Abstract
The purpose of this paper is to introduce a new generalization of asymptotically non-expansive set-valued mapping and to discuss its demi-closeness principle. Then, under certain conditions, we prove that the sequence defined by yn+1 = tn z+ (1-tn )un , un in Gn( yn ) converges strongly to some fixed point in reflexive Banach spaces. As an application, existence theorem for an iterative differential equation as well as convergence theorems for a fixed point iterative method designed to approximate this solution is proved
Highlights
It is well known that the concept of asymptotically nonexpansive by Goebel and Kirk [1] was introduced
One of the first results of iterative procedures was obtained by Browder [12]. He studied the iterative scheme for non-expansive mapping G: A → A
Na and Tang [16] proved the set-valued version of total asymptotic non-expansive mapping and some theorems about weak and strong convergence in uniformly convex Banach spaces for two steps iterative sequence which depending on the projection mapping
Summary
It is well known that the concept of asymptotically nonexpansive by Goebel and Kirk [1] was introduced. Every asymptotically nonexpansive mapping of a Banach space has a fixed point is proved.
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