International Journal of Computational Intelligence Systems

Volume 14, Issue 1, 2021, Pages 1856 - 1870

Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities

Authors
Muhammad Bilal Khan1, Muhammad Aslam Noor1, Khalida Inayat Noor1, Yu-Ming Chu2, *, ORCID
1Department of Mathematics, COMSATS University Islamabad, Islamabad, Pakistan
2Department of Mathematics, Huzhou University, Huzhou, P. R. China
*Corresponding author. Email: chuyuming@zjhu.edu.cn
Corresponding Author
Yu-Ming Chu
Received 14 February 2021, Accepted 31 May 2021, Available Online 28 June 2021.
DOI
10.2991/ijcis.d.210616.001How to use a DOI?
Keywords
Higher-order strongly preinvex fuzzy mapping; Invex fuzzy mappings; Fuzzy monotonicity; Fuzzy mixed variational-like inequalities
Abstract

A family of fuzzy mappings is called higher-order strongly preinvex fuzzy mappings (HOS-preinvex fuzzy mappings), which take the place of generalization of the notion of nonconvexity is introduced through the “fuzzy-max” order among fuzzy numbers. This family properly includes the family of preinvex fuzzy mappings and is included in the family of quasi preinvex fuzzy mappings. With the support of examples, we have discussed some special cases. Some properties are derived and relations among the HOS-preinvex fuzzy mappings, HOS-invex fuzzy mappings, and fuzzy HOS-monotonicity are obtained under some mild conditions. Then, we have shown that optimality conditions of generalized differentiable HOS-preinvex fuzzy mappings and for the sum of generalized differentiable (briefly, G-differentiable) preinvex fuzzy mappings and nongeneralized differentiable HOS-preinvex fuzzy mappings can be distinguished by HOS-fuzzy variational-like inequalities and HOS-fuzzy mixed variational-like inequalities, respectively which can be viewed as novel applications. These inequalities are very interesting outcome of our main results and appear to be new ones. Several exceptional cases are debated. Presented results in this paper can be considered and development of previously established results.

Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

In past few decades, the ideas of convexity and nonconvexity are well acknowledged in optimization concepts and gifted a vital role in operation research, economics, decision-making, and management. Hanson [1] initiated to introduce a generalized class of convexity which is known as an invex function. The invex function played a significant role in mathematical programing. A step forward, the invex set and preinvex function were introduced and studied by Israel and Mond [2]. Also, Noor [3] examined the optimality conditions of differentiable preinvex functions and proved that variational-like inequalities would characterize the minimum. Many classical convexity generalizations and extensions have been investigated by several authors. The concept of strongly convex functions on the convex set was studied and implemented by Polyak [4], which plays a crucial role in the theory of optimization. The unique existence of a solution to nonlinear complementary problems was explored by Karmardian [5] by the use of strongly convex functions. Qu and Li [6] and Rufián-Lizana et al. [7] studied a convergence analysis to solve equilibrium problems and variational inequalities with the help of strongly convex functions. Noor and Noor [811] derived the useful properties of strongly preinvex function and investigated its applications. For further study, we refer to the readers about applications and properties of the convex functions and generalized convex functions, see [1216] and the references therein. The notions of higher-order strongly (HOS)-preinvex functions were initiated by Lin and Fokushim [17], and utilized these notions in the analysis of mathematical programing with equilibrium constraints. HOS-preinvex functions and their generalization have many applications in different areas such as multilevel games, engineering design, and economical equilibrium. Some of the authors presented different types of HOS-preinvex functions and discussed their characterizations like Alabdali et al. [18], Noor and Noor [9,19,20], and Mohsen et al. [21] considered different classes and prove that the optimality condition of HOS-preinvex functions can be distinguished by different kinds of variational inequalities like strongly variational inequality, strongly variational-like inequality, HOS-variational inequality, and HOS-variational-like inequality.

Similarly, the notions of convexity and nonconvexity play a vital role in optimization under fuzzy domain because during characterization of the optimality condition of convexity, we obtain fuzzy variational inequalities so variational inequality theory and fuzzy complementary problem theory established powerful mechanism of the mathematical problems and they have a friendly relationship. Many authors contributed to this fascinating and interesting field. In 1989, Nanda and Kar [22] were initiated to introduce convex fuzzy mappings and characterized the notion of convex fuzzy mapping through the idea of epigraph. A step forward, Furukawa [23] and Syau [24] proposed and examined fuzzy mapping from space n to the set of fuzzy numbers, fuzzy valued Lipschitz continuity, logarithmic convex fuzzy mappings, and quasi-convex fuzzy mappings. Besides, Chang [25] discussed the idea of convex fuzzy mapping and find its optimality condition with the support of fuzzy variational inequality. Generalization and extension of fuzzy convexity play a vital and significant implementation in diverse directions. So let’s note that, one of the most considered classes of nonconvex fuzzy mapping is preinvex fuzzy mapping. Noor [26] introduced this idea and proved some results that distinguish the fuzzy optimality condition of differentiable fuzzy preinvex mappings by fuzzy variational-like inequality. Fuzzy variational inequality theory and complementary problem theory established a strong relationship with mathematical problems. Recently, Li and Noor [27] established an equivalence condition of preinvex fuzzy mapping and characterizations about preinvex fuzzy mappings under some mild conditions. With the support of examples, Wu and Xu [28] updated the definition of convex fuzzy mappings and established a new approach regarding to the existence of a fuzzy preinvex mapping under the condition of lower or upper semicontinuity. In 2012, Rufian-Lizana et al. [7] reviewed the existing literature and made appropriate modifications to the results obtained by Wu and Xu [28] regarding invex fuzzy mappings. For differentiable and twice differentiable preinvex fuzzy mappings, Rufian-Lizana et al. [29] given the required and sufficient conditions. They demonstrated the validity of characterizations with help of examples, and improved the previous results provided by Li and Noor [27]. We refer to the readers for further analysis of literature on the applications and properties of variational-like inequalities and generalized convex fuzzy mappings, see [2,9,16,3047] and the references therein.

Motivated and inspired by the ongoing research work, we note that convex and generalized convex fuzzy mappings play an important role in fuzzy optimization. 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 and considers a family of classes of nonconvex fuzzy mappings is called HOS-preinvex fuzzy mappings and investigates some properties. Section 4 derives some relations among the HOS-preinvex fuzzy mappings, HOS-invex fuzzy mappings, and HOS-monotonicity under some mild conditions. Section 5 introduces the new classes of fuzzy variational-like inequality which is known as HOS-fuzzy variational-like inequality and HOS-fuzzy mixed variational-like inequality. Several special cases are also discussed. These inequalities are an interesting outcome of our main results.

2. PRELIMINARIES

Let be the set of real numbers whose standard element is symbolized by u. A fuzzy set φ in is a mapping φ:0,1 and support of φ is denoted by suppφ and is defined as

supp(φ)=u|φ(u)>0.

Definition 2.1.

If φ be a fuzzy set in and γ0,1, then γ-level sets of φ is denoted and is defined as follows:

φγ=u|φuγ.(1)

Definition 2.2.

A fuzzy number φ is a fuzzy set in with the following properties:

  1. φ is normal, i.e., there exists u such that φu=1;

  2. φ is upper semi continiuous;

  3. φ1τu+τϑminφu,φϑ, u,ϑ, τ0,1;

  4. suppφ is compact.

The LR-fuzzy numbers first introduced by Dubois and Prade [34] are defined as follows:

Definition 2.3.

Let L,R:[0,1][0,1] be two decreasing and upper semicontinuous functions with L0=R0=1 and L1=R1=0. Then the fuzzy number is defined as

φu=Lcuρ,  cρu<d,1,  cu<d,Ludr,  cu<d+r,0,  otherwise,
where r,ρ>0, and dc.

Let J0 denotes the set of all fuzzy numbers and let φJ0 be fuzzy number, if and only if, γ-levels φγ is a nonempty compact convex set of . This is represented by

φγ=φ*γ,φ*γ
where
φ*γ=infu|φuγ,
φ*γ=supu|φuγ.

Since each ρ is also a fuzzy number, defined as

ρ˜u=1if u=ρ0if uρ.

Thus a fuzzy number φ can be identified by a parametrized triples

φ*γ,φ*γ,γ:γ0,1.

This leads the following characterization of a fuzzy number in terms of the two end point functions φ*γ and φ*γ.

Theorem 2.1.

[48] Suppose that φ*γ:0,1 and φ*γ:0,1 satisfy the following conditions:

  1. φ*γ is a nondecreasing function.

  2. φ*γ is a nonincreasing function.

  3. φ*1φ*1.

  4. φ*γ and φ*γ are bounded and left continuous on (0,1] and right continuous at γ=0.

Then φ:0,1, defined by

φu=supγ:φ*γuφ*γ,
is a fuzzy number parameterization is given by φ*γ,φ*γ,γ:γ0,1. Moreover, If φ:0,1 is a fuzzy number with parametrization given by φ*γ,φ*γ,γ:γ0,1, then function φ*γ and φ*γ find the conditions (1)–(4).

Let φ,ϕJ0 represented parametrically φ*γ,φ*γ,γ: γ0,1 and ϕ*γ,ϕ*γ,γ:γ0,1, respectively. We say that φϕ if for all γ(0,1], φ*γϕ*γ, and φ*γϕ*γ. If φϕ, then there exist γ(0,1] such that φ*γ<ϕ*γ or φ*γϕ*γ. We say comparable if for any φ,ϕJ0, we have φϕ or φϕ otherwise they are noncomparable. Some time we may write φϕ instead of ϕφ and note that, we may say that J0 is a partial-ordered set under the relation .

If φ,ϕJ0, there exist ψJ0 such that φ=ϕ+˜ψ, then by this result we have existence of Hukuhara difference of φ and ϕ, and we say that ψ is the H-difference of φ and ϕ, and denoted by φ˜ϕ, see [49]. If H-difference exists, then

ψ*γ=φ˜ϕ*γ=φ*γϕ*γ,ψ*γ=φ˜ϕ*γ=φ*γϕ*γ.

Now we discuss some properties of fuzzy numbers under addition and scalar multiplication, if φ,ϕJ0 and ρ then φ+˜ϕ and ρφ define as

φ+˜ϕ=φ*γ+ϕ*γ,φ*γ+ϕ*γ,γ:γ0,1,(2)
ρφ=ρφ*γ,ρφ*γ,γ:γ0,1.(3)

Remark 2.1.

Obviously, J0 is closed under addition and nonnegative scaler multiplication and above defined properties on J0 are equivalent to those derived from the usual extension principle. Furthermore, for each scalar number ρ,

φ+˜ρ=φ*γ+ρ,φ*γ+ρ,γ:γ0,1.(4)

Definition 2.4.

A mapping F:KJ0 is called fuzzy mapping. For each γ0,1, denote Fuγ=F*u,γ,F*u,γ. Thus a fuzzy mapping F can be identified by a parametrized triples

Fu=F*u,γ,F*u,γ,γ:γ0,1.

Definition 2.5.

Let F:KJ0 be a fuzzy mapping. Then Fu is said to be continuous at uK, if for each γ0,1, both end point functions F*u,γ and F*u,γ are continuous at uK.

Definition 2.6.

[30] Let L=m,n and uL. Then fuzzy mapping F:m,nJ0 is said to be a generalized differentiable (in short, G-differentiable) at u if there exists an element F,uJ0 such that for all 0<τ, sufficiently small, there exist Fu+τ˜Fu,Fu˜Fuτ and the limits (in the metric D)

limτ0+Fu+τ˜Fuτ=limτ0+Fu˜Fuττ=F,u or limτ0+Fu˜Fu+ττ=limτ0+Fuτ˜Fuτ=F,u

or limτ0+Fu+τ˜Fuτ=limτ0+Fuτ˜Fuτ=F,u or limτ0+Fu˜Fu+ττ=limτ0+Fu˜Fuττ=F,u

where the limits are taken in the metric space J0,D, for φ,ϕJ0

Dφ,ϕ=sup0γ1Hφγ,ϕγ,
and H denote the well-known Hausdorff metric on space of intervals.

Definition 2.7.

[26] A fuzzy mapping F:KJ0 is called convex on the convex set K if

F1τu+τϑ1τFu+˜τFϑu,ϑK,τ0,1.

Definition 2.8.

[26] A fuzzy mapping F:KJ0 is called quasi-convex on the convex set K if

F1τu+τϑmaxFu,Fϑu,ϑK,τ0,1.

Definition 2.9.

[26] A fuzzy mapping F:KJ0 is called preinvex on the invex set K w.r.t. bi-function if

Fu+τϑ,u1τFu+˜τFϑu,ϑK,τ0,1,
where :K×K.

Lemma 2.1.

[27] Let K be an invex set w.r.t. and let F:KJ0 be a fuzzy mapping parametrized by

Fu=F*u,γ,F*u,γ,γ:γ0,1,uK.

Then Fu is preinvex fuzzy mapping on K if and only if, for all γ0,1,F*u,γ and F*u,γ are preinvex functions w.r.t. on K.

Definition 2.10.

[26] A fuzzy mapping F:KJ0 is called quasi-preinvex on the invex set K w.r.t. if

Fu+τϑ,umaxFu,Fϑu,ϑK,τ0,1.

For further study, let K be a nonempty invex set in . Let F:KJ0 be a fuzzy mapping and :K×K be an arbitrary bifunction. We denote . and .,. be the norm and inner product, respectively. Furthermore, throughout in this article fuzzy mappings are discussed through the so-called “fuzzy-max” order among fuzzy numbers. As it is well-known, the fuzzy-max order is a partial order relation “” on the set of fuzzy numbers.

3. HIGHER-ORDER STRONGLY PREINVEX FUZZY MAPPINGS

In this section, we propose a family of nonconvex fuzzy mappings which is known as HOS-preinvex fuzzy mappings. We define some different classes of HOS-preinvex fuzzy mappings and investigate some basic properties.

Definition 3.1.

Let K be an invex set and let F:KJ0 be fuzzy mapping. Then Fu is said to be HOS-preinvex fuzzy mapping w.r.t. bi-function .,., if there exist a constant Ω>0 such that

Fu+τϑ,u1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up,(5)

u,ϑK,τ0,1, where :K×K and p1.

Similarly, Fu is said to be HOS-preconcave fuzzy mapping on K if inequality (5) is reversed.

Now we discuss some special cases of HOS-preinvex fuzzy mappings:

  • If p=2, then (5) becomes

    Fu+τϑ,u1τFu+˜τFϑΩτ1τϑ,u2,u,ϑK,τ0,1.

Which is called the strongly preinvex fuzzy mapping. This is itself a very interesting problem to study its applications in pure and applied science like fuzzy optimization.

  • If Ω=0, then HOS-preinvex fuzzy mapping becomes preinvex fuzzy mapping w.r.t. bi-function .,., i.e.,

    Fu+τϑ,u1τFu+˜τFϑ,u,ϑK,τ0,1.

  • If ϑ,u=ϑu and p1, then (5) becomes

    Fu+τϑ,u1τFu+˜τFϑΩτp1τ+τ1τpϑup,u,ϑK,τ0,1.

Which is called the HOS-convex fuzzy mapping. When p=2, then Fu is called strongly convex fuzzy mapping.

  • If ϑ,u=ϑu and Ω=0, then HOS-preinvex fuzzy mapping becomes convex fuzzy mapping, i.e.,

    Fu+τϑu1τFu+˜τFϑ,u,ϑK,τ0,1.

  • If τ=12, then (5) becomes

  • F2u+ϑ,u2Fu+˜Fϑ2˜12pΩϑ,up,u,ϑK.

The mapping Fu is called the HOS-Jensen preinvex (in short, J-preinvex) fuzzy mapping. We also define the HOS-affine J-preinvex fuzzy mapping.

Definition 3.2.

A fuzzy mapping F:KJ0 is said to be HOS-affine preinvex fuzzy mapping on K w.r.t. bi-function , if there exist Ω>0 such that

Fu+τϑ,u=1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up,(6)
for all u,ϑK,τ0,1, where :K×K and p1.

In other words, a fuzzy mapping F:KJ0 is said to be HOS-affine preinvex fuzzy mapping on K w.r.t. bi-function , if Fu is both HOS-preinvex fuzzy mapping and HOS-preincave fuzzy mapping w.r.t. same bi-function .

If τ=12, then we also say that Fu is HOS-affine J-preinvex fuzzy mapping such that

F2u+ϑ,u2=Fu+˜Fϑ2˜12pϑ,up,u,ϑK.

Remark 3.1.

The HOS-preinvex fuzzy mappings have some very nice properties similar to convex fuzzy mappings:

  • If Fu is HOS-preinvex fuzzy mapping, then σFu is also HOS-preinvex for σ0.

  • If Fu and Gu both are HOS-preinvex fuzzy mappings w.r.t. bi-function .,., then maxFu,Gu is also HOS-preinvex fuzzy mapping w.r.t. .

We now prove a special result for HOS-preinvex fuzzy mapping which establish a equivalence relation between HOS-preinvex fuzzy mapping Fu, and end point functions F*u,γ and F*u,γ.

Theorem 3.1.

Let K be an invex set w.r.t. and let F:KJ0 be a fuzzy mapping parametrized by

Fu=F*u,γ,F*u,γ,γ:γ0,1,uK.(7)

Then Fu is HOS-preinvex fuzzy mapping on K with modulus Ω, if and only if, for all γ0,1, F*u,γ and F*u,γ are HOS-preinvex functions w.r.t. with modulus Ω.

(8)

Proof.

Assume that for each γ0,1, F*u,γ and F*u,γ are HOS-preinvex functions w.r.t. and modulus Ω on K. Then from (5), we have

F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,u,ϑK,τ0,1.

And

F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,u,ϑK,τ0,1.

Then by (8), (2), (3) and (4), we obtain

Fu+τϑ,u=F*u+τϑ,u,γ,F*u+τϑ,u,γ,γ:γ0,1,1τF*u,γ,1τF*u,γ,γ:γ0,1+˜τF*ϑ,γ,τF*ϑ,γ,γ:γ0,1˜Ωτp1τ+τ1τpϑ,up,=1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up,u,ϑK,τ0,1.

Hence Fu is HOS-preinvex fuzzy mapping on K with modulus Ω.

Conversely, let Fu be a strongly preinvex fuzzy mapping on K with modulus Ω. Then for all u,ϑK and τ0,1, we have

Fu+τϑ,u1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up.

From (8), we have

Fu+τϑ,u=F*u+τϑ,u,γ,F*u+τϑ,u,γ,γ:γ0,1,u,ϑK,τ0,1.

From (8), (2), (3) and (4), we obtain

1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up=1τF*u,γ,1τF*u,γ,γ:γ0,1+˜τF*ϑ,γ,τF*ϑ,γ,γ:γ0,1˜Ωτp1τ+τ1τpϑ,up,(9)
for all u,ϑK and τ0,1. Then by HOS-preinvexity of Fu, we have for all u,ϑK and τ0,1 such that
F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,
and
F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,
for each γ0,1. Hence, the result follows.

Note that, If ξϑ,u=ϑu, then Theorem 3.1, reduce to following statement such as

“Let K be a convex set and let F:KF0 be a fuzzy mapping parametrized by

Fu=F*u,γ,F*u,γ,γ:γ0,1,uK

Then F is HOS-convex fuzzy mapping on K if and only if, for all γ0,1,

F*u,γ and F*u,γ are HOS-convex function on K.”

Example 3.1.

We consider the fuzzy mappings F:0,1J0 defined by

Fuσ=σu2,σ0,u22u2σu2,σ(u2,2u2]0,oτherwise.

Then, for each γ0,1, we have Fγu=γu2,2γu2. Since end point functions F*u,γ, F*u,γ are HOS-preinvex functions for each γ0,1. Hence Fu is HOS-preinvex fuzzy mapping w.r.t.

ϑ,u=ϑu,
with 0<Ω1 and p2. It can be easily seen that for each Ω0,1 and p2, there exist a HOS-preinvex fuzzy mapping and Fu is neither convex fuzzy mapping and nor preinvex fuzzy mapping w.r.t. bifunction ϑ,u=ϑu with 0<Ω1 and p2.

We now established a result for HOS-preinvex fuzzy mapping which shows that the difference of HOS-preinvex fuzzy mapping and HOS-affine preinvex fuzzy mapping is a preinvex fuzzy mapping.

Theorem 3.2.

Let fuzzy mapping J:KJ0 be a HOS-affine preinvex w.r.t. . Then Fu is HOS-preinvex fuzzy w.r.t same , if and only if, G=FJ is preinvex fuzzy mapping.

Proof.

The “If” part is obvious. To prove the “only if” assume that, J:KJ0 be a HOS-affine preinvex w.r.t. bi-function , then there exist Ω>0 such that

Ju+τϑ,u=1τJu+˜τJϑ˜Ωτp1τ+τ1τpϑ,up.

Therefore, for each γ0,1, we have

J*u+τϑ,u,γ=1τJ*u,γ+τJ*ϑ,γΩτp1τ+τ1τpϑ,up,J*u+τϑ,u,γ=1τJ*u,γ+τJ*ϑ,γΩτp1τ+τ1τpϑ,up.(10)

Since Fu is HOS-preinvex fuzzy mapping, then, for each γ0,1, we have

F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,(11)
from (10) and (11), we have
F*u+τϑ,u,γJ*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γ1τJ*u,γτJ*ϑ,γ,F*u+τϑ,u,γJ*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γ1τJ*u,γτJ*ϑ,γ,
F*u+τϑ,u,γJ*u+τϑ,u,γ1τF*u,γJ*u,γ+τF*ϑ,γJ*ϑ,γ,F*u+τϑ,u,γJ*u+τϑ,u,γ1τF*u,γJ*u,γ+τF*ϑ,γJ*ϑ,γ,
from which it follows that
G*u+τϑ,u,γ=F*u+τϑ,u,γJ*u+τϑ,u,γ,G*u+τϑ,u,γ=F*u+τϑ,u,γJ*u+τϑ,u,γ,
G*u+τϑ,u,γ1τF*u,γJ*u,γ+τF*ϑ,γJ*ϑ,γ,G*u+τϑ,u,γ1τF*u,γJ*u,γ+τF*ϑ,γJ*ϑ,γ,
which implies that
G*u+τϑ,u,γ1τG*u,γ+τG*ϑ,γ,G*u+τϑ,u,γ1τG*u,γ+τG*ϑ,γ,
i.e.,
Gu+τϑ,u1τGu+˜τGu.

Showing that G=FJ is preinvex fuzzy mapping.

Definition 3.3.

A fuzzy mapping F:KJ0 is said to be HOS-quasi-preinvex on K w.r.t. bi-function , if there exist a Ω>0 such that

Fu+τϑ,umaxFu,Fϑ˜Ωτp1τ+τ1τpϑ,up,(12)

u,ϑK,τ0,1, where :K×K and p1.

Similarly, Fu is said to be HOS-quasi-preconcave fuzzy mapping on K if inequality (12) is reversed.

Remark 3.2.

From Definitions 3.1 and 3.3, for each γ0,1, we have

F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,
F*u+τϑ,u,γmaxF*u,γ,F*ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γmaxF*u,γ,F*ϑ,γΩτp1τ+τ1τpϑ,up,
i.e.,
Fu+τϑ,umaxFu,Fϑ˜Ωτp1τ+τ1τpϑ,up.

It can be easily note that each HOS-preinvex fuzzy mapping on K is HOS-quasi-preinvex fuzzy mapping, when F is a fuzzy mapping.

Definition 3.4.

A fuzzy mapping F:KJ0 is called pseudo preinvex on K if there exist a b:K×KJ0 such that

FϑFuFu+τϑ,uFu+˜ττ1bu,ϑ,u,ϑK,τ0,1,where b.,.0˜.(13)

Theorem 3.3.

Let Fu be a HOS-preinvex fuzzy mapping on K such that FϑFu. Then fuzzy mapping Fu is HOS-pseudo preinvex w.r.t. same .

Proof.

Let FϑFu and Fu be a HOS-preinvex fuzzy mapping. Then there exist modulus Ω and for all u,ϑK, τ0,1, such that

Fu+τϑ,u1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up.

Therefore, for every γ0,1, we have

F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,
F*u+τϑ,u,γF*u,γ+τF*ϑ,γF*u,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γF*u,γ+τF*ϑ,γF*u,γΩτp1τ+τ1τpϑ,up.(14)

From (14), we have

F*u+τϑ,u,γ<F*u,γ+ττ1F*u,γF*ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γ<F*u,γ+ττ1F*u,γF*ϑ,γΩτp1τ+τ1τpϑ,up,
F*u+τϑ,u,γ<F*u,γ+ττ1b*u,ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γ<F*u,γ+ττ1b*u,ϑ,γΩτp1τ+τ1τpϑ,up,
i.e.,
Fu+τϑ,uFu+˜ττ1bu,ϑ˜Ωτp1τ+τ1τpϑ,up,
where bu,ϑ=Fu˜Fϑ. This prove that Fu is HOS-pseudo preinvex fuzzy mapping w.r.t. same .

4. HIGHER-ORDER STRONGLY INVEX FUZZY MAPPINGS AND FUZZY MONOTONICITY

We need the following assumption regarding the function , which plays an important role in G-differentiation of the main results.

Condition C. [22]

ϑ,u+τϑ,u=1τϑ,u,
u,u+τϑ,u=τϑ,u.

Clearly for τ=0, we have ξϑ,u=0 if and only if ϑ=u, for all u,ϑK. For the application of Condition C, see [7,8,22,26,28,29,50].

Definition 4.1.

A G-differentiable fuzzy mapping F:KJ0 is called HOS-invex w.r.t. , if there exist a constant Ω>0 such that

Fϑ˜FuF,u,ϑ,u+˜Ωϑ,up,u,ϑK.(15)

If Ω=0, then HOS” invex fuzzy mapping is known as invex fuzzy mapping.

Example 4.1.

We consider the fuzzy mappings F:0,1J0 defined by, Fγu=γu2,2γu2, as in Example 3.1, then Fu is HOS-invex fuzzy mapping w.r.t. bifunction ϑ,u=ϑu, with 0<Ω1 and p2, where uϑ. We have F*u,γ=γu2 and F*u,γ=2γu2. Now we computing the following

F*ϑ,γF*u,γ=γϑ2γu2,
while
F*,u,γ,ϑ,u+Ωϑ,up=2γϑu+Ωϑup.

And γϑ2γu22γϑu+Ωϑup, with 0<Ω1 and p2, where uϑ.

Similarly, it can be easily show that

F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up.

Hence, Fu is HOS” invex fuzzy mapping w.r.t. bifunction ϑ,u=ϑu, with 0<Ω1 and p2. It can be easily seen that Fu is not invex fuzzy mapping w.r.t. bifunction ϑ,u=ϑu.

Definition 4.2.

A G-differentiable fuzzy mapping F:KJ0 is called HOS-pseudo invex w.r.t. , if there exist a constant Ω>0 such that

F,u,ϑ,u+˜Ωϑ,up0˜                                        Fϑ˜Fu0˜,u,ϑK.(16)

If Ω=0, then HOS” pseudo invex fuzzy mapping is known as pseudo invex fuzzy mapping w.r.t. .

Definition 4.3.

A G-differentiable fuzzy mapping F:KJ0 is called HOS-quasi invex w.r.t. , if there exist a constant Ω>0 such that

FϑFuF,u,ϑ,u+˜Ωϑ,up0˜,u,ϑK.(17)

If Ω=0, then HOS” pseudo invex fuzzy mapping is known as quasi invex fuzzy mapping w.r.t. .

If ϑ,u=u,ϑ, then Definitions 4.14.3 reduce to known ones. All these definitions may play important role in fuzzy optimization problem and mathematical programing.

Example 4.2.

We consider the fuzzy mappings F:0,J0 defined by, Fγu=γu,54γu, then Fu is HOS-pseudo invex fuzzy mapping w.r.t. bifunction ϑ,u=ϑu, with 0Ω and p1, where uϑ. We have F*u,γ=γu and F*u,γ=54γu. Now we computing the following:

F*,u,γ,ϑ,u+Ωϑ,up=γϑu+Ωϑup0,
for all u,ϑK and γ0,1 with uϑ, 0Ω and p1; which implies that
F*ϑ,γ=γϑγu=F*u,γ,
  F*ϑ,γF*u,γ,(18)

Similarly, it can be easily show that

F*,u,γ,ϑ,u+Ωϑ,up=54γϑu+Ωϑup0,
for all u,ϑK and γ0,1 with uϑ, 0Ω and p1; that means
F*ϑ,γ=54γϑγu=F*u,γ,

From which, It follows that

F*ϑ,γF*u,γ.(19)

Hence, the fuzzy mapping Fγu=γu,54γu is HOS-pseudo invex fuzzy mapping w.r.t. ϑ,u=ϑu, with 0Ω and p1, where uϑ. it can be easily note that Fu is neither pseudo invex fuzzy mapping nor quasi invex fuzzy mapping w.r.t. .

Theorem 4.1.

Let F:KJ0 be a G-differentiable HOS-preinvex fuzzy mapping on invex set K and let condition C hold. Then Fu is HOS-preinvex fuzzy mapping, if and only if Fu is a HOS-invex fuzzy mapping.

Proof.

Let F:KJ0 be G-differentiable HOS-preinvex fuzzy mapping. Since Fu is HOS-preinvex fuzzy mapping then there exist a constant Ω>0, for all u,ϑK and τ0,1, we have

Fu+τϑ,u1τFu+˜τFϑ˜Ωτp1τ+τ1τpϑ,up,Fu+˜τFϑ˜Fu˜Ωτp1τ+τ1τpϑ,up,

Therefore, for every γ0,1, we have

F*u+τϑ,u,γF*u,γ+τF*ϑ,γF*u,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γF*u,γ+τF*ϑ,γF*u,γΩτp1τ+τ1τpϑ,up,
which implies that
τF*ϑ,γF*u,γF*u+τϑ,u,γF*u,γ+Ωτp1τ+τ1τpϑ,up,τF*ϑ,γF*u,γF*u+τϑ,u,γF*u,γ+Ωτp1τ+τ1τpϑ,up,
F*ϑ,γF*u,γF*u+τϑ,u,γF*u,γτ+Ωτp11τ+1τpϑ,up,F*ϑ,γF*u,γF*u+τϑ,u,γF*u,γτ+Ωτp11τ+1τpϑ,up.

Taking limit in the above inequality as τ0, we have

F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up,
F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up,
i.e.,
Fϑ˜FuF,u,ϑ,u+˜Ωϑ,up.(20)

Conversely, assume that Fu is a HOS-invex fuzzy mapping. Since K is an invex set then, we have, ϑτ=u+τϑ,uK, for all u,ϑK and τ0,1. Taking ϑ=ϑτ in (20), we get

F*ϑτ,γF*u,γF*,u,γ,ϑτ,u+Ωϑ,up,F*ϑτF*u,γF*,u,γ,ϑτ,u+Ωϑ,up,

Using Condition C, we have

F*ϑτ,γF*u,γ1τF*,u,γ,ϑ,u+Ω1τpϑ,up,F*ϑτ,γF*u,γ1τF*,u,γ,ϑ,u+Ω1τpϑ,up.(21)

In a similar way, we have

F*u,γF*ϑτ,γτF*,u,γ,ϑ,u+Ωτpϑ,up,F*u,γF*ϑτ,γτF*,u,γ,ϑ,u+Ωτpϑ,up.(22)

Multiplying (21) by τ and (22) by 1τ, and adding the resultant, we have

F*ϑτ,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,F*ϑτ,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,
which implies that
Fu+τϑ,u,γ1τFu,γ+˜τFϑ,γ˜Ωτp1τ+τ1τpϑ,up.

Hence, Fu is HOS-preinvex fuzzy mapping w.r.t. .

As special case of Theorem 4.2, when Ω=0, we have the following:

Corollary 4.1.

Let F:KJ0 be a G-differentiable fuzzy mapping on invex set K and let condition C hold. Then Fu is preinvex fuzzy mapping, if and only if Fu is a invex fuzzy mapping.

Theorem 4.2.

Let Fu be a G-differentiable HOS-preinvex fuzzy mapping on K and Condition C hold. If Fu is a HOS-invex fuzzy mapping, then

F,u,ϑ,u+˜F,ϑ,u,ϑ˜Ωϑ,up+u,ϑp,u,ϑK.(23)

Proof.

Assume that Fu is a HOS-invex fuzzy mapping. Then, for every γ0,1, we have

F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up,F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up.(24)

Replacing ϑ by u and u by ϑ in (24), we have

F*u,γF*ϑ,γF*,ϑ,γ,u,ϑ+Ωu,ϑp,F*u,γF*ϑ,γF*,ϑ,γ,u,ϑ+Ωu,ϑp.(25)

Adding (24) and (25), we have

F*,u,γ,ϑ,u+F*,ϑ,γ,u,ϑΩϑ,up+u,ϑp,F*,u,γ,ϑ,u+F*,ϑ,γ,u,ϑΩϑ,up+u,ϑp,
i.e.,
F,u,ϑ,u+˜F,ϑ,u,ϑ˜Ωϑ,up+u,ϑp.

Hence, the required result.

It can be easily noted that converse of above Theorem 4.2, is true only for p=2.

Theorems 4.1 and 4.2, enable us to define the followings new definitions.

Definition 4.4.

A G-differentiable fuzzy mapping F:KJ0 is said to be

  • fuzzy HOS-monotone w.r.t. if and only if, there exist a constant Ω>0 such that

    F,u,ϑ,u+˜F,ϑ,u,ϑ˜Ωϑ,up+u,ϑp,u,ϑK.(26)

  • fuzzy monotone w.r.t. bi-function if and only if,

    F,u,ϑ,u+˜F,ϑ,u,ϑ0˜,u,ϑK.(27)

  • fuzzy HOS-pseudomonotone w.r.t. if and only if, there exist a constant Ω>0 such that

    F,u,ϑ,u+˜Ωϑ,up0˜˜F,ϑ,u,ϑ0˜,u,ϑK.(28)

  • Strictly fuzzy monotone w.r.t. bi-function if and only if,

    F,u,ϑ,u+˜F,ϑ,u,ϑ0˜,u,ϑK.(29)

  • fuzzy pseudomonotone w.r.t. bi-function if and only if,

    F,u,ϑ,u0˜F,ϑ,u,ϑ0˜,u,ϑK.(30)

  • fuzzy quasimonotone w.r.t. bi-function if and only if,

    F,u,ϑ,u0˜F,ϑ,u,ϑ0˜,u,ϑK.(31)

  • strictly fuzzy pseudomonotone w.r.t. bi-function if and only if,

    F,u,ϑ,u0˜F,ϑ,u,ϑ0˜,u,ϑK.(32)

If ϑ,u=u,ϑ, then Definition 4.4, reduces to new ones.

Example 4.3.

We consider the fuzzy mappings F:0,J0 defined by

Fuσ=σ2u2,σ0,2u25u2σ3u2,σ(2u2,5u2]0,oτherwise.,

Then, for each γ0,1, we have Fγu=2γu2,53γu2, Fu is fuzzy HOS-pseudomonotone w.r.t. bifunction ϑ,u=uϑ, with 1Ω and p1, where ϑu. We have F*u,γ=2γu2 and F*u,γ=53γu2. Now we computing the following:

F*,u,γ,ϑ,u+Ωϑ,up=4γuuϑ+Ωuϑp0,
for all u,ϑK and γ0,1 with ϑu, 1Ω and p1; which implies that
F*,ϑ,γ,u,ϑ=4γuϑu=4γϑuϑ,0,u,ϑK,

Similarly, it can be easily show that

F*,u,γ,ϑ,u+Ωϑ,up=253γuuϑ+Ωuϑp0,
for all u,ϑK and γ0,1 with ϑu, 1Ω and p1; that means
F*,ϑ,γ,u,ϑ=253γuϑu=253γϑuϑ0,u,ϑK,

From which, it follows that

F*,ϑ,γ,u,ϑ0.

Hence, the G-differentiable fuzzy mapping Fγu=γu,54γu is fuzzy HOS-pseudo monotone w.r.t. ϑ,u=uϑ, with 1Ω and p1, where ϑu. it can be easily note that F,u is neither fuzzy pseudomonotone mapping nor fuzzy quasimonotone w.r.t. .

Theorem 4.3.

Let F:KJ0 be fuzzy mapping on K w.r.t. and Condition C hold. Let Fu is G-differentiable on K with following conditions:

  1. Fu+τϑ,uFϑ.

  2. F,. is a fuzzy HOS-monotone.

Then

Fϑ˜FuF,u,ϑ,u+˜2pΩϑ,up,u,ϑK.(33)

Proof.

Let F,. is fuzzy HOS-monotone. Then, from (26), we have

F,ϑ,u,ϑ˜F,u,ϑ,u˜Ωϑ,up+u,ϑp.

Therefore, for every γ0,1, we have

F*,ϑ,γ,u,ϑF*,u,γ,ϑ,uΩϑ,up+u,ϑp,F*,ϑ,γ,u,ϑF*,u,γ,ϑ,uΩϑ,up+u,ϑp,(34)

Since K is an invex set so we have, ϑτ=u+τϑ,uK for all u,ϑK and τ0,1. Taking ϑ=ϑτ in (34), we get

F*,u+τϑ,u,γ,u,u+τϑ,uF*,u,γ,u+τϑ,u,uΩu+τϑ,u,up+u,u+τϑ,up,F*,u+τϑ,u,γ,u,u+τϑ,uF*,u,γ,u+τϑ,u,uΩu+τϑ,u,up+u,u+τϑ,up,
by using Condition C, we have
F*,u+τϑ,u,γ,τϑ,uF*,u,γ,τϑ,u+2Ωτpϑ,up,F*,u+τϑ,u,γ,τϑ,uF*,u,γ,τϑ,u+2Ωτpϑ,up,
F*,u+τϑ,u,γ,ϑ,uF*,u,γ,ϑ,u+2Ωτp1ϑ,up,F*,u+τϑ,u,γ,ϑ,uF*,u,γ,ϑ,u+2Ωτp1ϑ,up,(35)

Let

H*τ=F*u+τϑ,u,γ,H*τ=F*u+τϑ,u,γ,
taking G-derivative w.r.t. τ, we get
H*,τ=F*,u+τϑ,u,γ.ϑ,u=F*,u+τϑ,u,γ,ϑ,u,H*,τ=F*,u+τϑ,u,γ.ϑ,u=F*,u+τϑ,u,γ,ϑ,u,
from which, using (35), we have
H*,τF*,u,γ,ϑ,u+2Ωτp1ϑ,up,H*,τF*,u,γ,ϑ,u+2Ωτp1ϑ,up.(36)

Integrating (36) over 0,1 w.r.t. τ, we get

H*1H*0F*,u,γ,ϑ,u+2pΩϑ,up,H*1H*0F*,u,γ,ϑ,u+2pΩϑ,up.
F*u+ϑ,u,γF*u,γF*,u,γ,ϑ,u+2pΩϑ,up,F*u+ϑ,u,γF*u,γF*,u,γ,ϑ,u+2pΩϑ,up.

From condition (i), we have

F*ϑ,γF*u,γF*,u,γ,ϑ,u+2pΩϑ,up,F*ϑ,γF*u,γF*,u,γ,ϑ,u+2pΩϑ,up.
i.e.,
Fϑ˜FuF,u,ϑ,u+˜2pΩϑ,up,u,ϑK.

Theorem 4.4.

Let F:KJ0 be fuzzy mapping on K w.r.t. and Condition C hold. Let Fu is G-differentiable on K with following conditions:

  1. Fu+τϑ,uFϑ.

  2. F,. is a fuzzy HOS-pseudomonotone.

Then F is a HOS-pseudo invex fuzzy mapping.

Proof.

Let F, be a fuzzy HOS-pseudomonotone. Then for all u,ϑK, we have

F,u,ϑ,u+˜Ωϑ,up0˜.

Therefore, for every γ0,1, we have

F*,u,γ,ϑ,u+Ωϑ,up0,F*,u,γ,ϑ,u+Ωϑ,up0.
which implies that
F*,ϑ,γ,u,ϑ0,F*,ϑ,γ,u,ϑ0.(37)

Since K is an invex set so we have, ϑτ=u+τϑ,uK for all u,ϑK and τ0,1. Taking ϑ=ϑτ in (37), we get

F*,u+τϑ,u,γ,u,u+τϑ,u0,F*,u+τϑ,u,γ,u,u+τϑ,u0.
by using Condition C, we have
F*,u+τϑ,u,γ,ϑ,u0,F*,u+τϑ,u,γ,ϑ,u0.(38)

Assume that

H*τ=F*u+τϑ,u,γ,H*τ=F*u+τϑ,u,γ,
taking G-derivative w.r.t. τ, then using (38), we have
H*,τ=F*,u+τϑ,u,γ,ϑ,u0,H*,τ=F*,u+τϑ,u,γ,ϑ,u0,(39)

Integrating (39) over 0,1 w.r.t. τ, we get

H*1H*00,H*1H*00,
which implies that
F*u+ϑ,u,γF*u,γ0,F*u+ϑ,u,γF*u,γ0.

From condition (i), we have

F*ϑ,γF*u,γ0,F*ϑ,γF*u,γ0,
i.e.,
Fϑ˜Fu0˜,u,ϑK.

Hence, Fu is a HOS-pseudo invex fuzzy mapping.

This result finds the necessary condition for HOS-pseudo invex fuzzy mapping.

As special case of Theorems 4.3 and 4.4, we have the following:

Corollary 4.2.

Let F:KJ0 be fuzzy mapping on K w.r.t. and Condition C hold. Let Fu is G-differentiable on K with following conditions:

  1. Fu+τϑ,uFϑ.

  2. F,. is a fuzzy pseudomonotone.

Then F is a pseudo invex fuzzy mapping.

Corollary 4.3.

Let F:KJ0 be fuzzy mapping on K w.r.t. and Condition C hold. Let Fu is G-differentiable on K with following conditions

  1. Fu+τϑ,uFϑ.

  2. F,. is a fuzzy quasimonotone.

Then F is a quasi invex fuzzy mapping.

We now discuss the fuzzy optimality condition for G-differentiable HOS-preinvex fuzzy mappings, which is main motivation of our results.

5. HIGHER-ORDER STRONGLY FUZZY MIXED VARIATIONAL-LIKE INEQUALITIES

A familiar reality in mathematical programing is that fuzzy variational inequality theory and complementary problem theory established strong relationship with mathematical problems.

Theorem 5.1.

Let F be a G-differentiable HOS-preinvex fuzzy mapping with modulus Ω>0. If uK is the minimum of the mapping F, then

Fϑ˜FuΩϑ,up,u,ϑK.(40)

Proof:

Let uK be a minimum of F. Then

FuFϑ,ϑK.

Therefore, for every γ0,1, we have

F*u,γF*ϑ,γ,F*u,γF*ϑ,γ.(41)

Since K is an invex set, for all u,ϑK, τ0,1, ϑτ=u+τϑ,uK. Taking ϑ=ϑτ in (41), we get

0F*u+τϑ,u,γF*u,γτ,0F*u+τϑ,u,γF*u,γτ.

Taking limit in the above inequality as τ0, we get

0F*,u,γ,ϑ,u,0F*,u,γ,ϑ,u.(42)

Since F:KJ0 is a G-differentiable HOS-preinvex fuzzy mapping, so

F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,F*u+τϑ,u,γ1τF*u,γ+τF*ϑ,γΩτp1τ+τ1τpϑ,up,
F*ϑ,γF*u,γF*u+τϑ,u,γF*u,γτ+Ωτp11τ+1τpϑ,up,F*ϑ,γF*u,γF*u+τϑ,u,γF*u,γτ+Ωτp11τ+1τpϑ,up,
again taking limit in the above inequality as τ0, we get
F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up,F*ϑ,γF*u,γF*,u,γ,ϑ,u+Ωϑ,up,
from which, using (42), we have
F*ϑ,γF*u,γΩϑ,up,F*ϑ,γF*u,γΩϑ,up,
i.e.,
Fϑ˜FuΩϑ,up.

Hence, the result follows.

Remark 5.1.

If Fu be a G-differentiable HOS-preinvex fuzzy mapping modulus Ω>0, and

F,u,ϑ,u+˜Ωϑ,up0˜,u,ϑK,(43)
then uK is the minimum of the mapping Fu. The inequality of the type (43) is called HOS-fuzzy variational-like inequality. It is very important to note that the optimality condition of preinvex fuzzy mappings can’t be obtained with the help of (43). So this idea inspires us to introduce a more general form of fuzzy variational-like inequality of which (43) is a special case. To be more unambiguous, for given fuzzy mapping Q, bi function .,. and a Ω>0, consider the problem of finding uK, such that
Qu,ϑ,u+˜Ωϑ,up0˜,ϑK,p1.(44)

This inequality is called HOS-fuzzy variational-like inequality.

We consider the functional Iϑ, defined as

Iϑ=Fϑ+˜Jϑ,ϑ,(45)
where Fu is a G-differentiable preinvex fuzzy mapping and Ju is a non G-differentiable HOS-preinvex fuzzy mapping.

We know show that the minimum of the functional Iϑ, can be characterized by a class of variational-like inequalities.

Theorem 5.2.

Let F:KJ0 be a G-differentiable preinvex fuzzy mapping and J:KJ0 be a non-G-differentiable HOS-preinvex fuzzy mapping. Then the functional Iϑ has minimum uK, if and only if uK satisfies

F,u,ϑ,u+˜Jϑ˜JuΩϑ,up,ϑK.(46)

Proof:

Let uK be the minimum of I then by definition, for all ϑK we have

IuIϑ.

Therefore, for every γ0,1, we have

I*u,γI*ϑ,γ,I*u,γI*ϑ,γ.(47)

Since K is an invex set so ϑτ=u+τϑ,u, for all u,ϑK and τ0,1. Replacing ϑ by ϑτ in (47), we get

I*u,γI*u+τϑ,u,γ,I*u,γI*u+τϑ,u,γ.
which implies that, using (45)
F*u,γ+J*u,γF*u+τϑ,u,γ+J*u+τϑ,u,γ,F*u,γ+J*u,γF*u+τϑ,u,γ+J*u+τϑ,u,γ.

Since J is HOS-preinvex fuzzy mapping then,

F*u,γ+J*u,γF*u+τϑ,u,γ+1τJ*u,γ+τJ*ϑ,γΩτp1τ+τ1τpϑ,up,F*u,γ+J*u,γF*u+τϑ,u,γ+1τJ*u,γ+τJ*ϑ,γΩτp1τ+τ1τpϑ,up,
0F*u+τϑ,u,γF*u,γ+τJ*ϑ,γJ*u,γΩτp1τ+τ1τpϑ,up,0F*u+τϑ,u,γF*u,γ+τJ*ϑ,γJ*u,γΩτp1τ+τ1τpϑ,up,

Now dividing by “τ” and taking limτ0, we have

0limτ0F*u+τϑ,u,γF*u,γτ+J*ϑ,γJ*u,γΩτp11τ+1τpϑ,up,0limτ0F*u+τϑ,u,γF*u,γτ+J*ϑ,γJ*u,γΩτp11τ+1τpϑ,up,
then
0F*,u,γ,ϑ,u+J*ϑ,γJ*u,γΩϑ,up,0F*,u,γ,ϑ,u+J*ϑ,γJ*u,γΩϑ,up,
i.e.,
Ωϑ,upF,u,ϑ,u+˜Jϑ˜Ju.

Conversely, let (46) be satisfy to prove uK is a minimum of I. Assume that for all ϑK we have

Iu˜Iϑ=Fu+˜Ju˜Fϑ˜Jϑ,=Fu˜Fϑ+˜Ju˜Jϑ,

Therefore, for every γ0,1, we have

I*u,γI*ϑ,γ=F*u,γF*ϑ,γ+J*u,γJ*ϑ,γ,I*u,γI*ϑ,γ=F*u,γF*ϑ,γ+J*u,γJ*ϑ,γ.
by Corollary 4.1, we have
I*u,γI*ϑ,γF*,u,γ,ϑ,u+J*ϑ,γJ*u,γ,I*u,γI*ϑ,γF*,u,γ,ϑ,u+J*ϑ,γJ*u,γ.
from which, using (46), we have
I*u,γI*ϑ,γΩϑ,up0,I*u,,γI*ϑ,γΩϑ,up0,
i.e.,
Iu˜Iϑ˜Ωϑ,up0˜,
hence, IuIϑ.

Note that the (46) is called HOS-fuzzy mixed variational-like inequalities. This result shows that the minimum of fuzzy functional Iϑ can be characterized by HOS-fuzzy mixed variational-like inequality. It is very important to observe that optimality conditions of preinvex fuzzy mappings and HOS-preinvex fuzzy mappings can’t be obtained with the help of (46). This idea encourage us to introduce a more general type of fuzzy variational-like inequality of which (46) is a particular case. In order to be more precise, for given fuzzy mappings Q, T, bi function .,. and a Ω>0, consider problem of finding uK, such that

Qu,ϑ,u+˜Tϑ˜Tu+˜Ωϑ,up0˜,ϑK,p1.(48)

This inequality is called HOS-fuzzy mixed variational-like inequality.

Now we discuss some special cases of HOS-fuzzy mixed variational-like inequalities:

If p=2, then (48) is called Strongly fuzzy mixed variational-like inequality such as

Qu,ϑ,u+˜Tϑ˜Tu+˜Ωϑ,u20˜,ϑK.

If ϑ,u=ϑu, then (48) is called strongly fuzzy mixed variational inequality such as

Qu,ϑu+˜Tϑ˜Tu+˜Ωϑu20˜,ϑK.

Similarly, we can obtain fuzzy variational inequality and variational-like inequality as special cases of (48). In a similar way, some special cases of HOS-fuzzy variational-like inequality can also be discussed.

Remark 5.2.

The inequalities (44) and (48), 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

Convex and nonconvex fuzzy mappings play an important role in fuzzy optimization. Therefore, by the importance of nonconvex fuzzy mappings, we introduced and consider a family of classes of nonconvex fuzzy mappings is called HOS-preinvex fuzzy mappings. It is illustrated that classical convexity and nonconvexity are special cases of HOS-preinvex fuzzy mappings. We have also introduced the notions of quasi-preinvex and log-preinvex fuzzy mappings and investigated some properties. Some relations among the HOS-preinvex fuzzy mappings, HOS-invex fuzzy mappings, and fuzzy HOS-monotonicities are derived under some mild conditions. We have proved that optimality conditions of G-differentiable HOS-preinvex fuzzy mappings and for the sum of G-differentiable preinvex fuzzy mappings and non G-differentiable HOS-preinvex fuzzy mappings can be characterized by HOS-fuzzy variational-like inequalities and HOS-fuzzy mixed variational-like inequalities, respectively. The inequalities (44) and (48) are the interesting outcome of our main results. It is itself an engaging problem to flourish some well-organized some numerical methods for solving HOS-fuzzy variational-like inequalities and HOS-fuzzy mixed variational-like inequalities together with applications in applied and pure sciences. In the future, we try to investigate the applications of HOS-fuzzy variational-like inequalities and HOS-fuzzy mixed variational-like inequalities in existence theory. We hope that these concepts and applications will be helpful for other authors to pay their roles in different fields of sciences.

CONFLICTS OF INTEREST

The authors declare that they have no competing interests.

AUTHORS' CONTRIBUTIONS

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

Funding Statement

The research is supported by the National Natural Science Foundation of China (Grant No. 61673169).

ACKNOWLEDGMENTS

The authors would like to thank the Rector, COMSATS University Islamabad, Islamabad, Pakistan, for providing excellent research and academic environments.

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Journal
International Journal of Computational Intelligence Systems
Volume-Issue
14 - 1
Pages
1856 - 1870
Publication Date
2021/06/28
ISSN (Online)
1875-6883
ISSN (Print)
1875-6891
DOI
10.2991/ijcis.d.210616.001How to use a DOI?
Copyright
© 2021 The Authors. Published by Atlantis Press B.V.
Open Access
This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

Cite this article

TY  - JOUR
AU  - Muhammad Bilal Khan
AU  - Muhammad Aslam Noor
AU  - Khalida Inayat Noor
AU  - Yu-Ming Chu
PY  - 2021
DA  - 2021/06/28
TI  - Higher-Order Strongly Preinvex Fuzzy Mappings and Fuzzy Mixed Variational-Like Inequalities
JO  - International Journal of Computational Intelligence Systems
SP  - 1856
EP  - 1870
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210616.001
DO  - 10.2991/ijcis.d.210616.001
ID  - Khan2021
ER  -