Convergence Of Iterative Algorithms For Quasivariational And Variational-Like Inequalities

In this paper, a new class of generalized nonlinear quasivaria-tional inequalities in Hilbert spaces are introduced and studied. By using the projection method, a perturbed three-step iterative algorithm is constructed. The existence of solution for the generalized nonlinear quasivariational inequality involving relaxed monotone, relaxed Lipschitz and strongly monotone mappings is established, and some convergence and stability results of the iterative sequences generated by the algorithm are proved. On the other hand, a new class of completely generalized strongly nonlinear variational-like inequalities in reflexive Banach spaces are considered and investigated. By utilizing a minimax inequality due to Ding and Tan, Banach fixed point theorem and the auxiliary variational-like inequality technique, the existence and uniqueness result of solution for the completely generalized strongly nonlinear variational-like inequality is established, and two kinds of new iterative algorithms for finding the approximate solutions of the completely generalized strongly nonlinear variational-like inequality are constructed, several convergence results of the iterative sequences generated by the algorithms are also discussed. These results presented in this paper extend, improve and unify the corresponding results in recent literature.