New Class of Preinvex Fuzzy Mappings and Related Inequalities

Abstract

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

1. 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 [35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57], 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 analysis has contributed significantly and fundamentally to the development of several practical and pure scientific domains. Similar to this, fuzzy convex analysis is regarded as the core concept in fuzzy optimization. In order to describe a fuzzy number, it is an interval’s generalized form (in crisp set theory). Fuzzy numbers were first defined by Zadeh [58], and subsequently, Dubois and Prade [59] expanded on this work by adding new requirements for fuzzy numbers. Additionally, Goetschel and Voxman [60] adjusted many conditions on fuzzy numbers to make them easier to manage. For example, in [61], the first criterion for a fuzzy number is that it is a continuous function, but in [60], the fuzzy number is upper semi-continuous. The objective is to make it simple to establish a metric for a collection of fuzzy numbers by relaxing the requirements on them, which will then enable us to examine certain fundamental topological space features. The concepts of fuzzy mapping (𝘍𝘔) from ${ℝ}^n$ to the set of fuzzy numbers, Lipschitz continuity of fuzzy values, fuzzy logarithmic convex and quasi-convex-FMs, and others were studied by Furukawa [61], Nanda, and Kar [62], and Syau [63]. Yan and Xu [64] proposed the notions of epigraphs and convexity of 𝘍𝘔s and detailed the characteristics of convex fuzzy and quasi-convex- 𝘍𝘔s based on the concept of ordering established by Goetschel and Voxman [65]. For more information, see [66,67,68,69,70,71,72,73,74,75,76] and the references therein.

The concept of fuzzy convexity has been broadly applied and expanded in several ways, with important applications in numerous fields. Preinvex-𝘍𝘔 should be mentioned as one of the convex-𝘍𝘔 generalizations that are most often used. By introducing and researching the concept of fuzzy preinvex mapping on the fuzzy invex set, Noor [77] showed that the fuzzy optimality requirements of differentiable fuzzy preinvex mappings may be identified by variational-like inequalities. Preinvex-local 𝘍𝘔s minimums are also global minimums on invex sets, and an invex set’s epigraph is a necessary and sufficient condition for an 𝘍𝘔 to be preinvex. The preinvex-𝘍𝘔 notion that Noor [77] introduced was further developed by Syau in [78]. Additionally, Syau and Lee [79] covered the terms for continuity and convexity using metric definitions based on fuzzy numbers and linear ordering. Extension of the Weierstrass theorem from real-valued functions to 𝘍𝘔s is also one of their key contributions in the literature. See [80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98] and its references for contemporary applications.

Inspired and motivated by current research efforts, as well as by the significance of the concepts of the invexity and preinvexity of mappings, in Section 2, we review various concepts that will be useful for further research, including fuzzy sets, fuzzy numbers, 𝘍𝘔s, convex-𝘍𝘔s, and preinvex-𝘍𝘔s. Section 3 examines the primary findings. The concepts of strongly $\alpha$-preinvex, quasi $\alpha$-preinvex, and log $\alpha$-preinvex-𝘍𝘔s are presented in Section 3, along with some of their features. Section 4 studies brand-new connections between distinct strongly $\alpha$-preinvex-𝘍𝘔 ideas. Then, as an intriguing byproduct of our primary findings, the minimum of strongly $\alpha$-preinvexity is characterized by strongly fuzzy $\alpha$-variational-like inequalities.

2. Preliminaries

A fuzzy set on $ℝ$ is a mapping $\psi :ℝ\to \left[0,1\right]$, for each fuzzy set and $\mathit{ĩ}\in \left(0, 1\right]$, then $\mathit{ĩ}$-level sets of $\psi$ are denoted and defined as follows ${\psi }_{\mathit{ĩ}}=\left\{\sigma \in ℝ| \psi \left(\sigma \right)\ge \mathit{ĩ}\right\}$. If $\mathit{ĩ}=0$, then $supp\left(\psi \right)=\left\{\sigma \in ℝ\left| \psi \left(\sigma \right)\right.〉0\right\}$ is considered support of $\psi$. By ${\left[\psi \right]}^0$, we define the closure of $supp\left(\psi \right)$.

Definition 1.

A fuzzy set is considered a fuzzy number with the following properties:

(a) 

$\psi$is normal, i.e., there exists $\sigma \in ℝ$ such that $\psi \left(\sigma \right)=1;$

(b) 

$\psi \left(\left(1-\mathfrak{s}\right)\sigma +\mathfrak{s}\varsigma \right)\ge min\left(\psi \left(\sigma \right), \psi \left(\varsigma \right)\right)$ for all $\sigma ,\varsigma \in ℝ$,$\mathfrak{s}\in \left[0,1\right]$;

(c) 

sup$\left(\psi \right)$ is compact.

${\mathbb{E}}_0$ denotes the set of all fuzzy numbers. For a fuzzy number, it is convenient to distinguish the following $\mathit{ĩ}$-levels,

$${\psi }_{\mathit{ĩ}}=\left\{\sigma \in ℝ| \psi \left(\sigma \right)\ge \mathit{ĩ}\right\},$$

From this definition, we have

$${\psi }_{\mathit{ĩ}}=\left[{\psi }_{*}\left(\mathit{ĩ}\right), {\psi }^{*}\left(\mathit{ĩ}\right)\right],$$

where

$${\psi }_{*}\left(\mathit{ĩ}\right)=inf\left\{\sigma \in ℝ| \psi \left(\sigma \right)\ge \mathit{ĩ}\right\}, {\psi }^{*}\left(\mathit{ĩ}\right)=sup\left\{\sigma \in ℝ| \psi \left(\sigma \right)\ge \mathit{ĩ}\right\}.$$

Since each $r\in ℝ$ is also a fuzzy number, it can be defined as

$$\tilde{r}\left(\sigma \right)=\left\{\begin{array}{c}1 \mathrm{if} \sigma =r\\ 0 \mathrm{if} \sigma \ne r \end{array}\right..$$

Now we discuss some properties of fuzzy numbers under addition and scalar multiplication. If ${\psi }_1,{\varphi }_1\in {\mathbb{E}}_0$ and $\rho \in ℝ$, then ${\psi }_1\widetilde{+}{\varphi }_1$ and $\rho {\psi }_1$can be defined as

$$\left({\psi }_1\widetilde{+}{\varphi }_1\right)\left(w\right)=su{p}_{\sigma +\varsigma =w}min\left\{\left({\psi }_1\right)\left(w\right), \left({\varphi }_1\right)\left(w\right)\right\},$$

$$\left(\rho {\psi }_1\right)\left(\sigma \right)=\left\{\begin{array}{c}{\psi }_1\left({\rho }^{-1}\sigma \right) \mathrm{if} \rho \ne 0\\ 0, \mathrm{if} \rho =0.\end{array}\right.$$

It is also well known that for any ${\psi }_1,{\varphi }_1\in {\mathbb{E}}_0$ and $\rho \in ℝ$

$${\left({\psi }_1\widetilde{+}{\varphi }_1\right)}^{*}\left(\mathit{ĩ}\right)={\psi }_1{}^{*}\left(\mathit{ĩ}\right)+{\varphi }_1{}^{*}\left(\mathit{ĩ}\right), {\left({\psi }_1\widetilde{+}{\varphi }_1\right)}_{*}\left(\mathit{ĩ}\right)={\psi }_1{}_{*}\left(\mathit{ĩ}\right)+{\varphi }_1{}_{*}\left(\mathit{ĩ}\right),$$

$${\left(\rho {\psi }_1\right)}_{*}\left(\mathit{ĩ}\right)=\left\{\begin{array}{c}\rho {\psi }_1{}_{*}\left(\mathit{ĩ}\right) if \rho \ge 0\\ \rho {\psi }_1{}^{*}\left(\mathit{ĩ}\right) if \rho <0 \end{array}\right., {(\rho {\psi }_1)}^{*}\left(\mathit{ĩ}\right)=\left\{\begin{array}{c}\rho {\psi }_1{}^{*}\left(\mathit{ĩ}\right) if \rho \ge 0\\ \rho {\psi }_1{}_{*}\left(\mathit{ĩ}\right) if \rho <0 \end{array}\right.,$$

for each $\mathit{ĩ}\in \left[0, 1\right].$ From this definition, we have

$${\psi }_1\widetilde{+}{\varphi }_1=\left\{\left({\psi }_1{}_{*}\left(\mathit{ĩ}\right)+{\varphi }_1{}_{*}\left(\mathit{ĩ}\right),{\psi }_1{}^{*}\left(\mathit{ĩ}\right)+{\varphi }_1{}^{*}\left(\mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}$$

(1)

$$\rho {\psi }_1=\left\{\left(\rho {\psi }_1{}_{*}\left(\mathit{ĩ}\right),\rho {\psi }_1{}^{*}\left(\mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}.$$

(2)

Remark 1.

Obviously,${\mathbb{E}}_0$. is closed under addition and nonnegative scalar multiplication. Furthermore, for each scalar number$r\in ℝ,$

$${\psi }_1\widetilde{+}r=\left[{\psi }_{*}\left(\mathit{ĩ}\right)+r, {\psi }^{*}\left(\mathit{ĩ}\right)+r\right].$$

(3)

For any ${\psi }_1,{\varphi }_1\in {\mathbb{E}}_0$, we say that ${\psi }_1⪯{\varphi }_1$ if for all $\mathit{ĩ}\in \left(0, 1\right]$, ${\psi }_1{}^{*}\left(\mathit{ĩ}\right)\le {\varphi }_1{}^{*}\left(\mathit{ĩ}\right)$, and ${\psi }_1{}_{*}\left(\mathit{ĩ}\right)\le {\varphi }_1{}_{*}\left(\mathit{ĩ}\right)$. If ${\psi }_1⪯{\varphi }_1,$ then there exist $\mathit{ĩ}\in \left(0, 1\right]$ such that ${\psi }_1{}^{*}\left(\mathit{ĩ}\right)<{\varphi }_1{}^{*}\left(\mathit{ĩ}\right)$ or ${\psi }_1{}_{*}\left(\mathit{ĩ}\right)\le {\varphi }_1{}_{*}\left(\mathit{ĩ}\right)$. We say comparable if for any ${\psi }_1,{\varphi }_1\in {\mathbb{E}}_0$, we have ${\psi }_1⪯{\varphi }_1$ or ${\psi }_1⪰{\varphi }_1$; otherwise, they are noncomparable. Sometimes we may write ${\psi }_1⪯{\varphi }_1$ instead of ${\varphi }_1⪰{\psi }_1$ and note that ${\mathbb{E}}_0$ is a partial ordered set under the relation $⪰$.

If ${\psi }_1,{\varphi }_1\in {\mathbb{E}}_0,$ there exist ${\omega }_1\in {\mathbb{E}}_0$ such that ${\psi }_1={\varphi }_1\widetilde{+}{\omega }_1,$ then by this result, we have the existence of the Hukuhara difference of ${\psi }_1$ and ${\varphi }_1$, and we say that ${\omega }_1$ is the H-difference of ${\psi }_1$ and ${\varphi }_1,$ denoted by ${\psi }_1\widetilde{-}{\varphi }_1$ (see [82]). If H-difference exists, then ${\left({\omega }_1\right)}^{*}\left(\mathit{ĩ}\right)={\left({\psi }_1\widetilde{-}{\varphi }_1\right)}^{*}\left(\mathit{ĩ}\right)={\psi }_1{}^{*}\left(\mathit{ĩ}\right)-{\varphi }_1{}^{*}\left(\mathit{ĩ}\right)$, ${\left({\omega }_1\right)}_{*}={\left({\psi }_1\widetilde{-}{\varphi }_1\right)}_{*}\left(\mathit{ĩ}\right)={\psi }_1{}_{*}\left(\mathit{ĩ}\right)-{\varphi }_1{}_{*}\left(\mathit{ĩ}\right).$

A mapping$\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered 𝘍𝘔. For each $\mathit{ĩ}, \in \left[0, 1\right]$associated with $\widetilde{{\rm Y}}$, we define the family of interval-valued functions ${\widetilde{{\rm Y}}}_{\mathit{ĩ}}:\mathcal{C}\to {Ҡ}_C$ defined by ${\widetilde{{\rm Y}}}_{\mathit{ĩ}}\left(\sigma \right)={\left[\widetilde{{\rm Y}}\left(\sigma \right)\right]}^{\mathit{ĩ}}$ and denoted by ${\left[\widetilde{{\rm Y}}\left(\sigma \right)\right]}^{\mathit{ĩ}}=\left[{\widetilde{{\rm Y}}}_{*}\left(\sigma ,\mathit{ĩ}\right), {\widetilde{{\rm Y}}}^{*}\left(\sigma ,\mathit{ĩ}\right)\right]$ Now, for any $\mathit{ĩ}\in \left[0, 1\right],$ the endpoint functions ${\widetilde{{\rm Y}}}_{*}\left(\sigma ,\mathit{ĩ}\right), {\widetilde{{\rm Y}}}^{*}\left(\sigma ,\mathit{ĩ}\right): \mathcal{C}\to ℝ$ are called lower and upper functions, respectively.

Definition 2.

[80] Let $I=\left(m, n\right)$ and $\sigma \in \left(m, n\right)$. Then 𝘍𝘔 $\widetilde{{\rm Y}}:\left(m, n\right)\to {\mathbb{E}}_0$ is considered a generalized differentiable (G-differentiable) at $\sigma$ if there exists an element${\widetilde{{\rm Y}}}^{,}\left(\sigma \right)\in {\mathbb{E}}_0$ such that for all $0<t$, sufficiently small, there exist $\widetilde{{\rm Y}}\left(\sigma +t\right) \widetilde{– }\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\sigma \right) \widetilde{– }\widetilde{{\rm Y}}\left(\sigma -t\right)$ and the limits (in the metric $D$)

$$\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma +t\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{t}=\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma -t\right)}{t}={\widetilde{{\rm Y}}}^{,}\left(\sigma \right)$$

or

$$\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma +t\right)}{-t}=\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma -t\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{-t}={\widetilde{{\rm Y}}}^{,}\left(\sigma \right)$$

or

$$\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma +t\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{t}=\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma -t\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{-t}={\widetilde{{\rm Y}}}^{,}\left(\sigma \right)$$

or

$$\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma +t\right)}{-t}=\underset{t\to 0^{+}}{\lim }\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma -t\right)}{t}={\widetilde{{\rm Y}}}^{,}\left(\sigma \right)$$

where the limits are taken in the metric space $\left(E,D\right),$ for ${\psi }_1, {\psi }_2\in {\mathbb{E}}_0$

$$D\left({\psi }_1, {\psi }_2\right)=\underset{0\le \mathit{ĩ}\le 1}{\sup }H\left({\psi }_1{}_{\mathit{ĩ}}, {\psi }_2{}_{\mathit{ĩ}}\right),$$

and $H$ denotes the well-known Hausdorff metric on the space of intervals ${Ҡ}_C$.

Definition 3.

[62] A 𝘍𝘔 $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered convex on the convex set $\mathcal{C}$ if

$$\widetilde{{\rm Y}}\left(\left(1-\mathfrak{s}\right)\sigma +\mathfrak{s}\varsigma \right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right) \forall \sigma ,\varsigma \in \mathcal{C}, \mathfrak{s}\in \left[0, 1\right].$$

(4)

Strictly fuzzy convex mapping if strict inequality holds for$\widetilde{{\rm Y}}\left(\sigma \right)\ne \widetilde{{\rm Y}}\left(\varsigma \right)$. $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$is considered a fuzzy concave mapping if$-\widetilde{{\rm Y}}$is convex on$\mathcal{C}.$Strictly fuzzy concave mapping if strict inequality holds for$\widetilde{{\rm Y}}\left(\sigma \right)\ne \widetilde{{\rm Y}}\left(\varsigma \right)$.

Definition 4.

[62] A 𝘍𝘔 $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered quasi-convex on the convex set $\mathcal{C}$ if

$$\widetilde{{\rm Y}}\left(\left(1-\mathfrak{s}\right)\sigma +\mathfrak{s}\varsigma \right)⪯max\left(\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\varsigma \right)\right) \forall \sigma ,\varsigma \in \mathcal{C}, \mathfrak{s}\in \left[0, 1\right].$$

(5)

Definition 5.

[27] The set $\mathcal{C}$ in $ℝ$ is considered an invex set pertaining to arbitrary bifunction $𝑗\left(.,.\right),$ if

$$\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\in \mathcal{C}, \forall \sigma ,\varsigma \in \mathcal{C}, \mathfrak{s}\in \left[0, 1\right].$$

(6)

The invex set $\mathcal{C}$ is also called the $𝑗$-connected set if $𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma .$ Note that, convex set $𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma$ is considered an invex set in the classical sense, but the converse is not valid. For instance, the following set $\mathcal{C}=\left[-7,-2\right]\cup \left[2,10\right]$ is an invex set pertaining to nontrivial bifunction $𝑗:ℝ\times ℝ\to ℝ$ given as

$$\begin{gathered} 𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma , \varsigma \ge 0, \sigma \ge 0, \\ 𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma , 0\ge \varsigma , 0\ge \sigma , \\ 𝑗\left(\varsigma ,\sigma \right)=-7-\sigma , \varsigma \ge 0\ge \sigma , \\ 𝑗\left(\varsigma ,\sigma \right)=2-\sigma , \sigma \ge 0\ge \varsigma . \end{gathered}$$

Definition 6.

[27] A 𝘍𝘔$\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered preinvex on invex set $\mathcal{C}$ pertaining to bifunction $𝑗$ if

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right) \forall \sigma ,\varsigma \in \mathcal{C}, \mathfrak{s}\in \left[0, 1\right],$$

(7)

where $𝑗:\mathcal{C}\times \mathcal{C}\to ℝ.$

Strictly fuzzy preinvex mapping if strict inequality holds for $\widetilde{{\rm Y}}\left(\sigma \right)\ne \widetilde{{\rm Y}}\left(\varsigma \right)$. $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered fuzzy preincave mapping if $-\widetilde{{\rm Y}}$ is preinvex on $\mathcal{C}.$ Strictly fuzzy preincave mapping if strict inequality holds for $\widetilde{{\rm Y}}\left(\sigma \right)\ne \widetilde{{\rm Y}}\left(\varsigma \right)$.

Definition 7.

[27] 𝘍𝘔 $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered quasi-preinvex on invex set $\mathcal{C}$ pertaining to $𝑗$ if

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯max\left(\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\varsigma \right)\right) \forall \sigma , \varsigma \in \mathcal{C}, \mathfrak{s}\in \left[0,1\right].$$

(8)

Definition 8.

[27] . A mapping $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ is considered fuzzy log-preinvex on invex set $\mathcal{C}$ pertaining to bifunction $𝑗$ if there exists a positive number $\omega$ such that

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯{\left(\widetilde{{\rm Y}}\left(\sigma \right)\right)}^{1-\mathfrak{s}}{\left(\widetilde{{\rm Y}}\left(\varsigma \right)\right)}^{\mathfrak{s}} \forall \sigma ,\varsigma \in \mathcal{C}, \mathfrak{s}\in \left[0, 1\right],$$

(9)

where $\widetilde{{\rm Y}}\left(.\right)\succ \tilde{0}.$

Definition 9.

[34]. The set ${\mathcal{C}}_{\alpha }$ is considered an $\alpha$-invex set pertaining to arbitrary bifunctions $𝑗\left(.,.\right)$ and $\alpha \left(.,.\right),$ if

$$\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\in {\mathcal{C}}_{\alpha }, \forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right],$$

(10)

where $𝑗:{\mathcal{C}}_{\alpha }\times {\mathcal{C}}_{\alpha }\to ℝ$ and $𝑗:{\mathcal{C}}_{\alpha }\times {\mathcal{C}}_{\alpha }\to ℝ\backslash 0$. The $\alpha$-invex set ${\mathcal{C}}_{\alpha }$ is also called the $\alpha 𝑗$-connected set. Note that the convex set with $\alpha \left(\varsigma ,\sigma \right)=1$ and $𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma$ is considered an invex set in the classical sense, but the converse is not valid. For example, in the following set ${\mathcal{C}}_{\alpha }={ℝ}^1-\left(\frac{-1}{2}, \frac{1}{2}\right)$,${\mathcal{C}}_{\alpha }$ is an invex set pertaining to nontrivial bifunction $𝑗:{\mathcal{C}}_{\alpha }⨯{\mathcal{C}}_{\alpha }\to {\mathcal{C}}_{\alpha }$ and $\alpha \left(\varsigma ,\sigma \right)=1,$ given as

$$𝑗\left(\varsigma ,\sigma \right)=\left\{\begin{array}{c}\varsigma -\sigma for \varsigma >0, \sigma >0 or \varsigma <0, \sigma <0\\ \sigma -\varsigma for \varsigma >0, \sigma <0 or \varsigma 〈0, \sigma 〉0.\end{array}\right.$$

Remark 2.

(i) 

If $\alpha \left(\varsigma ,\sigma \right)=1$, then the set ${\mathcal{C}}_{\alpha }$ is an invex set.

(ii) 

If $0<\alpha \left(\varsigma ,\sigma \right)<1$ and $𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma$, then the set ${\mathcal{C}}_{\alpha }$ is considered star-shaped.

(iii) 

If$\alpha \left(\varsigma ,\sigma \right)=1$ and $𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma ,$ then the set ${\mathcal{C}}_{\alpha }$ is considered convex.

3. Strongly $\alpha$-Preinvex Fuzzy Mappings

Let ${\mathcal{C}}_{\alpha }$ be a nonempty $\alpha$-invex subset of $ℝ$ pertaining to $𝑗, \alpha .$ Let $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ be continuous mapping and $𝑗:{\mathcal{C}}_{\alpha }\times {\mathcal{C}}_{\alpha }\to ℝ$ be an arbitrary continuous bifunction. Let $\alpha :{\mathcal{C}}_{\alpha }\times {\mathcal{C}}_{\alpha }\to ℝ\backslash 0$ be a bifunction. We denote ‖$.$‖ and 〈$., .$〉 as the norm and inner product, respectively.

Definition 10.

Let ${\mathcal{C}}_{\alpha }$ be an $\alpha$-invex set and $\omega$ be a positive number. Then 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ is considered strongly $\alpha$-preinvex pertaining to bifunctions $𝑗\left(.,.\right)$ and $\alpha \left(.,.\right)$ if

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2 \forall \sigma , \varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right],$$

(11)

strictly strongly $\alpha$-preinvex-𝘍𝘔 if strict inequality holds for $\widetilde{{\rm Y}}\left(\sigma \right)\ne \widetilde{{\rm Y}}\left(\varsigma \right)$ and $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ is considered strongly fuzzy preincave mapping if $-\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex on${\mathcal{C}}_{\alpha }.$ Strictly fuzzy preincave mapping if strict inequality holds for $\widetilde{{\rm Y}}\left(\sigma \right)\ne \widetilde{{\rm Y}}\left(\varsigma \right)$.

Remark 3.

Strongly α-preinvex-FMs have some very nice properties similar to fuzzy preinvex mapping,

(1) 

If $\widetilde{{\rm Y}}$is strongly$\alpha$-preinvex-𝘍𝘔, then ${\rm Y}\widetilde{{\rm Y}}$ is also strongly $\alpha$-preinvex for ${\rm Y}\ge 0$.

(2) 

If $\widetilde{{\rm Y}}$ and $\tilde{G}$ both are strongly $\alpha$-preinvex-𝘍𝘔s, then $max\left(\widetilde{{\rm Y}}\left(\sigma \right),\tilde{G}\left(\sigma \right)\right)$ is also strongly $\alpha$-preinvex-𝘍𝘔s.

Now we discuss some special cases of strongly $\alpha$-preinvex-𝘍𝘔s:

If $\omega =0,$ then strongly $\alpha$-preinvex-𝘍𝘔 becomes $\alpha$-preinvex-𝘍𝘔, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right), \forall \sigma , \varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

(12)

If $\alpha \left(\varsigma ,\sigma \right)=1,$ then strongly $\alpha$-preinvex-𝘍𝘔 becomes strongly preinvex-𝘍𝘔, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(13)

$$\forall \sigma , \varsigma \in {\mathrm{C}}_{\_\alpha }, s\in \left[0, 1\right]$$

If $\omega =0,$ then (13) reduces to

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right), \forall \sigma , \varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

(14)

The mapping $\widetilde{{\rm Y}}$ is considered preinvex-𝘍𝘔.

If $0<\alpha \left(\varsigma ,\sigma \right)<1 ,$ then strongly $\alpha$-preinvex-𝘍𝘔 is considered star-shaped strongly $\alpha$-preinvex-𝘍𝘔.

If $\alpha \left(\varsigma ,\sigma \right)=1$ and $𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma ,$ then strongly $\alpha$-preinvex-𝘍𝘔 becomes strongly convex-𝘍𝘔, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\left(\varsigma -\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\varsigma -{\sigma }^2,$$

(15)

$$\forall \sigma , \varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right]$$

If $\omega =0,$ then Equation (15) becomes

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\left(\varsigma -\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right), \forall \sigma , \varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

The mapping $\widetilde{{\rm Y}}$ is considered convex-𝘍𝘔.

If $0<\alpha \left(\varsigma ,\sigma \right)<1$and $\mathfrak{s}=\frac{1}{2}$, then Equation (11) becomes

$$\widetilde{{\rm Y}}\left(\frac{2\sigma +\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)}{2}\right)⪯\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\widetilde{{\rm Y}}\left(\varsigma \right)}{2}\widetilde{-}\omega \frac{1}{4}\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2 \forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

The mapping $\widetilde{{\rm Y}}$ is considered star-shaped J-strongly$\alpha$-preinvex-𝘍𝘔.

If $\mathfrak{s}=\frac{1}{2}$, then (11) becomes

$$\widetilde{{\rm Y}}\left(\frac{2\sigma +\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)}{2}\right)⪯\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\widetilde{{\rm Y}}\left(\varsigma \right)}{2}\widetilde{-}\omega \frac{1}{4}\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(16)

$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$

The 𝘍𝘔 $\widetilde{{\rm Y}}$ is considered J-strongly $\alpha$-preinvex. For $\alpha \left(\varsigma ,\sigma \right)=1$, Equation (16) reduces to

$$\widetilde{{\rm Y}}\left(\frac{2\sigma +𝑗\left(\varsigma ,\sigma \right)}{2}\right)⪯\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\widetilde{{\rm Y}}\left(\varsigma \right)}{2}\widetilde{-}\omega \frac{1}{4}\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2 ,$$

(17)

$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$. Then 𝘍𝘔 $\widetilde{{\rm Y}}$ is considered strongly J-preinvex. When $\omega =0$, then Equation (17) becomes

$$\widetilde{{\rm Y}}\left(\frac{2\sigma +𝑗\left(\varsigma ,\sigma \right)}{2}\right)⪯\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\widetilde{{\rm Y}}\left(\varsigma \right)}{2} \forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

Then 𝘍𝘔 $\widetilde{{\rm Y}}$ is considered J-preinvex.

We also define the affine strongly$\alpha$-preinvex mapping.

Definition 11.

A mapping$\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$is considered strongly affine$\alpha$-preinvex-𝘍𝘔 on$\alpha$-invex set${\mathcal{C}}_{\alpha }$pertaining to bifunction$𝑗, \alpha$if there exists a positive number$\omega$such that

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2 \forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

(18)

If $\mathfrak{s}=\frac{1}{2}$, then we also say that$\widetilde{{\rm Y}}$is strongly affine J-$\alpha$-preinvex-𝘍𝘔such that

$$\widetilde{{\rm Y}}\left(\frac{2\sigma +\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)}{2}\right)=\frac{\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\widetilde{{\rm Y}}\left(\varsigma \right)}{2}\widetilde{-}\frac{1}{4}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2 \forall \sigma ,\varsigma \in \mathcal{C}.$$

(19)

Theorem 1.

Let ${\mathcal{C}}_{\alpha }$ be an $\alpha$-invex set pertaining to $𝑗$ and let $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ be a 𝘍𝘔 parametrized by

$$\widetilde{{\rm Y}}\left(\sigma \right)=\left\{\left({\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right),{\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}, \forall \sigma \in {\mathcal{C}}_{\alpha }.$$

(20)

Then $\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex on ${\mathcal{C}}_{\alpha }$ with modulus $\omega$ if, and only if, for all $\mathit{ĩ}\in \left[0, 1\right],$${\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right)$ and ${\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right)$ are strongly $\alpha$-preinvex pertaining to $𝑗, \alpha ,$ and modulus $\omega .$

Proof. 

Assume that for each $\mathit{ĩ}\in \left[0, 1\right],$ ${\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right)$ and ${\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right)$ are strongly $\alpha$-preinvex pertaining to $𝑗, \alpha$ and modulus $\omega$ on ${\mathcal{C}}_{\alpha }.$ Then from Equation (21), we have

$${\widetilde{{\rm Y}}}_{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right)\le \left(1-\mathfrak{s}\right){\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right)+\mathfrak{s}{\widetilde{{\rm Y}}}_{*}\left(\varsigma , \mathit{ĩ}\right)-\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(21)

$$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right],$$

and

$${\widetilde{{\rm Y}}}^{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right)\le \left(1-\mathfrak{s}\right){\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right)+\mathfrak{s}{\widetilde{{\rm Y}}}^{*}\left(\varsigma , \mathit{ĩ}\right)-\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

Then by Equations (21), (1), and (2), we obtain

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=\left\{\left({\widetilde{{\rm Y}}}_{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right),{\widetilde{{\rm Y}}}^{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\},$$

$$⪯\left\{\left(\left(1-\mathfrak{s}\right){\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right),\left(1-\mathfrak{s}\right){\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}\widetilde{+}\left\{\left((\mathfrak{s}{\widetilde{{\rm Y}}}_{*}\left(\varsigma , \mathit{ĩ}\right),\mathfrak{s}{\widetilde{{\rm Y}}}^{*}\left(\varsigma , \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$=\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $\mathfrak{s}\in \left[0, 1\right]$. Hence, $\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex-𝘍𝘔 on ${\mathcal{C}}_{\alpha }$ with modulus $\omega .$

Conversely, let $\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex-𝘍𝘔 on ${\mathcal{C}}_{\alpha }$ with modulus $\omega .$ Then for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $\mathfrak{s}\in \left[0, 1\right],$ we have $\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$ From Equation (20), we have

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=\left\{\left({\widetilde{{\rm Y}}}_{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right),{\widetilde{{\rm Y}}}^{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}$$

$$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

From Equation (20) and Equations (1)–(3), we obtain

$$\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2=\left\{\left(\mathfrak{s}{\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right),\mathfrak{s}{\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}\widetilde{+}\left\{\left(\mathfrak{s}{\widetilde{{\rm Y}}}_{*}\left(\varsigma , \mathit{ĩ}\right), \mathfrak{s}{\widetilde{{\rm Y}}}^{*}\left(\varsigma , \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $\mathfrak{s}\in \left[0, 1\right].$ Then by strongly $\alpha$-preinvexity of $\widetilde{{\rm Y}}$, we have for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $\mathfrak{s}\in \left[0, 1\right]$such that

$${\widetilde{{\rm Y}}}_{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right)\le \left(1-\mathfrak{s}\right){\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right)+\mathfrak{s}{\widetilde{{\rm Y}}}_{*}\left(\varsigma , \mathit{ĩ}\right)-\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

And

$${\widetilde{{\rm Y}}}^{*}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right), \mathit{ĩ}\right)\le \left(1-\mathfrak{s}\right){\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right)+\mathfrak{s}{\widetilde{{\rm Y}}}^{*}\left(\varsigma , \mathit{ĩ}\right)-\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

for each $\mathit{ĩ}\in \left[0, 1\right].$ Hence, the result follows. □

Example 1.

We consider the𝘍𝘔s$\widetilde{{\rm Y}}:\left(0, \infty \right)\to {\mathbb{E}}_0$defined by,

$$\widetilde{{\rm Y}}\left(\sigma \right)\left(\sigma \right)=\left\{\begin{array}{c}\frac{\sigma }{{\sigma }^2}, \sigma \in \left[0, {\sigma }^2\right]\\ \frac{2{\sigma }^2-\sigma }{{\sigma }^2}, \sigma \in \left({\sigma }^2, 2{\sigma }^2\right] \\ 0 , otherwise\end{array}\right.,$$

Then, for each$\mathit{ĩ}\in \left[0, 1\right],$we have${\widetilde{{\rm Y}}}_{\mathit{ĩ}}\left(\sigma \right)=\left[\mathit{ĩ}{\sigma }^2,\left(2-\mathit{ĩ}\right){\sigma }^2\right]$. Since endpoint functions${\widetilde{{\rm Y}}}_{*}\left(\mathit{ĩ}\right),$ ${\widetilde{{\rm Y}}}^{*}\left(\mathit{ĩ}\right)$are strongly generalized preinvex for each$\mathit{ĩ}\in \left[0,1\right]$, then$\widetilde{{\rm Y}}$is strongly generalized preinvex-𝘍𝘔pertaining to

$$𝑗\left(\varsigma ,\sigma \right)=\varsigma -\sigma \mathrm{and} \alpha \left(\varsigma ,\sigma \right)=1,$$

and $\omega >0 .$

We now establish a result for strongly $\alpha$-preinvex-𝘍𝘔, which shows that the difference of strongly $\alpha$-preinvex-𝘍𝘔 and strong affine $\alpha$-preinvex-𝘍𝘔 is again a strongly $\alpha$-preinvex-𝘍𝘔.

Theorem 2.

Let𝘍𝘔 $\tilde{g}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$be a strongly affine$\alpha$-preinvex pertaining to$𝑗, \alpha$and$\omega >0$. Then$\widetilde{{\rm Y}}$is strongly$\alpha$-preinvex-𝘍𝘔 pertaining to the same$𝑗, \alpha$if, and only if,$\tilde{G}=\widetilde{{\rm Y}}\widetilde{-}\tilde{g}$is$\alpha$-preinvex-𝘍𝘔 pertaining to$𝑗, \alpha$.

Proof. 

The “If” part is obvious. To prove the “only if”, assume that $\tilde{g}:\mathcal{C}\to {\mathbb{E}}_0$ is a strongly affine $\alpha$-preinvex-𝘍𝘔 pertaining to $𝑗, \alpha$ and $\omega >0$. Then

$$\tilde{g}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=\left(1-\mathfrak{s}\right)\tilde{g}\left(\sigma \right)\widetilde{+}\mathfrak{s}\tilde{g}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2$$

(22)

Since $\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex-𝘍𝘔 pertaining to the same $𝑗, \alpha$, then

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(23)

From Equations (22) and (23), we have

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\tilde{g}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\left(1-\mathfrak{s}\right)\tilde{g}\left(\sigma \right)\widetilde{-}\mathfrak{s}\tilde{g}\left(\varsigma \right),$$

$$=\left(1-\mathfrak{s}\right)\left(\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\tilde{g}\left(\sigma \right)\right)\widetilde{+}\mathfrak{s}\left(\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\tilde{g}\left(\varsigma \right)\right),$$

from which it follows that

$$\tilde{G}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\tilde{g}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),$$

$$⪯\left(1-\mathfrak{s}\right)\left(\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\tilde{g}\left(\sigma \right)\right)\widetilde{+}\mathfrak{s}\left(\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\tilde{g}\left(\varsigma \right)\right),$$

$$=\left(1-\mathfrak{s}\right)\tilde{G}\left(\sigma \right)\widetilde{+}\mathfrak{s}\tilde{G}\left(\sigma \right),$$

Showing that $\tilde{G}=\widetilde{{\rm Y}}\widetilde{-}\tilde{g}$is strongly $\alpha$-preinvex-𝘍𝘔. □

Definition 12.

A𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$is considered strongly quasi$\alpha$-preinvex on$\alpha$-invex set${\mathcal{C}}_{\alpha }$pertaining to$𝑗, \alpha$if there exists a positive number$\omega$such that

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\max \left(\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\varsigma \right)\right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(24)

$$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

Similarly, A 𝘍𝘔 $\widetilde{{\rm Y}}$ is considered strongly quasi-preincave if $-\widetilde{{\rm Y}}$ is strongly quasi $\alpha$-preinvex on ${\mathcal{C}}_{\alpha }.$

If$\alpha \left(\varsigma ,\sigma \right)=1$, then we get the definition of strong quasi-preinvex-𝘍𝘔, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯\max \left(\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\varsigma \right)\right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(25)

$$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

If $\alpha \left(\varsigma ,\sigma \right)=1$ and $\omega =0$, then we get the definition of quasi-preinvex-𝘍𝘔 in the classical sense, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯\max \left(\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\varsigma \right)\right),\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right].$$

(26)

Theorem 3.

Let𝘍𝘔 $\widetilde{{\rm Y}}:\mathcal{C}\to {\mathbb{E}}_0$ be a strongly$\alpha$-preinvex and$\omega >0,$such that$\widetilde{{\rm Y}}\left(\varsigma \right)\prec \widetilde{{\rm Y}}\left(\sigma \right).$Then$\widetilde{{\rm Y}}$is strictly strongly quasi$\alpha$-preinvex-𝘍𝘔.

Proof. 

Let $\widetilde{{\rm Y}}\left(\varsigma \right)\prec \widetilde{{\rm Y}}\left(\sigma \right)$ and $\widetilde{{\rm Y}}$ be strongly $\alpha$-preinvex-𝘍𝘔. Then, for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $\mathfrak{s}\in \left[0, 1\right]$ we have

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

since $\widetilde{{\rm Y}}\left(\varsigma \right)\prec \widetilde{{\rm Y}}\left(\sigma \right),$ we have

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\prec \widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

Hence, $\widetilde{{\rm Y}}$ is strictly strongly quasi $\alpha$-preinvex-𝘍𝘔 with $\omega >0.$ □

Theorem 4.

Let ${\mathcal{C}}_{\alpha }$ be a $\alpha$-invex set pertaining to $𝑗, \alpha$ and let $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ be a 𝘍𝘔 parametrized by

$$\widetilde{{\rm Y}}\left(\sigma \right)=\left\{\left({\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right),{\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right),\mathit{ĩ}\right):\mathit{ĩ}\in \left[0, 1\right]\right\}, \forall \sigma \in {\mathcal{C}}_{\alpha }.$$

Then$\widetilde{{\rm Y}}$ is strongly quasi $\alpha$-preinvex on ${\mathcal{C}}_{\alpha }$ with modulus $\omega$ if, and only if, for all $\mathit{ĩ}\in \left[0, 1\right],$

${\widetilde{{\rm Y}}}_{*}\left(\sigma , \mathit{ĩ}\right)$ and ${\widetilde{{\rm Y}}}^{*}\left(\sigma , \mathit{ĩ}\right)$ are strongly quasi $\alpha$-preinvex pertaining to $𝑗$ and modulus $\omega .$

Proof. 

The demonstration is analogous to the demonstration of Theorem 1. □

Definition 13.

A𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$is considered strongly log$\alpha$-preinvex on$\alpha$-invex set${\mathcal{C}}_{\alpha }$pertaining to bifunctions$𝑗, \alpha ,$if there exists a positive number$\omega$such that

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯{\left(\widetilde{{\rm Y}}\left(\sigma \right)\right)}^{1-\mathfrak{s}}{\left(\widetilde{{\rm Y}}\left(\varsigma \right)\right)}^{\mathfrak{s}}\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2 ,$$

(27)

$$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right], \mathrm{where} \widetilde{{\rm Y}}\left(.\right)\succ \tilde{0}.$$

Similarly, A 𝘍𝘔 $\widetilde{{\rm Y}}$ is considered strongly log $\alpha$-preincave if $-\widetilde{{\rm Y}}$ is strongly log $\alpha$-preinvex on ${\mathcal{C}}_{\alpha }.$

If$\alpha \left(\varsigma ,\sigma \right)=1$, then we get the definition of log preinvex-𝘍𝘔, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)⪯{\left(\widetilde{{\rm Y}}\left(\sigma \right)\right)}^{1-\mathfrak{s}}{\left(\widetilde{{\rm Y}}\left(\varsigma \right)\right)}^{\mathfrak{s}}\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

(28)

$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right]$, where $\widetilde{{\rm Y}}\left(.\right)\succ \tilde{0}$. The mapping $\widetilde{{\rm Y}}$ is strongly log-preinvex-𝘍𝘔 pertaining to $, \alpha ,$.

If $\alpha \left(\varsigma ,\sigma \right)=1$ and $\omega =0$, then we get the definition of log-preinvex-𝘍𝘔 in the classical sense, that is

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯{\left(\widetilde{{\rm Y}}\left(\sigma \right)\right)}^{1-\mathfrak{s}}{\left(\widetilde{{\rm Y}}\left(\varsigma \right)\right)}^{\mathfrak{s}}.$$

$\forall \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }, \mathfrak{s}\in \left[0, 1\right]$, where $\widetilde{{\rm Y}}\left(.\right)\succ \tilde{0}.$ The mapping $\widetilde{{\rm Y}}$ is log-preinvex-𝘍𝘔 pertaining to $𝑗, \alpha$.

From Definition 13, we have

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯{\left(\widetilde{{\rm Y}}\left(\sigma \right)\right)}^{1-\mathfrak{s}}{\left(\widetilde{{\rm Y}}\left(\varsigma \right)\right)}^{\mathfrak{s}}\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$⪯\max \left(\widetilde{{\rm Y}}\left(\sigma \right), \widetilde{{\rm Y}}\left(\varsigma \right)\right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

It can easily be seen that strongly log $\alpha$-preinvex-𝘍𝘔$\Rightarrow$ strongly $\alpha$-preinvex-𝘍𝘔 $\Rightarrow$ strongly quasi $\alpha$-preinvex-𝘍𝘔.

For $\mathfrak{s}=1$, Definition 10 and Definition 13, reduces to:

Condition A.

$$\widetilde{{\rm Y}}\left(\sigma +\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\widetilde{{\rm Y}}\left(\varsigma \right) \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

Which plays an important role in studying the properties of $\alpha$-preinvex-𝘍𝘔s and 𝛼-invex-𝘍𝘔s. If $\alpha \left(\varsigma ,\sigma \right)=1$, then Condition A reduces to the following for preinvex-𝘍𝘔s.

Condition B.

$$\widetilde{{\rm Y}}\left(\sigma +𝑗\left(\varsigma ,\sigma \right)\right)⪯\widetilde{{\rm Y}}\left(\varsigma \right) \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

For the applications of Condition B, see [28,30,34,62].

Definition 14.

A𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$is considered pseudo$\alpha$-preinvex on the$\alpha$-invex set${\mathcal{C}}_{\alpha }$if there exists a strictly positive bifunction$b\left(.,.\right)$such that

$$\widetilde{{\rm Y}}\left(\varsigma \right)\prec \widetilde{{\rm Y}}\left(\sigma \right)⟹\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\prec \widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\left(\mathfrak{s}-1\right)b\left(\sigma ,\varsigma \right), \mathrm{for} \mathrm{all} ,\varsigma \in {\mathcal{C}}_{\alpha },t\in \left[1, 0\right].$$

Theorem 5.

Let$\widetilde{{\rm Y}}$be a strongly$\alpha$-preinvex-𝘍𝘔 on${\mathcal{C}}_{\alpha }$such that$\widetilde{{\rm Y}}\left(\varsigma \right)\prec \widetilde{{\rm Y}}\left(\sigma \right).$Then$\widetilde{{\rm Y}}$is strongly pseudo$\alpha$-preinvex-𝘍𝘔.

Proof. 

Let $\widetilde{{\rm Y}}\left(\varsigma \right)\prec \widetilde{{\rm Y}}\left(\sigma \right)$ and $\widetilde{{\rm Y}}$ be a strongly $\alpha$-preinvex-𝘍𝘔. Then

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2, \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha },t\in \left[1,0\right]$$

$$=\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\left(\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)\right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$\prec \widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\left(\mathfrak{s}-1\right)\left(\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\varsigma \right)\right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$=\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\left(\mathfrak{s}-1\right)b\left(\sigma ,\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

where $b\left(\sigma ,\varsigma \right)=\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\varsigma \right)$. This prove that $\widetilde{{\rm Y}}$ is strongly pseudo $\alpha$-preinvex-𝘍𝘔. □

4. G-Differentiable Strongly $\alpha$-Preinvex Fuzzy Mappings

In this section, we propose the concepts of strongly $\alpha 𝑗$-invex-𝘍𝘔s and strongly $\alpha 𝑗$-monotone fuzzy operators. With the help of these ideas, different properties of strongly $\alpha$-preinvex-𝘍𝘔s are characterized. At the end, it is proved that the minimum of strongly $\alpha$-preinvex-𝘍𝘔s can be distinguished by strongly fuzzy $\alpha$-variational-like inequalities.

Definition 15.

The G-differentiable 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ on $\alpha$-invex set ${\mathcal{C}}_{\alpha }$ is considered strongly $\alpha$-invex pertaining to $𝑗, \alpha$ if there exists a constant $\omega >0$ such that

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪯⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(29)

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha },$where ${\widetilde{{\rm Y}}}^{,},$ is G-differentiable of $\widetilde{{\rm Y}}$ at $\sigma .$

From Definition 15, it enables us to define the following new definitions:

Definition 16.

The G-differentiable 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ on ${\mathcal{C}}_{\alpha }$ is considered $\alpha$-invex pertaining to $𝑗, \alpha ,$ if there exists a constant $\omega >0$ such that

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉,$$

(30)

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha },$ where ${\widetilde{{\rm Y}}}^{,},$ is G-differentiable of $\widetilde{{\rm Y}}$ at $\sigma .$

Definition 17.

The G-differentiable 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ on ${\mathcal{C}}_{\alpha }$ is considered strongly pseudo $\alpha$-invex if there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2⪰\tilde{0}⟹\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\tilde{0},$$

(31)

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$

Definition 18.

The G-differentiable 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ on ${\mathcal{C}}_{\alpha }$ is considered pseudo $\alpha 𝑗$-invex if there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪰\tilde{0}⟹\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\tilde{0},$$

(32)

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$

Definition 19.

The G-differentiable 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ on ${\mathcal{C}}_{\alpha }$ is considered strongly quasi $\alpha 𝑗$-invex if there exists a constant $\omega >0$ such that

$$\widetilde{{\rm Y}}\left(\varsigma \right)⪯\widetilde{{\rm Y}}\left(\sigma \right)⟹⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2⪯\tilde{0},$$

(33)

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$

Definition 20.

The G-differentiable 𝘍𝘔 $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ on ${\mathcal{C}}_{\alpha }$ is considered quasi $\alpha 𝑗$-invex if there exists a constant $\omega >0$ such that

$$\widetilde{{\rm Y}}\left(\varsigma \right)⪯\widetilde{{\rm Y}}\left(\sigma \right)⟹⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪯\tilde{0},$$

(34)

for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$.

If $𝑗\left(\varsigma ,\sigma \right)=-𝑗\left(\sigma ,\varsigma \right),$ 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 $𝑗$,$\alpha$, which play an important role in the G-differentiation of the main results.

Condition C

[34]

$$𝑗\left(\varsigma ,\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=\left(1-\mathfrak{s}\right)𝑗\left(\varsigma ,\sigma \right),$$

$$𝑗\left(\sigma ,\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)=-\mathfrak{s}𝑗\left(\varsigma ,\sigma \right), \mathrm{for} \mathrm{all} \sigma , \varsigma \in {\mathcal{C}}_{\alpha },\mathfrak{s}\in \left[0, 1\right].$$

Clearly for $\mathfrak{s}=0,$ we have $𝑗\left(\sigma ,\sigma \right)=0$ for all $\sigma \in {\mathcal{C}}_{\alpha }.$

It is well known that each G-differentiable preinvex-𝘍𝘔 is invex-𝘍𝘔, but to prove its converse, we need a special condition.

It can easily be seen that if $\alpha \left(\varsigma ,\sigma \right)=1$, then Condition C collapses to the following condition:

Condition D

[27]

$$𝑗\left(\varsigma ,\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)=\left(1-\mathfrak{s}\right)𝑗\left(\varsigma ,\sigma \right),$$

$$𝑗\left(\sigma ,\sigma +\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)\right)=-\mathfrak{s}𝑗\left(\varsigma ,\sigma \right), \mathrm{for} \mathrm{all} \sigma , \varsigma \in {\mathcal{C}}_{\alpha },\mathfrak{s}\in \left[0, 1\right].$$

For the applications of Condition D, see [29,30,31,77,78].

Theorem 6.

Let$\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ be a G-differentiable strongly $\alpha$-preinvex-𝘍𝘔 pertaining to $𝑗, \alpha$. Let Condition C hold and $\alpha \left(\sigma ,w\right)=\alpha \left(\varsigma ,w\right)$ for all $\sigma , \varsigma , w\in {\mathcal{C}}_{\alpha }$. Then $\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex-𝘍𝘔 when, and only when, $\widetilde{{\rm Y}}$ is strongly α-invex-𝘍𝘔.

Proof. 

Let $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ be G-differentiable $\alpha$-preinvex-𝘍𝘔. Since $\widetilde{{\rm Y}}$ is $\alpha$-preinvex, then for each $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $t\in \left[1,0\right]$, we have

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$=\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\left(\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)\right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

which implies that

$$\mathfrak{s}\left(\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)\right)⪰\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\frac{\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{\mathfrak{s}}\widetilde{+}\omega \left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

taking the limit in the above inequality as $\mathfrak{s}\to 0$, we have

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

Conversely, let $\widetilde{{\rm Y}}$ be a 𝛼-invex-𝘍𝘔. Since ${\mathcal{C}}_{\alpha }$ is an $\alpha$-invex set, we have, ${\varsigma }_{\mathfrak{s}}=\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\in {\mathcal{C}}_{\alpha }$ for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $t\in \left[1,0\right]$. Taking $\varsigma$=${\varsigma }_{\mathfrak{s}}$ in (29), we get

$$\widetilde{{\rm Y}}\left({\varsigma }_{\mathfrak{s}}\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left({\varsigma }_{\mathfrak{s}},\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left({\varsigma }_{\mathfrak{s}},\sigma \right)〉\widetilde{+}\omega 𝑗{\left({\varsigma }_{\mathfrak{s}},\sigma \right)}^2,$$

using Condition C, we have

$$\widetilde{{\rm Y}}\left({\varsigma }_{\mathfrak{s}}\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\alpha \left({\varsigma }_{\mathfrak{s}},\sigma \right)⟨{\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left({\varsigma }_{\mathfrak{s}},\sigma \right)〉\widetilde{+}\omega {\left(1-\mathfrak{s}\right)}^2\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

(35)

In a similar way, we have

$$\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left({\varsigma }_{\mathfrak{s}}\right)⪰⟨ \alpha \left(\sigma ,{\varsigma }_{\mathfrak{s}}\right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\sigma ,{\varsigma }_{\mathfrak{s}}\right)〉$$

$$=-\mathfrak{s}\alpha \left(\varsigma ,{\varsigma }_{\mathfrak{s}}\right)⟨ {\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\varsigma ,\sigma \right)\widetilde{+}\omega {\mathfrak{s}}^2\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2〉$$

(36)

Multiplying (35) by $\mathfrak{s}$ and (36) by $\left(1-\mathfrak{s}\right)$, and adding the resultant, we have

$$\widetilde{{\rm Y}}\left({\varsigma }_{\mathfrak{s}}\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

which implies that

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2, \mathrm{for} \mathrm{all} \sigma , \varsigma \in {\mathcal{C}}_{\alpha }.$$

Hence, $\widetilde{{\rm Y}}$ is strongly $\alpha$-preinvex-𝘍𝘔 pertaining to $𝑗, \alpha$. □

Theorem 7.

Let $\widetilde{{\rm Y}}$ be a G-differentiable𝘍𝘔 on ${\mathcal{C}}_{\alpha }$. If the 𝘍𝘔 $\widetilde{{\rm Y}}$ is strongly 𝛼-invex, then

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪯\widetilde{-}\omega \left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right),$$

(37)

$$\mathrm{for} \mathrm{all} \sigma , \varsigma \in {\mathcal{C}}_{\alpha }.$$

Conversely, if Conditions A and C hold, and $\alpha \left(\varsigma ,\sigma \right)=\alpha \left(\sigma ,\varsigma \right)$ for all $\sigma , \varsigma \in {\mathcal{C}}_{\alpha }$, then $\widetilde{{\rm Y}}$ is strongly 𝛼-invex-𝘍𝘔 pertaining to $𝑗, \alpha$.

Proof. 

Let $\widetilde{{\rm Y}}$ is strongly 𝛼-invex-𝘍𝘔. Then,

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2, \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha },$$

(38)

replacing $\varsigma$ by $\sigma$ and $\sigma$ by $\varsigma$ in Equation (38), we get

$$\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\varsigma \right)⪰⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2, \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

(39)

Adding Equations (38) and (39), we have

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪯\widetilde{-}\omega \left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right), \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

Conversely, assume that (37) holds. Then

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪯\widetilde{-}⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉\widetilde{-}\omega \left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right), \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha },$$

which implies that

$$⟨{\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪯\widetilde{-}⟨{\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉\widetilde{-}\tilde{\omega }\left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right),$$

(40)

since $\alpha \left(\varsigma ,\sigma \right)=\alpha \left(\sigma ,\varsigma \right),$ where $\tilde{\omega }=\frac{\omega }{\alpha \left(\varsigma ,\sigma \right)}$.

Since ${\mathcal{C}}_{\alpha }$ is an $\alpha$-invex set, we have, ${\varsigma }_{\mathfrak{s}}=\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\in {\mathcal{C}}_{\alpha }$ for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $t\in \left[1,0\right]$. Taking $\varsigma$=${\varsigma }_{\mathfrak{s}}$ in (40), we get

$$⟨{\widetilde{{\rm Y}}}^{,}\left({\varsigma }_{\mathfrak{s}}\right), 𝑗\left(\sigma , {\varsigma }_{\mathfrak{s}}\right)〉⪯\widetilde{-}⟨{\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left({\varsigma }_{\mathfrak{s}},\sigma \right)〉\widetilde{-}\tilde{\omega }\left(𝑗{\left({\varsigma }_{\mathfrak{s}},\sigma \right)}^2+𝑗{\left(\sigma ,{\varsigma }_{\mathfrak{s}}\right)}^2\right),$$

by using Condition C, we have

$$⟨{\widetilde{{\rm Y}}}^{,}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)〉⪰⟨{\widetilde{{\rm Y}}}^{,}\left(\sigma \right),\mathfrak{s}𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}2\tilde{\omega }{\mathfrak{s}}^2\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(41)

Let

$$\tilde{h}\left(\mathfrak{s}\right)=\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),$$

taking G-derivative pertaining to $\mathfrak{s}$, we get

$${\tilde{h}}^{,}\left(\mathfrak{s}\right)=⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),𝑗\left(\varsigma ,\sigma \right)〉,$$

from which, using Equation (41), we have

$${\tilde{h}}^{,}\left(\mathfrak{s}\right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}2\tilde{\omega }\alpha \left(\varsigma ,\sigma \right)\mathfrak{s}\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

(42)

$$=⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}2\omega \mathfrak{s}\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

Integrating Equation (42) between 0 to1 pertaining to $\mathfrak{s}$, we get

$$\tilde{h}\left(1\right)\widetilde{-}\tilde{h}\left(0\right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

which implies that

$$\widetilde{{\rm Y}}\left(\sigma +\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

by using Condition, A

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right),𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2, \mathrm{for} \mathrm{all} \sigma , \varsigma \in {\mathcal{C}}_{\alpha }.$$

Showing that $\widetilde{{\rm Y}}$ is $\alpha$-invex-𝘍𝘔 on ${\mathcal{C}}_{\alpha }.$ □

With Theorem 6 and Theorem 7, we have the following new definitions.

Definition 21.

A G-differentiable mapping $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ is considered:

(i) 

Strongly $\alpha 𝑗$-monotone fuzzy operator when, and only when, there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)⪯\widetilde{-}\omega \left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right),$$

$for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }$.

(ii) 

$\alpha 𝑗$-monotone fuzzy operator when, and only when, we have

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}⟨{\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪯\tilde{0} for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }\mathbf{.}$$

(iii) 

Strongly $\alpha 𝑗$-pseudomonotone fuzzy operator when, and only when, there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right)⪰\tilde{0}⟹ \widetilde{-}\alpha \left(\sigma ,\varsigma \right)⟨{\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪰\tilde{0},$$

$for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }$.

(iv) 

Strongly relaxed $\alpha 𝑗$-pseudomonotone fuzzy operator when, and only when, there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪰\tilde{0}⟹\widetilde{-}⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉\widetilde{-}\omega \left(\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2+\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2\right)⪰\tilde{0},$$

$for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }$

(v) 

Strictly $\alpha 𝑗$-monotone fuzzy operator when, and only when, we have

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉\prec \tilde{0}, for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }.$$

(vi) 

$\alpha 𝑗$-pseudomonotone fuzzy operator when, and only when, there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪰\tilde{0}⟹ ⟨ \alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪯\tilde{0}, for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }\mathbf{.}$$

(vii) 

Quasi $\alpha 𝑗$-pseudomonotone fuzzy operator when, and only when, there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉>\tilde{0}⟹ ⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝒿\left(\sigma ,\varsigma \right)〉⪯\tilde{0}, for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }.$$

(viii) 

Strictly $\alpha 𝑗$-pseudomonotone fuzzy operator when, and only when, there exists a constant $\omega >0$ such that

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪰\tilde{0}⟹ ⟨ \alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉\prec \tilde{0}, for all \sigma , \varsigma \in {\mathcal{C}}_{\alpha }.$$

If $𝑗\left(\varsigma ,\sigma \right)=-𝑗\left(\sigma ,\varsigma \right),$ then Definition 21, reduce to new one.

As special case of Theorem 7, we have the following:

Corollary 1.

Let $\widetilde{{\rm Y}}$ be G-differentiable 𝘍𝘔 on${\mathcal{C}}_{\alpha }$ and let Condition C hold. Then $\widetilde{{\rm Y}}$ is strongly $\alpha$-invex-𝘍𝘔 if, and only if, ${\widetilde{{\rm Y}}}^{,}$ is strongly $\alpha 𝑗$-monotone fuzzy operator.

Theorem 8.

Let G-differential ${\widetilde{{\rm Y}}}^{,}$ of mapping $\widetilde{{\rm Y}}$ on ${\mathcal{C}}_{\alpha }$ be strongly $\alpha 𝑗$-pseudomonotone fuzzy operator, and let Conditions A and C hold. Then $\widetilde{{\rm Y}}$ is a strongly pseudo $\alpha 𝑗$-invex-𝘍𝘔.

Proof. 

Let ${\widetilde{{\rm Y}}}^{,}$ be a strongly $\alpha 𝑗$-pseudomonotone fuzzy operator. Then for all $\sigma , \varsigma \in \mathcal{C},$ we have

$$⟨\alpha \left(\varsigma , \sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉⪰\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

Then

$$\widetilde{-}⟨\alpha \left(\sigma ,\varsigma \right){\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪰\tilde{0},$$

which implies that

$$\widetilde{-}⟨{\widetilde{{\rm Y}}}^{,}\left(\varsigma \right), 𝑗\left(\sigma ,\varsigma \right)〉⪰\tilde{0}$$

(43)

since ${\mathcal{C}}_{\alpha }$ is an $\alpha$-invex set so we have, ${\varsigma }_{\mathfrak{s}}=\sigma +\mathfrak{s}\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\in {\mathcal{C}}_{\alpha }$ for all $\sigma ,\varsigma \in {\mathcal{C}}_{\alpha }$ and $t\in \left[1,0\right]$. Taking $\varsigma$=${\varsigma }_{\mathfrak{s}}$ in (4.15), we get

$$⟨\widetilde{-}{\widetilde{{\rm Y}}}^{,}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\right), 𝑗\left(\sigma , \sigma +\mathfrak{s}\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)〉⪰\tilde{0},$$

by using Condition C, we have

$$⟨{\widetilde{{\rm Y}}}^{,}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),𝑗\left(\varsigma ,\sigma \right)〉⪰\tilde{0},$$

(44)

assume that

$$\tilde{H}\left(\mathfrak{s}\right)=\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),$$

Taking G-derivative pertaining to $\mathfrak{s}$, then using (44), we have

$${\tilde{H}}^{,}\left(\mathfrak{s}\right)=\alpha \left(\varsigma , \sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\right),𝑗\left(\varsigma ,\sigma \right)⪰\tilde{0},$$

(45)

Integrating Equation (44) between 0 and 1 pertaining to $\mathfrak{s}$, we get

$$\tilde{H}\left(1\right)\widetilde{-}\tilde{H}\left(0\right)⪰\tilde{0},$$

$$\widetilde{{\rm Y}}\left(\sigma +\alpha \left(\varsigma , \sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\tilde{0},$$

By using Condition A, we have

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\tilde{0}, \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

Hence, $\widetilde{{\rm Y}}$ is a strongly pseudo $\alpha 𝑗$-invex-𝘍𝘔. □

Following results are special cases of Theorem 8, we have:

Corollary 2.

Let G-differential ${\widetilde{{\rm Y}}}^{,}$ of 𝘍𝘔 $\widetilde{{\rm Y}}$ on ${\mathcal{C}}_{\alpha }$ be $\alpha 𝑗$-pseudomonotone fuzzy operator. Then, $\widetilde{{\rm Y}}$ is a pseudo $\alpha 𝑗$-invex-𝘍𝘔.

Corollary 3.

Let G-differential ${\widetilde{{\rm Y}}}^{,}$ of 𝘍𝘔 $\widetilde{{\rm Y}}$ on ${\mathcal{C}}_{\alpha }$ be strongly quasi $\alpha 𝑗$-pseudomonotone fuzzy operator. Then $\widetilde{{\rm Y}}$ is a strongly quasi $\alpha 𝑗$-invex-𝘍𝘔.

Corollary 4.

Let G-differential ${\widetilde{{\rm Y}}}^{,}$ of 𝘍𝘔 $\widetilde{{\rm Y}}$ on ${\mathcal{C}}_{\alpha }$ be quasi $\alpha 𝑗$-pseudomonotone fuzzy operator, and let Condition C hold. Then $\widetilde{{\rm Y}}$ is a pseudo $\alpha 𝑗$-invex-𝘍𝘔.

We now discuss the fuzzy optimality condition for G-differentiable strongly $\alpha$-preinvex-𝘍𝘔s, which is main motivation of our next result.

Theorem 9.

Let $\widetilde{{\rm Y}}$ be a G-differentiable strongly $\alpha$-preinvex-𝘍𝘔 modulus $\omega >0$. If $\sigma \in {\mathcal{C}}_{\alpha }$ is the minimum of the $\widetilde{{\rm Y}}$, then

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2, \mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha }.$$

(46)

Proof. 

Let $\sigma \in {\mathcal{C}}_{\alpha }$ be a minimum of $\widetilde{{\rm Y}}$. Then

$$\widetilde{{\rm Y}}\left(\sigma \right)⪯\widetilde{{\rm Y}}\left(\varsigma \right)$$

(47)

for all $\varsigma \in {\mathcal{C}}_{\alpha }$.

Since ${\mathcal{C}}_{\alpha }$ is an $\alpha$-invex set, for all $\sigma , \varsigma \in {\mathcal{C}}_{\alpha }$, $\mathfrak{s}\in \left[0,1\right]$,

$${\varsigma }_{\mathfrak{s}}=\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\in {\mathcal{C}}_{\alpha }$$

Taking $\varsigma ={\varsigma }_{\mathfrak{s}}$ in (47), we get

$$\tilde{0}⪯\frac{\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{\mathfrak{s}},$$

taking limit in the above inequality as $\mathfrak{s}\to \infty$, we get

$$\tilde{0}⪯⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉,$$

(48)

since $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ is a G-differentiable strongly $\alpha$-preinvex-𝘍𝘔, so

$$\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)⪯\left(1-\mathfrak{s}\right)\widetilde{{\rm Y}}\left(\sigma \right)\widetilde{+}\mathfrak{s}\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\omega \mathfrak{s}\left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\frac{\widetilde{{\rm Y}}\left(\sigma +\mathfrak{s}\alpha \left(\varsigma ,\sigma \right)𝑗\left(\varsigma ,\sigma \right)\right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)}{\mathfrak{s}}\widetilde{+}\omega \left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

Again, taking limit in the above inequality as $\mathfrak{s}\to 0$, we get

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \left(1-\mathfrak{s}\right)\Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

from which, using Equation (48), we have

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2.$$

Hence, the result follows. □

Theorem 10.

Let $\widetilde{{\rm Y}}$ be a G-differentiable strongly $\alpha$-preinvex-𝘍𝘔 modulus $\omega >0,$ and

$$⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2⪰\tilde{0},\mathrm{for} \mathrm{all} \sigma ,\varsigma \in {\mathcal{C}}_{\alpha },$$

(49)

then $\sigma \in {\mathcal{C}}_{\alpha }$ is the minimum of the 𝘍𝘔 $\widetilde{{\rm Y}}.$

Proof. 

Let $\widetilde{{\rm Y}}:{\mathcal{C}}_{\alpha }\to {\mathbb{E}}_0$ be a G-differentiable strongly $\alpha$-preinvex-𝘍𝘔 and $\sigma \in {\mathcal{C}}_{\alpha }$ satisfies Equation (29), we have

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰⟨\alpha \left(\varsigma ,\sigma \right){\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)〉\widetilde{+}\omega \Vert 𝑗{\left(\varsigma ,\sigma \right)\Vert }^2,$$

from which, using Equation (32), we have

$$\widetilde{{\rm Y}}\left(\varsigma \right)\widetilde{-}\widetilde{{\rm Y}}\left(\sigma \right)⪰\tilde{0},$$

which implies that

$$\widetilde{{\rm Y}}\left(\varsigma \right)⪰\widetilde{{\rm Y}}\left(\sigma \right)\square$$

Remark 4.

The inequality of the type Equation (49) is considered strongly variational-like inequality. We would like to emphasize that the optimality conditions of the strongly$\alpha$-preinvex-𝘍𝘔 can be characterized by the following inequality

$${\widetilde{{\rm Y}}}^{,}\left(\sigma \right), 𝑗\left(\varsigma ,\sigma \right)⪰\tilde{0},$$

which is considered a variational-like inequality.

5. Conclusions

In this paper, we generalized the concepts of convex-𝘍𝘔s and preinvex-𝘍𝘔s pertaining to $𝑗$,$\alpha$, which is called strongly $\alpha$-preinvex-𝘍𝘔s pertaining to $𝑗$,$\alpha$. It is shown that strongly convex-𝘍𝘔s and strongly preinvex-𝘍𝘔s are special cases of strongly $\alpha$-preinvex-𝘍𝘔s. Under certain conditions, it is also proved that differential strongly $\alpha$-preinvex-𝘍𝘔s are strongly $\alpha$-invex-𝘍𝘔s and vice versa. It is also proved that the minimum of strongly $\alpha$-preinvex-𝘍𝘔s can be characterized by strong $\alpha$-variational-like inequalities and $\alpha$-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.