Some results on Rockafellar-type iterative algorithms for zeros of accretive operators

Jong Soo Jung

doi:10.1186/1029-242x-2013-255

Abstract

We provide a new proof technique to obtain strong convergence of the sequences generated by viscosity iterative methods for a Rockafellar-type iterative algorithm and a Halpern-type iterative algorithm to a zero of an accretive operator in Banach spaces. By using a new method different from previous ones, the main results improve and develop the recent well-known results in this area.

Highlights

Let E be a real Banach space with the norm · and the dual space E∗
The value of x∗ ∈ E∗ at y ∈ E is denoted by y, x∗ and the normalized duality mapping J from E into E∗ is defined by
The set of zeros of A is denoted by A, that is, A– := z ∈ D(A) : ∈ Az

Summary

Introduction

Let E be a real Banach space with the norm · and the dual space E∗. The value of x∗ ∈ E∗ at y ∈ E is denoted by y, x∗ and the normalized duality mapping J from E into E∗ is defined byJ (x) = x∗ ∈ E∗ : x, x∗ = x x∗ , x = x∗ , ∀x ∈ E.Recall that a (possibly multivalued) operator A : D(A) ⊂ E → E with the domain D(A) and the range R(A) in E is accretive if, for each xi ∈ D(A) and yi ∈ Axi (i = , ), there exists a j ∈ J (x – x ) such that y – y , j ≥ . (Here J is the normalized duality mapping.) In a Hilbert space, an accretive operator is called a monotone operator. ) for finding a zero of an accretive operator A in a uniformly convex Banach space E with a weakly continuous duality mapping Jφ with gauge function φ or with a uniformly Gâteaux differentiable norm: For any initial point x ∈ E and fixed point u ∈ E, xn+ = βnxn + ( – βn)Jrn αnu + ( – αn)xn , ∀n ≥

Results

Conclusion