Auxiliary principle and iterative algorithms for generalized mixed nonlinear variational-like inequalities

The aim of this paper is to study the solvability of a class of generalized mixed nonlinear variational- like inequalities in Hilbert spaces. Using the auxiliary principle technique, the Banach fixed-point theorem and an inequality due to Chang and Xiang, we construct two iterative algorithms for finding approximate solutions of the generalized mixed nonlinear variational-like inequality. Under some conditions we prove the existence and uniqueness of solution for the generalized mixed nonlinear variational-like inequality and establish the strong convergence of approximate solutions to the exact solution of the generalized mixed nonlinear variational-like inequality. Our results extend, improve and unify some known results in the literature.