Criterion for Generalized Weakly Fuzzy Invex Monotonocities

Khan, Meraj A.; Ahmad, Izhar; Aljohani, Abdulrahman

doi:https://doi.org/10.1155/2018/3749650

Advances in Fuzzy Systems

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

Research Article | Open Access

Volume 2018 | Article ID 3749650 | https://doi.org/10.1155/2018/3749650

Criterion for Generalized Weakly Fuzzy Invex Monotonocities

Meraj A. Khan,¹Izhar Ahmad,²and Abdulrahman Aljohani¹

Academic Editor: Pushpinder Singh

Received30 Nov 2017

Accepted07 Mar 2018

Published11 Apr 2018

Abstract

The present paper deals with the concepts of generalized fuzzy invex monotonocities and generalized weakly fuzzy invex functions. Some necessary conditions for weakly fuzzy invex monotonocities are presented. Moreover, the concept of fuzzy strong invex monotonocities and fuzzy strong invex functions are also discussed. To strengthen our definitions, we provide nontrivial examples of fuzzy invex monotonocities and weakly fuzzy invex functions.

1. Introduction

In the last few years, the conception of convexity and generalized convexity is well recognized in the optimization theory and accomplished a significant role in the computational economics, management, decision making, and operation research. Consequently, the generalized convexity and generalized monotonocities are fundamental tools in these areas of research.

Hanson [1] introduced the generalized version of convex function namely invex function. Further, generalized invex monotonocities have been explored by Ruiz-Garzon et al. [2] and Yang et al. [3]. A step forward Yang et al. [4] found that there were some errors in [2] and modified the results of Ruiz-Garzon et al. [2] and they also proposed the notion of strong pseudo invex monotonicity and quasi-invex monotonicity. In addition, Antczak [5, 6] and Suneja et al. [7] deliberated the properties and execution of preinvex functions and their generalizations for the nonlinear programming problems. Aghezzaf and Hachimi [8] presented the differentiable type I functions and derived the appropriate duality results for a Mond-Weir type dual. Gulati et al. [9] presented the more generalized class of convex function, called -type I functions, and derived sufficiency and duality results for a multiobjective programming problem. Recently, Ahmad et al. [10] discussed twice weakly differentiable and interval valued bonvex functions. Duality results are also discussed for Mangasarian type dual model.

The notion of fuzzy generalized convex functions has been investigated by several authors and presently, it is a fascinating and exciting area of research. In 1989, Nanda and Kar [11] presented the idea of convex fuzzy mappings and obtained the convex fuzzy mapping under the epigraph of convex function in a convex set. In [12], Furukawa proposed the concept of convexity and Lipschitz continuity for the category of fuzzy valued functions. Yan and Xu [13] investigated the convexity and quasi-convexity of fuzzy mapping involving the concept of ordering [14]. These concepts further generalized for fuzzy functions by Noor [15]. Qiu and Zhang [16] discussed the convexity invariance of fuzzy sets under the extension principles. In [17–21], authors proposed the perception of fuzzy convex mappings and presented the idea of invexity, pseudoconvexity, and pseudoinvexity for fuzzy mappings. Recently, Li et al. [22] introduced the fuzzy generalized convex mappings and discussed the properties. The optimality conditions and duality results are also presented under fuzzy weakly univex functions. Some new types of fuzzy starshapedness and their relationships and basic properties are investigated in [23]. Many authors have studied the applications of generalized fuzzy convex mappings to fuzzy optimization problems (see [24–28]). The purpose of the present paper is to develop the notion of generalized invex monotonocities for fuzzy valued mappings.

2. Notations and Preliminaries

Let be the universal set whose standard element is denoted by . A fuzzy set in is a function and support of is denoted by and is given by

Definition 1. If be a fuzzy set in and , then the cut of fuzzy set is defined as

Definition 2. A fuzzy number is a fuzzy set in and satisfies the following conditions:(i) is normal; that is, there exits such that ,(ii) is upper semicontinuous,(iii) is fuzzy convex that is, ,(iv) is compact.

The fuzzy numbers first introduced in [29] are defined as follows:

Definition 3. Let be two upper semicontinuous and decreasing functions and , and Then the fuzzy number is given bywhere and
Let be the set of fuzzy numbers and let be a fuzzy number if and only if is a nonempty compact convex subset of . This is denoted by for each A fuzzy number is resolved by the end points of the interval .
A real number is a particular case of fuzzy number and is specified byThe parametric form of a fuzzy number is presented as The following is the characterization of a fuzzy number in terms of the end point and

Lemma 4 (see [14]). Assume that and satisfy the following conditions:(i) is bounded increasing function,(ii) is bounded decreasing function,(iii),(iv)For , and ,(v) and For , , the fuzzy addition and scalar multiplication for are defined as We know that for any , , and , that is, for each , For any , if for each , and If and then . We say that , and there exists some , such that or For , if either or , then and are comparable, or else they are not comparable, where is a partial order relation on

Definition 5 (triangular fuzzy number). A fuzzy number is called a triangular fuzzy number if In addition a fuzzy number is said to be linear, if and are linear. A fuzzy triangular number is denoted by For the triangular fuzzy number , we have

Definition 6 (fuzzy mapping). Let and let be a fuzzy mapping. Then the cut of is given by , where and . Thus is represented by the two functions and ; these functions are defined from to . Moreover, is bounded increasing function of and is bounded decreasing function of and for each

Definition 7 (continuity of a fuzzy mapping). If is a fuzzy mapping, then is said to be continuous at , if for each both the end point functions and are continuous at .

Differentiability of the functions is one of the important tool of the generalized convexity and invex monotonocities. In this paper our aim is to study the concept of fuzzy invex monotone and fuzzy invex function, so we will discuss the notion of differentiability for fuzzy mappings. Puri and Ralescu [30] presented the concept of H-differentiability for fuzzy mappings. Further the concept of S-differentiability, G-differentiability, and weak differentiability was given by Seikkala [31] and Bede and Gal [32], respectively. Rufián-Lizana et al. [33] pointed that G-differentiability is more general than H-differentiability and S-differentiability for fuzzy mappings. In [34], Bede and Stefanini showed that a weak differentiability of fuzzy mappings is different from G-differentiability. Therefore, it will be worthwhile to study the generalized invex monotonocities and invex functions by using weakly differentiable fuzzy mappings.

Definition 8 (weakly differentiable function [26]). Let be a fuzzy mapping. If the derivatives of , with respect to for each exist and are denoted by , , respectively, then is said to be weakly differentiable.

3. Generalized Convexity and Invex Monotonocities

In this section, we collected some basic definitions of generalized invex functions and generalized invex monotonocities.

Definition 9. A nonempty set is said to be invex if there exists a vector valued function , such that for any ,
Hanson [1] introduced the generalized version of convex function, namely, invex function. A function is said to be invex if there exists a vector valued function such that the next inequality holds, for all

Definition 10 (pseudoinvex monotone [2]). Let be an invex set with respect to Then the function is said to be (strictly) pseudoinvex monotone on if for all

Definition 11 (pseudoinvex function [2]). A differentiable function is said to be (strictly) pseudoinvex function with respect to if for all

Definition 12 (pseudoinvex monotone [2]). Let be an invex set with respect to Then the function is said to be (strictly) pseudoinvex monotone on if for all

Definition 13 (quasi-invex monotone [2]). A function , defined on an open invex subset of , is said to be quasi-invex monotone with respect to on if for all

Definition 14 (quasi-invex function [2]). A function which is differentiable on an open invex subset of is said to be quasi-invex function with respect to on if for all

Now we have the following definitions of strong pseudoinvex monotonicity and strong pseudoinvex functions.

Definition 15 (strong pseudoinvex monotone [3]). A function defined on an open invex subset of is said to be strong pseudoinvex monotone with respect to on , if there exists a scalar , such that for all

Definition 16 (strong pseudoinvex function [3]). A function which is differentiable on an open invex subset of is said to be strong pseudoinvex function with respect to on if there exists a scalar such that for all

4. Weakly Fuzzy Pseudoinvex Monotonicity

In this section, we propose the new concept of weakly fuzzy pseudoinvex monotone and weakly fuzzy pseudoinvex function and we will prove a necessary condition for weakly fuzzy pseudoinvex monotonicity.

Definition 17. A fuzzy mapping is said to be (Strictly) weakly fuzzy pseudoinvex monotone with respect to on if for any

In support of above definition, we have the following example.

Example 18. The triangular fuzzy valued function is a weakly fuzzy pseudoinvex monotone with respect to , where and
The given function can be written as , where and and , where and Then it is easy to see that is a weakly fuzzy monotone.

Definition 19. A fuzzy mapping , which is weakly differentiable on is said to be (strictly) weakly fuzzy pseudoinvex function with respect to on if for any

Moreover, we have the following example for weakly fuzzy pseudoinvex function.

Example 20. The triangular fuzzy valued function is a weakly fuzzy pseudoinvex function with respect to and
The given function can be written as ; that means , , and then by some simple computation we can see the following: Hence is a weakly fuzzy pseudoinvex function.

Now we will define the Condition C as follows.

Condition C. For vector valued function the following equations are called Condition C: for any and Moreover, we have the following result for subsequent use.

Note 21 (see [2]). From Condition C, we have

Theorem 22. Let be a fuzzy mapping on an open invex set with respect to and satisfies Condition C. Let be weakly differentiable on with the following assumptions:(i) and , for some (ii) is weakly fuzzy pseudoinvex monotone with respect to on Then is a weakly fuzzy pseudoinvex function with respect to on

Proof. Suppose thatOur aim is to show that Assume to the contrary that By the assumption (i) for some
By Note 21 and (25), we have the following inequalities: By the assumption that is weakly fuzzy pseudo monotone with respect to , then it follows from (26) that where
Since , then by Note 21, (27) become The above inequalities contradict (21). Hence is a weakly fuzzy pseudoinvex function with respect to

Theorem 23. Let be a fuzzy mapping on an open invex set with respect to and satisfies the Condition C. Let be weakly differentiable on with the following assumptions:(i) and , for each and ,(ii) is strictly weakly fuzzy pseudoinvex monotone with respect to on Then is a strictly weakly fuzzy pseudoinvex function with respect to on

Proof. Suppose that Our aim is to show that Assume to the contrary that By the assumption (i), we have for some
By Condition C, we have Utilizing (33) in (32), we have for some
By the strictly weakly fuzzy pseudo monotonicity of with respect to and (34) Applying the Condition C and the fact that , then (35) takes the form The above inequalities contradict (29). Hence, is a strictly weakly fuzzy pseudoinvex function with respect to

5. Weakly Fuzzy Quasi-invex Monotonicity

In this section, necessary conditions for weakly fuzzy quasi-invex monotone are discussed.

Definition 24. A fuzzy mapping is said to be weakly fuzzy quasi-invex monotone with respect to on if for any

Example 25. The triangular fuzzy valued function is a weakly fuzzy quasi-invex monotone with respect to , where and , .
; we have Now we compute the following for all and
Similarly, we can compute and It is easy to see that is not weakly fuzzy pseudoinvex monotone function.

Definition 26. A weakly differentiable fuzzy mapping is said to be weakly fuzzy quasi-invex function with respect to on if for any

Example 27. The fuzzy valued function is a weakly fuzzy quasi-invex function with respect to and
The given function can be written as ; we have Now, it is easy to show the following for all Hence, the function is a weakly fuzzy quasi-invex function, but not weakly pseudoinvex function.

Theorem 28. Let be a fuzzy mapping on an open invex set with respect to and satisfying Condition C. Let be weakly differentiable on with the following conditions(i) and , for some (ii) is weakly fuzzy quasi-invex monotone with respect to on Then is a weakly fuzzy quasi-invex function with respect to on

Proof. Suppose that is not weakly fuzzy quasi-invex function; then ButBy the assumption (i) and (43) for some
By Condition C, we have By utilizing (46) in (45), it is easy to see the following for some
Since is weakly fuzzy quasi-invex monotone, then from (47) we have Again applying the Condition C and the fact , the above equation reduced to which contradicts (44). Hence is a weakly fuzzy quasi-invex function with respect to

6. Fuzzy Strong Pseudo Invex Monotonicity

In this section, we propose the idea of fuzzy strong pseudoinvex monotonicity and fuzzy strong pseudo invex functions. Finally, we prove a necessary condition for fuzzy strong pseudoinvex monotone.

Definition 29. A fuzzy mapping is said to be fuzzy strong pseudoinvex monotone with respect to on an invex set if there exists a scalar such that for any

Now we have the following example.

Example 30. The triangular fuzzy valued function is a fuzzy strong pseudoinvex monotone with respect to and , where
For , then we have and Now we compute the following for all
Similarly, for , we can show the followingfor all and , but is neither weakly fuzzy pseudoinvex monotone function nor weakly fuzzy quasi-invex monotone function.

Definition 31. A weakly differentiable fuzzy mapping is said to be fuzzy strong pseudoinvex function with respect to on an invex set if there exists a scalar such that for any

Example 32. The triangular fuzzy valued function is a fuzzy strong pseudoinvex function with respect to and , where
We can write , and then we have and . . Then we have for all and ; that means Similarly, it is easy to show that Hence the function is a fuzzy strong pseudoinvex function with respect to and , where It is easy see that is neither weakly fuzzy pseudoinvex function nor weakly fuzzy quasi-invex function.

Theorem 33. Let be a fuzzy mapping on an open invex set with respect to and satisfying the Condition C. Let is weakly differentiable on with the following conditions(i), for all (ii) is a fuzzy strong pseudoinvex monotone with respect to on . Then is a fuzzy pseudoinvex function with respect to on

Proof. Suppose for any
From the assumption that is open invex set with respect to and satisfying the Condition C, the above inequalities become Since is a fuzzy strong pseudoinvex monotone with respect to , then there exists such that Again by Condition C and the above inequalities take the form for all
Let and .
Then from (60) and (61) we have,Integrating from 0 to 1 After integration, we have the following or By the assumption (i), we acquire Hence, is a fuzzy strong pseudo invex function.

7. Conclusion

The concept of fuzzy optimization is well recognized in the literature and many authors are showing their interest in this direction. The idea of fuzzy convexity has been studied by several authors. The motive of the present paper is to project the concept of generalized weakly fuzzy monotonocities and generalized weakly fuzzy invex functions with nontrivial examples. Moreover, we tried to build up the relationship between generalized fuzzy monotones and generalized fuzzy invex functions. The results proved in the present paper generalize the existing results appearing in the literature. Findings of this paper can be used for fuzzy multiobjective programming problems.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

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

Acknowledgments

The research of the second author is financially supported by King Fahd University of Petroleum and Minerals, Saudi Arabia, under the Internal Research Project no. IN161058.

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Copyright © 2018 Meraj A. Khan 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.

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