On Strongly Generalized Preinvex Fuzzy Mappings

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.


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][7][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][14][15][16][17][18][19][20][21][22][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. e 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.
e 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 R n 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. e 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. e 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 eorem from real-valued functions to fuzzy mappings is also one of their significant contributions in the literature. For recent applications, see [40][41][42][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. Section 2 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. is inequality is itself an interesting outcome of our main results.

Preliminaries
Let R 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 c ∈ (0, 1]; then c-level sets of ψ are denoted and defined as follows:

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 ψ(u) � 1 (2) φ is upper semicontinuous; that is, for given u ∈ R, there exist ε > 0 and δ > 0 such that ∀u, ϑ ∈ R, τ ∈ [0, 1] (4) [ψ] 0 is compact F 0 denotes the set of all fuzzy numbers. For fuzzy number, it is convenient to distinguish following c-levels: From these definitions, we have where Each ρ ∈ R is also a fuzzy number, defined as us, a fuzzy number ψ can be identified by a parametrized triple: is leads to the following characterization of a fuzzy number in terms of the two end point functions ψ * (c) and ψ * (c).
We say that it is comparable if, for any ψ, ϕ ∈ F 0 , 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 F 0 is a partial ordered set under the relation ≤ .
If ψ, ϕ ∈ F 0 , there exists μ ∈ F 0 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 Now we discuss some properties of fuzzy numbers under addition and scaler multiplication; if ψ, ϕ ∈ F 0 and ρ ∈ R, then ψ+ ϕ and ρψ are defined as Remark 1. Obviously, F 0 is closed under addition and nonnegative scaler multiplication and the above-defined properties on F 0 are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number ρ ∈ R, us, a fuzzy mapping T can be identified by a parametrized triple: Definition 3 (see [40]). Let L � (m, n) and u ∈ L. en fuzzy mapping T: (m, n) ⟶ F 0 is said to be generalized differentiable (in short, G-differentiable) at u if there exists an element T , (u) ∈ F 0 . For all 0 < τ sufficiently small, there exist T(u + τ)− T(u), T(u)− T(u − τ), and the limits (in the metric D).
where the limits are taken in the metric space and H denotes the well-known Hausdorff metric on space of intervals.
Definition 4 (see [27]). A fuzzy mapping T: It is strictly convex fuzzy mapping if strict inequality holds for T(u) ≠ T(ϑ). T: K ⟶ F 0 is said to be concave fuzzy mapping if − T is convex on K. It is strictly concave fuzzy mapping if strict inequality holds for T(u) ≠ T(ϑ).
Definition 5 (see [27]). A fuzzy mapping T: Definition 6 (see [2]). e set K ξ in R is said to be invex set with respect to arbitrary bifunction ξ(., .), if Definition 7 (see [17]). A fuzzy mapping T: K ξ ⟶ F 0 is called preinvex on the invex set K ξ with respect to bifunction ξ if It is strictly preinvex fuzzy mapping if strict inequality holds for T(u) ≠ T(ϑ). T: K ξ ⟶ F 0 is said to be preconcave fuzzy mapping if − T is preinvex on K ξ . It is strictly preconcave fuzzy mapping if strict inequality holds for T(u) ≠ T(ϑ).

Lemma 1.
(see [42]). Let K ξ be an invex set with respect to ξ and let T: K ξ ⟶ F 0 be a fuzzy mapping parametrized by Definition 8 (see [17]). A comparable fuzzy mapping For further study, let K ξ be a nonempty closed invex set in R. Let T: K ξ ⟶ F 0 be a fuzzy mapping and let ξ: K ξ × K ξ ⟶ 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.

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. Let K ξ be an invex set and let T: K ξ ⟶ F 0 be fuzzy mapping. en 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 It is strictly strongly generalized preinvex fuzzy mapping if strict inequality holds for T(u) ≠ T(ϑ) and T: K ξ ⟶ F 0 is said to be strongly generalized preconcave fuzzy mapping if − T is strongly generalized preinvex on K ξ . It is strictly generalized preconcave fuzzy mapping if strict inequality holds for T(u) ≠ T(ϑ). Now we discuss some special cases of strongly generalized preinvex fuzzy mappings. If which is called the higher-ordered strongly preinvex fuzzy mapping. is 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.
If ω � 0, then higher-ordered strongly convex fuzzy mapping becomes convex fuzzy mapping; that is, Mapping T is called the strongly generalized Jensen preinvex (in short, J-preinvex) fuzzy mapping.
We also define the strongly generalized affine J-preinvex fuzzy mapping.
Definition 10. A mapping T: K ξ ⟶ F 0 is said to be strongly generalized affine preinvex fuzzy mapping on K ξ with respect to an arbitrary nonnegative function Ω and bifunction ξ, if, ∀u, ϑ ∈ K ξ , τ ∈ [0, 1], there exists a positive number ω such that where ξ: If τ � (1/2), then we also say that T is strongly generalized affine J-preinvex fuzzy mapping such that for allu, ϑ ∈ K ξ Remark 2. e strongly generalized preinvex fuzzy mappings have some very nice properties similar to preinvex fuzzy mapping: (i) If T is strongly generalized preinvex fuzzy mapping, then σT is also strongly generalized preinvex for σ ≥ 0 (ii) If T and G both are strongly generalized preinvex fuzzy mappings with respect to an arbitrary nonnegative function Ω(.) and bifunction ξ(., .), then max(T(u), G(u)) is also strongly generalized preinvex fuzzy mapping with respect to Ω and ξ Theorem 2. Let K ξ be an invex set with respect to ξ and let T: K ξ ⟶ F 0 be a fuzzy mapping parametrized by for all u ∈ K ξ , and then T is strongly generalized preinvex fuzzy mapping on K ξ with modulus ω if and only if, for all c ∈ [0, 1], T * (u, c) and T * (u, c) are strongly generalized preinvex functions with respect to Ω, ξ, and modulus ω.
Let K ξ be a convex set and let T: K ⟶ F 0 be a fuzzy mapping parametrized by and then F is a strongly generalized convex fuzzy mapping on K ξ if and only if, for all c ∈ [0, 1], T * (u, c) and T * (u, c) are strongly generalized convex functions on K ξ . □ Example 1. We consider the fuzzy mappings T: (0, ∞) ⟶ F 0 defined by Since end point functions T * (c), T * (c) are strongly generalized preinvex for each c ∈ [0, 1], T is a strongly generalized preinvex fuzzy mapping with respect to and 0 < ω ≤ 1 . From Example 1, 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 mapping J: K ξ ⟶ F 0 be strongly generalized affine preinvex with respect to Ω and ξ. en 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.

Remark 3. From Definition 12, we have
(46) It can be easily seen that each strongly generalized logpreinvex fuzzy mapping on K ξ is strongly generalized quasipreinvex fuzzy mapping and strongly generalized preinvex fuzzy mapping on K ξ is strongly generalized quasi-preinvex fuzzy mapping, when T is a comparable fuzzy mapping.
Definition 13. A fuzzy mapping T: K ξ ⟶ F 0 is called pseudo-preinvex on K ξ if there exists a strictly positive bifuzzy mapping b(., .) such that Theorem 4. Let T be a strongly generalized preinvex fuzzy mapping on K ξ such that T(ϑ)≺T(u). en fuzzy mapping T is strongly generalized pseudo-preinvex with respect to same nonnegative function Ω and ξ.
Proof. Let T(ϑ)≺T(u) and let T be a strongly generalized preinvex fuzzy mapping.
□ Definition 15. A G-differentiable fuzzy mapping T: K ξ ⟶ F 0 is called strongly generalized invex with respect to Ω and ξ if, ∀u, ϑ ∈ K ξ , there exists constant ω > 0 such that Definition 16. A G-differentiable fuzzy mapping T: K ξ ⟶ F 0 is called strongly generalized pseudo-invex with respect to Ω and ξ if, ∀u, ϑ ∈ K ξ , there exists constant ω > 0 such that Definition 17. A G-differentiable fuzzy mapping T: K ξ ⟶ F 0 is called strongly generalized quasi-invex with respect to Ω and ξ if, ∀u, ϑ ∈ K ξ , there exists constant ω > 0 such that Definition 19. A G-differentiable fuzzy mapping T: K ξ ⟶ F 0 is called quasi-invex with respect to Ω and ξ if, ∀u, ϑ ∈ K ξ such that If ξ(ϑ, u) � − ξ(u, ϑ), then definitions reduce to known ones. All Definitions 15-19 may play an important role in fuzzy optimization problem and mathematical programing. 8 Journal of Mathematics Theorem 6. Let T: K ξ ⟶ F 0 be a G-differentiable strongly generalized preinvex fuzzy mapping. en T is a strongly generalized invex fuzzy mapping.

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Taking limit in the above inequality as τ ⟶ 0, we have From that, we have and when ω � 0, eorem 6 reduces to the following result. □ Theorem 7. Let T: K ξ ⟶ F 0 be a G-differentiable preinvex fuzzy mapping. en, T is an invex fuzzy mapping.
To prove conversion of eorem 6, 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]: e following result finds the conversion of eorem 6.
As a special case of eorem 8, we have the following.
Theorem 9. Let T be a G-differentiable fuzzy mapping on K ξ 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. e proof is similar to that of eorem 8.

□
Theorem 10. Let G-differential T , of fuzzy mapping T on K ξ be a strongly generalized pseudomonotone fuzzy operator and Condition C holds. Let the function Ω be homogenous of degree 2 and even. en, T is a strongly generalized pseudoinvex fuzzy mapping.
that is Hence, T(u) is a strongly generalized pseudo-invex fuzzy mapping. □ Theorem 11. Let G-differential T , of fuzzy mapping T on K ξ be a pseudomonotone fuzzy operator and Condition C holds. en T is a pseudo-invex fuzzy mapping.
Theorem 12. Let G-differential T , of fuzzy mapping T on K ξ be a quasi-monotone fuzzy operator and Condition C holds. en 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.

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 where K ξ is a subset of R and T: K ξ ⟶ F 0 is a fuzzy mapping. A point u ∈ K ξ is called a feasible point. If u ∈ K ξ and no ϑ ∈ K ξ , T(u)≺T(ϑ), then u is called an optimal solution, a global optimal solution, or simply a solution to the fuzzy optimization problem. Theorem 13. Let T be a G-differentiable strongly generalized preinvex fuzzy mapping modulus ω > 0. If u ∈ K ξ is the minimum of the mapping T, then T(ϑ)− T(u) ≽ ωΩ(ξ(ϑ, u)), ∀ u, ϑ ∈ K ξ (111) Proof. Let u ∈ K ξ be a minimum of T. en, ∀ϑ ∈ K ξ , we have T(u) ≼ (ϑ).