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The purpose of this paper is to introduce and study two new hybrid proximal-point algorithms for finding a common element of the set of solutions to a generalized equilibrium problem and the sets of zeros of two maximal monotone operators in a uniformly smooth and uniformly convex Banach space. We established strong and weak convergence theorems for these two modified hybrid proximal-point algorithms, respectively.

We discuss the minimax problems for set-valued mappings with several hierarchical structures, and scalar hierarchical minimax theorems, hierarchical minimax theorems, hierarchical minimax inequalities for set-valued mappings, and the existence of cone-saddle points.

In this paper, we introduce and analyze a multi-step hybrid steepest-descent extragradient algorithm and multi-step composite Mann-type viscosity iterative algorithm for finding a solution of triple hierarchical variational inequalities defined over the common set of solutions of mixed equilibrium problems, variational inclusions, variational inequalities, and fixed point problems. Under appropriate assumptions, we prove that the proposed algorithms converge strongly to a common element of the fixed point set of a strict pseudocontractive mapping, a solution set of finitely many generalized mixed equilibrium problems, a solution set of finitely many variational inclusions, and a solution set of a general system of variational inequalities. Such an element is a unique solution of a triple hierarchical variational inequality problem. In addition, we also consider as an application the proposed algorithm to solve a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, and a general system of variational inequalities. The results obtained in this paper improve and extend the corresponding results announced by many other authors.

We study new types for minimax theorems with a couple of set-valued mappings, and we propose several versions for minimax theorems in topological vector spaces setting. These problems arise naturally from some minimax theorems in the vector settings. Both the types of scalar minimax theorems and the set minimax theorems are discussed. Furthermore, we propose three versions of minimax theorems for the last type. Some examples are also proposed to illustrate our theorems.