Exponentially Preinvex Fuzzy Mappings and Fuzzy Exponentially Mixed Variational-Like Inequalities

Muhammad Bilal Khan ,Khalida İnayat Noor ,Hassan Almusawa ,Muhammad Aslam Noor ,Kottakkaran Sooppy Nisar

doi:10.28924/2291-8639-19-2021-518

Muhammad Bilal Khan , Khalida İnayat Noor + Show 3 more

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https://doi.org/10.28924/2291-8639-19-2021-518

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Abstract

In this article, our aim is to consider a class of nonconvex fuzzy mapping known as exponentially preinvex fuzzy mapping. With the support of some examples, the notions of exponentially preinvex fuzzy mappings are explored and discussed in some special cases. Some properties are also derived and relations among the exponentially preinvex fuzzy mappings (exponentially preinvex-FMs), exponentially invex fuzzy mappings (exponentially-IFMs), and exponentially monotonicity are established under some mild conditions. In the end, using the fact that fuzzy optimization and fuzzy variational inequalities have close relationships, we have proven that the optimality conditions of exponentially preinvex fuzzy mapping can be distinguished by exponentially fuzzy variational-like inequality and exponentially fuzzy mixed variational-like inequality. These inequalities render the very interesting outcomes of our main results and appear to be the new ones. Presented results in this paper can be considered and the development of previously obtained results.

Highlights

In the last few decades, the ideas of convexity and nonconvexity are well-acknowledged in optimization concepts and gifted a vital role in operation research, economics, decision making, and management
In the end, using the fact that fuzzy optimization and fuzzy variational inequalities have close relationships, we have proven that the optimality conditions of exponentially preinvex fuzzy mapping can be distinguished by exponentially fuzzy variational-like inequality and exponentially fuzzy mixed variational-like inequality
Motivated and inspired by the ongoing research work, we note that convex and generalized convex-fuzzy mappings (FMs) play an important role in fuzzy optimization

Summary

Introduction

In the last few decades, the ideas of convexity and nonconvexity are well-acknowledged in optimization concepts and gifted a vital role in operation research, economics, decision making, and management. Noor [23] examined the optimality conditions of differentiable preinvex functions and proved that variational-like inequalities would characterize the minimum. Noor and Noor [28] generalized the exponentially convex functions and defined a class of nonconvex functions which is known as exponentially preinvex function and proved that the minimum of a differentiable exponentially convex function is distinguished by exponentially variational-like inequality. A step forward, Furukawa [14] and Syau [31] proposed and examined FM from space Rn to the set of fuzzy numbers, fuzzy valued Lipschitz continuity, logarithmic convex-FMs and quasi-convex-FMs. Besides, Chang [12] discussed the idea of convex-FM and find its optimality condition with the support of fuzzy variational inequality. Noor [22] introduced this idea and proved some results that distinguish the fuzzy optimality condition of differentiable fuzzy preinvex mappings by fuzzy variational-like inequality. We refer to the readers for further analysis of literature on the applications and properties of variational-like inequalities and generalized convex-FMs, see ([8], [13], [15], [24], [32]) and the references therein

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