Strong convergence of some iterative algorithms for a general system of variational inequalities

Abstract

In this paper, we introduce two iterative algorithms (one implicit algorithm and one explicit algorithm) for finding a common element of the solution set of a general system of variational inequalities for continuous monotone mappings and the fixed point set of a continuous pseudocontractive mapping in a Hilbert space. First, this system of variational inequalities is proven to be equivalent to a fixed point problem of nonexpansive mapping. Then we establish strong convergence of the sequence generated by the proposed iterative algorithms to a common element of the solution set and the fixed point set, which is the unique solution of a certain variational inequality. c 2017 All rights reserved.Keywords: Composite iterative algorithm, general system of variational inequatlites, continuous monotone mapping, continuous peudocontractive mapping, ρ-Lipschitzian, η-strongly monotone mapping, variational inequality, strongly positive bounded linear operator, fixed points. 2010 MSC: 47J20, 47H05, 47H09, 47H10, 49J40, 49M05.

General iterative methods for systems of variational inequalities with the constraints of generalized mixed equilibria and fixed point problem of pseudocontractions

“…Compared with Proposition 3.3, Theorem 3.4, and Theorem 3.7 in [ 11 ], respectively, our Theorems 3.1 , 3.2 , and 3.3 improve and develop them in the following aspects: GSVI ( 1.3 ) with solutions being also fixed points of a continuous pseudocontinuous mapping in [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7] is extended to GSVI ( 1.3 ) with solutions being also common solutions of a finite family of generalized mixed equilibrium problems (GMEPs) and fixed points of a continuous pseudocontinuous mapping in our Theorems 3.1 , 3.2 , and 3.3 ; in the argument process of our Theorems 3.1 , 3.2 , and 3.3 , we use the variable parameters and (resp., and ) in place of the fixed parameters λ and ν in the proof of [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7], and additionally deal with a pool of variable parameters (resp., ) involving a finite family of GMEPs; the iterative schemes in our Theorems 3.1 , 3.2 , and 3.3 are more advantageous and more flexible than the iterative schemes in [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7], because they can be applied to solving three problems (i.e., GSVI ( 1.3 ), a finite family of GMEPs, and the fixed point problem of a continuous pseudocontractive mapping) and involve much more parameter sequences; it is worth emphasizing that our general implicit iterative scheme ( 3.1 ) is very different from Jung’s composite implicit iterative scheme in [ 12 ], because the term “ ” in Jung’s implicit scheme is replaced by the term “ ” in our implicit scheme ( 3.1 ). Moreover, the term “ ” in Jung’s explicit scheme is replaced by the term “ ” in our explicit scheme ( 3.3 ).…”

“…the iterative schemes in our Theorems 3.1 , 3.2 , and 3.3 are more advantageous and more flexible than the iterative schemes in [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7], because they can be applied to solving three problems (i.e., GSVI ( 1.3 ), a finite family of GMEPs, and the fixed point problem of a continuous pseudocontractive mapping) and involve much more parameter sequences;…”

“…GSVI ( 1.3 ) with solutions being also fixed points of a continuous pseudocontinuous mapping in [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7] is extended to GSVI ( 1.3 ) with solutions being also common solutions of a finite family of generalized mixed equilibrium problems (GMEPs) and fixed points of a continuous pseudocontinuous mapping in our Theorems 3.1 , 3.2 , and 3.3 ;…”

“…in the argument process of our Theorems 3.1 , 3.2 , and 3.3 , we use the variable parameters and (resp., and ) in place of the fixed parameters λ and ν in the proof of [ 12 , Proposition 3.3, Theorem 3.4, and Theorem 3.7], and additionally deal with a pool of variable parameters (resp., ) involving a finite family of GMEPs;…”

“…In the meantime, inspired by Ceng et al [ 1 ], Jung [ 12 ] introduced a general system of variational inequalities (GSVI) for two continuous monotone mappings and of finding such that where are two constants. In order to find an element of , he proposed one implicit algorithm generating a net : with and , and an explicit algorithm generating a sequence : with , , , and any initial guess, where for , and for .…”