Extended Multi-valued Pseuducontractive Mappings and Extended Mann and Ishikawa Iteration Schemes for Finite Family of Mappings

Abstract

In this paper, new classes of {\it extended} multi-valued pseudocontractive mappings are introduced. It is established that the type-one subclass of the extended strictly pseudocontractive mappings is more closely related to the class of single-valued strictly pseudocontractive mappings in the sense that the possession of L-Lipschitzian and demiclosedness properties as well as closed and convex set of fixed points are guaranteed. Also, we introduce an extended Mann and an extended Ishikawa iteration schemes for approximating a common fixed point of a finite family of mappings. Furthermore, using the extended Mann and the extended Ishikawa iteration schemes, we prove weak and strong convergence theorems for our new classes of extended strictly pseudocontractive and pseudocontractive mappings, respectively. Numerical examples are also included to illustrate our results. The results obtained improve, complement and extend the results on multi-valued and single-valued mappings in the contemporary literature.