Convergence of Infinite Family of Multivalued Quasi-Nonexpansive Mappings Using Multistep Iterative Processes

Renu Chugh,Sanjay Kumar

doi:10.1155/2014/781704

Abstract

We prove strong and weak convergence results using multistep iterative sequences for countable family of multivalued quasi-nonexpansive mappings by using some conditions in uniformly convex real Banach space. The results presented extended and improved the corresponding result of Zhang et al. (2013), Bunyawat and Suantai (2012), and some others from finite family, one countable family, and two countable families tok-number of countable families of multivalued quasi-nonexpansive mappings. Also we used a numerical example in C++ computational programs to prove that the iterative scheme we used has better rate of convergence than other existing iterative schemes.

Highlights

I=1 where the sequence {αn,i}∞ i=0 ⊂ [0, 1) satisfying ∑∞ i=0 αn,i = 1 and xn,i ∈ Tixn such that d(p, xn,i) = d(p, Tixn) for i ∈ N. They proved the convergence of iterative processes to common fixed point of countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach space
Motivated by [6, 20,21,22], in this paper, we extended the result of Zhang et al [21] from two countable families to knumber of countable families and proved weak and strong convergence results of two new multistep iterative processes to common fixed point of countable family of multivalued quasi-nonexpansive mappings in a uniformly convex Banach space
(4) We prove weak and strong convergence results for new multistep iterative scheme (7)

Summary

Introduction

I=1 where the sequence {αn,i}∞ i=0 ⊂ [0, 1) satisfying ∑∞ i=0 αn,i = 1 and xn,i ∈ Tixn such that d(p, xn,i) = d(p, Tixn) for i ∈ N. i=1 where the sequence {αn,i}∞ i=0 ⊂ [0, 1) satisfying ∑∞ i=0 αn,i = 1 and xn,i ∈ Tixn such that d(p, xn,i) = d(p, Tixn) for i ∈ N They proved the convergence of iterative processes to common fixed point of countable family of multivalued quasi-nonexpansive mappings in uniformly convex Banach space. Motivated by [6, 20,21,22], in this paper, we extended the result of Zhang et al [21] from two countable families to knumber of countable families and proved weak and strong convergence results of two new multistep iterative processes to common fixed point of countable family of multivalued quasi-nonexpansive mappings in a uniformly convex Banach space.

Main Results

Strong and Weak Convergence Results for New Multistep

Numerical Computation

Conclusion