Construction of a common solution of a finite family of variational inequality problems for monotone mappings

Volume 9, Issue 4, pp 1645--1657 http://dx.doi.org/10.22436/jnsa.009.04.21

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Authors

Mohammed Ali Alghamdi - Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Naseer Shahzad - Operator Theory and Applications Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P. O. Box 80203, Jeddah 21589, Saudi Arabia. Habtu Zegeye - Department of Mathematics, University of Botswana, Pvt. Bag 00704 Gaborone, Botswana.

Abstract

Let C be a nonempty, closed and convex subset of a real Hilbert space H. Let \(A_i : C \rightarrow H\), for \(i = 1; 2\); be two \(L_i\)-Lipschitz monotone mappings and let \(f : C \rightarrow C\) be a contraction mapping. It is our purpose in this paper to introduce an iterative process for finding a point in \(V I(C;A_1) \cap V I(C;A_2) \)under appropriate conditions. As a consequence, we obtain a convergence theorem for approximating a common solution of a finite family of variational inequality problems for Lipschitz monotone mappings. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.

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Keywords

• Fixed points of a mapping
• monotone mapping
• strong convergence
• variational inequality.

MSC

•  47H09
•  47J20
•  65J15

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