Iterative algorithm for finding approximate solutions of mixed quasi-variational-like inclusions

In the present paper, we introduce the concept of η-relaxed strong convexity of a differentiable functional and extend Ding and Yao's auxiliary variational inequality technique [X.P. Ding, J.C. Yao, Existence and algorithm of solutions for mixed quasi-variational-like inclusions in Banach spaces, Computers and Mathematics with Applications, 49 (2005), 857-869] to develop iterative algorithms for finding the approximate solutions to the mixed quasi-variational-like inclusion problem (in short, MQVLIP) in a Banach space. On the one hand, we establish a result on the existence of a solution to the equilibrium problem by virtue of well-known Brouwer's fixed-point theorem. Moreover, by using this result we derive the existence and uniqueness of a solution to the MQVLIP and the existence of the approximate solutions generated by the algorithm for the MQVLIP. On the other hand, we use the concepts of η-relaxed strong convexity of a differentiable functional and η-cocoercivity of a composite map to prove the strong convergence of the approximate solutions to the unique solution of the MQVLIP.