General iterative algorithms for solving mixed quasi-variational-like inclusions

In this paper, we extend the auxiliary variational inequality technique due to Ding and Yao [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Comput. Math. Appl. 49 (2005) 857–869] to develop iterative algorithms for finding the approximate solutions of a mixed quasi-variational-like inclusion problem (in short, MQVLIP) in the setting of Banach spaces. We first establish a result on the existence of a solution of the equilibrium problem by virtue of the Fan–KKM lemma. Then by using this result and a result by Ding and Tan [X.P. Ding, K.K. Tan, A minimax inequality with applications to existence of equilibrium point and fixed point theorems, Colloq. Math. 63 (2) (1992) 233–247], we derive the existence of a unique solution of MQVLIP and the existence of approximate solutions generated by the proposed algorithms. Moreover, we also provide the new criteria for convergence of approximate solutions to the exact solution of MQVLIP.