and Applied Analysis 3 for all n ∈ N, where x1 x ∈ C and {αn} is a sequence of 0, 1 . Strong convergence of this type of iterative sequence has beenwidely studied: for instance, see 13, 14 and the references therein. In 2006, Martinez-Yanes and Xu 15 have adapted Nakajo and Takahashi’s 10 idea to modify the process 1.5 for a nonexpansive mapping T in a Hilbert space: x0 x ∈ C, yn αnx0 1 − αn Tx...

Let X be a Banach space with closed unit ball B. Given k ∈ N, X is said to be k-β, repectively, (k + 1)-nearly uniformly convex ((k + 1)-NUC), if for every ε > 0, there exists δ, 0 < δ < 1, so that for every x ∈ B, and every ε-separated sequence (xn) ⊆ B, there are indices (ni) k i=1, respectively, (ni) k+1 i=1 , such that 1 k+1 ‖x + ∑k i=1 xni‖ ≤ 1 − δ, respectively, 1 k+1 ‖ ∑k+1 i=1 xni‖ ≤ 1−...

Let X be a real strictly convex and Kadec Banach space and G a nonempty closed relatively boundedly weakly compact subset of X : Let BðX Þ (resp. KðXÞ) be the family of nonempty bounded closed (resp. compact) subsets of X endowed with the Hausdorff distance and let BGðXÞ denote the closure of the set fAABðX Þ : A-G 1⁄4 |g and KGðX Þ 1⁄4 BGðX Þ-KðXÞ: We introduce the admissible family A of BðX Þ...

Let X be a Banach space, C a closed subset of X , and T : C -+ C a nonexpansive mapping. Conditions are given which assure that if the fixed point set F ( T ) of T has nonempty interior then the Picard iterates of the mapping T always converge to a point of F ( T ) . If T is asymptotically regular, it suffices to assume that the closed subsets of X are densely proximinal and that nested spheres...

Let Ω be a connected compact Hausdorff space and X a Banach space for which the strong Banach-Stone property is valid. We give a complete characterization of the generalized bi-circular projection on the Banach spaces C(Ω) and C(Ω, X). We show that in each case generalized bi-circular projections are bi-contractive. We also give some results concerning projections in the convex hull of two or m...

Let K be a nonempty closed convex subset of a reflexive and strictly convex Banach space E with a uniformly Gâteaux differentiable norm, F {T h : h ≥ 0} a generalized asymptotically nonexpansive self-mapping semigroup of K, and f : K → K a fixed contractive mapping with contractive coefficient β ∈ 0, 1 . We prove that the following implicit and modified implicit viscosity iterative schemes {xn}...

for each * in X. For any Banach space X and any element * of Xt J(x) is a nonempty closed convex subset of the sphere of radius ||x|| about zero in X*. If X* is strictly convex, J is a singlevalued mapping of X into X* and is continuous from the strong topology of X to the weak* topology of X*. J is continuous in the strong topologies if and only if the norm in X is C on the complement of the o...

The purpose of this paper is to introduce a general iterative method for finding solutions of a general system of variational inclusions with Lipschitzian relaxed cocoercive mappings. Strong convergence theorems are established in strictly convex and 2-uniformly smooth Banach spaces. Moreover, we apply our result to the problem of finding a common fixed point of a countable family of strict pse...

Suppose X and Y are locally compact Hausdorff spaces, E and F are Banach spaces and F is strictly convex. We show that every linear isometry T from C0(X, E) into C0(Y, F) is essentially a weighted composition operator T f (y) = h(y)(f (ϕ(y))).