The single-valued extension property for bilateral operator weighted shifts

In this paper, we give necessary and sufficient conditions for a bilateral operator weighted shift to enjoy the single-valued extension property.

     

Approximate fixed point sequences and convergence theorems for Lipschitz pseudocontractive maps

Let be a nonempty closed convex subset of a real Banach space and be a Lipschitz pseudocontractive self-map of with . An iterative sequence is constructed for which as . If, in addition, is assumed to be bounded, this conclusion still holds without the requirement that Moreover, if, in addition, has a uniformly Gâteaux differentiable norm and is such that every closed bounded convex subset of has the fixed point property for nonexpansive self-mappings, then the sequence converges strongly to a fixed point of . Our iteration method is of independent interest.

     

Generalized projection and approximation of fixed points of nonself maps

Let K be a nonempty closed convex proper subset of a real uniformly convex and uniformly smooth Banach space E; T:K→E be an asymptotically weakly suppressive, asymptotically weakly contractive or asymptotically nonextensive map with F(T){xK: Tx=x}≠. Using the notion of generalized projection, iterative methods for approximating fixed points of T are studied. Convergence theorems with estimates of convergence rates are proved. Furthermore, the stability of the methods with respect to perturbations of the operators and with respect to perturbations of the constraint sets are also established.     

Convergence of paths and approximation of fixed points of asymptotically nonexpansive mappings

Let be a real Banach space with a uniformly Gâteaux differentiable norm possessing uniform normal structure, be a nonempty closed convex and bounded subset of , be an asymptotically nonexpansive mapping with sequence . Let be fixed, be such that , , and . Define the sequence iteratively by , n= 0, 1, 2, ..._. $"> It is proved that, for each integer , there is a unique such that If, in addition, and , then converges strongly to a fixed point of .

     

Convergence theorems for fixed points of uniformly continuous generalized Φ-hemi-contractive mappings

Let E be a real normed linear space, K be a nonempty subset of E and be a uniformly continuous generalized Φ-hemi-contractive mapping, i.e., , and there exist x^∗∈F(T) and a strictly increasing function , Φ(0)=0 such that for all x∈K, there exists j(x−x^∗)∈J(x−x^∗) such that

〈Tx−x^∗,j(x−x^∗)〉?‖x−x^∗‖2−Φ(‖x−x^∗‖).     

General auxiliary problem and algorithm for a generalized multi-valued variational-like inequality problem in reflexive Banach spaces

In this article, some generalized multi-valued variational-like inequality problems and their related general auxiliary problems in real reflexive Banach spaces are introduced. An existence theorem for one of the general auxiliary problems is proved. Furthermore, by exploiting this theorem, an algorithm for the corresponding generalized multi-valued variational-like inequality problem is constructed. The theorems of this article generalize, improve and unify many known corresponding results.     

Approximation of Solutions of Nonlinear Equations of Monotone and Hammerstein Type

Let X be a real uniformly smooth and uniformly convex Banach space with dual X ^*. Let A: X → X ^* be a bounded uniformly submonotone map. It is proved that a Mann-type approximation sequence converges strongly to Jx ^* where x ^* ∈ N(A). Furthermore, as an application of this result an iterative sequence which converges strongly to a solution of the Hammerstein equation u+KFu = 0 is constructed where, F:X→X^* and K:X^*→X are monotone-type mappings. No invertibility assumption is imposed on K. Moreover, neither K nor F need be compact. Finally, our method is of independent interest.     

Iterative Methods for Fixed Points of Asymptotically Weakly Contractive Maps

Suppose K is a closed convex nonexpansive retract of a real uniformly smooth Banach space E with P as the nonexpansive retraction. Suppose T : K → E is an asymptotically d-weakly contractive map with sequence {k_n }, k_n ≥ 1, lim k_n = 1 and with F(T) _n int (K) ≠ ø F(T):= {x ∈ K: Tx = x}. Suppose {x _n } is iteratively defined by x _n+1 = P((l ? k_nα_n )x _n +k _n α _n T(PT) ^n?l x_n ), n = 1,2,...,x ₁ ∈ K, where α_n∈ (0,l) satisfies lim α_n = 0 and Σα_n = ∞. It is proved that {x _n } converges strongly to some x ^* ∈ F(T)∩ int K. Furthermore, if K is a closed convex subset of an arbitrary real Banach space and T is, in addition uniformly continuous, with F(T) ≠ ø, it is proved that {x_n } converges strongly to some x ^* ∈ F(T).