On Invariant Approximations in Modular Spaces

This article is devoted to presenting results on invariant approximations over a non-star-shsped weakly compact subset of a complete modular space by introduced a new notion called S-star-shaped with center f: if be a mapping and , . Then the existence of common invariant best approximation is proved for Banach operator pair of mappings by combined the hypotheses with Opial’s condition or demi-closeness condition

Some Results on Iterative Proximal Convergence and Chebyshev Center

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair M , N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M ∪ N satisfying T M ⊆ M and T N ⊆ N , to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M ∪ N satisfying T N ⊆ N and T M ⊆ M , Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M . Some illustrative examples are provided to support our results.

Convergence theorems forI-nonexpansive mapping

We establish the weak convergence of a sequence of Mann iterates of anI-nonexpansive map in a Banach space which satisfies Opial's condition.

A 3-space problem related to the fixed point property

We study some properties which imply weak normal structure and thus the fixed point property. We investigate whether the latter two properties are inherited by spaces obtained by direct sum with a finite dimensional space. We exhibit a space X which satisfies Opial's condition, X ⊕ ℝ does not have weak normal structure but X ⊕ ℝ has the fixed point property.

On $ R $-subcommuting maps and best approximations in Banach spaces

A result on best approximation is proved in Banach spaces satisfying Opial's condition. Recent results of Jungck and Sessa [Math. Japonica, 42(1995), 249-252.] are extended to a new class of noncommuting maps.

Asymptotic behaviour for an almost-orbit of nonexpansive semigroups in Banach spaces

In this paper, by using the technique of product nets, we are able to prove a weak convergence theorem for an almost-orbit of right reversible semigroups of nonexpansine mappings in a general Banach space X with Opial's condition. This includes many well known results as special cases. Let C be a weakly compact subset of a Banach space X with Opial's condition. Let G be a right reversible semitopological semigroup,  = {T (t): t ∈ G} a nonexpansive semigroup on C, and u (·) an almost-orbit of . Then {u (t): t ∈ G} is weakly convergent (to a common fixed point of ) if and only if it is weakly asymptotically regular (that is, {u (ht) − u (t)} converges to 0 weakly for every h ∈ G).

Duality map characterisations for Opial conditions

We characterise Opial's condition, the non-strict Opial condition, and the uniform Opial condition for a Banach space X in terms of properties of the duality mapping from X into X*.

Fixed points of mappings satisfying semicontractivity conditions

Let A be a subset of a Banach space E. A mapping T: A →A is called asymptoically semicontractive if there exists a mapping S: A×A→A and a sequence (kn) in [1, ∞] such that Tx=S(x, x) for all x ∈A while for each fixed x ∈A, S(., x) is asymptotically nonexpansive with sequence (kn) and S(x,.) is strongly compact. Among other things, it is proved that each asymptotically semicontractive self-mpping T of a closed bounded and convex subset A of a uniformly convex Banach space E which satisfies Opial's condition has a fixed point in A, provided s has a certain asymptoticregurity property.