Abstract Monotone Operators Representable by Abstract Convex Functions

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1 Applied Mathematical Sciences, Vol. 6, 2012, no. 113, Abstract Monotone Operators Representable by Abstract Convex Functions H. Mohebi and A. R. Sattarzadeh Department of Mathematics of Shahid Bahonar University of Kerman and Kerman Graduate University of Technology, Kerman, Iran Abstract In this paper, we present the relation between abstract monotone polarity and abstract monotone closure. Finally, we investigate the relation between abstract Fitzpatrick set and abstract monotone polarity. Mathematics Subject Classification: 47H05, 47H04, 52A01, 26A51, 26B25 Keywords: Monotone operator, abstract monotonicity, abstract convex function 1 Introduction Several approaches to the theory of monotone operators have established links between maximal monotone operators and convex functions (see [1, 2, 3]). Recently, a theory of monotone operators has been developed in the framework of abstract convexity (see [2]). In this paper, by a similar argument as in [3] we give a representation for abstract monotone operators by abstract convex functions, and we present the relation between abstract Fitzpatrick set and abstract monotone polarity, which generalize the results obtained in [1] and [3]. 2 Preliminaries Let X be a set and L be a set of real valued functions l : X R, which will be called abstract linear. For each l L and c R, consider the shift h l,c of l on the constant c : h l,c (x) :=l(x) c, (x X). The function h l,c is called L-affine. Recall (see [4]) that the set L is called a set of abstract

2 5650 H. Mohebi and A. R. Sattarzadeh 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. Now, we consider the coupling function.,. : X L R is defined by x, l := l(x) for all x X and all l L. For an extended real valued function f define the Fenchel-Moreau L-conjugate fl of f (see [4]) by f L (l) := sup x X(l(x) f(x)), l L. Now, we give some definitions and results relative to abstract monotonicity (see [2]). (i). A set valued mapping T : X 2 L is called L-monotone operator (or, abstract monotone operator) if l(x) l(x ) l (x)+l (x ) 0 for all l Tx, l Tx and all x, x X. (ii). A set valued mapping T : X 2 L is called maximal L-monotone operator (or, maximal abstract monotone operator) if T is L-monotone and T = T for any L-monotone operator T : X 2 L such that G(T ) G(T ). (iii). Let T : X 2 L be a set valued mapping. Correspondence to the mapping T define the L-Fitzpatrick function (or, abstract Fitzpatrick function) ϕ T : X L R by ϕ T (x, l) := sup l Tx,x X[l(x )+l (x) l (x ) l(x)]+l(x) for all x X and all l L. From now on, let (X, +) be an additive group. Let L be an additive group of real valued abstract linear functions defined on X such that each l L is an additive function. Assume that L is equipped with the point-wise operation + of functions such that for each l L, define ( l)(x) := l(x) for all x X. Let K X L be any non-empty additive group such that (x, l) :=( x, l) K and 0 := (0, 0) K. Define L := {(l, x) L X :(x, l) K} L X. It is clear that L with the point-wise operation + of functions is an additive group. Define the coupling function.,. : K L R by (x,l ), (l, x) := l(x )+l (x), (x,l ) K; (l, x) L. 3 Abstract Monotone Polarity In this section, we assume that X, L, K and L are as in Section 2. First, we define the reflexive and symmetric binary relation μ on K by (x, l)μ(x,l ) l(x) l(x ) l (x)+l (x ) 0 for all (x, l), (x,l ) K. Now, we define the abstract monotone polarity as in the classical setting (see [3]). Definition 3.1. The abstract monotone polar of a subset S of K is defined by S μ := {(x, l) K :(x, l)μ(x,l ), (x,l ) S}. Definition 3.2. For a subset S of K, we call S μμ the μ-closure of S. We also say that a set S K is μ-closed if and only if S = S μμ. The proof of the following results about the abstract monotone polarity is similar to the one of in [3] in the classical setting, and therefore we omit their proofs.

3 Abstract monotone operators 5651 Lemma 3.1. Let S K be a set and {S i } i I be a collection of subsets of K. Then the following assertions are true. (i) ( i I S i) μ = i I Sμ. (ii) S S μμ. (iii) S μμμ = S μ. (iv) S S = S μ S μ. (v) μ = K. Lemma 3.2. Let S K be an L-monotone set. Then, (x, l) S μ if and only if S {(x, l)} is an L-monotone set. Proposition 3.1. Let S K be a set. equivalent. (i) S is L-monotone. (ii) S S μ. (iii) S μμ S μ. (iv) S μμ is L-monotone. Then the following assertions are Theorem 3.1. Let S be a subset of K. Then, S is maximal L-monotone if and only if S = S μ. Proof: Suppose that S is maximal L-monotone. Then, by Proposition 3.1, one has S S μ. Now, let (x, l) S μ be arbitrary. By Lemma 3.2, we have S (x, l) is an L-monotone set which contains S. Since S is maximal, we deduce that (x, l) S, and hence S μ S. For the converse, assume that S = S μ. Then in view of Proposition 3.1 one has S is L-monotone. Suppose that S is any L-monotone set such that S S. Thus, by Proposition 3.1 and Lemma 3.1 we have S S μ S μ = S, and so S = S. Therefore, S is a maximal L-monotone set. 4 Abstract Fitzpatrick Set In this section, we obtain some results on Fitzpatrick set in the framework of abstract convexity. We assume that X and L are as in Section 2, and K := X L and L := L X. Definition 4.1. Let f : K R {+ } be an H L -convex function. Then the abstract Fitzpatrick set associated with f is defined by G f = {(x, l) K : (l, x) L f(x, l)}. Lemma 4.1. Let A be a non-empty L-monotone subset of K. Then, A G ϕa. Moreover, A = G ϕa whenever A is a maximal L-monotone set. Proof: This is an immediate consequence of the definition of ϕ A and Fenchel- Young equality.

4 5652 H. Mohebi and A. R. Sattarzadeh Lemma 4.2. Let f : K R {+ } be an H L -convex function and let (x, l) K be arbitrary. Then, (x, l) G f if and only if 1f(x, l)+ 1f 2 2 L (l, x) = l(x). Proof: This is an immediate consequence of Fenchel-Young equality. Proposition 4.1. Let A K be an L-monotone set. Then, G ϕa A μ. Proof: Since A is an L-monotone set, it follows from Lemma 4.1 that A G ϕa. Also, we have G ϕa is an L-monotone set, and so by Proposition 3.1 and by using the maximal L-monotone extension of G ϕa one has G ϕa A μ. Theorem 4.1. Let A K be an L-monotone set. Then, G ϕa co K,L (A). Proof: Assume if possible that there exists (x, l) G ϕa \ co K,L (A). Then we have co K,L (A) =supp(t A ; K), where T A (l, x) = sup (a,t) A (a, t), (l, x). Therefore, (x, l) / supp(t A ; K). Hence there exists (y, l ) K such that (x, l), (l,y) > sup (a, t), (l,y). (4.1) (a,t) A On the other hand, we have (x, l) G ϕa, and so (l, x) L ϕ A (x, l). Thus for (l,y) L one has ϕ A (x + y, l + l ) ϕ A (x, l) + (x, l), (l,y). By the definition of Fitzpatrick function there exists (a 0,t 0 ) A such that x+y, t 0 + a 0,l+ l a 0,t 0 ϕ A (x, l)+ (x, l), (l,y). Therefore in view of (4.1) we have ϕ A (x, l) x + y, t 0 + a 0,l+ l a 0,t 0 (x, l), (l,y) < x + y, t 0 + a 0,l+ l a 0,t 0 sup (a, t), (l,y) (a,t) A x + y, t 0 + a 0,l+ l a 0,t 0 (a 0,t 0 ), (l,y) = t 0 (x)+l(a 0 ) t 0 (a 0 ) ϕ A (x, l). This implies that ϕ A (x, l) <ϕ A (x, l), which is a contradiction. Hence, G ϕa co K,L (A). Theorem 4.2. Let A be an L-monotone subset of K such that A = A μ co K,L (A). Then, G ϕa = A. Proof: Since A G ϕa A μ co K,L (A), it follows that A = G ϕa. Acknowledgments. This research was supported partially by Kerman Graduate University of Technology and Mahani Mathematical Research Center.

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