New Classes of Generalized Invex Monotonicity

Variational inequalities theory has been widely used in many fields, such as economics, physics, engineering, optimization and control, transportation [1, 4]. Like convexity to mathematical programming problem (MP), monotonicity plays an important role in solving variational inequality (VI). To investigate the variational inequality, many kinds of monotone mappings have been introduced in the literature, see Karamardian and Schaible [5], for example. In [2], Crouzeix, et al. introduced the concepts of monotone plusmappings and proved the important role in the convergence of cutting-planemethod for solving variational inequities. In [14], Zhu andMarcotte introduced the classes of generalized cocoercive mapping and related them to classes previously introduced. Zhu and Marcotte [15] investigate iterative schemes for solving nonlinear variational inequalities under cocoercive assumption. Variational-like inequality problem (VLIP) or prevariational inequalities (PVI) ismore general problem than VIP, which is first introduced by Parida et al. [9]. Invex monotonicity, which is a generalization of classical monotonicity, is investigated widely by many researchers for studying invex function, which is generalization of convex function [6– 8, 12, 13], and solving VLIP [3, 9–11]. Ruiz-Garzón et al. [10] introduce some generalized invex monotonicity which are also discussed in [13], mentioned as generalized invariant monotonicity. The purpose of this paper is to introduce new classes of generalized invex monotone plus mappings and generalized invex cocoercive mappings and analyze their properties and relationships with respect to other concepts of invex monotonicity. Some examples,