Some Novel Aspects of Biconvex Functions and Biequilibrium Problems

In this chapter, we introduce and study several kinds of relative strongly biconvex functions with respect to an arbitrary function k and the bifunctions $$\beta (.,.), \quad \varphi (.),$$ β ( . , . ) , φ ( . ) , which are called the relative strongly k-biconvex functions. These strongly functions are nonconvex functions and include the biconvex function, convex functions, uniformly convex functions, $$\theta $$ θ -convex functions, k-convex and their variant forms as special cases. We study some properties of k-biconvex functions. Several parallelograms laws for inner product spaces are obtained as novel applications of the strongly k-biconvex functions. It is shown that the minimum of k-biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the biequilibrium problems. The auxiliary technique is used to suggest several new inertial type methods for solving the strongly biequilibrium problems and related problems. Convergence analysis of the proposed methods is considered under suitable weaker conditions. Several important special cases are obtained as novel applications of the results. Some open problems are also suggested for future research.