Some criteria for maximal abstract monotonicity

In this paper, we develop a theory of monotone operators in the framework of abstract convexity. First, we provide a surjectivity result for a broad class of abstract monotone operators. Then, by using an additivity constraint qualification, we prove a generalization of Fenchel’s duality theorem in the framework of abstract convexity and give some criteria for maximal abstract monotonicity. Finally, we present necessary and sufficient conditions for maximality of abstract monotone operators.


Introduction
and G(F) := {(x, y) ∈ X × Y : y ∈ F(x)}, respectively. The inverse of F is the set valued mapping F −1 : Y −→ 2 X defined by F −1 (y) := {x ∈ X : y ∈ F(x)}. Now, let X be a set and L be a set of real valued abstract linear functions l : X −→ R defined on X. For each l ∈ L and c ∈ R, consider the shift h l,c of l on the constant c : The function h l,c is called L-affine. Recall (see [22]) that the set L is called a set of abstract linear functions if h l,c / ∈ L for all l ∈ L and all c ∈ R\{0}. The set of all L-affine functions will be denoted by H L . If L is the set of abstract linear functions, then h l,c = h l 0 ,c 0 if and only if l = l 0 and c = c 0 .
If L is a set of abstract linear functions, then the mapping (l, c) −→ h l,c is a one-to-one correspondence. In this case, we identify h l,c with (l, c), in other words, we consider an element (l, c) ∈ L × R as a function defined on X by x −→ l(x) − c(x ∈ X ). Recall (see [22]) that a function f ∈ F (X ) is . We say that (see [7,12] Similarly, the Fenchel-Moreau X -conjugate g * X of an extended real valued function g defined on L is given by The function f * * L ,X := ( f * L ) * X is called the second conjugate (or biconjugate) of f, and by definition we have f * * L ,X (x) := sup l∈L (l(x) − f * (l)), x ∈ X.
The following properties of the conjugate function follow directly from the definition.
(i) Fenchel-Young's inequality: if f ∈ F (X ), then (ii) For f 1 and f 2 ∈ F (X ), we have If X is a set on which an addition + is defined, then we say that a function f ∈ F (X ) is additive if Let f : X −→ (−∞, +∞] be a function and x 0 ∈ dom f. Recall (see [22]) that an element l ∈ L is called an L-subgradient of f at The set ∂ L f (x 0 ) of all L-subgradients of f at x 0 is called L-subdifferential of f at x 0 . The subdifferential ∂ L f (x 0 ) (see [ Recall (see [7]) that for proper functions f, g ∈ F (X ), the infimal convolution of f with g is denoted by f ⊕ g : X −→ (−∞, +∞] and is defined by The infimal convolution of f with g is said to be exact provided the above infimum is achieved for every x ∈ X (see [7]). Now, assume that X is a set and L is a set of real valued abstract linear functions l : X −→ R defined on X, with the coupling function ., . : X × L −→ R defined by x, l := l(x) for all x ∈ X and all l ∈ L . In the following, we present some definitions and properties of abstract monotone operators (see [5,10,15,20]).
(i) A set valued mapping T : X −→ 2 L is called L-monotone operator (or, abstract monotone operator) if for all l ∈ T x, l ∈ T x and all x, x ∈ X.
It is worth noting that if X is a Banach space with dual space X * and L := X * , then T is a monotone operator in the classical sense. (ii) A set valued mapping T : X −→ 2 L is called maximal L-monotone (or maximal abstract monotone) if T is L-monotone and T = T for any L-monotone operator T : (iv) A subset S of X × L is called maximal L-monotone (or, maximal abstract monotone) if S is L-monotone and S = S for any L-monotone set S such that S ⊆ S . (v) Let T : X −→ 2 L be a set valued mapping. Corresponding to the mapping T define the L-Fitzpatrick function (or, abstract Fitzpatrick function) ϕ T : X × L −→R by for all x ∈ X and all l ∈ L .
There exist examples of abstract convex functions such that their L-subdifferentials are maximal L-monotone operators (for more details see [15,16]).
In the following, we gather some results which will be used later.
Lemma 2.1 [15]. Let T : X −→ 2 L be a maximal L-monotone operator. Then Then the following assertions are equivalent: (i) The mapping supp (., H L ) is additive in f and g.

A surjectivity result
Let U be an arbitrary set and L be an additive group of abstract linear functions on U . We define the coupling between U × L and L × U as We will say that A : X −→ 2 L is L-monotone if so is its extension to U obtained by assigning empty images to the elements in U \X. Similarly, a function h : Given an L-monotone operator A : X −→ 2 L , consider the Fitzpatrick family of abstract convex representations of A belongs to H A l 0 . Notice that, for any (m, u) ∈ L × U , If A : X −→ 2 L is an L-monotone operator and h ∈ H A , denote by A h : U −→ 2 L the operator defined by In particular, when h = ϕ A , we will simply write A, instead of A ϕ A , for ease of notation. According to the following proposition, A is an extension of A, i.e. G(A) ⊆ G A .
Therefore, for all (x, m) ∈ G(A), one obtains i.e. (x, m) ∈ G A . Thus, A is an extension of A.
be H L×U -convex functions. We call an abstract skewed Fenchel functional for f and g any (m, u) ∈ L × U such that Remark 3.1 If U is an additive set and the elements of L are odd functions, then, defining the function 2 : X × L → X × L by 2 (x, l) = (x, −l) for all (x, l) ∈ X × L, the existence of an abstract skewed Fenchel functional for f and g is equivalent to the existence of an abstract Fenchel functional for f and g • 2 , i.e. an element (m, u) ∈ L × U such that The proof of this fact is immediate, given that, for all (m, u) ∈ L × U , Then one concludes Thus, i.e., as a consequence of the arbitrariness of l 0 ∈ L, λF is a closed subspace of X . Indeed, in this case [21,Corollary 4.3] guarantees the existence of a Fenchel functional for h w * and k • 2 , for all w * ∈ X * . Then, identifying X with its image through the canonical inclusion in X * * , setting L := X * , U := X * * and taking Remark 3.1 into account, the previous theorem applies. (b) Let X, Y be reflexive Banach spaces and t : X → Y be an injective and continuous function. Define and, for all l ∈ L, set It is easy to check that the definition of · L does not depend on the choice of y * and that (L , · L ) is a normed space. Setting U := L * , then (t, Id) : X × L → Y × L is a continuous and injective function, L × L * can be taken as a set of abstract linear functions on X × L and the H L×L * -convex functions will be called hidden convex functions [27]. Moreover, one can prove that the function ζ : X → L * defined by for any x ∈ X , is injective. It does indeed take values in L * , given that ζ(x) is linear and for all x ∈ X and l ∈ L, and its injectivity is a direct consequence of that of t. As a consequence of [7,Corollary 5.4], if A, B : X −→ 2 L are maximal L-monotone operators and the abstract Fitzpatrick function of B, ϕ B : X × L → (−∞, +∞], is continuous on X × L, then, for all l 0 ∈ L, there exists a Fenchel functional m, m * ∈ L × L * for (ϕ A ) l 0 and ϕ B • ρ 2 . Therefore, if the functions in L are odd, identifying X with ζ(X ) and taking Remark 3.1 into account, then the surjectivity condition (3.1) holds for the extensions A and B.

Some results on abstract monotonicity
In this section, we present a generalization of Fenchel duality theorem in the framework of abstract convexity, and by using this theorem, we give criteria for maximal abstract monotonicity and obtain some other related results. Let X be a set with an operation + having the following properties: There exists a unique element 0 ∈ X such that 0 Let L be a set of real valued additive abstract linear functions defined on X. Assume that L is equipped with the point-wise operation + of functions such that (L , +) satisfies the properties (A 1 ), (A 2 ) and (A 3 ), where for each l ∈ L , define (−l)(x) := −l(x) for all x ∈ X, and define the function 0 ∈ L by 0(x) := 0 for all x ∈ X. We consider the coupling function ., . : X × L −→ R defined by x, l := l(x) for all x ∈ X and all l ∈ L .
x ∈ X and all l ∈ L . Indeed, assume that l ∈ L and x ∈ X are arbitrary. Then Let K ⊆ X × L be any non-empty set such that K satisfies the properties ( We can consider an element (l, x) ∈ L * as the function defined on K by : and an element (x, l) ∈ K as a function is defined on L * by : Note that the coupling function ., . * is symmetric, that is It is easy to check that L * and K are sets of real valued abstract linear functions. Indeed, Since (0, 0) ∈ K , put x = 0 and l = 0 in (4.2). Thus, we have c 0 = 0. This is a contradiction, because c 0 = 0. Hence, h (l,x),c / ∈ L * for all (l, x) ∈ L * and all c ∈ R\{0}. Therefore, L * is a set of abstract linear functions. By a similar argument, K is also a set of abstract linear functions.
and T is a maximal L-monotone operator (for more details see [5]).

Denote by P(H
Remark 4.2 Let S be any non-empty subset of K . Then the restriction to K of the L-Fitzpatrick function ϕ S : X × L −→R associated with S, defined by is an H L * -convex function. Indeed, by definition, we have for all (x, l) ∈ K , and hence the result follows.
Remark 4.4 Note that for a maximal L-monotone subset S of K , it follows from Lemma 2.1 that Then, for arbitrary (x, l) ∈ K , we have In the sequel, we we will use the following assumption.

Assumption (D):
Assume that there exists a function γ ∈ P(H L * ) such that In view of Remark 4.3, the condition γ ∈ P(H L * ) is automatically satisfied by any function Notice that, in the case when X is a Banach space with the dual space X * and L := X * , The following two results have been proved in [5].
Then, k (x 0 ,l 0 ) is an H L * -convex function and Lemma 4.2 [5,Theorem 4.1]. Let f and g ∈ P(H L * ) be such that dom f ∩ domg = ∅, and supp (., H L * ) be additive in f and g. Assume that f + g ≥ λ on K (λ ∈ R). Then there exists (l, x) ∈ L * such that In the following, we give some examples of a function which satisfies the Assumption (D).
Let X := Q n ⊂ R n , where Q is the set of all rational numbers endowed with the ordinary addition. It is clear that X satisfies the properties (A 1 ), (A 2 ) and (A 3 ). Let p : X −→ R be an additive function. Let a ∈ R and y ∈ X be arbitrary. Define the function l y,a : X −→ R by where [., .] is an inner product on R n . The function l y,a (a ∈ R, y ∈ X ) has the following properties.
(1) l y,a is an additive function for each a ∈ R and each y ∈ X.
In view of (1), (2) and (3), it is easy to check that L is a set of real valued additive abstract linear functions defined on X which satisfies the properties (A 1 ), (A 2 ) and (A 3 ). Define Since p is additive and p(−x) = −p(x) for all x ∈ X, it is easy to see that K and L * satisfy the properties (A 1 ), (A 2 ) and (A 3 ). Define the coupling function ., . * on K × L * as in (4.1). Now, we present the following example.
The following example is a special case of Example 4.3.
Therefore, by a similar argument as in Example 4.3, one can show that γ satisfies the Assumption (D).
Example 4.5 Let ., . be an inner product on R 2 . Define the function T : It is clear that T is a continuous linear operator and Let K := G(T ) and L * := G(T ). Define the coupling function ., . * : Therefore, γ satisfies the Assumption (D) (for more details see [5]).
If supp (., H L * ) is additive in h and γ, then there exists (x 0 , l 0 ) ∈ K such that Proof This is an immediate consequence of Lemma 4.2 and the Assumption (D)(ii) and (v).

Remark 4.6
Note that necessary and sufficient conditions for additivity of the mapping supp (., H L ) have been given in [7]. Also, in [7] it has given some examples of sets of abstract linear functions and functions f and g such that the mapping supp (., H L ) is additive in functions f and g.
In the following, we give an example such that the mapping supp (., H L * ) is additive in ϕ S and δ.
Therefore we have and hence Note that δ = 2ϕ S on K .
Finally, We show that supp (., H L * ) is additive in ϕ S and δ. It is easy to check that For the converse inclusion, suppose that f ∈ H L * and f ≤ ϕ S + δ. Let f 1 := 1 3 f on K , and f 2 := 2 3 f on K . Since f ∈ H L * and λ(l y , x) ∈ L * = L × X for all (l y , x) ∈ L * and all λ ∈ Q, it is easy to check that f 1 , f 2 ∈ H L * , f 1 ≤ ϕ S and f 2 ≤ δ. Thus, we have f 1 ∈ supp (ϕ S , H L * ) and f 2 ∈ supp (δ, H L * ). Also, we have f = f 1 + f 2 , and hence Now, let S be a non-empty subset of K . Define the function ψ S : It is clear that (4.8)

Theorem 4.2 Suppose that the Assumption (D) holds. Let S be a non-empty subset of K , and supp (., H L * ) be additive in ϕ S and γ. If S is maximal L-monotone, then there exists
Proof By maximality of S we conclude that ψ S (x, l) ≥ 0 for all (x, l) ∈ K . Therefore, by Assumption (D) and (4.8) we deduce that Since by Remark 4.2 ϕ S is an H L * -convex function, it follows from Theorem 4.1 that there exists (x 0 , l 0 ) ∈ K such that But, by Remark 4.4 we have (ϕ S ) * L * • t ≥ ϕ S on K , and hence it follows from (4.9) that This implies that This, together with the definition of ψ S , implies that Thus, we have and so because δ(x 0 , l 0 ) ≥ 0. Since S is a maximal L-monotone set, it follows from (4.11) that (x 0 , l 0 ) ∈ S. Therefore, in view of (4.10) we conclude that δ(x 0 , l 0 ) ≤ 0, and hence δ(x 0 , l 0 ) = 0, which completes the proof.
In the following, we give an example of a non-empty set S such that the mapping supp (., H L * ) is additive in ϕ S and γ. and let g ∈ P(H L * ) be arbitrary. We show that supp (., H L * ) is additive in ϕ S and g. Hence, in particular, supp (., H L * ) is additive in ϕ S and γ. We have Therefore, one has It is easy to see that For the converse inclusion, let f ∈ H L * and f ≤ ϕ S + g. Let f 1 := ϕ S . Then, f 1 ∈ H L * , and so f 1 ∈ supp (ϕ S , H L * ). Define By using the properties of L * , and since f ∈ H L * and f ≤ ϕ S + g, it is easy to show that f 2 ∈ H L * and f 2 ≤ g. Thus, f 2 ∈ supp (g, H L * ). Also, we have f = f 1 + f 2 , and hence the proof is complete.
Let S be a subset of K and (x 0 , l 0 ) ∈ K . Define the translation of S by  Proof Since S is L-monotone, it is easy to check that S t is an L-monotone set. Now, we show that S t is maximal. To this end, let (x 0 , l 0 ) ∈ K be arbitrary and (4.12) Let (x 1 , l 1 ) ∈ S be arbitrary and let x = x 1 − x and l = l 1 − l. Then we have (x , l ) = (x 1 , l 1 ) − (x, l) ∈ K , and therefore it follows from (4.12) that and so Since S is maximal L-monotone, in view of (4.13) we conclude that (x + x 0 , l + l 0 ) ∈ S, and hence (x 0 , l 0 ) ∈ S t , which completes the proof.
The following theorem gives criteria for maximal abstract monotonicity, which is a generalization of [29,Theorem 10.3].

Theorem 4.3 Suppose that the Assumption (D) holds. Let S be an L-monotone subset of K .
Then the following assertions are true:

(1) Let supp (., H L * ) be additive in ϕ S t and γ, where S t is a translation of S. If S is a maximal L-monotone set, then for each
is a maximal L-monotone set.
Proof (1) Assume that S is a maximal L-monotone set and (x, l) ∈ K is given. Then, by Lemma 4.3, S t := S − (x, l) is a maximal L-monotone set. Therefore, since by the hypothesis supp (., H L * ) is additive in ϕ S t and γ, in view of Theorem 4.2 there exists (x 1 , l 1 ) ∈ S t such that δ(x 1 , l 1 ) = 0. (4.14) Since (x 1 , l 1 ) ∈ S t , it follows that there exists (x 0 , l 0 ) ∈ S such that x 1 = x 0 − x and l 1 = l 0 − l. Thus, the result follows from (4.14).
(2) We have by the hypothesis S is an L-monotone set. Now, we show that S is a maximal L-monotone subset of K . To this end, let (x 0 , l 0 ) ∈ K and By the hypothesis for this Therefore, by the definition of δ and (4.15) we obtain This implies that γ (x 0 − x 1 , l 0 − l 1 ) = 0, and so by Assumption (D)(iv), we deduce that x 0 = x 1 and l 0 = l 1 . Consequently, (x 0 , l 0 ) = (x 1 , l 1 ) ∈ S. Hence, S is a maximal L-monotone subset of K . This completes the proof.
In the following, we give an example of a non-empty set S such that the mapping supp (., H L * ) is additive in ϕ S t and γ, where S t is any translation of S.
and T is a maximal L-monotone operator. Let K := G(T ) ⊆ X × L , and It is easy to check that K and L * satisfy the properties (A 1 ), (A 2 ) and (A 3 ). Define the coupling function ., . * on K × L * as in (4.1). Assume that γ is defined on K as in Example 4.4. Thus, we have γ (x, l x ) = x 2 = l x (x) for all x ∈ X. Moreover, γ satisfies the Assumption (D). Let S := G(T ). Since T is a maximal L-monotone operator, we deduce that S is a maximal L-monotone set. Thus, it follows from Lemma 2.1 that Now, let (x 0 , l x 0 ) ∈ K be arbitrary and S t := S − (x 0 , l x 0 ). Therefore, we have Now, we show that supp (., H L * ) is additive in ϕ S t and γ. It is clear that Let f ∈ supp (ϕ S t + γ, H L * ) be arbitrary. Then, f ∈ H L * and f ≤ ϕ S t + γ. Let f 1 = f 2 := 1 2 f. Therefore, by using the properties of L * and since f ∈ H L * , we conclude that

Necessary and sufficient conditions for maximal abstract monotonicity
In this section, we give a generalization of [11, Theorem 2.1] and, by using this generalization, we obtain necessary and sufficient conditions for maximality of abstract monotone operators.
Let X, L , K and L * be as in Sect. 4, which satisfy the properties (A 1 ), (A 2 ) and (A 3 ). Define the coupling function ., . * on K × L * as in (4.1). Suppose that the Assumption (D) holds. Let (x 0 , l 0 ) ∈ K be arbitrary. Define the function k (x 0 ,l 0 ) : K −→ R by In view of Lemma 4.1, we have k (x 0 ,l 0 ) ∈ P(H L * ) and Also, we define the function α : It is easy to see, by Assumption (D)(i), (ii) and (v), that In the following, we give an assumption.
The following example shows that for a function α and a function k (x 0 ,l 0 ) the infimal convolution α ⊕ k (x 0 ,l 0 ) of α with k (x 0 ,l 0 ) is exact and an H L * -convex function for each (x 0 , l 0 ) ∈ K . Example 5.1 Let X, L , K , L * and γ be as in Example 4.4. Let (x 0 , l y 0 ) ∈ K be arbitrary. By (5.3), one has α(x, l y ) = γ (x, −l y ) = 1 2 (x 2 + y 2 ) for all (x, l y ) ∈ K . Also, in view of (5.1) and Example 4.7, we have Therefore, one has It is not difficult to see that the above infimum is achieved at That is, the infimal convolution α ⊕ k (x 0 ,l y 0 ) is exact for each (x 0 , l y 0 ) ∈ K . Also, by using the properties of L * , one has where t 0 = − 1 2 (x 0 + y 0 ) ∈ X. Therefore the infimal convolution α ⊕ k (x 0 ,l y 0 ) is an H L * -convex function for each (x 0 , l y 0 ) ∈ K .
In the sequel, let X ⊆ X be any non-empty set such that X satisfies the properties It is clear that G(d) = D. Also, define the set valued mapping −d : for each x ∈ X . Therefore, we have In the following, we give an example of a function d such that dom (d) = X. Moreover, d is single-valued. (Note that −l y = l −y .) Therefore, for each x ∈ X, we have This implies that dom (d) = X, and also, d is single-valued. Moreover, one has d −1 (l y ) = {y} for each y ∈ X, that is, dom (d −1 ) = L and d −1 is single-valued. In view of Fenchel-Young inequality and the Assumption (D)(ii) we conclude that This, together with (5.11) and (5.12) implies that which completes the proof. (5.14) Define the function k (x 0 ,l 0 ) on K as in (5.1), and the function α on K as in (5.3). Since Therefore, it follows from Lemma 4.2 and Theorem 5.1 that there exists ( we conclude from (5.15) that This implies that and hence δ(x 1 , −l 1 ) = 0. That is, (x 1 , l 1 ) ∈ G(d). Therefore, it follows from (5.14) that On the other hand, since δ(x 1 , −l 1 ) = 0, we conclude from (5.16) that δ(x 0 −x 1 , l 0 −l 1 ) = 0. Thus, by using (5.17) and the definition of δ we have and so γ (x 0 − x 1 , l 0 − l 1 ) = 0. This, together with the Assumption (D)(iv) implies that x 0 = x 1 and l 0 = l 1 . Consequently, (x 0 , l 0 ) = (x 1 , l 1 ) ∈ G(d), and hence the proof is complete.
In the sequel, for any non-empty subset S of K , define The following theorem is a generalization of [11, Theorem 2.1].
Theorem 5.2 Suppose that the Assumptions (C) and (D) hold. Let A : X −→ 2 L be any L -monotone operator. Assume that supp (., H L * ) is additive in ϕ A and h S , where A is any operator whose graph is a translation of G(A), h S is defined by (5.18) and S is any non-empty subset of K . Then the following assertions are equivalent: (1) A is a maximal L -monotone operator.
Proof (1) ⇒ (2). Assume that (1) holds and B : X −→ 2 L is any maximal Lmonotone operator such that ϕ B is finite-valued. Let (x 0 , l 0 ) ∈ K be arbitrary. Consider the set valued mapping A : X −→ 2 L such that G(A ) := G(A) − (x 0 , l 0 ). Since A is a maximal L -monotone operator, in view of Lemma 4.3 we conclude that A is also a maximal L -monotone operator. Let S := G(B). Then, ϕ S is finite-valued and by Remark 4.2 we have h S ∈ P(H L * ). Since A and B are maximal L -monotone operators, it follows from Lemma 2.1 that Since by the hypothesis supp (., H L * ) is additive in ϕ A and h S , it follows from Lemma 4.2 that there exists (l,x) ∈ L * := L × X such that Therefore, it follows from (5.19) that Then, we have This completes the proof of the implication (1) ⇒ (2).
(2) ⇒ (3). Suppose that (2) holds. Let B := d, where d is defined by (5.9). Since the Assumptions (C) and (D) hold, then in view of Proposition 5.2 we conclude that B is a maximal L -monotone operator. Also, we have Thus, by Assumption (D)(i) and (ii), one has This implies that ϕ B is finite-valued. Therefore, by the hypothesis (2) because if γ (x, −l) = 0, then by Assumption (D)(iv) we have (x, l) = (0, 0), which is a contradiction.
In the following, we give an example of an L-monotone operator A : X −→ 2 L and a non-empty subset S of K := X × L such that the mapping supp (., H L * ) is additive in ϕ A and h G(d) , where A is any operator whose graph is a translation of G(A) and h S is defined by (5.18). Let g ∈ P(H L * ) and S ⊆ K be arbitrary. We show that supp (., H L * ) is additive in ϕ A and g. Hence, in particular, supp (., H L * ) is additive in ϕ A and h S . It is clear that A is an L-monotone operator. Let (x 0 , l y 0 ) ∈ K be arbitrary. Then, one has It is easy to see that supp (ϕ G(A) t , H L * ) + supp (g, H L * ) ⊆ supp (ϕ G(A) t + g, H L * ).
For the converse inclusion, let f ∈ H L * and f ≤ ϕ A + g. Let f 1 := ϕ A . Then, f 1 ∈ H L * , and so f 1 ∈ supp (ϕ A , H L * ). Define By using the properties of L * , and since f ∈ H L * , −ϕ A ∈ H L * and f ≤ ϕ A + g, it is easy to check that f 2 ∈ H L * and f 2 ≤ g. Thus, f 2 ∈ supp (g, H L * ). Also, we have f = f 1 + f 2 , and hence the proof is complete.