Approximate solution of zero point problem involving H-accretive maps in Banach spaces and applications

Rajat Vaish,Mohd Sarfaraz,Md Kalimuddin Ahmad,Kaleem Raza Kazmi

doi:10.1186/s13660-019-2163-y

Abstract

In this manuscript, we introduce two iterative methods for finding the common zeros of two H-accretive mappings in uniformly smooth and uniformly convex Banach spaces. The proposed iterative methods are based on Mann and Halpern iterative methods and viscosity approximation method. Strong convergence results are established for iterative algorithms. Applications based on convex minimization problem, variational inequality problem and equilibrium problem are derived from the main result. Numerical implementation of the main results and application are demonstrated by some examples. Our results extend, generalize, and unify the previously known results given in literature.

Highlights

3 Main results we demonstrate strong convergence of the sequences acquired from the proposed iterative methods for finding a common zero of two H-accretive mappings
These iterative methods are based on Mann and Halpern iterative methods and viscosity approximation method
It is easy to observe that for κn = 0, n ∈ N, our proposed iterative methods consist of method of alternating resolvents, and our work extends the methods developed by Bauschke et al [2], Boikanyo et al [4], and Liu et al [19]

Summary

Introduction

Where g : C → C is a contraction mapping They established an outcome which ensures the strong convergence of the sequence {xn} acquired from (1.5) to a zero of an m-accretive mapping Φ in a uniformly smooth Banach space. They proved that the sequence {xn} obtained from (1.7) is weakly convergent to a fixed point of Jμ,Φ Jμ,Ψ This iterative method is based on alternating resolvents method which is an extension of alternating projections method introduced and studied by von Neumann [33] and Bregmann [5]. Under appropriate restrictions on control sequences, they established the strong convergence of the sequence obtained from (1.8) to a common zero point of maximal monotone mappings Φ and Ψ in a Hilbert space. Generalize, and unify the results given by Bauschke et al [2], Kim et al [17], Qin et al [22], Chen et al [9], and Boikanyo et al [4]

Preliminaries

Conclusion