Iterative Approximations for Zeros of Sum of Accretive Operators in Banach Spaces

Yekini Shehu

doi:10.1155/2016/5973468

Abstract

We study the approximation of zero for sum of accretive operators using a modified Mann type forward-backward splitting algorithm and obtain strong convergence of the sequence generated by our scheme to the zero of sum of accretive operators in uniformly convex real Banach spaces which are also uniformly smooth. Our result is new and complements many recent and important results in this direction in the literature.

Highlights

Introduction and PreliminariesLet E be a real Banach space
We study the approximation of zero for sum of accretive operators using a modified Mann type forward-backward splitting algorithm and obtain strong convergence of the sequence generated by our scheme to the zero of sum of accretive operators in uniformly convex real Banach spaces which are uniformly smooth
Wei and Duan [25] studied the problem of finding zeros of the sum of finitely many m-accretive operators and finitely many α-inversely strongly accretive operators in a real smooth and uniformly convex Banach space (i.e., find x ∈ E such that 0 ∈ ∑Ni=1(Aix + Bix))

Summary

Introduction

Introduction and PreliminariesLet E be a real Banach space. The modulus of convexity δE : [0, 2] → [0, 1] is defined as δE (ε) = inf {1 − 󵄩󵄩󵄩󵄩x + 2 y󵄩󵄩󵄩󵄩 : ‖x‖ 󵄩󵄩󵄩󵄩y󵄩󵄩󵄩󵄩 y󵄩󵄩󵄩󵄩. In [4], it was shown that if E is uniformly smooth and if D is the fixed point Journal of Function Spaces set of a nonexpansive mapping from C into itself, there is a unique sunny nonexpansive retraction from C onto D.

Results

Conclusion