A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces
- PDF / 500,587 Bytes
- 24 Pages / 439.37 x 666.142 pts Page_size
- 38 Downloads / 191 Views
Journal of Fixed Point Theory and Applications
A strong convergence theorem for approximation of a zero of the sum of two maximal monotone mappings in Banach spaces Getahun B. Wega and Habtu Zegeye Abstract. The purpose of this paper is to study the method of approximation for a zero of the sum of two maximal monotone mappings in Banach spaces and prove strong convergence of the proposed method under suitable conditions. The method of proof is of independent interest. In addition, we give some applications to the minimization problems. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings. Mathematics Subject Classification. 47H05, 47J25, 49M27, 90C25. Keywords. Firmly nonexpansive, Hilbert spaces, maximal monotone mapping, strong convergence, zero points.
1. Introduction ∗
Let E be a Banach space with its dual E ∗ . Let A : E → 2E be a mapping, we denote the domain of A by Dom(A) = {x ∈ E : Ax = ∅} and the range of A by ran(A) = {Ax : x ∈ Dom(A)}. A is called monotone if x∗ − y ∗ , x − y ≥ 0, ∀(x, x∗ ), (y, y ∗ ) ∈ Gph(A), where Gph(A) = {(x, x∗ ) ∈ E × E ∗ : x∗ ∈ Ax} which is called the graph of A. It is also called maximal monotone if its graph is not properly contained in the graph of any other monotone mapping. ∗ Let A, B : E → 2E be monotone mappings. Consider the problem of finding: x∗ ∈ E such that 0 ∈ Ax∗ + Bx∗ .
(1.1)
We denote the solution set of (1.1) by (A + B)−1 (0). This problem has been studied by many authors in Hilbert spaces (see, for instance, [2,8,9,17, 19] and the references therein). 0123456789().: V,-vol
57
Page 2 of 24
G. B. Wega and H. Zegeye
In 1979, Passty [19] introduced the following forward–backward splitting method given by xn+1 = (I + λn B)−1 (I − λn A)xn , n ≥ 1,
(1.2)
where {λn } is a sequence of positive numbers, A and B are maximal monotone mappings with Dom(A) ⊆ Dom(B), and A is single-valued mapping. He proved weak convergence of his sequence to the solution of problem (1.1) under certain conditions provided that A + B is maximal monotone. In 2008, Eckstein and Svaiter [10] constructed a splitting algorithm method which starts by reformulating (1.1) as the problem of locating a point in a certain extended solution set Se (A, B) ⊂ H × H and they proved weak convergence of their sequence to the solution of problem (1.1) under certain conditions. The extended solution set for the problem (1.1), which is the subset of H × H is defined by: Se (A, B) = {(z, w) ∈ H × H : w ∈ B(z), −w ∈ A(z)}.
(1.3)
We note that (see, e.g., [33]) finding a point in Se (A, B) is equivalent to solving (1.1) in the sense that 0 ∈ A(z) + B(z) ⇔ ∃w ∈ H : (z, w) ∈ Se (A, B).
(1.4)
If the monotone mappings A and B are maximal monotone, it has been shown that (see, [33]) the corresponding extended solution set Se (A, B) is closed and convex in H × H. With regard to a strong convergence, several authors have studied different iterative schemes (see for example, [12,21,34–39] and the references therein) for a zero of the sum of mo
Data Loading...
