Abstract
In this paper, we introduce and study some new classes of biconvex functions with respect to an arbitrary function and a bifunction, which are called the higher order strongly biconvex functions. These functions are nonconvex functions and include the biconvex function, convex functions, and k-convex as special cases. We study some properties of the higher order strongly biconvex functions. Several parallelogram laws for inner product spaces are obtained as novel applications of the higher order strongly biconvex affine functions. It is shown that the minimum of generalized biconvex functions on the k-biconvex sets can be characterized by a class of equilibrium problems, which is called the higher order strongly biequilibrium problems. Using the auxiliary technique involving the Bregman functions, several new inertial type methods for solving the higher order strongly biequilibrium problem are suggested and investigated. Convergence analysis of the proposed methods is considered under suitable conditions. Several important special cases are obtained as novel applications of the derived results. Some open problems are also suggested for future research.
Highlights
Variational inequality theory, which was introduced and studied by Stampacchia [1] can be viewed as an important and significant extension of the variational principles, the origin of which can be traced back to Euler, Lagrange, Newton, and Bernoulli brothers
We prove that the minimum of the differential k-biconvex functions on the kbiconvex sets can be characterized by a class of equilibrium problems
We suggest and analyze some iterative methods for higher order strongly biequilibrium problems (5.6) using the auxiliary principle technique coupled with Bregman functions as developed by Noor [28] [30]
Summary
Variational inequality theory, which was introduced and studied by Stampacchia [1] can be viewed as an important and significant extension of the variational principles, the origin of which can be traced back to Euler, Lagrange, Newton, and Bernoulli brothers. Noor et al [13] [14] [15] [16] and Zhu and Marcote [17] used the auxiliary principle involving the Bregman functions They have suggested and analyzed a wide class of iterative method for solving variational inequalities and equilibrium problems. They have shown [2] [3] that the Bregman functions have some practical important types of functions such as Burg entropy, which is very important in information theory and Shannon entropy, having applications in several areas of applied mathematics such as machine learning.
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