Auxiliary Principle And Iterative Algorithms For A Generalized Mixed Nonlinear Variational-like Inequality

Variational inequality theory has become a very powerful tool for studying a rich kind of problems arising in many different fields of pure mathematics and applied sciences. Recent years,variational inequality theory provides a large mathematics framework for discussing many interesting problems in the field of optimization, equilibrium, elasticity, etc. Generalized mixed nonlinear variational-like inequality is a useful and important part of variational inequality.This paper introduces and studies a new class of generalized mixed nonlinear variational-like inequalities in Hilbert spaces. The generalized mixed variational-like inequality in this paper includes a number of known variational inequalities as special case, The auxiliary principle technique is extended to study the existence and uniqueness of solution for the generalized mixed nonlinear variational-like inequality.First, by KKM theory, the existence results of solutions of the auxiliary problems with respect to the generalized mixed nonlinear variational-like inequality are shown., in this part, two theorems are proved in different ways.Second, in order to find out the approximate solutions of the generalized mixed nonlinear variational-like inequality, we suggest two iterative algorithms. The two iterative algorithms, which are one-step iterative with errors and two-step Ishikawa iterative with errors, respectively, for solving the generalized mixed nonlinear variational-like inequality are constructed by using the existence results.Finally, not only the existence results of solutions of the generalized mixed nonlinear variational-like inequality are shown by the Banach fixed-point theorem, but also the convergence results of iterative sequences generated by the two algorithms are proved. This part is very important, four theorems are shown for the existence results of solutions of the generalized mixed nonlinear variational-like inequality and the convergence results of iterative sequences in different ways by the two algorithms.The results presented in this paper extend, improve and unify several known results in the literature.