Algorithms of common solutions to quasi variational inclusion and fixed point problems

Abstract

The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper not only improve and extend the main results in Korpelevich (Ekonomika i Matematicheskie Metody, 1976, 12(4):747-756), but also extend and replenish the corresponding results obtained by Iiduka and Takahashi (Nonlinear

Convergence of iterative sequences for fixed points of an infinite family of nonexpansive mappings based on a hybrid steepest descent methods

“…[34,35]Let C be a nonempty closed convex subset of a real Hilbert space H. Let T 1 ,T 2 ,..., be an infinite family of nonexpansive mappings of C into itself such that…”

An extragradient approximation method for variational inequality problem on fixed point problem of nonexpensive mappings and monotone mappings

“…P C is call the metric projection of H onto C. Lemma 2.1 (see Zhang, Lee and Chan [25]). The metric projection P C has the following properties:…”

Iterative algorithms for finding a common solution of system of the set of variational inclusion problems and the set of fixed point problems

“…(see [11]) u H is a solution of variational inclusion (1.4) if and only if u = J M, l (u -lBu), ∀l >0, i.e., Further, if l (0, 2a], then V I(H, B, M) is closed convex subset in H. Lemma 2.9. (see [22]) The resolvent operator J M,l associated with M is single-valued, nonexpansive for all l >0 and 1-inverse-strongly monotone.…”

“…We know that a mapping B : H H is said to be monotone, if for each x, y H, we have [11][12][13][14][15][16]). The set of the solution of (1.4) is denoted by V I(H, B, M).…”