Implicit Relaxed and Hybrid Methods with Regularization for Minimization Problems and Asymptotically Strict Pseudocontractive Mappings in the Intermediate Sense

Lu-Chuan Ceng,Qamrul Hasan Ansari,Ching-Feng Wen

doi:10.1155/2013/854297

Abstract

We first introduce an implicit relaxed method with regularization for finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mappingSin the intermediate sense and the set of solutions of the minimization problem (MP) for a convex and continuously Frechet differentiable functional in the setting of Hilbert spaces. The implicit relaxed method with regularization is based on three well-known methods: the extragradient method, viscosity approximation method, and gradient projection algorithm with regularization. We derive a weak convergence theorem for two sequences generated by this method. On the other hand, we also prove a new strong convergence theorem by an implicit hybrid method with regularization for the MP and the mappingS. The implicit hybrid method with regularization is based on four well-known methods: the CQ method, extragradient method, viscosity approximation method, and gradient projection algorithm with regularization.

Highlights

In 1972, Goebel and Kirk [1] established that every asymptotically nonexpansive mapping S : C → C defined on a nonempty closed convex bounded subset of a uniformly convex Banach space, that is, there exists a sequence {kn} such that limn → ∞ kn = 1 and󵄩󵄩󵄩󵄩Snx − Sny󵄩󵄩󵄩󵄩 ≤ kn 󵄩󵄩󵄩󵄩x − y󵄩󵄩󵄩󵄩, ∀n ≥ 1, ∀x, y ∈ C (1)has a fixed point in C
We prove a new strong convergence theorem by an implicit hybrid method with regularization for the minimization problem (MP) and the mapping S
The main aim of this paper is to propose some iterative schemes for finding a common solution of fixed point set of an asymptotically κ-strict pseudocontractive mapping and the solution set of the minimization problem

Summary

Introduction

The main aim of this paper is to propose some iterative schemes for finding a common solution of fixed point set of an asymptotically κ-strict pseudocontractive mapping and the solution set of the minimization problem. We introduce an implicit relaxed algorithm with regularization for finding a common element of the fixed point set Fix(S) of an asymptotically κ-strict pseudocontractive mapping S and the solution set Γ of minimization problem (8). This implicit relaxed method with regularization is based on three well-known methods, namely, the extragradient method [6], viscosity approximation method, and gradient projection algorithm with regularization. The weak and strong convergence results of these two algorithms are established, respectively

Preliminaries

Weak Convergence Theorem

Strong Convergence Theorem