Iterative methods for a system of nonlinear generalized mixed implicit equilibrium problems in Banach spaces

A new System of nonlinear generalized mixed implicit equilibrium problems involving non-monotone set-valued mappings with non-compact values in g-uniformly smooth Banach spaces is introduced. By using the Yosida approximation, a System of generalized Wiener-Hopf equations is considered. The equivalence between the System of nonlinear generalized mixed implicit equilibrium problems and a System of generalized Wiener-Hopf equations is established. By using this equivalence, a fixed point formulation is derived. Two new iterative algorithms with mixed errors are proposed and the existence theorems for solutions of the aforesaid Systems are established. Under some suitable conditions, the convergence analysis of the sequences generated by the proposed iterative algorithms is discussed. Some fatal errors in the results in [20] are pointed out and the correct versions of these results are presented. Some comments related to the work in [20] are given at the end. The results presented in this paper extend and improve some known results in the literature.