Abstract
It is a familiar fact that convex and non-convex fuzzy mappings play a critical role in the study of fuzzy optimization. Due to the behavior of its definition, the idea of convexity plays a significant role in the subject of inequalities. The concepts of convexity and symmetry have a tight connection. We may use whatever we learn from one to the other, thanks to the significant correlation that has developed between both in recent years. Our aim is to consider a new class of fuzzy mappings (FMs) known as strongly preinvex fuzzy mappings (strongly preinvex-FMs) on the invex set. These FMs are more general than convex fuzzy mappings (convex-FMs) and preinvex fuzzy mappings (preinvex-FMs), and when generalized differentiable (briefly, G-differentiable), strongly preinvex-FMs are strongly invex fuzzy mappings (strongly invex-FMs). Some new relationships among various concepts of strongly preinvex-FMs are established and verified with the support of some useful examples. We have also shown that optimality conditions of G-differentiable strongly preinvex-FMs and the fuzzy functional, which is the sum of G-differentiable preinvex-FMs and non G-differentiable strongly preinvex-FMs, can be distinguished by strongly fuzzy variational-like inequalities and strongly fuzzy mixed variational-like inequalities, respectively. In the end, we have established and verified a strong relationship between the Hermite–Hadamard inequality and strongly preinvex-FM. Several exceptional cases are also discussed. These inequalities are a very interesting outcome of our main results and appear to be new ones. The results in this research can be seen as refinements and improvements to previously published findings.
Highlights
Many generalizations and extensions have been studied for classical convexity
We introduced and studied a new class of preinvex-fuzzy mappings (FMs) called strongly preinvex-FMs
To characterize the optimality condition of the sum of preinvex-FMs and strongly preinvex-FMs, we introduced the strong fuzzy mixed variational-like inequality
Summary
Many generalizations and extensions have been studied for classical convexity. Nanda and Kar [27], Syau [28] and Furukawa [29] introduced the concept of convexFMs from Rn to the set of fuzzy numbers They defined different type of convex-FMs, such as logarithmic convex-FMs and quasi-convex-FMs, as well studying. The concept of fuzzy preinvex mapping on the invex set was introduced and studied by Noor [32] He demonstrated that variational inequalities may be used to specify the fuzzy optimality conditions of differentiable fuzzy preinex mappings. Syau and Lee [34] examined various aspects of fuzzy optimization and discussed continuity and convexity through linear ordering and metrics defined on fuzzy integers They extended the Weirstrass theorem from real-valued functions to FMs. For recent applications, see [35,36,37,38,39] and the references therein.
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