Existence and iterative approximations of solutions for strongly nonlinear variational-like inequalities

Abstract

Abstract. In this paper, we introduce and study a new class of stronglynonlinear variational-like inequalities. Under suitable conditions, we provethe existence of solutions for the class of strongly nonlinear variational-like inequalities. By making use of the auxiliary principle technique, wesuggest an iterative algorithm for the strongly nonlinear variational-likeinequality and give the convergence criteria of the sequences generatedby the iterative algorithm. 1. IntroductionIt is well known that there are lots of iterative type algorithms for ndingthe approximate solutions of various variational inequalities in Hilbert spaces[3] and [8-13]. Among the most e ective numerical technique is the projectionmethod and its variant forms. However, the standard projection technique canno longer be applied to suggest the iterative type algorithm for variational-likeinequalities. This fact motivated Gowinski, Lions and Tremoliers [7] to developthe auxiliary principle technique, which does not depend on the projection.Ding [4,5] and Ding and Tan [6] extended the auxiliary principle technique tosuggest several iterative algorithms to compute approximate solutions for someclasses of general nonlinear mixed variational inequalities and variational-likeinequalities.In this paper, we introduce and study a new class of strongly nonlinearvariational-like inequalities. By applying a result due to Chang [1], we provethe existence of solutions for the class of strongly nonlinear variational-likeinequalities. Using the auxiliary principle technique, we suggest and analyzea new three-step iterative algorithm for solving the class of strongly nonlinearvariational-like inequalities. The convergence criteria of the sequence generatedby the iterative algorithm are given.