Abstract
In this article, we introduce a new notion of generalized convex fuzzy mapping known as strongly generalized preinvex fuzzy mapping on the invex set. Firstly, we have investigated some properties of strongly generalized preinvex fuzzy mapping. In particular, we establish the equivalence among the strongly generalized preinvex fuzzy mapping, strongly generalized invex fuzzy mapping, and strongly generalized monotonicity. We also prove that the optimality conditions for the sum of G-differentiable preinvex fuzzy mappings and non-G-differentiable strongly generalized preinvex fuzzy mappings can be characterized by strongly generalized fuzzy mixed variational-like inequalities, which can be viewed as a novel and innovative application. Several special cases are discussed. Results obtained in this paper can be viewed as improvement and refinement of previously known results.
Highlights
Convexity plays an essential role in many areas of mathematical analysis and, due to its vast applications in diverse areas, many authors extensively generalized and extended this concept using novel and different approaches
Mohan and Neogy [3] further extended their work by proving that, subject to certain conditions, a preinvex function defined on the invex set is an invex function and vice versa and they showed that quasi-invex function is quasi-preinvex function
Noor et al [6,7,8] studied the optimality conditions of differentiable preinvex functions on the invex set that can be characterized by variational inequalities
Summary
Convexity plays an essential role in many areas of mathematical analysis and, due to its vast applications in diverse areas, many authors extensively generalized and extended this concept using novel and different approaches. Noor et al [6,7,8] studied the optimality conditions of differentiable preinvex functions on the invex set that can be characterized by variational inequalities. An important and significant generalization of convex function is strongly convex function, which is introduced by Polyak [9], which plays major role in optimization theory and related areas. Ammar and Metz [27] studied different types of convexity and defined generalized convexity of fuzzy sets They formulated a general fuzzy nonlinear programing problem with application of the concept of convexity. E idea of fuzzy preinvex mapping on the fuzzy invex set was introduced and studied by Noor [37] and they verified that fuzzy optimality conditions of differentiable fuzzy preinvex mappings can be distinguished by variational-like inequalities. Several special cases are discussed. is inequality is itself an interesting outcome of our main results
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