Convergence theorem for common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings
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Convergence theorem for common fixed points of a finite family of multi-valued Bregman relatively nonexpansive mappings Naseer Shahzad1* and Habtu Zegeye2 * Correspondence: [email protected] 1 Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia Full list of author information is available at the end of the article
Abstract In this paper, it is our purpose to introduce an iterative process for the approximation of a common fixed point of a finite family of multi-valued Bregman relatively nonexpansive mappings. We prove that the sequence of iterates generated converges strongly to a common fixed point of a finite family of multi-valued Bregman nonexpansive mappings in reflexive real Banach spaces. MSC: 47H05; 47H09; 47H10; 47J25; 49J40; 90C25 Keywords: Bregman projection; Legendre function; multi-valued Bregman nonexpansive mapping; relatively nonexpansive multi-valued mapping; single-valued Bregman nonexpansive mapping; strong convergence
1 Introduction Let E be a reflexive real Banach space E, and E∗ its dual. Let f : E → (–∞, ∞] be a proper convex and lower semicontinuous function. The subdifferential of f at x ∈ E is the convex set defined by ∂f (x) = x∗ ∈ E∗ : f (x) + x∗ , y – x ≤ f (y), ∀y ∈ E .
(.)
The Fenchel conjugate of f is the function f ∗ : E∗ → (–∞, +∞] defined by f ∗ (y) = sup{y, x – f (x) : x ∈ E}. It is not difficult to check that when f is proper and lower semicontinuous, so is f ∗ . The function f is said to be essentially smooth if ∂f is both locally bounded and singlevalued on its domain. It is called essentially strictly convex, if (∂f )– is locally bounded on its domain and f is strictly convex on every convex subset of dom ∂f . f is said to be Legendre, if it is both essentially smooth and essentially strictly convex. Let dom f = {x ∈ E : f (x) < ∞}. Then for any x ∈ int(dom f ) and y ∈ E, the right-hand derivative of f at x in the direction of y is defined by f ◦ (x, y) := lim+ t→
f (x + ty) – f (x) . t
(.)
If the limit in (.) exists then f is called Gâteaux differentiable at x. In this case, f ◦ (x, y) coincides with ∇f (x), the value of the gradient ∇f of f at x. The function f is called Gâteaux ©2014Shahzad and Zegeye; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Shahzad and Zegeye Fixed Point Theory and Applications 2014, 2014:152 http://www.fixedpointtheoryandapplications.com/content/2014/1/152
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differentiable if it is Gâteaux differentiable for any x ∈ int(dom f ). The function f called Fréchet differentiable at x if the limit in (.) is attained uniformly for all y ∈ E such that y = and f is said to be uniformly Fréchet differentiable on a subset C of E if the limit is attained uniformly for x ∈ C and y = . When the subdifferential of f is single-val
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