# On Strongly Generalized Preinvex Fuzzy Mappings

Authors: Peide Liu; Muhammad Bilal Khan; Muhammad Aslam Noor; Khalida Inayat Noor
Journal: Journal of Mathematics
 (2021)

---

## Abstract

In this article, we introduce a new notion of generalized convex fuzzy mapping known as strongly generalized preinvex fuzzy mapping on the invex set. Firstly, we have investigated some properties of strongly generalized preinvex fuzzy mapping. In particular, we establish the equivalence among the strongly generalized preinvex fuzzy mapping, strongly generalized invex fuzzy mapping, and strongly generalized monotonicity. We also prove that the optimality conditions for the sum of G-differentiable preinvex fuzzy mappings and non-G-differentiable strongly generalized preinvex fuzzy mappings can be characterized by strongly generalized fuzzy mixed variational-like inequalities, which can be viewed as a novel and innovative application. Several special cases are discussed. Results obtained in this paper can be viewed as improvement and refinement of previously known results.


---

## Body

## 1. Introduction

Convexity plays an essential role in many areas of mathematical analysis and, due to its vast applications in diverse areas, many authors extensively generalized and extended this concept using novel and different approaches. In [1], Hasnon introduced a useful generalization of convex function that is an invex function and proved the validity of significant results that hold for both convex and invex functions under few conditions. Ben-Israel and Mond [2] introduced the concept of preinvex functions on the invex sets. Later on, Mohan and Neogy [3] further extended their work by proving that, subject to certain conditions, a preinvex function defined on the invex set is an invex function and vice versa and they also showed that quasi-invex function is quasi-preinvex function. Noor [4] and Weir and Mond [5] showed that the preinvex functions preserved some important properties of convex functions. Furthermore, Noor et al. [6–8] studied the optimality conditions of differentiable preinvex functions on the invex set that can be characterized by variational inequalities. Similarly, an important and significant generalization of convex function is strongly convex function, which is introduced by Polyak [9], which plays major role in optimization theory and related areas. Karmardian [10] discussed the unique existence of a solution of the nonlinear complementarity problems by using strongly convex functions. With the support of strongly convex functions, Qu and Li [11] and Nikodem and Pales [12] investigated the convergence analysis for solving equilibrium problems and variational inequalities. For further study, we refer the readers to [13–23] and the references therein for applications and properties of the strongly convex and preinvex functions.In [24], the enormous research work fuzzy set and system has been dedicated to development of different fields and it plays an important role in the study of a wide class of problems arising in pure mathematics and applied sciences including operation research, computer science, managements sciences, artificial intelligence, control engineering, and decision sciences. The convex analysis has played an important and fundamental part in development of various fields of applied and pure science. Similarly, fuzzy convex analysis is used as the fundamental theory in fuzzy optimization. Fuzzy convex sets have been widely discussed by many authors, Liu [25] investigated some properties of convex fuzzy sets and, with the support of examples, he modified the concept of Zadeh [24] about shadow of fuzzy sets. Lowen [26] gathered some elementary well-known results about convex sets and proved separation theorem for convex fuzzy sets. Ammar and Metz [27] studied different types of convexity and defined generalized convexity of fuzzy sets. Furthermore, they formulated a general fuzzy nonlinear programing problem with application of the concept of convexity. In addition, the properties of convex fuzzy sets attracted the attention of a wide range of authors in [3, 28, 29] and the references therein. To discuss fuzzy number, it is a generalized form of an interval (in crisp set theory). Zadeh [24] defined fuzzy numbers and then Dubois and Prade [30] extended the work of Zadeh using new conditions on fuzzy number. Further, Goetschel and Voxman [31] modified different condition on fuzzy numbers so that fuzzy numbers can be easily handled. For instance, in [30], on fuzzy number, one condition is that this is a continuous function but, in [31], fuzzy number is upper semicontinuous. The goal is that, by the relaxation of conditions on fuzzy number, we can easily define a metric for collection of fuzzy numbers; then this metric allows us to study some basic properties of topological space. Furukawa [32], Nanda and Kar [33], and Syau [34] worked on the concept of fuzzy mapping from ℝn to the set of fuzzy numbers, Lipschitz continuity of fuzzy mapping, and fuzzy logarithmic convex and quasi-convex fuzzy mappings. Based on the concept of ordering illustrated by Goetschel and Voxman [35], Yan and Xu [36] presented the concepts of epigraphs and convexity of fuzzy mappings and described the characteristics of convex fuzzy and quasi-convex fuzzy mappings. The idea of fuzzy convexity has been generalized and extended in diversity of directions, which has significant implementation in many areas. It is worth mentioning that one of the most considered generalizations of convex fuzzy mapping is preinvex fuzzy mapping. The idea of fuzzy preinvex mapping on the fuzzy invex set was introduced and studied by Noor [37] and they verified that fuzzy optimality conditions of differentiable fuzzy preinvex mappings can be distinguished by variational-like inequalities. Moreover, any local minimum of a preinvex fuzzy mapping is a global minimum on invex set and a necessary and sufficient condition for fuzzy mapping is to be preinvex if its epigraph is an invex set. In [38], Syau further modified the concept of preinvex fuzzy mapping that was presented by Noor [37]. Syau and Lee [39] also discussed the terminologies of continuity and convexity through linear ordering and metric defined on fuzzy numbers. Extension in the Weierstrass Theorem from real-valued functions to fuzzy mappings is also one of their significant contributions in the literature. For recent applications, see [40–43] and the references therein.Motivated and inspired by the ongoing research work and by the importance of the idea of invexity and preinvexity of fuzzy mappings, the paper is organized as follows. Section2 recalls some basic definitions, preliminary notations, and results which will be helpful for further study. Section 3 introduces the notions of strongly generalized preinvex, quasi-preinvex, and log-preinvex fuzzy mappings and investigates some properties. Section 4 studies the new relationships among various concepts of strongly preinvex fuzzy mappings and establishes the equivalence among the strongly generalized preinvex fuzzy mapping, strongly generalized invex fuzzy mapping, and strongly generalized monotonicity. We also introduce several new concepts of strongly generalized invex fuzzy mappings and strongly generalized monotonicities and then discuss their relation. Section 5 introduces the new class of fuzzy variational-like inequality, which is known as strongly generalized fuzzy mixed variational-like inequality. Several special cases are discussed. This inequality is itself an interesting outcome of our main results.


## 2. Preliminaries

Letℝ be the set of real numbers. A fuzzy subset A of ℝ is characterized by a mapping ψ:ℝ⟶0,1 called the membership function, for each fuzzy set and γ∈0,1; then γ-level sets of ψ are denoted and defined as follows: ψγ=u∈ℝ|ψu≥γ. If γ=0, then suppψ=u∈ℝ|ψu>0 is called support of ψ. By ψ0, we define the closure of suppψ.Definition 1.
A fuzzy set is said to be fuzzy number with the following properties:(1)
ψ is normal; that is, there exists u∈ℝ such that ψu=1(2)
φ is upper semicontinuous; that is, for given u∈ℝ, there exist ε>0 and δ>0 such that φu−φϑ<ε for all ϑ∈ℝ with u−ϑ<δ(3)
ψ is fuzzy convex; that is, ψ1−τu+τϑ≥minψu,ψϑ∀u,ϑ∈ℝ, τ∈0,1(4)
ψ0 is compact
F0 denotes the set of all fuzzy numbers. For fuzzy number, it is convenient to distinguish following γ-levels:(1)ψγ=u∈ℝ|ψu≥γ.
From these definitions, we have(2)ψγ=ψ∗γ,ψ∗γ,where(3)ψ∗γ=infu∈ℝ|ψu≥γ,ψ∗γ=supu∈ℝ|ψu≥γ.
Eachρ∈ℝ is also a fuzzy number, defined as(4)ρ˜u=1,ifu=ρ,0,ifu≠ρ.
Thus, a fuzzy numberψ can be identified by a parametrized triple:(5)ψ∗γ,ψ∗γ,γ:γ∈0,1.
This leads to the following characterization of a fuzzy number in terms of the two end point functionsψ∗γ and ψ∗γ.Theorem 1 (see [35]).
Suppose thatψ∗γ:0,1⟶ℝ and ψ∗γ:0,1⟶ℝ satisfy the following conditions: (1)
ψ∗γ is a nondecreasing function(2)
ψ∗γ is a nonincreasing function(3)
ψ∗1≤ψ∗1(4)
ψ∗γ and ψ∗γ are bounded and left continuous on 0,1 and right continuous at γ=0
Thenψ:ℝ⟶0,1, defined by(6)ψu=supγ:ψ∗γ≤u≤ψ∗γ,is a fuzzy number and its parameterization form is given by ψ∗γ,ψ∗γ,γ:γ∈0,1. Moreover, if ψ:ℝ⟶0,1 is a fuzzy number with parametrization given by ψ∗γ,ψ∗γ,γ:γ∈0,1, then functions ψ∗γ and ψ∗γ find conditions (1)–(4).
Letψ,ϕ∈F0 represent parametrically ψ∗γ,ψ∗γ,γ:γ∈0,1 and ϕγ,ϕ∗γ,γ:γ∈0,1, respectively. We say that ψ∗ϕ if, for all γ∈0,1, ψ∗γ≤ϕ∗γ and ψ∗γ≤ϕ∗γ. If ψ∗ϕ, then there exists γ∈0,1 such that ψ∗γ<ϕ∗γ or ψ∗γ≤ϕ∗γ. We say that it is comparable if, for any ψ,ϕ∈F0, we have ψ≤ϕ or ψ≥ϕ; otherwise, they are noncomparable. Sometimes we may write ψ≤ϕ instead of ψ≥ϕ and note that we may say that F0 is a partial ordered set under the relation ≤.
Ifψ,ϕ∈F0, there exists μ∈F0 such that ψ=ϕ+˜μ, and then by this result we have the existence of Hukuhara difference of ψ and ϕ, and we say that μ is the H-difference of ψ and ϕ, denoted by ψ−˜ϕ; see [43]. If H-difference exists, then(7)μ∗γ=ψ−˜ϕ∗γ=ψvγ−ϕ∗γ,(8)μ∗=ψ−˜ϕ∗γ=ψ∗γ−ϕ∗γ.
Now we discuss some properties of fuzzy numbers under addition and scaler multiplication; ifψ,ϕ∈F0 and ρ∈ℝ, then ψ+˜ϕ and ρψ are defined as(9)ψ+˜ϕ=ψ∗γ+ϕ∗γ,ψ∗γ+ϕ∗γ,γ:γ∈0,1,(10)ρψ=ρψ∗γ,ρψ∗γ,γ:γ∈0,1.Remark 1.
Obviously,F0 is closed under addition and nonnegative scaler multiplication and the above-defined properties on F0 are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number ρ∈ℝ,(11)ψ+˜ρ=ψ∗γ+ρ,ψ∗γ+ρ,γ:γ∈0,1.Definition 2.
A mappingT:γ⟶F0 is called fuzzy mapping. For each γ∈0,1, denote Tuγ=T∗u,γ,T∗u,γ. Thus, a fuzzy mapping T can be identified by a parametrized triple:(12)Tu=T∗u,γ,T∗u,γ,γ:γ∈0,1.Definition 3 (see [40]).
LetL=m,n and u∈L. Then fuzzy mapping T:m,n⟶F0 is said to be generalized differentiable (in short, G-differentiable) at u if there exists an element T,u∈F0.
For all0<τ sufficiently small, there exist Tu+τ−˜Tu,Tu−˜Tu−τ, and the limits (in the metric D). (13)limτ⟶0+Tu+τ−˜Tuτ=limτ⟶0+Tu−˜Tu−ττ=T,u,orlimτ⟶0+Tu−˜Tu+τ−τ=limτ⟶0+Tu−τ−˜Tu−τ=T,u,orlimτ⟶0+Tu+τ−˜Tuτ=limτ⟶0+Tu−τ−˜Tu−τ=T,u,orlimτ⟶0+Tu−˜Tu+τ−τ=limτ⟶0+Tu−˜Tu−ττ=T,u,where the limits are taken in the metric space F0,D, for ψ,ϕ∈F0.(14)Dψ,ϕ=sup0≤γ≤1Hψγ,ϕγ,and H denotes the well-known Hausdorff metric on space of intervals.Definition 4 (see [27]).
A fuzzy mappingT:K⟶F0 is called convex on the convex set K if(15)T1−τu+τϑ≼1−τTu+˜τTϑ,∀u,ϑ∈Kτ∈0,1.
It is strictly convex fuzzy mapping if strict inequality holds forTu≠Tϑ. T:K⟶F0 is said to be concave fuzzy mapping if −T is convex on K. It is strictly concave fuzzy mapping if strict inequality holds for Tu≠Tϑ.Definition 5 (see [27]).
A fuzzy mappingT:K⟶F0 is called quasi-convex on the convex set K if(16)T1−τu+τϑ≼maxTu,Tϑ,∀u,ϑ∈K,τ∈0,1.Definition 6 (see [2]).
The setKξ in ℝ is said to be invex set with respect to arbitrary bifunction ξ.,., if(17)u+τξϑ,u∈Kξ,∀u,ϑ∈Kξ,τ∈0,1.
The invex setKξ is also called ξ-connected set. Note that convex set ξϑ,u=ϑ−u is called an invex set in classical sense, but the converse is not valid. For instance, the set Kξ=−7,−2∪2,10 is an invex set with respect to nontrivial bifunction ξ:ℝ×ℝ⟶ℝ given as(18)ξϑ,u=ϑ−u,ϑ≥0,u≥0,ξϑ,u=ϑ−u,0≥ϑ,0≥u,ξϑ,u=−7−u,ϑ≥0≥u,ξϑ,u=2−u,u≥0≥ϑ.Definition 7 (see [17]).
A fuzzy mappingT:Kξ⟶F0 is called preinvex on the invex set Kξ with respect to bifunction ξ if(19)Tu+τξϑ,u≼1−τTu+˜τTϑ,∀u,ϑ∈Kξ, τ∈0,1, where ξ:Kξ×Kξ⟶ℝ.
It is strictly preinvex fuzzy mapping if strict inequality holds forTu≠Tϑ. T:Kξ⟶F0 is said to be preconcave fuzzy mapping if −T is preinvex on Kξ. It is strictly preconcave fuzzy mapping if strict inequality holds for Tu≠Tϑ.Lemma 1.
(see [42]). Let Kξ be an invex set with respect to ξ and let T:Kξ⟶F0 be a fuzzy mapping parametrized by(20)Tu=T∗u,γ,T∗u,γ,γ:γ∈0,1,∀u∈Kξ. Then T is preinvex on Kξ if and only if, for all γ∈0,1,T∗u,γ and T∗u,γ are preinvex with respect to ξ on Kξ.Definition 8 (see [17]).
A comparable fuzzy mappingT:Kξ⟶F0 is called quasi-preinvex on the invex set Kξ with respect to ξ if, ∀u,ϑ∈Kξ, τ∈0,1 such that(21)Tu+τξϑ,u≼maxTu,Tϑ.
For further study, letKξ be a nonempty closed invex set in ℝ. Let T:Kξ⟶F0 be a fuzzy mapping and let ξ:Kξ×Kξ⟶ℝ be an arbitrary bifunction. Let Ω. be a nonnegative function, defined as Ω:ℝ⟶ℝ+. We let . and .,. be the norm and inner product, respectively.


## 3. Strongly Generalized Preinvex Fuzzy Mappings

In this section, we propose the new concept of nonconvex fuzzy mappings known as strongly generalized preinvex fuzzy mappings. We define some different types of nonconvex fuzzy mappings and investigate some basic properties.Definition 9.
LetKξ be an invex set and let T:Kξ⟶F0 be fuzzy mapping. Then T is said to be strongly generalized preinvex fuzzy mapping with respect to an arbitrary nonnegative function Ω. and bifunction ξ.,., if there exists a constant ω such that(22)Tu+τξϑ,u≤1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,∀u,ϑ∈Kξ,τ∈0,1.ξ:Kξ×Kξ⟶ℝ and Ω:ℝ⟶ℝ+.
It is strictly strongly generalized preinvex fuzzy mapping if strict inequality holds forTu≠Tϑ and T:Kξ⟶F0 is said to be strongly generalized preconcave fuzzy mapping if −T is strongly generalized preinvex on Kξ. It is strictly generalized preconcave fuzzy mapping if strict inequality holds for Tu≠Tϑ.
Now we discuss some special cases of strongly generalized preinvex fuzzy mappings.
IfΩϑ,u=ξϑ,uρ, ρ>0, then, for allu,ϑ∈Kξ,τ∈0,1, (4) becomes(23)Tu+τξϑ,u≤1−τTu+˜τTϑ−ωτ1−τξϑ,uρ,which is called the higher-ordered strongly preinvex fuzzy mapping. This is itself a very interesting problem to study its applications in pure and applied science like fuzzy optimization. When ρ=2, T is called strongly preinvex fuzzy mapping with respect to bifunction ξ.
Ifω=0, then strongly generalized preinvex fuzzy mapping becomes preinvex fuzzy mapping with respect to bifunction ξ.,.; that is,(24)Tu+τξϑ,u≤1−τTu+˜τTϑ,∀u,ϑ∈Kξ,τ∈0,1.
IfΩξϑ,u=ϑ−uρ, with ξϑ,u=ϑ−u and ρ>0, then∀u,ϑ∈Kξ,τ∈0,1, and inequality (4) becomes(25)Tu+τξϑ,u≼1−τTu+˜τTϑ−ωτ1−τϑ−uρ,and this is called the higher-ordered strongly convex fuzzy mapping. When ρ=2, T is called strongly convex fuzzy mapping.
Ifω=0, then higher-ordered strongly convex fuzzy mapping becomes convex fuzzy mapping; that is,(26)Tu+τϑ−u≼1−τTu+˜τTϑ.∀u,ϑ∈Kξ,τ∈0,1.
Ifτ=1/2, then (22) becomes(27)T2u+ξϑ,u2≼Tu+˜Tϑ2−˜14ωΩξϑ,u,∀u,ϑ∈Kξ.
MappingT is called the strongly generalized Jensen preinvex (in short, J-preinvex) fuzzy mapping.
We also define the strongly generalized affineJ-preinvex fuzzy mapping.Definition 10.
A mappingT:Kξ⟶F0 is said to be strongly generalized affine preinvex fuzzy mapping on Kξ with respect to an arbitrary nonnegative function Ω and bifunction ξ, if, ∀u,ϑ∈Kξ,τ∈0,1, there exists a positive number ω such that(28)Tu+τξϑ,u=1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,where ξ:Kξ×Kξ⟶ℝ and Ω:ℝ⟶ℝ+.
Ifτ=1/2, then we also say that T is strongly generalized affine J-preinvex fuzzy mapping such that(29)T2u+ξϑ,u2=Tu+˜Tϑ2−˜14ωΩξϑ,u,for allu,ϑ∈KξRemark 2.
The strongly generalized preinvex fuzzy mappings have some very nice properties similar to preinvex fuzzy mapping:(i)
IfT is strongly generalized preinvex fuzzy mapping, then σT is also strongly generalized preinvex for σ≥0(ii)
IfT and G both are strongly generalized preinvex fuzzy mappings with respect to an arbitrary nonnegative function Ω. and bifunction ξ.,., then maxTu,Gu is also strongly generalized preinvex fuzzy mapping with respect to Ω and ξTheorem 2.
LetKξ be an invex set with respect to ξ and let T:Kξ⟶F0 be a fuzzy mapping parametrized by(30)Tu=T∗u,γ,T∗u,γ,γ:γ∈0,1,for all u∈Kξ, and then T is strongly generalized preinvex fuzzy mapping on Kξ with modulus ω if and only if, for all γ∈0,1,T∗u,γ and T∗u,γ are strongly generalized preinvex functions with respect to Ω, ξ, and modulus ω.Proof.
Assume that, for eachγ∈0,1,T∗u,γ and T∗u,γ are strongly generalized preinvex functions with respect to Ω,ξ, and modulus ω on Kξ. Then, from (4), for all u,ϑ∈Kξ,τ∈0,1, we have(31)T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u.
Then, by (30), (9), (10), and (11), we obtain(32)Tu+τξϑ,u=T∗u+τξϑ,u,γ,T∗u+τξϑ,u,γ,γ:γ∈0,1,≼1−τT∗u,γ,1−τT∗u,γ,γ:γ∈0,1+˜τT∗ϑ,γ,τT∗ϑ,γ,γ:γ∈0,1−˜ωτ1−τΩξϑ,u,=1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,∀u,ϑ∈Kξ,τ∈0,1.
Hence,Tu is a strongly generalized preinvex fuzzy mapping on Kξ with modulus ω.
Conversely, letTu be a strongly generalized preinvex fuzzy mapping on Kξ with modulus ω. Then, for all u,ϑ∈Kξ and τ∈0,1, we have(33)Tu+τξϑ,u≼1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u.
From (30), we have(34)Tu+τξϑ,u=T∗u+τξϑ,u,γ,T∗u+τξϑ,u,γ,γ:γ∈0,1,∀u,ϑ∈Kξ,τ∈0,1.
From (30), (9), (10), and (11), we obtain(35)1−τTu+˜τTϑ−˜Ωτp1−τ+τ1−τpξϑ,up=1−τT∗u,γ,1−τT∗u,γ,γ:γ∈0,1+˜τT∗ϑ,γ,τT∗ϑ,γ,γ:γ∈0,1−˜ωτ1−τΩξϑ,u,for all u,ϑ∈Kξ and τ∈0,1. Then, by strongly generalized preinvexity of Tu, we have, for all u,ϑ∈Kξ and τ∈0,1such that(36)T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,for each γ∈0,1. Hence, the result follows.
Note that ifξϑ,u=ϑ−u, then Theorem 2 reduces to the following statement.
LetKξ be a convex set and let T:K⟶F0 be a fuzzy mapping parametrized by(37)Tu=T∗u,γ,T∗u,γ,γ:γ∈0,1,∀u∈Kξ,and then F is a strongly generalized convex fuzzy mapping on Kξ if and only if, for all γ∈0,1,T∗u,γ and T∗u,γ are strongly generalized convex functions on Kξ.Example 1.
We consider the fuzzy mappingsT:0,∞⟶F0 defined by(38)Tuσ=σu2,σ∈0,u2,2u2−σu2,σ∈u2,2u2,0,otherwise.
Then, for eachγ∈0,1, we have Tγu=γu2,2−γu2. Since end point functions T∗γ,T∗γ are strongly generalized preinvex for each γ∈0,1, T is a strongly generalized preinvex fuzzy mapping with respect to(39)Ωξϑ,u=ξϑ,u2andξϑ,u=ϑ−u,and 0<ω≤1.
From Example1, it can be easily seen that, for each ω∈0,1, there exists a strongly generalized preinvex fuzzy mapping.
We now established a result for strongly generalized preinvex fuzzy mapping, which shows that the difference of strongly generalized preinvex fuzzy mapping and strongly generalized affine preinvex fuzzy mapping is again a strongly generalized preinvex fuzzy mapping.Theorem 3.
Let fuzzy mappingJ:Kξ⟶F0 be strongly generalized affine preinvex with respect to Ω and ξ. Then T is strongly generalized preinvex fuzzy with respect to same Ω and ξ if and only if G=T−˜J is a preinvex fuzzy mapping.Proof.
The “If” part is obvious. To prove the “only if” part, assume thatJ:Kξ⟶F0 is strongly generalized affine preinvex with respect to nonnegative function Ω and bifunction ξ; then there exists ω>0 such that(40)Ju+τξϑ,u=1−τJu+˜τJϑ−˜ωτ1−τΩξϑ,u,and since T is a strongly generalized preinvex fuzzy mapping, we have(41)Tu+τξϑ,u≼1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,and, from (40) and (41), we have(42)Tu+τξϑ,u−˜Ju+τξϑ,u≼1−τTu+˜τTϑ−˜1−τJu−˜τJϑ,=1−τTu−˜Ju+˜τTϑ−˜Jϑ,from which it follows that(43)Gu+τξϑ,u=Tu+τξϑ,u−˜J1−τu+τϑ,≼1−τTu−˜Ju+˜τTϑ−˜Jϑ,=1−τGu+˜τGu.
It is shown thatG=T−˜ is a strongly generalized preinvex fuzzy mapping.Definition 11.
A comparable fuzzy mappingT:Kξ⟶F0 is said to be strongly generalized quasi-preinvex on Kξ with respect to nonnegative function Ω and bifunction ξ if, ∀u,ϑ∈Kξ,τ∈0,1, there exists a positive number ω such that(44)Tu+τξϑ,u≼maxTu,Tϑ−˜ωτ1−τΩξϑ,u,where ξ:Kξ×Kξ⟶ℝ and Ω:ℝ⟶ℝ+.Definition 12.
A fuzzy mappingT:Kξ⟶F0 is called strongly generalized log-preinvex on Kξ with respect to nonnegative function Ω and bifunction ξ, if, ∀u,ϑ∈Kξ,τ∈0,1, there exists a positive number ω such that(45)Tu+τξϑ,u≼Tu1−τTϑτ−˜ωτ1−τΩξϑ,uwhere T.≻0˜, ξ:Kξ×Kξ⟶ℝ and Ω:ℝ⟶ℝ+.Remark 3.
From Definition12, we have(46)Tu+τξϑ,u≼Tu1−τTϑτ−˜ωτ1−τΩξϑ,u,≼1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,≼maxTu,Tϑ−˜ωτ1−τΩξϑ,u.
It can be easily seen that each strongly generalized log-preinvex fuzzy mapping onKξ is strongly generalized quasi-preinvex fuzzy mapping and strongly generalized preinvex fuzzy mapping on Kξ is strongly generalized quasi-preinvex fuzzy mapping, when T is a comparable fuzzy mapping.Definition 13.
A fuzzy mappingT:Kξ⟶F0 is called pseudo-preinvex on Kξ if there exists a strictly positive bifuzzy mapping b.,. such that(47)Tϑ≺Tu⇒Tu+τξϑ,u≺Tu+˜ττ−1bu,ϑ,for all u,ϑ∈Kξ,τ∈0,1, where b:Kξ×Kξ⟶F0.Theorem 4.
LetT be a strongly generalized preinvex fuzzy mapping on Kξ such that Tϑ≺Tu. Then fuzzy mapping T is strongly generalized pseudo-preinvex with respect to same nonnegative function Ω and ξ.Proof.
LetTϑ≺Tu and let T be a strongly generalized preinvex fuzzy mapping. Then, for all ,ϑ∈Kξ,τ∈0,1, there exists modulus ω such that(48)Tu+τξϑ,u≼1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,
and, therefore, for everyγ∈0,1, we have(49)T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u.
From the above equation, we have(50)T∗u+τξϑ,u,γ≤T∗u,γ+τT∗ϑ,γ−T∗u,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ≤T∗u,γ+τT∗ϑ,γ−T∗u,γ−ωτ1−τΩξϑ,u.
From (50), we have(51)T∗u+τξϑ,u,γ<T∗u,γ+ττ−1T∗u,γ−T∗ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ<T∗u,γ+ττ−1T∗u,γ−T∗ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ<T∗u,γ+ττ−1b∗u,ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ<T∗u,γ+ττ−1b∗u,ϑ,γ−ωτ1−τΩξϑ,u,that is(52)Tu+τξϑ,u≺Tu+˜ττ−1bu,ϑ−˜ωτ1−τΩξϑ,u,where bu,ϑ=Tu−˜Tϑ. This completes the proof.


## 4. G-Differentiable Strongly Generalized Preinvex Fuzzy Mappings

Throughout this section, we assume that the nonnegative functionΩ is homogenous of degree two and even; that is, ρu=ρ2Ωu,Ω−u=Ωu, for all u∈Kξ, ρ∈ℝ, unless otherwise specified.Definition 14.
A fuzzy mappingT:Kξ⟶F0 is called sharply strongly generalized pseudo-preinvex with respect to Ω and ξon Kξ if, ∀u,ϑ∈Kξ,τ∈0,1, there exists ω>0 such that(53)T,u,ξϑ,u≽0˜⇒Tϑ+τξϑ,u+˜ωτ1−τΩξϑ,u≼ϑ,where T,, is G-differentiable of T at u.Theorem 5.
LetT be a sharply strongly generalized pseudo-preinvex fuzzy mapping on Kξ with ω>0. Then,(54)−˜T,ϑ,ξϑ,u≽ωΩξϑ,u,∀u,ϑ∈Kξ.Proof.
SinceT is a sharply strongly generalized pseudo-preinvex fuzzy mapping on Kξ, ∀u,ϑ∈Kξ,τ∈0,1, there exists ω>0 such that(55)T,u,ξϑ,u≽0˜,which implies that(56)Tϑ≽ϑ+τξϑ,u+˜ωτ1−τΩξϑ,u,
and, therefore, for everyγ∈0,1, we have(57)T∗ϑ,γ≤T∗ϑ+τξϑ,u,γ+ωτ1−τΩξϑ,u,T∗ϑ,γ≤T∗ϑ+τξϑ,u,γ+ωτ1−τΩξϑ,u,from which we get(58)T∗ϑ+τξϑ,u−T∗ϑτ+ω1−τΩξϑ,u≤0,T∗ϑ+τξϑ,u−T∗ϑτ+ω1−τΩξϑ,u≤0,and, taking limit in the above inequality as τ⟶0, we get(59)−˜T,ϑ,ξϑ,u≽ωΩξϑ,u,∀u,ϑ∈Kξ.
Hence, the result follows.Definition 15.
A G-differentiable fuzzy mappingT:Kξ⟶F0 is called strongly generalized invex with respect to Ω and ξ if, ∀u,ϑ∈Kξ, there exists constant ω>0 such that(60)Tϑ−˜Tu≽T,u,ξϑ,u+˜ωΩξϑ,u.Definition 16.
A G-differentiable fuzzy mappingT:Kξ⟶F0 is called strongly generalized pseudo-invex with respect to Ω and ξ if, ∀u,ϑ∈Kξ, there exists constant ω>0 such that(61)T,u,ξϑ,u+˜ωΩξϑ,u≽0˜⇒Tϑ−˜Tu≽0˜.Definition 17.
A G-differentiable fuzzy mappingT:Kξ⟶F0 is called strongly generalized quasi-invex with respect to Ω and ξ if, ∀u,ϑ∈Kξ, there exists constant ω>0 such that(62)Tϑ≼u⇒T,u,ξϑ,u+˜ωΩξϑ,u≼0˜.Definition 18.
A G-differentiable fuzzy mappingT:Kξ⟶F0 is called pseudo-invex with respect to Ω and ξ if, ∀u,ϑ∈Kξ such that(63)T,u,ξϑ,u≽0˜⇒Tϑ−˜Tu≻¯0˜.Definition 19.
A G-differentiable fuzzy mappingT:Kξ⟶F0 is called quasi-invex with respect to Ω and ξ if, ∀u,ϑ∈Kξ such that(64)Tϑ≼u⇒T,u,ξϑ,u≼0˜.
Ifξϑ,u=−ξu,ϑ, then definitions reduce to known ones. All Definitions 15–19 may play an important role in fuzzy optimization problem and mathematical programing.Theorem 6.
LetT:Kξ⟶F0 be a G-differentiable strongly generalized preinvex fuzzy mapping. Then T is a strongly generalized invex fuzzy mapping.Proof.
LetT:Kξ⟶F0 be a G-differentiable strongly generalized preinvex fuzzy mapping. Since T is a strongly generalized preinvex fuzzy mapping, there exists constant ω>0, and, for all u,ϑ∈Kξ and τ∈0,1, we have(65)Tu+τξϑ,u≼1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u,≼u+˜τTϑ−˜Tu−˜ωτ1−τΩξϑ,u,
and, therefore, for everyγ∈0,1, we have(66)T∗u+τξϑ,u,γ≤T∗u,γ+τT∗ϑ,γ−T∗u,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ≤T∗u,γ+τT∗ϑ,γ−T∗u,γ−ωτ1−τΩξϑ,u,which implies that(67)τT∗ϑ,γ−T∗u,γ≥T∗u+τξϑ,u,γ−T∗u,γ+ωτ1−τΩξϑ,u,τT∗ϑ,γ−T∗u,γ≥T∗u+τξϑ,u,γ−T∗u,γ+ωτ1−τΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗u+τξϑ,u,γ−T∗u,γτ+ω1−τΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗u+τξϑ,u,γ−T∗u,γτ+ω1−τΩξϑ,u.
Taking limit in the above inequality asτ⟶0, we have(68)T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u.
From that, we have(69)Tϑ−˜Tu≽T,u,ξϑ,u+˜ωΩξϑ,u,and when ω=0, Theorem 6 reduces to the following result.Theorem 7.
LetT:Kξ⟶F0 be a G-differentiable preinvex fuzzy mapping. Then, T is an invex fuzzy mapping.
To prove conversion of Theorem6, we need the following assumption regarding the function ξ, which plays an important role in G-differentiation of the main results.
Condition C is as follows [33]:(70)ξϑ,u+τξϑ,u=1−τξϑ,u,ξu,u+τξϑ,u=−τξϑ,u.
The following result finds the conversion of Theorem6.Theorem 8.
LetT be a G-differentiable fuzzy mapping on Kξ and Condition C holds. Let the function Ω be homogenous of degree 2 and even. Then the following are equivalent:(a)
T is a strongly generalized preinvex fuzzy mapping:(71)bTϑ−˜Tu≽T,u,ξϑ,u+˜ωΩξϑ,u,(72)cT,u,ξϑ,u+˜T,ϑ,ξu,ϑ≼−˜ωΩξϑ,u+˜Ωξu,ϑ,∀u,ϑ∈Kξ.Proof.
(a) implies (b).
The demonstration is analogous to the demonstration of Theorem6.
(b) implies (c). Assume that (71) holds. Then,(73)Tϑ−˜Tu≽T,u,ξϑ,u+˜ωΩξϑ,u.
Then, for everyγ∈0,1, we have(74)T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u.
Replacingϑ by u and u by ϑ in (21), we have(75)T∗u,γ−T∗ϑ,γ≥T∗,ϑ,γ,ξu,ϑ+ωΩξu,ϑ,T∗u,γ−T∗ϑ,γ≥T∗,ϑ,γ,ξu,ϑ+ωΩξu,ϑ.
Adding (74) and (75), we have(76)T∗,u,γ,ξϑ,u+T∗,ϑ,γ,ξu,ϑ≤−ωΩξϑ,u+Ωξu,ϑ,T∗,u,γ,ξϑ,u+T∗,ϑ,γ,ξu,ϑ≤−ωΩξϑ,u+Ωξu,ϑ,that is(77)T,u,ξϑ,u+˜T,ϑ,ξu,ϑ≼−˜ωΩξϑ,u+˜Ωξu,ϑ,and (c) implies (b). Assume that (72) holds. Then,(78)T,ϑ,ξu,ϑ≼−˜T,u,ξϑ,u−˜ωΩξϑ,u+˜Ωξu,ϑ.
Therefore, for everyγ∈0,1, we have(79)T∗,ϑ,γ,ξu,ϑ≤−T∗,u,γ,ξϑ,u−ωΩξϑ,u+Ωξu,ϑ,T∗,ϑ,γ,ξu,ϑ≤−T∗,u,γ,ξϑ,u−ωΩξϑ,u+Ωξu,ϑ.
SinceKξ is an invex set, we have ϑτ=u+τξϑ,u∈Kξ for all u,ϑ∈Kξ and τ∈0,1. Taking ϑ=ϑτ in (23), we get (80)T∗,u+τξϑ,u,γ,ξu,u+τξϑ,u≤−T∗,u,γ,ξu+τξϑ,u−ωΩu+τξϑ,u,u+Ωξu,u+τξϑ,u,T∗,u+τξϑ,u,γ,ξu,u+τξϑ,uT∗,u,γ,ξu+τξϑ,u−ωΩu+τξϑ,u,u+Ωξu,u+τξϑ,u.
By using Condition C and the fact thatΩ is homogenous of degree 2 and even, we have(81)T∗,u+τξϑ,u,γ,τξϑ,u≥T∗,u,γ,τξϑ,u+2ωτ2Ωξϑ,u,T∗,u+τξϑ,u,γ,τξϑ,u≥T∗,u,γ,τξϑ,u+2ωτ2Ωξϑ,u,T∗,u+τξϑ,u,γ,ξϑ,u≥T∗,u,γ,ξϑ,u+2ωτΩξϑ,u,T∗,u+τξϑ,u,γ,ξϑ,u≥T∗,u,γ,ξϑ,u+2ωτΩξϑ,u.
Let(82)H∗τ=T∗u+τξϑ,u,γ,H∗τ=T∗u+τξϑ,u,γ,and, taking G-derivative with respect to τ, we get(83)H∗,τ=T∗,u+τξϑ,u,γ.ξϑ,u=T∗,u+τξϑ,u,γ,ξϑ,u,H∗,τ=T∗,u+τξϑ,u,γ.ξϑ,u=T∗,u+τξϑ,u,γ,ξϑ,u,from which, using (81), we have(84)H∗,τ≥T∗,u,γ,ξϑ,u+2ωτΩξϑ,u,H∗,τ≥T∗,u,γ,ξϑ,u+2ωτΩξϑ,u.
Integrating (84) over 0,1 with respect to τ, we get(85)H∗1−H∗0≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,H∗1−H∗0≥T∗,u,γ,ξϑ,u+ωΩξϑ,u.T∗u+ξϑ,u,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,T∗u+ξϑ,u,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u.
SinceT is a strongly generalized preinvex fuzzy mapping, for τ=1, we have(86)T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u.
From that, we have(87)Tϑ−˜Tu≽T,u,ξϑ,u+˜ωΩξϑ,u.
(b) implies (a). Assume that (71) holds. Since Kξ is an invex set, we have ϑτ=u+τξϑ,u∈Kξ for all u,ϑ∈Kξ and τ∈0,1. Taking ϑ=ϑτ in (19) for each γ∈0,1, we get(88)T∗ϑτ,γ−T∗u,γ≥T∗,u,γ,ξϑτ,u+ωΩξϑτ,u,T∗ϑτ−T∗u,γ≥T∗,u,γ,ξϑτ,u+ωΩξϑτ,u.
Using Condition C, we have(89)T∗ϑτ,γ−T∗u,γ≥1−τT∗,u,γ,ξϑ,u+ω1−τ2Ωξϑ,u,T∗ϑτ,γ−T∗u,γ≥1−τT∗,u,γ,ξϑ,u+ω1−τ2Ωξϑ,u.
In a similar way, we obtain(90)T∗u,γ−T∗ϑτ,γ≥−τT∗,u,γ,ξϑ,u+ωτ2Ωξϑ,u,T∗u,γ−T∗ϑτ,γ≥−τT∗,u,γ,ξϑ,u+ωτ2Ωξϑ,u.
Multiplying (89) by τ and (90) by 1−τ and adding the resultant, we have(91)T∗ϑτ,γ≤1−τT∗u,γ+τT∗ϑ,γ−Ωωτ1−τΩξϑ,u,T∗ϑτ,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,that is(92)Tu+τξϑ,u≼1−τTu+˜τTϑ−˜ωτ1−τΩξϑ,u.
Hence,T is a strongly generalized preinvex fuzzy mapping with respect to Ω and ξ.
Theorems6 and 8 enable us to define the following new definitions.Definition 20.
A G-differentiable fuzzy mappingT:Kξ⟶F0 is said to be the following: (i)
Strongly generalized monotone fuzzy operator with respect toΩ and ξ if and only if, ∀u,ϑ∈Kξ, there exists constant ω>0 such that(93)T,u,ξϑ,u+˜T,ϑ,ξu,ϑ≼−˜Ωξϑ,u+˜Ωξu,ϑ.(ii)
Monotone fuzzy operator with respect to bifunctionξ if and only if, ∀u,ϑ∈Kξ,(94)T,u,ξϑ,u+˜T,ϑ,ξu,ϑ≼0˜.(iii)
Strongly generalized pseudomonotone fuzzy operator with respect toΩ and ξ if and only if, ∀u,ϑ∈Kξ, there exists constant ω>0 such that(95)T,u,ξϑ,u+˜ωΩξϑ,u≽0˜⇒−˜T,ϑ,ξu,ϑ≽0˜.(iv)
Strictly monotone fuzzy operator with respect to bifunctionξ if and only if(96)T,u,ξϑ,u+˜T,ϑ,ξu,ϑ≺0˜,∀u,ϑ∈Kξ.(v)
Pseudomonotone fuzzy operator with respect to bifunctionξ if and only if(97)T,u,ξϑ,u≽0˜⇒T,ϑ,ξu,ϑ≼0˜,∀u,ϑ∈Kξ.(vi)
Quasi-monotone fuzzy operator with respect to bifunctionξ if and only if(98)T,u,ξϑ,u≻0˜⇒T,ϑ,ξu,ϑ≼0˜,∀u,ϑ∈Kξ.(vii)
Strictly pseudomonotone fuzzy operator with respect to bifunctionξ if and only if(99)T,u,ξϑ,u≽0˜⇒T,ϑ,ξu,ϑ≺0˜,∀u,ϑ∈Kξ.
Ifξϑ,u=−ξu,ϑ, then Definition 20 reduces to a new one.
As a special case of Theorem8, we have the following.Theorem 9.
LetT be a G-differentiable fuzzy mapping on Kξ and Condition C holds. If the function Ω is homogenous of degree 2 and even, then T is a strongly generalized ξ-invex fuzzy mapping if and only if T, is a strongly generalized ξ-monotone fuzzy operator.Proof.
The proof is similar to that of Theorem8.Theorem 10.
Let G-differentialT, of fuzzy mapping T on Kξ be a strongly generalized pseudomonotone fuzzy operator and Condition C holds. Let the function Ω be homogenous of degree 2 and even. Then, T is a strongly generalized pseudo-invex fuzzy mapping.Proof.
LetT, be fuzzy strongly generalized pseudomonotone. Then, for all u,ϑ∈Kξ, we have(100)T,u,ξϑ,u+˜ωΩξϑ,u≽0˜⇒−˜T,ϑ,ξu,ϑ≽0˜.
Therefore, for everyγ∈0,1, we have(101)−T∗,ϑ,γ,ξu,ϑ≥0,−T∗,ϑ,γ,ξu,ϑ≥0.
SinceKξ is an invex set, we have ϑτ=u+τξϑ,u∈Kξ for all u,ϑ∈Kξ and τ∈0,1. Taking ϑ=ϑτ in (101), we get(102)−T∗,u+τξϑ,u,γ,ξu,u+τξϑ,u≥0,−T∗,u+τξϑ,u,γ,ξu,u+τξϑ,u≥0.By using Condition C, we have(103)T∗,u+τξϑ,u,γ,ξϑ,u≥0,T∗,u+τξϑ,u,γ,ξϑ,u≥0.
Assume that(104)H∗τ=T∗u+τξϑ,u,γ,H∗τ=T∗u+τξϑ,u,γ,and, taking G-derivative with respect to τ, using (103), we have(105)H∗,τ=T∗,u+τξϑ,u,γ,ξϑ,u≥0,H∗,τ=T∗,u+τξϑ,u,γ,ξϑ,u≥0.
Integrating (105) over 0,1 with respect to τ, we get(106)H∗1−H∗0≥0,H∗1−H∗0≥0,which implies that(107)T∗u+ξϑ,u,γ−T∗u,γ≥0,T∗u+ξϑ,u,γ−T∗u,γ≥0.
From condition (i), we have(108)T∗ϑ,γ−T∗u,γ≥0,T∗ϑ,γ−T∗u,γ≥0,that is(109)Tϑ−˜Tu≽0˜,∀ϑ∈Kξ.
Hence,Tu is a strongly generalized pseudo-invex fuzzy mapping.Theorem 11.
Let G-differentialT, of fuzzy mapping T on Kξ be a pseudomonotone fuzzy operator and Condition C holds. Then T is a pseudo-invex fuzzy mapping.Theorem 12.
Let G-differentialT, of fuzzy mapping T on Kξ be a quasi-monotone fuzzy operator and Condition C holds. Then T is a quasi-invex fuzzy mapping.
We now discuss the fuzzy optimality condition for G-differentiable strongly generalized preinvex fuzzy mappings, which is the main motivation of our results.


## 5. Strongly Generalized Fuzzy Mixed Variational-Like Inequalities

A well-known fact in mathematical programing is that variational inequality problem has a close relationship with the optimization problem. Similarly, the fuzzy variational inequality problem also has a close relationship with the fuzzy optimization problem.Consider the unconstrained fuzzy optimization problem(110)minu∈KξTu,where Kξ is a subset of ℝ and T:Kξ⟶F0 is a fuzzy mapping.A pointu∈Kξ is called a feasible point. If u∈Kξ and no ϑ∈Kξ, Tu≺Tϑ, then u is called an optimal solution, a global optimal solution, or simply a solution to the fuzzy optimization problem.Theorem 13.
LetT be a G-differentiable strongly generalized preinvex fuzzy mapping modulus ω>0. If u∈Kξ is the minimum of the mapping T, then(111)Tϑ−˜Tu≽ωΩξϑ,u,∀u,ϑ∈KξProof.
Letu∈Kξ be a minimum of T. Then, ∀ϑ∈Kξ, we have(112)Tu≼ϑ.
Therefore, for everyγ∈0,1, we have(113)T∗u,γ≤T∗ϑ,γ,T∗u,γ≤T∗ϑ,γ.
SinceKξ is an invex set, for all u,ϑ∈Kξ, τ∈0,1, ϑτ=u+τξϑ,u∈Kξ. Taking ϑ=ϑτ in (113), we get(114)0≤T∗u+τξϑ,u,γ−T∗u,γτ,0≤T∗u+τξϑ,u,γ−T∗u,γτ.
Taking limit in the above inequality asτ⟶0, we get(115)0≤T∗,u,γ,ξϑ,u,0≤T∗,u,γ,ξϑ,u,0˜≼T,u,ξϑ,u,and since T:Kξ⟶F0 is a G-differentiable strongly generalized preinvex fuzzy mapping,(116)T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,T∗u+τξϑ,u,γ≤1−τT∗u,γ+τT∗ϑ,γ−ωτ1−τΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗u+τξϑ,u,γ−T∗u,γτ+ω1−τΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗u+τξϑ,u,γ−T∗u,γτ+ω1−τΩξϑ,u,and, again taking limit in the above inequality as τ⟶0, we get(117)T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,T∗ϑ,γ−T∗u,γ≥T∗,u,γ,ξϑ,u+ωΩξϑ,u,from which, using (115), we have(118)T∗ϑ,γ−T∗u,γ≥ωΩξϑ,u,T∗ϑ,γ−T∗u,γ≥ωΩξϑ,u,that is(119)Tϑ−˜Tu≽ωΩξϑ,u.This completes the proof.Remark 4.
IfT is a G-differentiable strongly generalized preinvex fuzzy mapping modulus ω>0, and(120)T,u,ξϑ,u+˜ωΩξϑ,u≽0˜,∀u,ϑ∈Kξ, then u∈Kξ is the minimum of the mapping T.
The inequality of the type in (120) is called strongly generalized fuzzy variational-like inequality. We would like to emphasize that the optimality conditions of the strongly generalized preinvex fuzzy mapping can be characterized by the following inequality:(121)T,u,ξϑ,u≽0˜,and, ∀u,ϑ∈Kξ, this is called variational-like inequality.
We consider the functionalIϑ, defined as(122)Iϑ=Tϑ+˜Jϑ,∀ϑ∈ℝ, where T is a G-differentiable preinvex fuzzy mapping and J is a non-G-differentiable strongly preinvex fuzzy mapping.
We show that the minimum of the functionalIϑ can be characterized by a class of variational-like inequalities.Theorem 14.
LetT:Kξ⟶F0 be a G-differentiable preinvex fuzzy mapping and let J:Kξ⟶F0 be a non-G-differentiable strongly generalized preinvex fuzzy mapping. Then the functional Iϑ has minimum u∈Kξ if and only if u∈Kξ satisfies(123)T,u,ξϑ,u+˜Jϑ−˜Ju≽ωΩξϑ,u,∀ϑ∈Kξ.Proof.
Letu∈Kξ be the minimum of I; then, by definition, for all ϑ∈Kξ, we have(124)Iu≼Iϑ,and since Kξ is an invex set, ϑτ=u+τξϑ,u, for all u,ϑ∈Kξ and τ∈0,1. Replacing ϑ by ϑτ in (124), we get(125)Iu≼Iu+τξϑ,u,
and, therefore, for everyγ∈0,1, we have(126)I∗u,γ≤I∗u+τξϑ,u,γ,I∗u,γ≤I∗u+τξϑ,u,γ.which implies that, using (122),(127)T∗u,γ+J∗u,γ≤T∗u+τξϑ,u,γ+J∗u+τξϑ,u,γ,T∗u,γ+J∗u,γ≤T∗u+τξϑ,u,γ+J∗u+τξϑ,u,γ.Since J is a strongly generalized preinvex fuzzy mapping,(128)0≤T∗u+τξϑ,u,γ−T∗u,γ+τJ∗ϑ,γ−J∗u,γ−ωτ1−τΩξϑ,u,0≤T∗u+τξϑ,u,γ−T∗u,γ+τJ∗ϑ,γ−J∗u,γ−ωτ1−τΩξϑ,u.
Now, dividing by “τ” and taking limτ⟶0, we have(129)0≤limτ⟶0T∗u+τξϑ,u,γ−T∗u,γτ+J∗ϑ,γ−J∗u,γ−ω1−τΩξϑ,u,0≤limτ⟶0T∗u+τξϑ,u,γ−T∗u,γτ+J∗ϑ,γ−J∗u,γ−ω1−τΩξϑ,u,
Sincelimτ⟶0T∗u+τξϑ,u,γ−T∗u,γ/τ=T∗,u,γ,ξϑ,u and limτ⟶0T∗u+τξϑ,u,γ−T∗u,γ/τ=T∗,u,γ,ξϑ,u, from the above inequalities, we have(130)0≤T∗,u,γ,ξϑ,u+J∗ϑ,γ−J∗u,γ−ωΩξϑ,u,0≤T∗,u,γ,ξϑ,u+J∗ϑ,γ−J∗u,γ−ωΩξϑ,u,
SinceT is a fuzzy mapping, from the above inequalities, we have(131)0˜≼T,u,ξϑ,u+˜Jϑ−˜Ju−˜ωΩξϑ,u.
Conversely, let (123) be satisfied to prove that u∈Kξ is a minimum of I. Assume that, for all ϑ∈Kξ, we have(132)Iu−˜Iϑ=Tu+˜Ju−˜Tϑ−˜Jϑ.
Therefore, for everyγ∈0,1, we have(133)I∗u,γ−I∗ϑ,γ=T∗u,γ−T∗ϑ,γ+J∗u,γ−J∗ϑ,γ,I∗u,γ−I∗ϑ,γ=T∗u,γ−T∗ϑ,γ+J∗u,γ−J∗ϑ,γ.By Theorem 7, we have(134)I∗u,γ−I∗ϑ,γ≤−T∗,u,γ,ξϑ,u+J∗ϑ,γ−J∗u,γ,I∗u,γ−I∗ϑ,γ≤−T∗,u,γ,ξϑ,u+J∗ϑ,γ−J∗u,γ.
By using (123), we have(135)I∗u,γ−I∗ϑ,γ≤−ωΩξϑ,u≤0,I∗u,γ−I∗ϑ,γ≤−ωΩξϑ,u≤0,and, hence, Iu≼Iϑ.
Note that (123) is called a strongly generalized fuzzy mixed variational-like inequality. This result shows the characterization of the minimum of strongly generalized preinvex fuzzy mapping with the help of strongly generalized fuzzy mixed variational-like inequalities.
Now, we discuss some special cases of strongly generalized fuzzy mixed variational-like inequalities:(i)
IfΩξϑ,u=ξϑ,u2, then (123) is called a strongly fuzzy mixed variational-like inequality such as(136)T,u,ξϑ,u+˜Jϑ−˜Ju−˜ωϑ,u2≽0˜,∀ϑ∈Kξ.(ii)
Ifξϑ,u=ϑ−u, then (136) is called a strongly fuzzy mixed variational inequality such as(137)T,u,ϑ−u+˜Jϑ−˜Ju−˜ωϑ,u≽0˜,∀ϑ∈Kξ.(iii)
Ifω=0, then (136) is called a fuzzy mixed variational-like inequality such as(138)T,u,ξϑ,u+˜Jϑ−˜Ju≽0˜,∀ϑ∈Kξ.(iv)
Ifω=0, then (137) is called a fuzzy mixed variational inequality such as(139)T,u,ϑ−u+˜Jϑ−˜Ju≽0˜,∀ϑ∈Kξ.(v)
IfJ=0, then (138) is known as a fuzzy variational-like inequality.(140)T,u,ξϑ,u≽0˜,∀ϑ∈Kξ.(vi)
IfJ=0, then (139) is known as a fuzzy variational inequality.(141)T,u,ϑ−u≽0˜,∀ϑ∈Kξ.Theorem 15.
LetT:Kξ⟶F0 be a G-differentiable invex fuzzy mapping and let J:Kξ⟶F0 be a non-G-differentiable strongly preinvex fuzzy mapping. Then functional Iϑ has minimum u∈Kξ if and only if u∈Kξ satisfies(142)T,u,ξϑ,u+˜Jϑ−˜Ju−˜ωΩξϑ,u≽0˜,∀ϑ∈Kξ.Proof.
The proof is similar to that of Theorem14; it is omitted here.Remark 5.
Inequality (123) shows that the variational-like inequalities arise naturally in connection with the minimization of the G-differentiable preinvex fuzzy mappings subject to certain constraints.


## 6. Conclusion

In this paper, we introduced and studied a new class of fuzzy nonconvex mappings with respect to nonnegative function and bifunction known as strongly generalized preinvex fuzzy mappings. We proved that convex and preinvex fuzzy mappings are special cases of strongly generalized preinvex fuzzy mappings. In case of G-differentiable preinvex fuzzy mappings and non-G-differential strongly generalized preinvex fuzzy mappings, we introduced the new class of fuzzy variational-like inequalities is known as strongly generalized fuzzy mixed variational-like inequalities characterizing the optimality conditions for the sum of G-differentiable preinvex fuzzy mappings and non-G-differentiable strongly generalized preinvex fuzzy mappings. We hope that this link between these fields may lead to significant, new, and innovative result.



---

*Source: 6657602-2021-04-01.xml*