On Strongly Generalized Preinvex Fuzzy Mappings

Liu, Peide; Bilal Khan, Muhammad; Noor, Muhammad Aslam; Noor, Khalida Inayat

doi:https://doi.org/10.1155/2021/6657602

Journal of Mathematics

On this page

Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2021 | Article ID 6657602 | https://doi.org/10.1155/2021/6657602

On Strongly Generalized Preinvex Fuzzy Mappings

Peide Liu,¹Muhammad Bilal Khan,²Muhammad Aslam Noor,²and Khalida Inayat Noor²

Academic Editor: Kaleem R. Kazmi

Received14 Dec 2020

Revised12 Mar 2021

Accepted15 Mar 2021

Published01 Apr 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.

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 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. 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. This inequality is itself an interesting outcome of our main results.

2. Preliminaries

Let be the set of real numbers. A fuzzy subset of is characterized by a mapping called the membership function, for each fuzzy set and ; then -level sets of are denoted and defined as follows: . If , then is called support of . By , we define the closure of .

Definition 1. A fuzzy set is said to be fuzzy number with the following properties: (1) is normal; that is, there exists such that (2) is upper semicontinuous; that is, for given there exist and such that for all with (3) is fuzzy convex; that is, , (4) is compact denotes the set of all fuzzy numbers. For fuzzy number, it is convenient to distinguish following -levels:From these definitions, we havewhereEach is also a fuzzy number, defined asThus, a fuzzy number can be identified by a parametrized triple:This leads to the following characterization of a fuzzy number in terms of the two end point functions and

Theorem 1 (see [35]). Suppose that and satisfy the following conditions: (1) is a nondecreasing function(2) is a nonincreasing function(3)(4) and are bounded and left continuous on and right continuous at Then , defined byis a fuzzy number and its parameterization form is given by Moreover, if is a fuzzy number with parametrization given by then functions and find conditions (1)–(4).
Let represent parametrically and , respectively. We say that if, for all , and . If then there exists such that or . We say that it is comparable if, for any , we have or ; otherwise, they are noncomparable. Sometimes we may write instead of and note that we may say that is a partial ordered set under the relation .
If there exists such that and then by this result we have the existence of Hukuhara difference of and , and we say that is the H-difference of and denoted by ; see [43]. If H-difference exists, thenNow we discuss some properties of fuzzy numbers under addition and scaler multiplication; if and , then and are defined as

Remark 1. Obviously, is closed under addition and nonnegative scaler multiplication and the above-defined properties on are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number

Definition 2. A mapping is called fuzzy mapping. For each denote . Thus, a fuzzy mapping can be identified by a parametrized triple:

Definition 3 (see [40]). Let and . Then fuzzy mapping is said to be generalized differentiable (in short, G-differentiable) at if there exists an element .
For all sufficiently small, there exist , and the limits (in the metric ). where the limits are taken in the metric space , for .and denotes the well-known Hausdorff metric on space of intervals.

Definition 4 (see [27]). A fuzzy mapping is called convex on the convex set ifIt is strictly convex fuzzy mapping if strict inequality holds for . is said to be concave fuzzy mapping if is convex on It is strictly concave fuzzy mapping if strict inequality holds for .

Definition 5 (see [27]). A fuzzy mapping is called quasi-convex on the convex set if

Definition 6 (see [2]). The set in is said to be invex set with respect to arbitrary bifunction ifThe invex set is also called -connected set. Note that convex set is called an invex set in classical sense, but the converse is not valid. For instance, the set is an invex set with respect to nontrivial bifunction given as

Definition 7 (see [17]). A fuzzy mapping is called preinvex on the invex set with respect to bifunction if, where
It is strictly preinvex fuzzy mapping if strict inequality holds for . is said to be preconcave fuzzy mapping if is preinvex on It is strictly preconcave fuzzy mapping if strict inequality holds for .

Lemma 1. (see [42]). Let be an invex set with respect to and let be a fuzzy mapping parametrized by Then is preinvex on if and only if, for all and are preinvex with respect to on .

Definition 8 (see [17]). A comparable fuzzy mapping is called quasi-preinvex on the invex set with respect to if, , such thatFor further study, let be a nonempty closed invex set in Let be a fuzzy mapping and let be an arbitrary bifunction. Let be a nonnegative function, defined as . 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. Let be an invex set and let be fuzzy mapping. Then is said to be strongly generalized preinvex fuzzy mapping with respect to an arbitrary nonnegative function and bifunction , if there exists a constant such that and
It is strictly strongly generalized preinvex fuzzy mapping if strict inequality holds for and is said to be strongly generalized preconcave fuzzy mapping if is strongly generalized preinvex on It is strictly generalized preconcave fuzzy mapping if strict inequality holds for .
Now we discuss some special cases of strongly generalized preinvex fuzzy mappings.
If , , then, for all (4) becomeswhich 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 , is called strongly preinvex fuzzy mapping with respect to bifunction
If then strongly generalized preinvex fuzzy mapping becomes preinvex fuzzy mapping with respect to bifunction ; that is,If with and , then and inequality (4) becomesand this is called the higher-ordered strongly convex fuzzy mapping. When , is called strongly convex fuzzy mapping.
If , then higher-ordered strongly convex fuzzy mapping becomes convex fuzzy mapping; that is,If , then (22) becomesMapping 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 is said to be strongly generalized affine preinvex fuzzy mapping on with respect to an arbitrary nonnegative function and bifunction if, there exists a positive number such thatwhere and
If , then we also say that is strongly generalized affine J-preinvex fuzzy mapping such thatfor all

Remark 2. The strongly generalized preinvex fuzzy mappings have some very nice properties similar to preinvex fuzzy mapping:(i)If is strongly generalized preinvex fuzzy mapping, then is also strongly generalized preinvex for (ii)If and both are strongly generalized preinvex fuzzy mappings with respect to an arbitrary nonnegative function and bifunction , then is also strongly generalized preinvex fuzzy mapping with respect to and

Theorem 2. Let be an invex set with respect to and let be a fuzzy mapping parametrized byfor all and then is strongly generalized preinvex fuzzy mapping on with modulus if and only if, for all and are strongly generalized preinvex functions with respect to , , and modulus

Proof. Assume that, for each and are strongly generalized preinvex functions with respect to , and modulus on Then, from (4), for all , we haveThen, by (30), (9), (10), and (11), we obtainHence, is a strongly generalized preinvex fuzzy mapping on with modulus
Conversely, let be a strongly generalized preinvex fuzzy mapping on with modulus Then, for all and we haveFrom (30), we haveFrom (30), (9), (10), and (11), we obtainfor all and Then, by strongly generalized preinvexity of , we have, for all and such thatfor each Hence, the result follows.
Note that if then Theorem 2 reduces to the following statement.
Let be a convex set and let be a fuzzy mapping parametrized byand then is a strongly generalized convex fuzzy mapping on if and only if, for all and are strongly generalized convex functions on .

Example 1. We consider the fuzzy mappings defined byThen, for each we have . Since end point functions are strongly generalized preinvex for each , is a strongly generalized preinvex fuzzy mapping with respect toand
From Example 1, it can be easily seen that, for each , 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 be strongly generalized affine preinvex with respect to and . Then is strongly generalized preinvex fuzzy with respect to same and if and only if is a preinvex fuzzy mapping.

Proof. The “If” part is obvious. To prove the “only if” part, assume that is strongly generalized affine preinvex with respect to nonnegative function and bifunction ; then there exists such thatand since is a strongly generalized preinvex fuzzy mapping, we haveand, from (40) and (41), we havefrom which it follows thatIt is shown that is a strongly generalized preinvex fuzzy mapping.

Definition 11. A comparable fuzzy mapping is said to be strongly generalized quasi-preinvex on with respect to nonnegative function and bifunction if, , there exists a positive number such thatwhere and

Definition 12. A fuzzy mapping is called strongly generalized log-preinvex on with respect to nonnegative function and bifunction , if, there exists a positive number such thatwhere , and

Remark 3. From Definition 12, we haveIt can be easily seen that each strongly generalized log-preinvex fuzzy mapping on is strongly generalized quasi-preinvex fuzzy mapping and strongly generalized preinvex fuzzy mapping on is strongly generalized quasi-preinvex fuzzy mapping, when is a comparable fuzzy mapping.

Definition 13. A fuzzy mapping is called pseudo-preinvex on if there exists a strictly positive bifuzzy mapping such thatfor all , where

Theorem 4. Let be a strongly generalized preinvex fuzzy mapping on such that Then fuzzy mapping is strongly generalized pseudo-preinvex with respect to same nonnegative function and .

Proof. Let and let be a strongly generalized preinvex fuzzy mapping. Then, for all , there exists modulus such thatand, therefore, for every , we haveFrom the above equation, we haveFrom (50), we havethat iswhere . This completes the proof.

4. G-Differentiable Strongly Generalized Preinvex Fuzzy Mappings

Throughout this section, we assume that the nonnegative function is homogenous of degree two and even; that is, , for all , , unless otherwise specified.

Definition 14. A fuzzy mapping is called sharply strongly generalized pseudo-preinvex with respect to and on if, , there exists such thatwhere is G-differentiable of at

Theorem 5. Let be a sharply strongly generalized pseudo-preinvex fuzzy mapping on with Then,

Proof. Since is a sharply strongly generalized pseudo-preinvex fuzzy mapping on , , there exists such thatwhich implies thatand, therefore, for every , we havefrom which we getand, taking limit in the above inequality as , we getHence, the result follows.

Definition 15. A G-differentiable fuzzy mapping is called strongly generalized invex with respect to and if, , there exists constant such that

Definition 16. A G-differentiable fuzzy mapping is called strongly generalized pseudo-invex with respect to and if, , there exists constant such that

Definition 17. A G-differentiable fuzzy mapping is called strongly generalized quasi-invex with respect to and if, , there exists constant such that

Definition 18. A G-differentiable fuzzy mapping is called pseudo-invex with respect to and if, such that

Definition 19. A G-differentiable fuzzy mapping is called quasi-invex with respect to and if, such thatIf then definitions reduce to known ones. All Definitions 15–19 may play an important role in fuzzy optimization problem and mathematical programing.

Theorem 6. Let be a G-differentiable strongly generalized preinvex fuzzy mapping. Then is a strongly generalized invex fuzzy mapping.

Proof. Let be a G-differentiable strongly generalized preinvex fuzzy mapping. Since is a strongly generalized preinvex fuzzy mapping, there exists constant , and, for all and , we haveand, therefore, for every , we havewhich implies thatTaking limit in the above inequality as , we haveFrom that, we haveand when , Theorem 6 reduces to the following result.

Theorem 7. Let be a G-differentiable preinvex fuzzy mapping. Then, is an invex fuzzy mapping.
To prove conversion of Theorem 6, we need the following assumption regarding the function which plays an important role in G-differentiation of the main results.
Condition C is as follows [33]:The following result finds the conversion of Theorem 6.

Theorem 8. Let be a G-differentiable fuzzy mapping on and Condition C holds. Let the function be homogenous of degree 2 and even. Then the following are equivalent:(a) is a strongly generalized preinvex fuzzy mapping:

Proof. () implies ().
The demonstration is analogous to the demonstration of Theorem 6.
() implies (). Assume that (71) holds. Then,Then, for every , we haveReplacing by and by in (21), we haveAdding (74) and (75), we havethat isand () implies (). Assume that (72) holds. Then,Therefore, for every , we haveSince is an invex set, we have for all and . Taking in (23), we get By using Condition C and the fact that is homogenous of degree 2 and even, we haveLetand, taking G-derivative with respect to , we getfrom which, using (81), we haveIntegrating (84) over with respect to , we getSince is a strongly generalized preinvex fuzzy mapping, for , we haveFrom that, we have() implies (). Assume that (71) holds. Since is an invex set, we have for all and . Taking in (19) for each , we getUsing Condition C, we haveIn a similar way, we obtainMultiplying (89) by and (90) by and adding the resultant, we havethat isHence, is a strongly generalized preinvex fuzzy mapping with respect to and .
Theorems 6 and 8 enable us to define the following new definitions.

Definition 20. A G-differentiable fuzzy mapping is said to be the following: (i)Strongly generalized monotone fuzzy operator with respect to and if and only if, , there exists constant such that(ii)Monotone fuzzy operator with respect to bifunction if and only if, ,(iii)Strongly generalized pseudomonotone fuzzy operator with respect to and if and only if, , there exists constant such that(iv)Strictly monotone fuzzy operator with respect to bifunction if and only if(v)Pseudomonotone fuzzy operator with respect to bifunction if and only if(vi)Quasi-monotone fuzzy operator with respect to bifunction if and only if(vii)Strictly pseudomonotone fuzzy operator with respect to bifunction if and only ifIf , then Definition 20 reduces to a new one.
As a special case of Theorem 8, we have the following.

Theorem 9. Let be a G-differentiable fuzzy mapping on and Condition C holds. If the function is homogenous of degree 2 and even, then is a strongly generalized -invex fuzzy mapping if and only if is a strongly generalized -monotone fuzzy operator.

Proof. The proof is similar to that of Theorem 8.

Theorem 10. Let G-differential of fuzzy mapping on be a strongly generalized pseudomonotone fuzzy operator and Condition C holds. Let the function be homogenous of degree 2 and even. Then, is a strongly generalized pseudo-invex fuzzy mapping.

Proof. Let be fuzzy strongly generalized pseudomonotone. Then, for all , we haveTherefore, for every , we haveSince is an invex set, we have for all and . Taking in (101), we getBy using Condition C, we haveAssume thatand, taking G-derivative with respect to , using (103), we haveIntegrating (105) over with respect to , we getwhich implies thatFrom condition (i), we havethat isHence, is a strongly generalized pseudo-invex fuzzy mapping.

Theorem 11. Let G-differential of fuzzy mapping on be a pseudomonotone fuzzy operator and Condition C holds. Then is a pseudo-invex fuzzy mapping.

Theorem 12. Let G-differential of fuzzy mapping on be a quasi-monotone fuzzy operator and Condition C holds. Then 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 problemwhere is a subset of and is a fuzzy mapping.

A point is called a feasible point. If and no , , then is called an optimal solution, a global optimal solution, or simply a solution to the fuzzy optimization problem.

Theorem 13. Let be a G-differentiable strongly generalized preinvex fuzzy mapping modulus . If is the minimum of the mapping , then

Proof. Let be a minimum of . Then, we haveTherefore, for every , we haveSince is an invex set, for all , , . Taking in (113), we getTaking limit in the above inequality as , we getand since is a G-differentiable strongly generalized preinvex fuzzy mapping,and, again taking limit in the above inequality as , we getfrom which, using (115), we havethat isThis completes the proof.

Remark 4. If is a G-differentiable strongly generalized preinvex fuzzy mapping modulus and, then is the minimum of the mapping .
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:and, , this is called variational-like inequality.
We consider the functional , defined as where is a G-differentiable preinvex fuzzy mapping and is a non-G-differentiable strongly preinvex fuzzy mapping.
We show that the minimum of the functional can be characterized by a class of variational-like inequalities.

Theorem 14. Let be a G-differentiable preinvex fuzzy mapping and let be a non-G-differentiable strongly generalized preinvex fuzzy mapping. Then the functional has minimum if and only if satisfies

Proof. Let be the minimum of ; then, by definition, for all , we haveand since is an invex set, , for all and . Replacing by in (124), we getand, therefore, for every , we havewhich implies that, using (122),Since is a strongly generalized preinvex fuzzy mapping,Now, dividing by “” and taking , we haveSince and from the above inequalities, we haveSince is a fuzzy mapping, from the above inequalities, we haveConversely, let (123) be satisfied to prove that is a minimum of . Assume that, for all , we haveTherefore, for every , we haveBy Theorem 7, we haveBy using (123), we haveand, hence,
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)If , then (123) is called a strongly fuzzy mixed variational-like inequality such as(ii)If , then (136) is called a strongly fuzzy mixed variational inequality such as(iii)If then (136) is called a fuzzy mixed variational-like inequality such as(iv)If then (137) is called a fuzzy mixed variational inequality such as(v)If then (138) is known as a fuzzy variational-like inequality.(vi)If then (139) is known as a fuzzy variational inequality.

Theorem 15. Let be a G-differentiable invex fuzzy mapping and let be a non-G-differentiable strongly preinvex fuzzy mapping. Then functional has minimum if and only if satisfies

Proof. The proof is similar to that of Theorem 14; 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.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

All authors contributed equally to the writing of this paper and all of them read and approved the final manuscript.

Acknowledgments

The authors acknowledge the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments. This study was funded by the National Natural Science Foundation of China (no. 71771140).

References

M. A. Hanson, “On sufficiency of the Kuhn-Tucker conditions,” Journal of Mathematical Analysis and Applications, vol. 80, pp. 545–550, 1980.
View at: Google Scholar
A. Ben-Isreal and B. Mond, “What is invexity,” The Anziam Journal, vol. 28, pp. 1–9, 1986.
View at: Google Scholar
S. R. Mohan and S. K. Neogy, “On invex sets and preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 189, no. 3, pp. 901–908, 1995.
View at: Publisher Site | Google Scholar
M. A. Noor, “Generalized convex functions,” Panamerican Mathematical Journal, vol. 4, pp. 73–89, 1994.
View at: Google Scholar
T. Weir and B. Mond, “Preinvex functions in multiobjective optimization,” Journal of Mathematical Analysis and Applications, vol. 136, pp. 29–38, 1986.
View at: Google Scholar
M. A. Noor, “Variational-like inequalities,” Optimization, vol. 30, pp. 323–330, 1994.
View at: Google Scholar
M. A. Noor and K. I. Noor, “On strongly generalized preinvex functions,” Journal of Inequalities in Pure and Applied Mathematics, vol. 6, p. 102, 2005.
View at: Google Scholar
M. A. Noor and K. I. Noor, “Some characterizations of strongly preinvex functions,” Journal of Mathematical Analysis and Applications, vol. 316, no. 2, pp. 697–706, 2006.
View at: Publisher Site | Google Scholar
B. T. Polyak, “Existence theorems and convergence of minimizing sequences in extremum problems with restrictions,” Soviet Mathematics Doklady, vol. 7, pp. 2–75, 1966.
View at: Google Scholar
S. Karamardian, “The nonlinear complementarity problem with applications, Part 2,” Journal of Optimization Theory and Applications, vol. 4, no. 3, pp. 167–181, 1969.
View at: Publisher Site | Google Scholar
G. Qu and N. Li, “On the exponential stability of primal-dual gradient dynamics,” IEEE Control Systems Letters, vol. 3, no. 1, pp. 43–48, 2019.
View at: Publisher Site | Google Scholar
K. Nikodem and Z. Pales, “Characterizations of inner product spaces by strongly convex functions,” Banach Journal of Mathematical Analysis, vol. 5, no. 1, pp. 83–87, 2011.
View at: Publisher Site | Google Scholar
M. Adamek, “On a problem connected with strongly convex functions,” Mathematical Inequalities & Applications, vol. 19, no. 4, pp. 1287–1293, 2016.
View at: Publisher Site | Google Scholar
N. I. Akhiezer, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, Edinburgh, Scotland, 1965.
M. U. Awan, M. A. Noor, V. N. Mishra, and K. I. Noor, “Some characterizations of general preinvex functions,” International Journal of Analysis and Applications, vol. 15, pp. 46–56, 2017.
View at: Google Scholar
M. V. Jovanovic, “A note on strongly convex and strongly quasiconvex functions,” Mathematics Notes, vol. 60, pp. 584-585, 1966.
View at: Google Scholar
Z.-Y. Peng, X.-J. Long, and Z. Lin, “Some new properties of strongly convex fuzzy sets,” Advances in Intelligent and Soft Computing, vol. 2, pp. 687–693, 2009.
View at: Publisher Site | Google Scholar
A. Iqbal and I. Ahmad, “Strong geodesic convex functions of order m,” Numerical Functional Analysis and Optimization, vol. 40, no. 15, pp. 1840–1846, 2019.
View at: Publisher Site | Google Scholar
M. A. Khan, I. Ahmad, and A. Aljohani, “Criterion for generalized weakly fuzzy invex monotonocities,” Advances in Fuzzy Systems, vol. 2018, Article ID 3749650, 2018.
View at: Publisher Site | Google Scholar
Z. Y. Peng, “Semistrict, G-preinvexity and its application,” Journal of Inequalities and Applications, vol. 198, no. 1, pp. 1–11, 2012.
View at: Publisher Site | Google Scholar
Z. Y. Peng and S. S. Chang, “Some properties of semi-G-preinvex functions,” Taiwanese Journal of Mathematics, vol. 17, no. 3, pp. 873–884, 2013.
View at: Publisher Site | Google Scholar
Z.-Y. Peng, K.-K. Li, and J.-T. Zhou, “Semi-prequasi-invex type multiobjective optimization and generalized fractional programming problems,” Journal of Nonlinear Sciences and Applications, vol. 9, no. 12, pp. 6142–6152, 2016.
View at: Publisher Site | Google Scholar
M. B. Khan, M. A. Noor, K. I. Noor, A. T. Ab Ghani, and L. Abdullah, “Extended perturbed mixed variational-like inequalities for fuzzy mappings,” Journal of Mathematics, vol. 2021, Article ID 6652930, 2021.
View at: Publisher Site | Google Scholar
L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 3, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
Y.-M. Liu, “Some properties of convex fuzzy sets,” Journal of Mathematical Analysis and Applications, vol. 111, no. 1, pp. 119–129, 1985.
View at: Publisher Site | Google Scholar
R. Lowen, “Convex fuzzy sets,” Fuzzy Sets and Systems, vol. 3, no. 3, pp. 291–310, 1980.
View at: Publisher Site | Google Scholar
E. Ammar and J. Metz, “On fuzzy convexity and parametric fuzzy optimization,” Fuzzy Sets and Systems, vol. 49, no. 2, pp. 135–141, 1992.
View at: Publisher Site | Google Scholar
A. K. Katsaras and D. B. Liu, “Fuzzy vector spaces and fuzzy topological vector spaces,” Journal of Mathematical Analysis and Applications, vol. 58, no. 1, pp. 135–146, 1977.
View at: Publisher Site | Google Scholar
M. A. Noor and K. I. Noor, “Generalized preinvex functions and their properties,” International Journal of Stochastic Analysis, vol. 2006, p. 12736, 2006.
View at: Publisher Site | Google Scholar
D. Dubois and H. Prade, “Operations on fuzzy numbers,” International Journal of Systems Science, vol. 9, no. 6, pp. 613–626, 1978.
View at: Publisher Site | Google Scholar
R. Goetschel and W. Voxman, “Topological properties of fuzzy numbers,” Fuzzy Sets and Systems, vol. 10, no. 1–3, pp. 87–99, 1983.
View at: Publisher Site | Google Scholar
N. Furukawa, “Convexity and local Lipschitz continuity of fuzzy-valued mappings,” Fuzzy Sets and Systems, vol. 93, no. 1, pp. 113–119, 1998.
View at: Publisher Site | Google Scholar
S. Nanda and K. Kar, “Convex fuzzy mappings,” Fuzzy Sets and Systems, vol. 48, no. 1, pp. 129–132, 1992.
View at: Publisher Site | Google Scholar
Y.-R. Syau, “On convex and concave fuzzy mappings,” Fuzzy Sets and Systems, vol. 103, no. 1, pp. 163–168, 1999.
View at: Publisher Site | Google Scholar
R. Goetschel and W. Voxman, “Elementary fuzzy calculus,” Fuzzy Sets and Systems, vol. 18, no. 1, pp. 31–43, 1986.
View at: Publisher Site | Google Scholar
H. Yan and J. Xu, “A class of convex fuzzy mappings,” Fuzzy Sets and Systems, vol. 129, no. 1, pp. 47–56, 2002.
View at: Publisher Site | Google Scholar
M. A. Noor, “Fuzzy preinvex functions,” Fuzzy Sets and Systems, vol. 64, no. 1, pp. 95–104, 1994.
View at: Publisher Site | Google Scholar
Y.-R. Syau, “Preinvex fuzzy mappings,” Computers & Mathematics with Applications, vol. 37, no. 3, pp. 31–39, 1999.
View at: Publisher Site | Google Scholar
Y.-R. Syau and E. S. Lee, “Fuzzy Weirstrass theorem and convex fuzzy mappings,” Computers & Mathematics with Applications, vol. 51, no. 12, pp. 1741–1750, 2006.
View at: Publisher Site | Google Scholar
B. Bede and S. G. Gal, “Generalizations of the differentiability of fuzzy-number-valued functions with applications to fuzzy differential equations,” Fuzzy Sets and Systems, vol. 151, no. 3, pp. 581–599, 2005.
View at: Publisher Site | Google Scholar
X. P. Ding and J. Y. Park, “A new class of generalized nonlinear implicit quasivariational inclusions with fuzzy mappings,” Journal of Computational and Applied Mathematics, vol. 138, no. 2, pp. 243–257, 2002.
View at: Publisher Site | Google Scholar
J. Li and M. A. Noor, “On characterizations of preinvex fuzzy mappings,” Computers & Mathematics with Applications, vol. 59, no. 2, pp. 933–940, 2010.
View at: Publisher Site | Google Scholar
L. Stefanini and B. Bede, “Generalized Hukuhara differentiability of interval-valued functions and interval differential equations,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 3-4, pp. 1311–1328, 2009.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2021 Peide Liu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

245

Downloads

458

Citations