Asymptotic Behaviour of ϕ-Nonexpansive Sequences and Mappings in Banach Spaces

Abstract

In this paper, we introduce the notion of $\varphi$-nonexpansive sequences and maps in a Banach space E, and prove an ergodic theorem, a fixed point theorem, and some weak and strong convergence theorems for such sequences and maps in E, with rather mild assumptions on the Banach space E, weaker than those assumed by previous authors for nonexpansive maps. These results give a partial answer to the still open problem raised in 1981 by Djafari Rouhani for nonexpansive sequences in a Banach space.

Keywords:

• Asymptotic behavior
• asymptotically regular sequence
• strong convergence
• weak convergence
• $\varphi$-nonexpansive sequence

AMS Classification Number:

• Primary 47H25
• 47H09
• 47H10
• 47J25. Secondary 37A30
• 47H20
• 47H30
• 47H05

Acknowledgement

The authors are grateful to the Editor and the reviewers for their valuable comments leading to the improvement of the paper. This work is done while the third author was visiting the University of Texas at El Paso. The third author would like to thank Professor Djafari Rouhani and the Department of Mathematical Sciences for their kind hospitality at the University of Texas at El Paso during his visit.