Compositions and convex combinations of Bregman weakly relatively nonexpansive operators in reflexive Banach spaces
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Journal of Fixed Point Theory and Applications
Compositions and convex combinations of Bregman weakly relatively nonexpansive operators in reflexive Banach spaces Eskandar Naraghirad Abstract. In this paper, we investigate the right and left Bregman weakly relatively nonexpansive operators in reflexive Banach spaces. We first introduce the notions of convex hull demi-closedness principle, strictly strongly convex hull demi-closedness principle, strongly demi-closedness principle, and composition demi-closedness principle of a family of nonlinear mappings using another new notions of convex asymptotic fixed points, and composition asymptotic fixed points for such mappings in a Banach space E. We analyze, in particular, compositions and convex combinations of Bregman weakly relatively nonexpansive operators, and prove the convergence of the Mann iterative method for operators of these types. Finally, we use our results to approximate common zeros of maximal monotone mappings and solutions to convex feasibility problems. To support our results, we include nontrivial examples in the paper. Therefore, our results improve and generalize many known results in the current literature. Mathematics Subject Classification. 47H10, 37C25. Keywords. Bregman distance, Bergman weakly relatively nonexpansive operator, Legendre function, maximal monotone operator, nonexpansive operator, reflexive Banach space, resolvent.
1. Introduction Throughout this paper, we denote the set of real numbers and the set of positive integers by R and N, respectively. Let X be a Banach space with the norm . and the dual space X ∗ . For any x ∈ X, we denote the value of x∗ ∈ X ∗ at x by x, x∗ . Let {xn }n∈N be a sequence in X, we denote the strong convergence of {xn }n∈N to x ∈ X as n → ∞ by xn → x and the weak convergence by xn x. The modulus δ of convexity of X is denoted by: x + y : x ≤ 1, y ≤ 1, x − y ≥ δ() = inf 1 − 2 0123456789().: V,-vol
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for every with 0 < ≤ 2. A Banach space X is said to be uniformly convex if δ() > 0 for every > 0. Let SX = {x ∈ X : x = 1}. The norm of X is said to be G a ˆ teaux differentiable if, for each x, y ∈ SX , the limit x + ty − x lim t→0 t exists. In this case, X is called smooth. If the limit is attained uniformly for all x, y ∈ SX , then X is called uniformly smooth. The Banach space X is said to be strictly convex if x+y 2 < 1 whenever x, y ∈ SX and x = y. It is well known that X is uniformly convex if and only if X ∗ is uniformly smooth. It is also known that if X is reflexive, then X is strictly convex if and only if X ∗ is smooth; for more details, see [32,40,41]. 1.1. Some facts about gradients and conjugate functions For any convex function f : X → (−∞, +∞], we denote the domain of f by dom f = {x ∈ X : f (x) < ∞}. For any x ∈ int dom f and any y ∈ X, we denote by f o (x, y) the right-hand derivative of f at x in the direction y; that is: f (x + ty) − f (x) . f o (x, y) = lim t↓0 t (x) The function f is said to be G a ˆteaux differentiable at
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