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

Muhammad Bilal Khan; Muhammad Aslam Noor; Khalida Inayat Noor; Yu-Ming Chu

doi:10.2991/ijcis.d.210616.001

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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^{, *}^,

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.210616.001 How 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 [8–11] 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 [12–16] 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 ℝn 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 [2,9,16,30–47] 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 ℝ be the set of real numbers whose standard element is symbolized by u. A fuzzy set φ in ℝ is a mapping φ:ℝ→0,1 and support of φ is denoted by suppφ and is defined as

supp(φ)=u∈ℝ|φ(u)>0.

Definition 2.1.

If φ be a fuzzy set in ℝ and γ∈0,1, then γ-level sets of φ is denoted and is defined as follows:

φγ=u∈ℝ|φu≥γ.(1)

Definition 2.2.

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

φ is normal, i.e., there exists u∈ℝ such that φu=1;
φ is upper semi continiuous;
φ1−τu+τϑ≥minφu,φϑ, ∀u,ϑ∈ℝ, τ∈0,1;
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=Lc−uρ, c−ρ≤u<d,1, c≤u<d,Lu−dr, c≤u<d+r,0, otherwise,

where r,ρ>0, and d≥c.

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

φγ=φ*γ,φ*γ

where

φ*γ=infu∈ℝ|φu≥γ,

φ*γ=supu∈ℝ|φu≥γ.

Since each ρ∈ℝ is also a fuzzy number, defined as

ρ˜u=1if u=ρ0if u≠ρ.

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

φ*γ,φ*γ,γ:γ∈0,1.

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→ℝ and φ*γ:0,1→ℝ satisfy the following conditions:

φ*γ is a nondecreasing function.
φ*γ is a nonincreasing function.
φ*1≤φ*1.
φ*γ and φ*γ are bounded and left continuous on (0,1] and right continuous at γ=0.

Then φ:ℝ→0,1, defined by

φu=supγ:φ*γ≤u≤φ*γ,

is a fuzzy number parameterization is given by φ*γ,φ*γ,γ:γ∈0,1. Moreover, If φ:ℝ→0,1 is a fuzzy number with parametrization given by φ*γ,φ*γ,γ:γ∈0,1, then function φ*γ and φ*γ find the conditions (1)–(4).

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

If φ,ϕ∈J0, there exist ψ∈J0 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*γ−ϕ*γ,ψ*γ=φ−˜ϕ*γ=φ*γ−ϕ*γ.

Now we discuss some properties of fuzzy numbers under addition and scalar multiplication, if φ,ϕ∈J0 and ρ∈ℝ then φ+˜ϕ and ρφ define as

φ+˜ϕ=φ*γ+ϕ*γ,φ*γ+ϕ*γ,γ:γ∈0,1,(2)

ρφ=ρφ*γ,ρφ*γ,γ:γ∈0,1.(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∈ℝ,

φ+˜ρ=φ*γ+ρ,φ*γ+ρ,γ:γ∈0,1.(4)

Definition 2.4.

A mapping F:K⊂ℝ→J0 is called fuzzy mapping. For each γ∈0,1, 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:γ∈0,1.

Definition 2.5.

Let F:K⊂ℝ→J0 be a fuzzy mapping. Then Fu is said to be continuous at u∈K, if for each γ∈0,1, both end point functions F*u,γ and F*u,γ are continuous at u∈K.

Definition 2.6.

[30] Let L=m,n and u∈L. Then fuzzy mapping F:m,n→J0 is said to be a generalized differentiable (in short, G-differentiable) at u if there exists an element F,u∈J0 such that for all 0<τ, sufficiently small, there exist Fu+τ−˜Fu,Fu−˜Fu−τ and the limits (in the metric D)

limτ→0+Fu+τ−˜Fuτ=limτ→0+Fu−˜Fu−ττ=F,u or limτ→0+Fu−˜Fu+τ−τ=limτ→0+Fu−τ−˜Fu−τ=F,u

or limτ→0+Fu+τ−˜Fuτ=limτ→0+Fu−τ−˜Fu−τ=F,u or limτ→0+Fu−˜Fu+τ−τ=limτ→0+Fu−˜Fu−ττ=F,u

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

Dφ,ϕ=sup0≤γ≤1Hφγ,ϕγ,

and H denote the well-known Hausdorff metric on space of intervals.

Definition 2.7.

[26] A fuzzy mapping F:K→J0 is called convex on the convex set K if

F1−τu+τϑ≼1−τFu+˜τFϑ∀u,ϑ∈K,τ∈0,1.

Definition 2.8.

[26] A fuzzy mapping F:K→J0 is called quasi-convex on the convex set K if

F1−τu+τϑ≼maxFu,Fϑ∀u,ϑ∈K,τ∈0,1.

Definition 2.9.

[26] A fuzzy mapping F:K∂→J0 is called preinvex on the invex set K∂ w.r.t. bi-function ∂ if

Fu+τ∂ϑ,u≼1−τFu+˜τFϑ∀u,ϑ∈K∂,τ∈0,1,

where ∂:K∂×K∂→ℝ.

Lemma 2.1.

[27] Let K∂ be an invex set w.r.t. ∂ and let F:K∂→J0 be a fuzzy mapping parametrized by

Fu=F*u,γ,F*u,γ,γ:γ∈0,1,∀u∈K∂.

Then Fu is preinvex fuzzy mapping on K∂ if and only if, for all γ∈0,1,F*u,γ and F*u,γ are preinvex functions w.r.t. ∂ on K∂.

Definition 2.10.

[26] A fuzzy mapping F:K∂→J0 is called quasi-preinvex on the invex set K∂ w.r.t. ∂ if

Fu+τ∂ϑ,u≼maxFu,Fϑ∀u,ϑ∈K∂,τ∈0,1.

For further study, let K∂ be a nonempty invex set in ℝ. Let F:K∂→J0 be a fuzzy mapping and ∂:K∂×K∂→ℝ 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:K∂→J0 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+τ∂ϑ,u≼1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp∂‖ϑ,u‖p,(5)

∀u,ϑ∈K∂,τ∈0,1, where ∂:K∂×K∂→ℝ and p≥1.

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+τ∂ϑ,u≼1−τFu+˜τFϑ−Ωτ1−τ‖∂ϑ,u‖2,∀u,ϑ∈K∂,τ∈0,1.

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+τ∂ϑ,u≼1−τFu+˜τFϑ,∀u,ϑ∈K∂,τ∈0,1.
If ∂ϑ,u=ϑ−u and p≥1, then (5) becomes
Fu+τ∂ϑ,u≼1−τFu+˜τFϑ−Ωτp1−τ+τ1−τp‖ϑ−u‖p,∀u,ϑ∈K∂,τ∈0,1.

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

If ∂ϑ,u=ϑ−u and Ω=0, then HOS-preinvex fuzzy mapping becomes convex fuzzy mapping, i.e.,
Fu+τϑ−u≼1−τFu+˜τFϑ,∀u,ϑ∈K∂,τ∈0,1.
If τ=12, then (5) becomes
F2u+∂ϑ,u2≼Fu+˜Fϑ2−˜12pΩ‖∂ϑ,u‖p,∀u,ϑ∈K∂.

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:K∂→J0 is said to be HOS-affine preinvex fuzzy mapping on K∂ w.r.t. bi-function ∂, if there exist Ω>0 such that

Fu+τ∂ϑ,u=1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,(6)

for all u,ϑ∈K∂,τ∈0,1, where ∂:K∂×K∂→ℝ and p≥1.

In other words, a fuzzy mapping F:K∂→J0 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−˜12p‖∂ϑ,u‖p,∀u,ϑ∈K∂.

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 σ≥0.
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:K∂→J0 be a fuzzy mapping parametrized by

Fu=F*u,γ,F*u,γ,γ:γ∈0,1,∀u∈K∂.(7)

Then Fu is HOS-preinvex fuzzy mapping on K∂ with modulus Ω, if and only if, for all γ∈0,1, F*u,γ and F*u,γ are HOS-preinvex functions w.r.t. ∂ with modulus Ω.

(8)

Proof.

Assume that for each γ∈0,1, 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+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,∀u,ϑ∈K∂,τ∈0,1.

And

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,∀u,ϑ∈K∂,τ∈0,1.

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

Fu+τ∂ϑ,u=F*u+τ∂ϑ,u,γ,F*u+τ∂ϑ,u,γ,γ:γ∈0,1,≼1−τF*u,γ,1−τF*u,γ,γ:γ∈0,1+˜τF*ϑ,γ,τF*ϑ,γ,γ:γ∈0,1−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,=1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,∀u,ϑ∈K∂,τ∈0,1.

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,ϑ∈K∂ and τ∈0,1, we have

Fu+τ∂ϑ,u≼1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.

From (8), we have

Fu+τ∂ϑ,u=F*u+τ∂ϑ,u,γ,F*u+τ∂ϑ,u,γ,γ:γ∈0,1,∀u,ϑ∈K∂,τ∈0,1.

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

1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p=1−τF*u,γ,1−τF*u,γ,γ:γ∈0,1+˜τF*ϑ,γ,τF*ϑ,γ,γ:γ∈0,1−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,(9)

for all u,ϑ∈K∂ and τ∈0,1. Then by HOS-preinvexity of Fu, we have for all u,ϑ∈K∂ and τ∈0,1 such that

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

and

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

for each γ∈0,1. Hence, the result follows.

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

“Let K∂ be a convex set and let F:K→F0 be a fuzzy mapping parametrized by

Fu=F*u,γ,F*u,γ,γ:γ∈0,1,∀u∈K∂

Then F is HOS-convex fuzzy mapping on K∂ if and only if, for all γ∈0,1,

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

Example 3.1.

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

Fuσ=σu2,σ∈0,u22u2−σu2,σ∈(u2,2u2]0,oτherwise.

Then, for each γ∈0,1, we have Fγu=γu2,2−γu2. Since end point functions F*u,γ, F*u,γ are HOS-preinvex functions for each γ∈0,1. Hence Fu is HOS-preinvex fuzzy mapping w.r.t.

∂ϑ,u=ϑ−u,

with 0<Ω≤1 and p≥2. It can be easily seen that for each Ω∈0,1 and p≥2, 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=ϑ−u with 0<Ω≤1 and p≥2.

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:K∂→J0 be a HOS-affine preinvex w.r.t. ∂. Then Fu is HOS-preinvex fuzzy w.r.t same ∂, if and only if, G=F−J is preinvex fuzzy mapping.

Proof.

The “If” part is obvious. To prove the “only if” assume that, J:K∂→J0 be a HOS-affine preinvex w.r.t. bi-function ∂, then there exist Ω>0 such that

Ju+τ∂ϑ,u=1−τJu+˜τJϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.

Therefore, for each γ∈0,1, we have

J*u+τ∂ϑ,u,γ=1−τJ*u,γ+τJ*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,J*u+τ∂ϑ,u,γ=1−τJ*u,γ+τJ*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.(10)

Since Fu is HOS-preinvex fuzzy mapping, then, for each γ∈0,1, we have

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,(11)

from (10) and (11), we have

F*u+τ∂ϑ,u,γ−J*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−1−τJ*u,γ−τJ*ϑ,γ,F*u+τ∂ϑ,u,γ−J*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ1−τ−J*u,γ−τJ*ϑ,γ,

F*u+τ∂ϑ,u,γ−J*u+τ∂ϑ,u,γ≤1−τF*u,γ−J*u,γ+τF*ϑ,γ−J*ϑ,γ,F*u+τ∂ϑ,u,γ−J*u+τ∂ϑ,u,γ≤1−τF*u,γ−J*u,γ+τF*ϑ,γ−J*ϑ,γ,

from which it follows that

G*u+τ∂ϑ,u,γ=F*u+τ∂ϑ,u,γ−J*u+τ∂ϑ,u,γ,G*u+τ∂ϑ,u,γ=F*u+τ∂ϑ,u,γ−J*u+τ∂ϑ,u,γ,

G*u+τ∂ϑ,u,γ≤1−τF*u,γ−J*u,γ+τF*ϑ,γ−J*ϑ,γ,G*u+τ∂ϑ,u,γ≤1−τF*u,γ−J*u,γ+τF*ϑ,γ−J*ϑ,γ,

which implies that

G*u+τ∂ϑ,u,γ≤1−τG*u,γ+τG*ϑ,γ,G*u+τ∂ϑ,u,γ≤1−τG*u,γ+τG*ϑ,γ,

i.e.,

Gu+τ∂ϑ,u≼1−τGu+˜τGu.

Showing that G=F−J is preinvex fuzzy mapping.

Definition 3.3.

A fuzzy mapping F:K∂→J0 is said to be HOS-quasi-preinvex on K∂ w.r.t. bi-function ∂, if there exist a Ω>0 such that

Fu+τ∂ϑ,u≼maxFu,Fϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,(12)

∀u,ϑ∈K∂,τ∈0,1, where ∂:K∂×K∂→ℝ and p≥1.

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 γ∈0,1, we have

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

F*u+τ∂ϑ,u,γ≤maxF*u,γ,F*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤maxF*u,γ,F*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

i.e.,

Fu+τ∂ϑ,u≼maxFu,Fϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.

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:K∂→J0 is called pseudo preinvex on K∂ if there exist a b:K∂×K∂→J0 such that

Fϑ≺Fu⇒Fu+τ∂ϑ,u≺Fu+˜ττ−1bu,ϑ,∀u,ϑ∈K∂,τ∈0,1,where b.,.≻0˜.(13)

Theorem 3.3.

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

Proof.

Let Fϑ≺Fu and Fu be a HOS-preinvex fuzzy mapping. Then there exist modulus Ω and for all u,ϑ∈K∂, τ∈0,1, such that

Fu+τ∂ϑ,u≼1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.

Therefore, for every γ∈0,1, we have

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

F*u+τ∂ϑ,u,γ≤F*u,γ+τF*ϑ,γ−F*u,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤F*u,γ+τF*ϑ,γ−F*u,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.(14)

From (14), we have

F*u+τ∂ϑ,u,γ<F*u,γ+ττ−1F*u,γ−F*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ<F*u,γ+ττ−1F*u,γ−F*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

F*u+τ∂ϑ,u,γ<F*u,γ+ττ−1b*u,ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ<F*u,γ+ττ−1b*u,ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

i.e.,

Fu+τ∂ϑ,u≺Fu+˜ττ−1bu,ϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

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+τ∂ϑ,u=1−τ∂ϑ,u,

∂u,u+τ∂ϑ,u=−τ∂ϑ,u.

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

Definition 4.1.

A G-differentiable fuzzy mapping F:K∂→J0 is called HOS-invex w.r.t. ∂, if there exist a constant Ω>0 such that

Fϑ−˜Fu≽F,u,∂ϑ,u+˜Ω‖∂ϑ,u‖p,∀u,ϑ∈K∂.(15)

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

Example 4.1.

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

F*ϑ,γ−F*u,γ=γϑ2−γu2,

while

F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p=2γϑ−u+Ω‖ϑ−u‖p.

And γϑ2−γu2≥2γϑ−u+Ω‖ϑ−u‖p, with 0<Ω≤1 and p≥2, where u≤ϑ.

Similarly, it can be easily show that

F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p.

Hence, Fu is HOS” invex fuzzy mapping w.r.t. bifunction ∂ϑ,u=ϑ−u, with 0<Ω≤1 and p≥2. It can be easily seen that Fu is not invex fuzzy mapping w.r.t. bifunction ∂ϑ,u=ϑ−u.

Definition 4.2.

A G-differentiable fuzzy mapping F:K∂→J0 is called HOS-pseudo invex w.r.t. ∂, if there exist a constant Ω>0 such that

F,u,∂ϑ,u+˜Ω‖∂ϑ,u‖p≽0˜⇒ Fϑ−˜Fu≽0˜,∀u,ϑ∈K∂.(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:K∂→J0 is called HOS-quasi invex w.r.t. ∂, if there exist a constant Ω>0 such that

Fϑ≼Fu⇒F,u,∂ϑ,u+˜Ω∂ϑ,up≼0˜,∀u,ϑ∈K∂.(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 4.1–4.3 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,5−4γu, then Fu is HOS-pseudo invex fuzzy mapping w.r.t. bifunction ∂ϑ,u=ϑ−u, with 0≤Ω and p≥1, where u≤ϑ. We have F*u,γ=γu and F*u,γ=5−4γu. Now we computing the following:

F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p=γϑ−u+Ω‖ϑ−u‖p≥0,

for all u,ϑ∈K∂ and γ∈0,1 with u≤ϑ, 0≤Ω and p≥1; which implies that

F*ϑ,γ=γϑ≥γu=F*u,γ,

F*ϑ,γ≥F*u,γ,(18)

Similarly, it can be easily show that

F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p=5−4γϑ−u+Ω‖ϑ−u‖p≥0,

for all u,ϑ∈K∂ and γ∈0,1 with u≤ϑ, 0≤Ω and p≥1; that means

F*ϑ,γ=5−4γϑ≥γu=F*u,γ,

From which, It follows that

F*ϑ,γ≥F*u,γ.(19)

Hence, the fuzzy mapping Fγu=γu,5−4γu is HOS-pseudo invex fuzzy mapping w.r.t. ∂ϑ,u=ϑ−u, with 0≤Ω and p≥1, where u≤ϑ. 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:K∂→J0 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:K∂→J0 be G-differentiable HOS-preinvex fuzzy mapping. Since Fu is HOS-preinvex fuzzy mapping then there exist a constant Ω>0, for all u,ϑ∈K∂ and τ∈0,1, we have

Fu+τ∂ϑ,u≼1−τFu+˜τFϑ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,≼Fu+˜τFϑ−˜Fu−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

Therefore, for every γ∈0,1, we have

F*u+τ∂ϑ,u,γ≤F*u,γ+τF*ϑ,γ−F*u,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤F*u,γ+τF*ϑ,γ−F*u,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

which implies that

τF*ϑ,γ−F*u,γ≥F*u+τ∂ϑ,u,γ−F*u,γ+Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,τF*ϑ,γ−F*u,γ≥F*u+τ∂ϑ,u,γ−F*u,γ+Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

F*ϑ,γ−F*u,γ≥F*u+τ∂ϑ,u,γ−F*u,γτ+Ωτp−11−τ+1−τp‖∂ϑ,u‖p,F*ϑ,γ−F*u,γ≥F*u+τ∂ϑ,u,γ−F*u,γτ+Ωτp−11−τ+1−τp‖∂ϑ,u‖p.

Taking limit in the above inequality as τ→0, we have

F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p,

i.e.,

Fϑ−˜Fu≽F,u,∂ϑ,u+˜Ω‖∂ϑ,u‖p.(20)

Conversely, assume that Fu is a HOS-invex fuzzy mapping. Since K∂ is an invex set then, we have, ϑτ=u+τ∂ϑ,u∈K∂, for all u,ϑ∈K∂ and τ∈0,1. Taking ϑ=ϑτ in (20), we get

F*ϑτ,γ−F*u,γ≥F*,u,γ,∂ϑτ,u+Ω‖∂ϑ,u‖p,F*ϑτ−F*u,γ≥F*,u,γ,∂ϑτ,u+Ω‖∂ϑ,u‖p,

Using Condition C, we have

F*ϑτ,γ−F*u,γ≥1−τF*,u,γ,∂ϑ,u+Ω1−τp‖∂ϑ,u‖p,F*ϑτ,γ−F*u,γ≥1−τF*,u,γ,∂ϑ,u+Ω1−τp‖∂ϑ,u‖p.(21)

In a similar way, we have

F*u,γ−F*ϑτ,γ≥−τF*,u,γ,∂ϑ,u+Ωτp‖∂ϑ,u‖p,F*u,γ−F*ϑτ,γ≥−τF*,u,γ,∂ϑ,u+Ωτp‖∂ϑ,u‖p.(22)

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

F*ϑτ,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*ϑτ,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

which implies that

Fu+τ∂ϑ,u,γ≼1−τFu,γ+˜τFϑ,γ−˜Ωτp1−τ+τ1−τp‖∂ϑ,u‖p.

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:K∂→J0 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≼−˜Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p,∀u,ϑ∈K∂.(23)

Proof.

Assume that Fu is a HOS-invex fuzzy mapping. Then, for every γ∈0,1, we have

F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p,F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p.(24)

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

F*u,γ−F*ϑ,γ≥F*,ϑ,γ,∂u,ϑ+Ω‖∂u,ϑ‖p,F*u,γ−F*ϑ,γ≥F*,ϑ,γ,∂u,ϑ+Ω‖∂u,ϑ‖p.(25)

Adding (24) and (25), we have

F*,u,γ,∂ϑ,u+F*,ϑ,γ,∂u,ϑ≤−Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p,F*,u,γ,∂ϑ,u+F*,ϑ,γ,∂u,ϑ≤−Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p,

i.e.,

F,u,∂ϑ,u+˜F,ϑ,∂u,ϑ≼−˜Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p.

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:K∂→J0 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≼−˜Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p,∀u,ϑ∈K∂.(26)
fuzzy monotone w.r.t. bi-function ∂ if and only if,
F,u,∂ϑ,u+˜F,ϑ,∂u,ϑ≼0˜,∀u,ϑ∈K∂.(27)
fuzzy HOS-pseudomonotone w.r.t. ∂ if and only if, there exist a constant Ω>0 such that
F,u,∂ϑ,u+˜Ω‖∂ϑ,u‖p≽0˜⇒−˜F,ϑ,∂u,ϑ≽0˜,∀u,ϑ∈K∂.(28)
Strictly fuzzy monotone w.r.t. bi-function ∂ if and only if,
F,u,∂ϑ,u+˜F,ϑ,∂u,ϑ≺0˜,∀u,ϑ∈K∂.(29)
fuzzy pseudomonotone w.r.t. bi-function ∂ if and only if,
F,u,∂ϑ,u≽0˜⇒F,ϑ,∂u,ϑ≼0˜,∀u,ϑ∈K∂.(30)
fuzzy quasimonotone w.r.t. bi-function ∂ if and only if,
F,u,∂ϑ,u≻0˜⇒F,ϑ,∂u,ϑ≼0˜,∀u,ϑ∈K∂.(31)
strictly fuzzy pseudomonotone w.r.t. bi-function ∂ if and only if,
F,u,∂ϑ,u≽0˜⇒F,ϑ,∂u,ϑ≺0˜,∀u,ϑ∈K∂.(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,σ∈0,2u25u2−σ3u2,σ∈(2u2,5u2]0,oτherwise.,

Then, for each γ∈0,1, we have Fγu=2γu2,5−3γu2, Fu is fuzzy HOS-pseudomonotone w.r.t. bifunction ∂ϑ,u=u−ϑ, with 1≤Ω and p≥1, where ϑ≤u. We have F*u,γ=2γu2 and F*u,γ=5−3γu2. Now we computing the following:

F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p=4γuu−ϑ+Ω‖u−ϑ‖p≥0,

for all u,ϑ∈K∂ and γ∈0,1 with ϑ≤u, 1≤Ω and p≥1; which implies that

−F*,ϑ,γ,∂u,ϑ=−4γuϑ−u=4γϑu−ϑ,≥0,∀u,ϑ∈K∂,

Similarly, it can be easily show that

F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p=25−3γuu−ϑ+Ω‖u−ϑ‖p≥0,

for all u,ϑ∈K∂ and γ∈0,1 with ϑ≤u, 1≤Ω and p≥1; that means

−F*,ϑ,γ,∂u,ϑ=−25−3γuϑ−u=25−3γϑu−ϑ≥0,∀u,ϑ∈K∂,

From which, it follows that

−F*,ϑ,γ,∂u,ϑ≥0.

Hence, the G-differentiable fuzzy mapping Fγu=γu,5−4γu is fuzzy HOS-pseudo monotone w.r.t. ∂ϑ,u=u−ϑ, with 1≤Ω and p≥1, where ϑ≤u. it can be easily note that F,u is neither fuzzy pseudomonotone mapping nor fuzzy quasimonotone w.r.t. ∂.

Theorem 4.3.

Let F:K∂→J0 be fuzzy mapping on K∂ w.r.t. ∂ and Condition C hold. Let Fu is G-differentiable on K∂ with following conditions:

Fu+τ∂ϑ,u≼Fϑ.
F,. is a fuzzy HOS-monotone.

Then

Fϑ−˜Fu≽F,u,∂ϑ,u+˜2pΩ‖∂ϑ,u‖p,∀u,ϑ∈K∂.(33)

Proof.

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

F,ϑ,∂u,ϑ≼−˜F,u,∂ϑ,u−˜Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p.

Therefore, for every γ∈0,1, we have

F*,ϑ,γ,∂u,ϑ≤−F*,u,γ,∂ϑ,u−Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p,F*,ϑ,γ,∂u,ϑ≤−F*,u,γ,∂ϑ,u−Ω‖∂ϑ,u‖p+‖∂u,ϑ‖p,(34)

Since K∂ is an invex set so we have, ϑτ=u+τ∂ϑ,u∈K∂ for all u,ϑ∈K∂ and τ∈0,1. Taking ϑ=ϑτ in (34), we get

F*,u+τ∂ϑ,u,γ,∂u,u+τ∂ϑ,u≤−F*,u,γ,∂u+τ∂ϑ,u,u−Ω‖∂u+τ∂ϑ,u,u‖p+‖∂u,u+τ∂ϑ,u‖p,F*,u+τ∂ϑ,u,γ,∂u,u+τ∂ϑ,u≤−F*,u,γ,∂u+τ∂ϑ,u,u−Ω‖∂u+τ∂ϑ,u,u‖p+‖∂u,u+τ∂ϑ,u‖p,

by using Condition C, we have

F*,u+τ∂ϑ,u,γ,τ∂ϑ,u≥F*,u,γ,τ∂ϑ,u+2Ωτp‖∂ϑ,u‖p,F*,u+τ∂ϑ,u,γ,τ∂ϑ,u≥F*,u,γ,τ∂ϑ,u+2Ωτp‖∂ϑ,u‖p,

F*,u+τ∂ϑ,u,γ,∂ϑ,u≥F*,u,γ,∂ϑ,u+2Ωτp−1‖∂ϑ,u‖p,F*,u+τ∂ϑ,u,γ,∂ϑ,u≥F*,u,γ,∂ϑ,u+2Ωτp−1‖∂ϑ,u‖p,(35)

Let

H*τ=F*u+τ∂ϑ,u,γ,H*τ=F*u+τ∂ϑ,u,γ,

taking G-derivative w.r.t. τ, we get

H*,τ=F*,u+τ∂ϑ,u,γ.∂ϑ,u=F*,u+τ∂ϑ,u,γ,∂ϑ,u,H*,τ=F*,u+τ∂ϑ,u,γ.∂ϑ,u=F*,u+τ∂ϑ,u,γ,∂ϑ,u,

from which, using (35), we have

H*,τ≥F*,u,γ,∂ϑ,u+2Ωτp−1‖∂ϑ,u‖p,H*,τ≥F*,u,γ,∂ϑ,u+2Ωτp−1‖∂ϑ,u‖p.(36)

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

H*1−H*0≥F*,u,γ,∂ϑ,u+2pΩ‖∂ϑ,u‖p,H*1−H*0≥F*,u,γ,∂ϑ,u+2pΩ‖∂ϑ,u‖p.

F*u+∂ϑ,u,γ−F*u,γ≥F*,u,γ,∂ϑ,u+2pΩ‖∂ϑ,u‖p,F*u+∂ϑ,u,γ−F*u,γ≥F*,u,γ,∂ϑ,u+2pΩ‖∂ϑ,u‖p.

From condition (i), we have

F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+2pΩ‖∂ϑ,u‖p,F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+2pΩ‖∂ϑ,u‖p.

i.e.,

Fϑ−˜Fu≽F,u,∂ϑ,u+˜2pΩ‖∂ϑ,u‖p,∀u,ϑ∈K∂.

Theorem 4.4.

Let F:K∂→J0 be fuzzy mapping on K∂ w.r.t. ∂ and Condition C hold. Let Fu is G-differentiable on K∂ with following conditions:

Fu+τ∂ϑ,u≼Fϑ.
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,ϑ∈K∂, we have

F,u,∂ϑ,u+˜Ω‖∂ϑ,u‖p≽0˜.

Therefore, for every γ∈0,1, we have

F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p≥0,F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p≥0.

which implies that

−F*,ϑ,γ, ∂u,ϑ≥0,−F*,ϑ,γ, ∂u,ϑ≥0.(37)

Since K∂ is an invex set so we have, ϑτ=u+τ∂ϑ,u∈K∂ for all u,ϑ∈K∂ and τ∈0,1. Taking ϑ=ϑτ in (37), we get

−F*,u+τ∂ϑ,u,γ,∂u,u+τ∂ϑ,u≥0,−F*,u+τ∂ϑ,u,γ,∂u,u+τ∂ϑ,u≥0.

by using Condition C, we have

F*,u+τ∂ϑ,u,γ,∂ϑ,u≥0,F*,u+τ∂ϑ,u,γ,∂ϑ,u≥0.(38)

Assume that

H*τ=F*u+τ∂ϑ,u,γ,H*τ=F*u+τ∂ϑ,u,γ,

taking G-derivative w.r.t. τ, then using (38), we have

H*,τ=F*,u+τ∂ϑ,u,γ,∂ϑ,u≥0,H*,τ=F*,u+τ∂ϑ,u,γ,∂ϑ,u≥0,(39)

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

H*1−H*0≥0,H*1−H*0≥0,

which implies that

F*u+∂ϑ,u,γ−F*u,γ≥0,F*u+∂ϑ,u,γ−F*u,γ≥0.

From condition (i), we have

F*ϑ,γ−F*u,γ≥0,F*ϑ,γ−F*u,γ≥0,

i.e.,

Fϑ−˜Fu≽0˜,∀u,ϑ∈K∂.

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:K∂→J0 be fuzzy mapping on K∂ w.r.t. ∂ and Condition C hold. Let Fu is G-differentiable on K∂ with following conditions:

Fu+τ∂ϑ,u≼Fϑ.
F,. is a fuzzy pseudomonotone.

Then F is a pseudo invex fuzzy mapping.

Corollary 4.3.

Let F:K∂→J0 be fuzzy mapping on K∂ w.r.t. ∂ and Condition C hold. Let Fu is G-differentiable on K∂ with following conditions

Fu+τ∂ϑ,u≼Fϑ.
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 u∈K∂ is the minimum of the mapping F, then

Fϑ−˜Fu≽Ω‖∂ϑ,u‖p,∀u,ϑ∈K∂.(40)

Proof:

Let u∈K∂ be a minimum of F. Then

Fu≼Fϑ,∀ϑ∈K∂.

Therefore, for every γ∈0,1, we have

F*u,γ≤F*ϑ,γ,F*u,γ≤F*ϑ,γ.(41)

Since K∂ is an invex set, for all u,ϑ∈K∂, τ∈0,1, ϑτ=u+τ∂ϑ,u∈K∂. Taking ϑ=ϑτ in (41), we get

0≤F*u+τ∂ϑ,u,γ−F*u,γτ,0≤F*u+τ∂ϑ,u,γ−F*u,γτ.

Taking limit in the above inequality as τ→0, we get

0≤F*,u,γ,∂ϑ,u,0≤F*,u,γ,∂ϑ,u.(42)

Since F:K∂→J0 is a G-differentiable HOS-preinvex fuzzy mapping, so

F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u+τ∂ϑ,u,γ≤1−τF*u,γ+τF*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

F*ϑ,γ−F*u,γ≥F*u+τ∂ϑ,u,γ−F*u,γτ+Ωτp−11−τ+1−τp‖∂ϑ,u‖p,F*ϑ,γ−F*u,γ≥F*u+τ∂ϑ,u,γ−F*u,γτ+Ωτp−11−τ+1−τp‖∂ϑ,u‖p,

again taking limit in the above inequality as τ→0, we get

F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p,F*ϑ,γ−F*u,γ≥F*,u,γ,∂ϑ,u+Ω‖∂ϑ,u‖p,

from which, using (42), we have

F*ϑ,γ−F*u,γ≥Ω‖∂ϑ,u‖p,F*ϑ,γ−F*u,γ≥Ω‖∂ϑ,u‖p,

i.e.,

Fϑ−˜Fu≽Ω‖∂ϑ,u‖p.

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+˜Ω‖∂ϑ,u‖p≽0˜,∀u,ϑ∈K∂,(43)

then u∈K∂ 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 u∈K∂, such that

Qu,∂ϑ,u+˜Ω‖∂ϑ,u‖p≽0˜,∀ϑ∈K∂,p≥1.(44)

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

We consider the functional Iϑ, defined as

Iϑ=Fϑ+˜Jϑ,∀ϑ∈ℝ,(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:K∂→J0 be a G-differentiable preinvex fuzzy mapping and J:K∂→J0 be a non-G-differentiable HOS-preinvex fuzzy mapping. Then the functional Iϑ has minimum u∈K∂, if and only if u∈K∂ satisfies

F,u,∂ϑ,u+˜Jϑ−˜Ju≽Ω‖∂ϑ,u‖p,∀ϑ∈K∂.(46)

Proof:

Let u∈K∂ be the minimum of I then by definition, for all ϑ∈K∂ we have

Iu≼Iϑ.

Therefore, for every γ∈0,1, we have

I*u,γ≤I*ϑ,γ,I*u,γ≤I*ϑ,γ.(47)

Since K∂ is an invex set so ϑτ=u+τ∂ϑ,u, for all u,ϑ∈K∂ and τ∈0,1. Replacing ϑ by ϑτ in (47), we get

I*u,γ≤I*u+τ∂ϑ,u,γ,I*u,γ≤I*u+τ∂ϑ,u,γ.

which implies that, using (45)

F*u,γ+J*u,γ≤F*u+τ∂ϑ,u,γ+J*u+τ∂ϑ,u,γ,F*u,γ+J*u,γ≤F*u+τ∂ϑ,u,γ+J*u+τ∂ϑ,u,γ.

Since J is HOS-preinvex fuzzy mapping then,

F*u,γ+J*u,γ≤F*u+τ∂ϑ,u,γ+1−τJ*u,γ+τJ*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,F*u,γ+J*u,γ≤F*u+τ∂ϑ,u,γ+1−τJ*u,γ+τJ*ϑ,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

0≤F*u+τ∂ϑ,u,γ−F*u,γ+τJ*ϑ,γ−J*u,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,0≤F*u+τ∂ϑ,u,γ−F*u,γ+τJ*ϑ,γ−J*u,γ−Ωτp1−τ+τ1−τp‖∂ϑ,u‖p,

Now dividing by “τ” and taking limτ→0, we have

0≤limτ→0F*u+τ∂ϑ,u,γ−F*u,γτ+J*ϑ,γ−J*u,γ−Ωτp−11−τ+1−τp‖∂ϑ,u‖p,0≤limτ→0F*u+τ∂ϑ,u,γ−F*u,γτ+J*ϑ,γ−J*u,γ−Ωτp−11−τ+1−τp‖∂ϑ,u‖p,

then

0≤F*,u,γ,∂ϑ,u+J*ϑ,γ−J*u,γ−Ω‖∂ϑ,u‖p,0≤F*,u,γ,∂ϑ,u+J*ϑ,γ−J*u,γ−Ω‖∂ϑ,u‖p,

i.e.,

Ω‖∂ϑ,u‖p≼F,u,∂ϑ,u+˜Jϑ−˜Ju.

Conversely, let (46) be satisfy to prove u∈K∂ is a minimum of I. Assume that for all ϑ∈K∂ we have

Iu−˜Iϑ=Fu+˜Ju−˜Fϑ−˜Jϑ,=Fu−˜Fϑ+˜Ju−˜Jϑ,

Therefore, for every γ∈0,1, we have

I*u,γ−I*ϑ,γ=F*u,γ−F*ϑ,γ+J*u,γ−J*ϑ,γ,I*u,γ−I*ϑ,γ=F*u,γ−F*ϑ,γ+J*u,γ−J*ϑ,γ.

by Corollary 4.1, we have

I*u,γ−I*ϑ,γ≤−F*,u,γ,∂ϑ,u+J*ϑ,γ−J*u,γ,I*u,γ−I*ϑ,γ≤−F*,u,γ,∂ϑ,u+J*ϑ,γ−J*u,γ.

from which, using (46), we have

I*u,γ−I*ϑ,γ≤−Ω‖∂ϑ,u‖p≤0,I*u,,γ−I*ϑ,γ≤−Ω‖∂ϑ,u‖p≤0,

i.e.,

Iu−˜Iϑ≼−˜Ω‖∂ϑ,u‖p≼0˜,

hence, Iu≼Iϑ.

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 u∈K∂, such that

Qu,∂ϑ,u+˜Tϑ−˜Tu+˜Ω‖∂ϑ,u‖p≽0˜,∀ϑ∈K∂,p≥1.(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+˜Ω‖∂ϑ,u‖2≽0˜,∀ϑ∈K∂.

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

Qu,ϑ−u+˜Tϑ−˜Tu+˜Ω‖ϑ−u‖2≽0˜,∀ϑ∈K∂.

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.210616.001 How to use a DOI?
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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  -

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