k·k ∗ ( h S (cid:3) h T )( x, x ∗ ) , isa strong representative function of S || T and the parallel sum S || T is a maximal monotone operator of Gosseztype (D).Proof. The result follows directly Theorem 3.3 and Theorem 3.4, by noticing that the interiority-type condition inthese two statements becomes(0 , , ∈ ic (dom h T × pr X ∗ (dom h S ) − X × ∆ X ∗ ) = X × ic (pr X ∗ (dom h S ) × pr X ∗ (dom h S ) − ∆ X ∗ )or, equivalently, (0 , ∈ ic (pr X ∗ (dom h S ) × pr X ∗ (dom h S ) − ∆ X ∗ ) . According to Lemma 1.1, the latter relation is equivalent to0 ∈ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) . The next result follows from Theorem 3.3 and Theorem 4.1(i).
Theorem 4.2.
Let S : X ⇒ X ∗ and T : X ⇒ X ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, such that pr X ∗ (dom h S ) ∩ pr X ∗ (dom h T ) = ∅ .(a) Then it holds: ic ( R ( S ) − R ( T )) ⊆ ic (co R ( S ) − co R ( T )) ⊆ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) = ri(pr X ∗ (dom h S ) − pr X ∗ (dom h T )) . (b) If ∈ ic ( R ( S ) − R ( T )) or ∈ ic (co R ( S ) − co R ( T )) , then the extended parallel sum S || T is a maximal monotone operator of Gossez type (D).Proof. As (b) is a direct consequence of Theorem 4.1(i) and statement (a), we will turn our attention to the proofof the latter. Concerning it, one can easily notice that the inclusion ic ( R ( S ) − R ( T )) ⊆ ic (co R ( S ) − co R ( T ))follows directly from the definition of the intrinsic relative algebraic interior, while the equality ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) = ri(pr X ∗ (dom h S ) − pr X ∗ (dom h T ))is a direct consequence of [40, Theorem 2.7.2], applied to the proper, convex and lower semicontinuous functionΦ : X × X × X ∗ × X ∗ → R , Φ( x, u, x ∗ , u ∗ ) = h S ( x, x ∗ + u ∗ ) + h T ( u, x ∗ ) , by taking into account that (we consider the projection on the fourth component of the product space X × X × X ∗ × X ∗ ) pr X ∗ (dom Φ) = pr X ∗ (dom h S ) − pr X ∗ (dom h T ) . What it remained to be shown, namely that ic (co R ( S ) − co R ( T )) ⊆ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) , follows according to Lemma 1.1 and Theorem 3.3. Indeed, when u ∗ ∈ ic (co R ( S ) − co R ( T )) or, equivalently,0 ∈ ic (co R ( S ) − u ∗ − co R ( T )), one has that( u ∗ , ∈ ic (co R ( T ) × co R ( S ) − ∆ X ∗ ) ⊆ ic (pr X ∗ (dom h T ) × pr X ∗ (dom h S ) − ∆ X ∗ )and from here, again via Lemma 1.1, it follows u ∗ ∈ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )).Theorem 3.4 and Theorem 4.1(ii) give rise to the following result.14 heorem 4.3. Let S : X ⇒ X ∗ and T : X ⇒ X ∗ be two maximal monotone operators of Gossez type (D) withstrong representative functions h S and h T , respectively, such that pr X ∗ (dom h S ) ∩ pr X ∗ (dom h T ) = ∅ and D ( S ) ⊆ X (or, D ( T ) ⊆ X ).(a) Then it holds: ic ( R ( S ) − R ( T )) = ri ( R ( S ) − R ( T )) = ic (co R ( S ) − co R ( T )) = ri (co R ( S ) − co R ( T )) = ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) = ri(pr X ∗ (dom h S ) − pr X ∗ (dom h T )) . (b) One has the following sequence of equivalencies ∈ ic ( R ( S ) − R ( T )) ⇔ ∈ ri ( R ( S ) − R ( T )) ⇔ ∈ ic (co R ( S ) − co R ( T )) ⇔ ∈ ri (co R ( S ) − co R ( T )) ⇔ ∈ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) ⇔ ∈ ri(pr X ∗ (dom h S ) − pr X ∗ (dom h T )) and each of these conditions guarantees that the parallel sum S || T is a maximal monotone operator of Gosseztype (D).Proof. We will only prove statement (a), as (b) is a direct consequence of it and Theorem 4.1(ii).For an arbitrary u ∗ ∈ ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) one has, via Lemma 1.1, that( u ∗ , ∈ ic (pr X ∗ (dom h T ) × pr X ∗ (dom h S ) − ∆ X ∗ ) . Further, by Theorem 3.4 it follows ( u ∗ , ∈ ( R ( T ) × R ( S ) − ∆ X ∗ ), implying that u ∗ ∈ R ( S ) − R ( T ). Consequently, ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) ⊆ R ( S ) − R ( T ) . If ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) is empty, then there is nothing to be proved. Otherwise, the conclusion follows,by using that ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T )) ⊆ R ( S ) − R ( T ) ⊆ co R ( S ) − co R ( T ) ⊆ pr X ∗ (dom h S ) − pr X ∗ (dom h T )and aff( ic (pr X ∗ (dom h S ) − pr X ∗ (dom h T ))) = aff(pr X ∗ (dom h S ) − pr X ∗ (dom h T )). Remark 4.1.
In the setting of reflexive Banach spaces several interority-type regularity conditions ensuring themaximal monotonicity of the parallel sum S || T of two maximal monotone operators S and T have been introducedin the literature. While in [1] the condition int( R ( S )) ∩ R ( T ) = ∅ was considered, in [28] it has been assumed thatcone( R ( S ) − R ( T )) = X ∗ . Further, in a Hilbert space context, in [21] the conditioncone( R ( S ) − R ( T )) is a closed linear subspace of X ∗ has been stated, while in [26], in reflexive Banach spaces, the conditioncone(co R ( S ) − co R ( T )) is a closed linear subspace of X ∗ was proposed.Taking into account that an operator T : X ⇒ X ∗ is maximal monotone if and only if T − : X ∗ ⇒ X is maximalmonotone and that D ( T − ) = R ( T ), one can easily observe that all these interiority-type regularity conditionsensuring that S || T is maximal monotone, provided S and T are maximal monotone, are the counterpart of somemeanwhile classical ones stated for the maximal monotonicity of the sum S − + T − (see, for instance, [27, 31, 35])and can be easily derived from them.For interiority-type regularity conditions guaranteeing the maximal monotonicity of Gossez type (D) of theparallel sum and the extended parallel sum of two maximal monotone operators of Gossez type (D) in generalBanach spaces we refer to [33]. These results have been obtained as particular instances of some correspondingones formulated for the generalized parallel sum defined via a linear continuous mapping S || A T .15 xample 4.1. With this example we want to emphasize that there exist maximal monotone operators with amaximal monotone parallel sum and for which the interiority-type regularity condition (a) in Theorem 4.1 is notfulfilled, while the closedness-type condition (b) in Theorem 4.1 holds.Consider the proper, sublinear and lower semicontinuous functions f, g : R −→ R , f ( x , x ) = k ( x , x ) k + δ R ( x , x ) , where k · k denotes the Euclidean norm on R , and g ( x , x ) = √ / x + 1 / x + δ − R ( x , x ). Thenthe multivalued operators S := ∂f and T := ∂g are maximal monotone and their only representative functions are h S (( x , x ) , ( x ∗ , x ∗ )) = f ( x , x ) + f ∗ ( x ∗ , x ∗ ) and h T (( x , x ) , ( x ∗ , x ∗ )) = g ( x , x ) + g ∗ ( x ∗ , x ∗ ), respectively. Onecan easily verify that f ∗ = δ cl B R − R , where B R denotes the open unit ball of R , and g ∗ = δ [ √ / , + ∞ ) × [1 / , + ∞ ) .Obviously, pr R (dom h S ) ∩ pr R (dom h T ) = (cl B R − R ) ∩ [ √ / , + ∞ ) × [1 / , + ∞ ) = ∅ , where the projection is taken onto the second component of the product space R × R .We also have { ( u ∗ , v ∗ , u ∗∗ + v ∗∗ , r ) ∈ R × R × R × R : h ∗ S ( u ∗ , u ∗∗ ) + h ∗ T ( v ∗ , v ∗∗ ) ≤ r } = (cid:0) cl B R − R (cid:1) × [ √ / , + ∞ ) × [1 / , + ∞ ) × { ( x + y , x + y , r ) ∈ R × R : k ( x , x ) k + √ / y + 1 / y ≤ r } , which is obviously a closed set. Hence, condition (b) in Theorem 4.1 is fulfilled and S || T is maximal monotone.On the other hand, one can notice that condition (a) in Theorem 4.1 fails. Otherwise, one would have accordingto Theorem 4.3(b) that (0 , ∈ ri(pr R (dom h S ) − pr R (dom h T ))or, equivalently, (cid:0) B R − int R (cid:1) ∩ ( √ / , + ∞ ) × (1 / , + ∞ ) = ∅ , which would lead to a contradiction. A ∗ T A
For the second particular instance, we treat in this section, we stay in the same setting as in Section 3, but assumethat S : X ⇒ X ∗ is the multivalued operator with G ( S ) = { } × X ∗ , which is obviously maximal monotone ofGossez type (D). Its extension to the bidual, S : X ∗∗ ⇒ X ∗ , fulfills G ( S ) = { } × X ∗ , which means that theextended generalized parallel sum S || A T and the generalized parallel sum S || A T coincide (see also the proof ofTheorem 3.2) and fulfill S || A T ( x ) = S || A T ( x ) = A ∗ T A ( x ) ∀ x ∈ X. Since ϕ S = ψ S = δ { }× X ∗ , by Proposition 1.1 it follows that the only representative function of S is h S = δ { }× X ∗ .Since h ∗ S = δ X ∗ ×{ } , h S is actually a strong representative function of S .Having h T : Y × Y ∗ −→ R a representative function T , the extended infimal convolutions h S (cid:13) A h T and h ∗ S (cid:13) A h ∗ T of h S and h T become in this situation h AT : X × X ∗ → R , h AT ( x, x ∗ ) = inf { h T ( Ax, v ∗ ) : v ∗ ∈ Y ∗ , A ∗ v ∗ = x ∗ } and h ∗ AT : X ∗ × X ∗∗ → R , h ∗ AT ( x ∗ , x ∗∗ ) = inf { h ∗ T ( v ∗ , A ∗∗ x ∗∗ ) : v ∗ ∈ Y ∗ , A ∗ v ∗ = x ∗ } , respectively.Noticing that dom h T × pr X ∗ (dom h S ) − Im A × ∆ A ∗ Y ∗ = (pr Y (dom h T ) − Im A ) × Y ∗ × X ∗ , Theorem 3.2 givesrise to the following result. Theorem 4.4.
Let T : Y ⇒ Y ∗ be a maximal monotone operators of Gossez type (D) with strong representativefunction h T and A : X −→ Y a linear continuous mapping such that pr Y (dom h T ) ∩ Im A = ∅ . Assume that oneof the following conditions is fulfilled:(a) ∈ ic (pr Y (dom h T ) − Im A ) ;(b) the set { ( A ∗ v ∗ , v ∗∗ , r ) : r ∈ R , h ∗ T ( v ∗ , v ∗∗ ) ≤ r } is closed regarding X ∗ × Im A ∗∗ × R in ( X ∗ , w ∗ ) × ( Y ∗∗ , w ∗ ) × R .Then the function h : X × X ∗ −→ R , h ( x, x ∗ ) = cl k·k×k·k ∗ h AT ( x, x ∗ ) , is a strong representative function of A ∗ T A and A ∗ T A is a maximal monotone operator of Gossez type (D). G ( T ) × R ( S ) − Im A × ∆ A ∗ Y ∗ = ( D ( T ) − Im A ) × Y ∗ × X ∗ , via Theorem 3.4 and Corollary 3.2 we obtainthe following statement. Theorem 4.5.
Let T : Y ⇒ Y ∗ be a maximal monotone operators of Gossez type (D) with strong representativefunction h T and A : X −→ Y a linear continuous mapping such that pr Y (dom h T ) ∩ Im A = ∅ .(a) Then it holds: ic ( D ( T ) − Im A ) = ri ( D ( T ) − Im A ) = ic (co D ( T ) − Im A ) = ri (co D ( T ) − Im A ) = ic (pr Y (dom h T ) − Im A ) = ri(pr Y (dom h T ) − Im A ) . (b) One has the following sequence of equivalencies ∈ ic ( D ( T ) − Im A ) ⇔ ∈ ri ( D ( T ) − Im A ) ⇔ ∈ ic (co D ( T ) − Im A ) ⇔ ∈ ri (co D ( T ) − Im A ) ⇔ ∈ ic (pr Y (dom h T ) − Im A ) ⇔ ∈ ri(pr Y (dom h T ) − Im A ) and each of these conditions guarantees that A ∗ T A is a maximal monotone operator of Gossez type (D).
Remark 4.2.
Using as a starting point Theorem 4.4 and Theorem 4.5 and by employing the techniques usedin [12], one can further provide interiority- and closedness-type regularity conditions for the maximal monotonicityof Gossez type (D) of the sum of two maximal monotone operators of Gossez type (D), but also for the sum of amaximal monotone operator of Gossez type (D) with the composition of another maximal monotone operator ofGossez type (D) with a linear continuous mapping (for the latter one will thereby rediscover the statements givenin [37, Theorem 16]).
Acknowledgements.
The authors are thankful to E.R. Csetnek for pertinent comments and suggestions on anearlier draft of the article.
References [1] H. Attouch, Z. Chbani, A. Moudafi,
Une notion d’op´erateur de r´ecession pour les maximaux monotones ,S´eminaire d’Analyse Convexe, Montpellier, Expos´e No. 12, 37 pp, 1992.[2] H.H. Bauschke,
Fenchel duality, Fitzpatrick functions and the extension of firmly nonexpansive mappings ,Proceedings of the American Mathematical Society 135(1), 135–139, 2007.[3] J.M. Borwein,
Maximality of sums of two maximal monotone operators in general Banach space , Proceedingsof the American Mathematical Society 135(12), 3917–3924, 2007.[4] R.I. Bot¸,
Conjugate Duality in Convex Optimization , Springer-Verlag, Berlin Heidelberg, 2010.[5] R.I. Bot¸, E.R. Csetnek,
An application of the bivariate inf-convolution formula to enlargments of monotoneoperators , Set-Valued Analysis 16(7-8), 983–997, 2008.[6] R.I. Bot¸, S.-M. Grad, G. Wanka,
Maximal monotonicity for the precomposition with a linear operator , SIAMJournal on Optimization 17(4), 1239–1252, 2006.[7] R.I. Bot¸, S.-M. Grad, G. Wanka,
Weaker constraint qualifications in maximal monotonicity , Numerical Func-tional Analysis and Optimization 28(1-2), 27–41, 2007.[8] R.S. Burachik, S. Fitzpatrick,
On a family of convex functions associated to subdifferentials , Journal of Non-linear and Convex Analysis 6(1), 165–171, 2005.[9] R.S. Burachik, B.F. Svaiter,
Maximal monotonicity, conjugation and duality product , Proceedings of the Amer-ican Mathematical Society 131(8), 2379–2383, 2003.[10] R.S. Burachik, B.F. Svaiter,
Maximal monotone operators, convex functions and a special family of enlarge-ments , Set-Valued Analysis 10(4), 297–316, 2002.[11] S. Fitzpatrick,
Representing monotone operators by convex functions , in: “Workshop/Miniconference on Func-tional Analysis and Optimization (Canberra, 1988)”, Proceedings of the Centre for Mathematical Analysis 20,Australian National University, Canberra, 59–65, 1988.1712] Y. Garcia, M. Lassonde, J. Revalski,
Extended sums and extended compositions of monotone operators , Journalof Convex Analysis 13(3), 721–738, 2006.[13] J.-P. Gossez,
On the extensions to the bidual of a maximal monotone operator , Proceedings of the AmericanMathematical Society 62(1), 67-71, 1977.[14] J.-P. Gossez,
Op´erateurs monotones nonlin´eaires dans les espaces de Banach non r´eflexifs , Journal of Mathe-matical Analysis and Applications 34, 371–395, 1971.[15] R.B. Holmes,
Geometric Functional Analysis and its Applications , Springer-Verlag, Berlin, 1975.[16] M. Marques Alves, B.F. Svaiter,
A new old class of maximal monotone operators , Journal of Convex Analysis16(3-4), 881–890, 2009.[17] M. Marques Alves, B.F. Svaiter,
Bronsted-Rockafellar property and maximality of monotone operators rep-resentable by convex functions in non-reflexive Banach spaces , Journal of Convex Analysis 15(4), 693–706,2008.[18] M. Marques Alves, B.F. Svaiter,
On Gossez type (D) maximal monotone operators , Journal of Convex Analysis17(3-4), 1077–1088, 2010.[19] J.E. Mart´ınez-Legaz, B.F. Svaiter,
Monotone operators representable by l.s.c. convex functions , Set-ValuedAnalysis 13(1), 1–46, 2005.[20] J.E. Mart´ınez-Legaz, M. Th´era,
A convex representation of maximal monotone operators , Journal of Nonlinearand Convex Analysis 2(2), 243–247, 2001.[21] A. Moudafi,
On the stability of the parallel sum of maximal monotone operators , Journal of MathematicalAnalysis and Applications 199, 478–488, 1996.[22] J.B. Passty,
The parallel sum of nonlinear monotone operators , Nonlinear Analysis: Theory, Methods &Applications 10(3), 215-227, 1986.[23] J.P. Penot,
A representation of maximal monotone operators by closed convex functions and its impact oncalculus rules , Comptes Rendus Math´ematique. Acad´emie des Sciences Paris 338(11), 853–858, 2004.[24] J.P. Penot,
Is convexity useful for the study of monotonicity? , in: R.P. Agarwal, D. O’Regan (eds.), “NonlinearAnalysis and Applications”, Kluwer, Dordrecht, Vol. 1-2, 807–822, 2003.[25] J.P. Penot,
The relevance of convex analysis for the study of monotonicity , Nonlinear Analysis: Theory,Methods & Applications 58(7-8), 855–871, 2004.[26] J.P. Penot, C. Z˘alinescu,
Convex analysis can be helpful for the asymptotic analysis of monotone operators ,Mathematical Programming 116(1-2), 481-498, 2009.[27] J.P. Penot, C. Z˘alinescu,
Some problems about the representation of monotone operators by convex functions ,ANZIAM Journal 47, 1–20, 2005[28] H. Riahi,
About the inverse operations on the hyperspace of nonlinear monotone operators , Extracta Matem-aticae 8(1), 68–74, 1993.[29] R.T. Rockafellar,
Conjugate duality and optimization , Conference Board of the Mathematical Sciences RegionalConference Series in Applied Mathematics 16, Society for Industrial and Aplied Mathematics, Philadelphia,1974.[30] R.T. Rockafellar,
On the maximal monotonicity of subdifferential mappings , Pacific Journal of Mathematics33(1), 209–216, 1970.[31] S. Simons,
From Hahn-Banach to Monotonicity , Springer-Verlag, Berlin, 2008.[32] S. Simons,
Minimax and Monotonicity , Springer-Verlag, Berlin, 1998.[33] S. Simons,
Quadrivariate existence theorems and strong representability , arXiv:0809.0325v2, 2011.[34] S. Simons,
The range of a monotone operator , Journal of Mathematical Analysis and Applications 199, 176–201, 1996. 1835] S. Simons, C. Z˘alinescu,
Fenchel duality, Fitzpatrick functions and maximal monotonicity , Journal of Nonlinearand Convex Analysis 6(1), 1–22, 2005.[36] M.D. Voisei,
Calculus rules for maximal monotone operators in general Banach spaces , Journal of ConvexAnalysis 15(1), 73–85, 2008.[37] M.D. Voisei, C. Z˘alinescu,
Strongly-representable monotone operators , Journal of Convex Analysis 16(3-4),1011–1033, 2009.[38] L. Yao,
An affirmative answer to a problem posed by Z˘alinescu , Journal of Convex Analysis 18(3), 2011.[39] C. Z˘alinescu,
A new proof of the maximal monotonicity of the sum using the Fitzpatrick function , in: F.Giannessi, A. Maugeri (eds.), “Variational Analysis and Applications”, Nonconvex Optimization and its Ap-plications 79, Springer, New York, 1159-1172, 2005.[40] C. Z˘alinescu,
Convex Analysis in General Vector Spaces , World Scientific, Singapore, 2002.[41] C. Z˘alinescu,