New Class of Preinvex Fuzzy Mappings and Related Inequalities

This study aims to consider new kinds of generalized convex fuzzy mappings (convexFM), which are called strongly α-preinvex fuzzy mappings. We investigated the characterization of preinvex-FMs using α-preinvex-FMs, which can be viewed as a novel and innovative application. Some different types of strongly α-preinvex-FMs are introduced, and their properties are investigated. Under appropriate conditions, we establish the relationship between strongly α-invex-FMs and strongly αj-monotone fuzzy operators. Then, the minimum of strongly α-preinvex-FMs are characterized by strongly fuzzy α-variational-like inequalities. Results obtained in this paper can be viewed as a refinement and improvement of previously known results.


Introduction
For classical convexity, several generalizations and extensions have recently been researched. Strongly convex functions on convex sets are a key concept in optimization theory and related fields, developed and investigated by Polyak [1]. Karmardian [2] discussed how when utilizing highly convex functions, there is only one way to solve nonlinear complementarity issues. Zhao and Wang [3], Qu and Li [4], and Nikodem and Pales [5] researched convergence analysis for resolving equilibrium issues and variational inequalities with the use of strongly convex functions. Regarding the characteristics and applications of strongly convex functions, we recommend the reader to [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25] and its references for further information. Hanson [26] proposed invex functions for differentiable functions, and they played an important role in mathematical programming. Israel and Mond proposed and researched the idea of invex sets and preinvex functions. Differential preinvex functions are invex functions, as is common knowledge. The opposite is likewise true in light of Condition C [27]. Furthermore, Noor [28] has shown that the minimum may be described by variational inequalities by researching the optimality criteria of differentiable preinvex functions. Noor et al. [29][30][31][32][33] examined the uses of strongly preinvex functions and their characteristics. Jeyakumar and Mond [34] established a different class of V-invex mapping on the V-invex set (nonconvex function), which has important applications in multiobjective optimization and extended convex programming. See , as well as the references therein, for other uses and characteristics of strongly preinvex functions.
Operations research, computer science, management sciences, artificial intelligence, control engineering, and decision sciences are just a few of the applied sciences and pure mathematics problems that are studied in [58] as a result of the extensive research on fuzzy sets and systems that has been performed on the development of various fields. Convex

Definition 2. [80]
Let I = (m, n) and σ ∈ (m, n). Then FM Y : (m, n) → E 0 is considered a generalized differentiable (G-differentiable) at σ if there exists an element Y , (σ) ∈ E 0 such that for all 0 < t, sufficiently small, there exist Y(σ + t) -Y(σ), Y(σ) -Y(σ − t) and the limits (in the metric D) where the limits are taken in the metric space (E, D), for ψ 1 , and H denotes the well-known Hausdorff metric on the space of intervals Ł C .
Strictly fuzzy convex mapping if strict inequality holds for Y(σ) = Y(ς). Y : C → E 0 is considered a fuzzy concave mapping if − Y is convex on C. Strictly fuzzy concave mapping if strict inequality holds for Y(σ) = Y(ς).
The invex set C is also called the j-connected set if j(ς, σ) = ς − σ. Note that, convex set j(ς, σ) = ς − σ is considered an invex set in the classical sense, but the converse is not valid. For instance, the following set C = [−7, −2] ∪ [2,10] is an invex set pertaining to nontrivial bifunction j : R × R → R given as where j : C × C → R.
Definition 8. [27]. A mapping Y : C → E 0 is considered fuzzy log-preinvex on invex set C pertaining to bifunction j if there exists a positive number ω such that where Y(.) 0.
Definition 9. [34]. The set C α is considered an α-invex set pertaining to arbitrary bifunctions j(., .) and α(., .), if where j : C α × C α → R and j : C α × C α → R\0 . The α-invex set C α is also called the αj-connected set. Note that the convex set with α(ς, σ) = 1 and j(ς, σ) = ς − σ is considered an invex set in the classical sense, but the converse is not valid. For example, in the following set
Then FM Y is considered J-preinvex.
We also define the affine strongly α-preinvex mapping.
We now establish a result for strongly α-preinvex-FM, which shows that the difference of strongly α-preinvex-FM and strong affine α-preinvex-FM is again a strongly α-preinvex-FM. Theorem 2. Let FM g : C α → E 0 be a strongly affine α-preinvex pertaining to j, α and ω > 0. Then Y is strongly α-preinvex-FM pertaining to the same j, α if, and only if, G = Y − g is α-preinvex-FM pertaining to j, α.

Definition 12.
A FM Y : C α → E 0 is considered strongly quasi α-preinvex on α-invex set C α pertaining to j, α if there exists a positive number ω such that Similarly, A FM Y is considered strongly quasi-preincave if − Y is strongly quasi α-preinvex on C α .
Proof. The demonstration is analogous to the demonstration of Theorem 1.

Definition 13.
A FM Y : C α → E 0 is considered strongly log α-preinvex on α-invex set C α pertaining to bifunctions j, α, if there exists a positive number ω such that If α(ς, σ) = 1, then we get the definition of log preinvex-FM, that is The mapping Y is strongly log-preinvex-FM pertaining to , α,.
If α(ς, σ) = 1 and ω = 0, then we get the definition of log-preinvex-FM in the classical sense, that is The mapping Y is log-preinvex-FM pertaining to j, α. From Definition 13, we have It can easily be seen that strongly log α-preinvex-FM ⇒ strongly α-preinvex-FM ⇒ strongly quasi α-preinvex-FM. For s = 1, Definition 10 and Definition 13, reduces to: Which plays an important role in studying the properties of α-preinvex-FMs and α-invex-FMs. If α(ς, σ) = 1, then Condition A reduces to the following for preinvex-FMs.

G-Differentiable Strongly α-Preinvex Fuzzy Mappings
In this section, we propose the concepts of strongly αj-invex-FMs and strongly αj-monotone fuzzy operators. With the help of these ideas, different properties of strongly α-preinvex-FMs are characterized. At the end, it is proved that the minimum of strongly α-preinvex-FMs can be distinguished by strongly fuzzy α-variational-like inequalities.
From Definition 15, it enables us to define the following new definitions: for all σ, ς ∈ C α , where Y , , is G-differentiable of Y at σ.
If j(ς, σ) = −j(σ, ς), then the Definitions 15-20 reduce to known ones. These definitions may play an important role in the fuzzy optimization problem and mathematical programming.
We need the following assumption regarding the bifunctions j,α, which play an important role in the G-differentiation of the main results.
With Theorem 6 and Theorem 7, we have the following new definitions.

Definition 21.
A G-differentiable mapping Y : C α → E 0 is considered: (i) Strongly αj-monotone fuzzy operator when, and only when, there exists a constant ω > 0 such that (ii) αj-monotone fuzzy operator when, and only when, we have (iii) Strongly αj-pseudomonotone fuzzy operator when, and only when, there exists a constant ω > 0 such that f or all σ, ς ∈ C α .
(iv) Strongly relaxed αj-pseudomonotone fuzzy operator when, and only when, there exists a constant ω > 0 such that (v) Strictly αj-monotone fuzzy operator when, and only when, we have (vi) αj-pseudomonotone fuzzy operator when, and only when, there exists a constant ω > 0 such that (vii) Quasi αj-pseudomonotone fuzzy operator when, and only when, there exists a constant ω > 0 such that (viii) Strictly αj-pseudomonotone fuzzy operator when, and only when, there exists a constant ω > 0 such that If j(ς, σ) = −j(σ, ς), then Definition 21, reduce to new one.
As special case of Theorem 7, we have the following: Let Y be G-differentiable FM on C α and let Condition C hold. Then Y is strongly α-invex-FM if, and only if, Y , is strongly αj-monotone fuzzy operator.
Theorem 8. Let G-differential Y , of mapping Y on C α be strongly αj-pseudomonotone fuzzy operator, and let Conditions A and C hold. Then Y is a strongly pseudo αj-invex-FM.
Following results are special cases of Theorem 8, we have: Corollary 2. Let G-differential Y , of FM Y on C α be αj-pseudomonotone fuzzy operator. Then, Y is a pseudo αj-invex-FM.
Corollary 3. Let G-differential Y , of FM Y on C α be strongly quasi αj-pseudomonotone fuzzy operator. Then Y is a strongly quasi αj-invex-FM.
Corollary 4. Let G-differential Y , of FM Y on C α be quasi αj-pseudomonotone fuzzy operator, and let Condition C hold. Then Y is a pseudo αj-invex-FM.
We now discuss the fuzzy optimality condition for G-differentiable strongly α-preinvex-FMs, which is main motivation of our next result.

Conclusions
In this paper, we generalized the concepts of convex-FMs and preinvex-FMs pertaining to j,α, which is called strongly α-preinvex-FMs pertaining to j,α. It is shown that strongly convex-FMs and strongly preinvex-FMs are special cases of strongly α-preinvex-FMs. Under certain conditions, it is also proved that differential strongly α-preinvex-FMs are strongly α-invex-FMs and vice versa. It is also proved that the minimum of strongly α-preinvex-FMs can be characterized by strong α-variational-like inequalities and α-variational-like inequalities. This idea in fuzzy convex and nonconvex theory needs to be studied more thoroughly. Furthermore, multiobjective fuzzy optimization has large and significant applications. In the future, we will try to explore this concept for integral inequalities by using fuzzy Reimann and fuzzy Reimann-Liouville fractional integrals.