The purpose of this paper is to introduce an implicit iteration process for approximating common fixed points of two asymptotically nonexpansive mappings and to prove strong convergence theorems in uniformly convex Banach spaces.

and Applied Analysis 3 Let {xn} be a bounded sequence in a CAT 0 space X. For x ∈ X, one sets r x, {xn} lim sup n→∞ d x, xn . 2.5 The asymptotic radius r {xn} of {xn} is given by r {xn} inf x∈X {r x, {xn} }, 2.6 the asymptotic radius rC {xn} of {xn}with respect to C ⊂ X is given by rC {xn} inf x∈C {r x, {xn} }, 2.7 the asymptotic center A {xn} of {xn} is the set A {xn} {x ∈ X : r x, {xn} r {xn}...

and Applied Analysis 3 Recall that the Bregman projection [13] of x ∈ int domf onto the nonempty, closed, and convex subset C of domf is the necessarily unique vector projf C (x) ∈ C satisfying D f (projf C (x) , x) = inf{D f (y, x) : y ∈ C} . (12) Let f : E → (−∞, +∞] be a convex and Gâteaux differentiable function. The function f is said to be totally convex at x ∈ int domf if its modulus of ...

The purpose of this article is to prove strong convergence theorems for fixed points of closed hemirelatively nonexpansive mappings. In order to get these convergence theorems, the monotone hybrid iteration method is presented and is used to approximate those fixed points. Note that the hybrid iteration method presented by S. Matsushita and W. Takahashi can be used for relatively nonexpansive m...

Let C be a nonempty closed convex subset of a Hilbert spaceH, T a self-mapping of C. Recall that T is said to be nonexpansive if ‖Tx − Ty‖ ≤ ‖x − y‖, for all x, y ∈ C. Construction of fixed points of nonexpansive mappings via Mann’s iteration 1 has extensively been investigated in literature see, e.g., 2–5 and reference therein . But the convergence about Mann’s iteration and Ishikawa’s iterati...

A point x ∈ C is a fixed point of T provided Tx = x. Denote by F(T) the set of fixed points of T ; that is, F(T)= {x ∈ C : Tx = x}. It is assumed throughout the paper that T is a nonexpansive mapping such that F(T) =∅. One classical way to study nonexpansive mappings is to use contractions to approximate a nonexpansive mapping [1, 9]. More precisely, take t ∈ (0,1) and define a contraction Tt :...

and Applied Analysis 3 In this paper, we generalize and modify the iteration of Abbas et al. 7 from two mapping to the infinite family mappings {Ti : i ∈ N} of multivalued quasi-nonexpansive mapping in a uniformly convex Banach space. Let {Ti} be a countable family of multivalued quasi-nonexpansive mapping from a bounded and closed convex subset K of a Banach space into P K with F : ⋂∞ i 1 F Ti...

and Applied Analysis 3 Let X be a smooth Banach space. We always use φ : X × X → R to denote the Lyapunov functional defined by φ ( x, y ) ‖x‖ − 2〈x, Jy〉 ∥∥y∥∥2, ∀x, y ∈ X. 1.7 It is obvious from the definition of the function φ that (‖x‖ − ∥∥y∥∥)2 ≤ φ(x, y) ≤ (‖x‖ ∥∥y∥∥)2, ∀x, y ∈ X. 1.8 Following Alber 4 , the generalized projection ΠC : X → C is defined by ΠC x arg inf y∈C φ ( y, x ) , ∀x ∈ ...

In this paper, we define and study convergence of an Ishikawa type iteration scheme with a new and weaker control condition for two nonself asymptotically quasi-nonexpansive mappings on a uniformly convex Banach space.