International Journal of Analysis and Applications 

Volume 19, Number 4 (2021), 518-541 

URL: https://doi.org/10.28924/2291-8639 

DOI: 10.28924/2291-8639-19-2021-518 

 

 

Received March 16th, 2021; accepted April 14th, 2021; published May 11th, 2021. 

2010 Mathematics Subject Classification. 49J40. 

Key words and phrases. exponentially preinvex fuzzy mapping; exponentially invex fuzzy mappings; exponentially 

monotone fuzzy mappings; exponentially fuzzy mixed variational-like inequalities. 

©2021 Authors retain the copyrights 

of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 

518 

 

EXPONENTIALLY PREINVEX FUZZY MAPPINGS AND FUZZY EXPONENTIALLY 

MIXED VARIATIONAL-LIKE INEQUALITIES 

 

MUHAMMAD BILAL KHAN1, MUHAMMAD ASLAM NOOR1, KHALIDA INAYAT 

NOOR1, HASSAN ALMUSAWA2, KOTTAKKARAN SOOPPY NISAR3,* 

 

1Department of Mathematics, COMSATS University Islamabad, Pakistan 

2Department of Mathematics, College of Sciences, Jazan University, Jazan 45142, Saudi Arabia  

3Department of Mathematics, College of Arts and Sciences, Wadi Aldawaser, Prince Sattam bin 

Abdulaziz University, Saudi Arabia  

 

*Corresponding author: n.sooppy@psau.edu.sa; ksnisar1@gmail.com 

 

ABSTRACT. In this article, our aim is to consider a class of nonconvex fuzzy mapping known as exponentially preinvex 

fuzzy mapping. With the support of some examples, the notions of exponentially preinvex fuzzy mappings are 

explored and discussed in some special cases. Some properties are also derived and relations among the 

exponentially preinvex fuzzy mappings (exponentially preinvex-FMs), exponentially invex fuzzy mappings 

(exponentially-IFMs), and exponentially monotonicity are established under some mild conditions. In the end, using 

the fact that fuzzy optimization and fuzzy variational inequalities have close relationships, we have proven that the 

optimality conditions of exponentially preinvex fuzzy mapping can be distinguished by exponentially fuzzy 

variational-like inequality and exponentially fuzzy mixed variational-like inequality. These inequalities render the 

very interesting outcomes of our main results and appear to be the new ones. Presented results in this paper can be 

considered and the development of previously obtained results. 

 

https://doi.org/10.28924/2291-8639
https://doi.org/10.28924/2291-8639-19-2021-518


Int. J. Anal. Appl. 19 (4) (2021) 519 

 

1. INTRODUCTION 

In the last 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 [16] initiated to introduce a generalized class of convexity which is 

known as an invex function. The invex function played a significant role in mathematical 

programming. A step forward, the invex set invex function and preinvex function were 

introduced and studied by Israel and Mond [9], and Mohen and Neogy [20]. Also, Noor [23] 

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, see ([1], [6], [7], [15]) 

and the references therein. It is well known that logarithmically convex functions have serious 

importance in convex theory because by using these functions we can derive more accurate 

inequalities as compare to convex functions. Some of the authors discussed different classes of 

convex and nonconvex functions. These functions have a very close relationship with 

logarithmically convex functions. Bernstain [10], initiated to introduce exponentially convex 

function. A move forward, the concept of an r-convex function was introduced by Avriel [5], 

while Antczak [4] studied the notions of (r, p)-convex functions. For further details on r-convex 

function, we refer the reader to study the properties of r-convex function, see [34] and the 

reference therein. Exponentially convex functions and their generalizations have many 

applications in different fields such as information theory, data analysis, statistics, and machine 

learning; see [2-5]. Noor and Noor [25-27] have recently considered and characterized another 

type of exponentially convex functions and characterized its properties, and have demonstrated 

that convex functions are different from the exponentially convex functions defined by 

Bernstein [10]. Furthermore, Noor and Noor [28] generalized the exponentially convex 

functions and defined a class of nonconvex functions which is known as exponentially preinvex 

function and proved that the minimum of a differentiable exponentially convex function is 

distinguished by exponentially 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 



Int. J. Anal. Appl. 19 (4) (2021) 520 

 

problem theory established powerful mechanism of the mathematical problems and they have a 

friendly relationship. Many authors contributed to this fascinating and interesting field. The 

concept of fuzzy mappings (FMs) was proposed by Chang and Zadeh [11] in 1972. In 1989, 

Nanda and Kar [21] were initiated to introduce convex-FMs and characterized the notion of 

convex-FM through the idea of epigraph. A step forward, Furukawa [14] and Syau [31] 

proposed and examined FM from space ℝ𝑛 to the set of fuzzy numbers, fuzzy valued Lipschitz 

continuity, logarithmic convex-FMs and quasi-convex-FMs. Besides, Chang [12] discussed the 

idea of convex-FM 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-FM is preinvex-FM. Noor [22] 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 [19] 

established an equivalence condition of preinvex-FM and characterizations about preinvex-FMs 

under some mild conditions. With the support of examples, Wu and Xu [33] updated the 

definition of convex-FMs and established a new approach regarding the existence of a fuzzy 

preinvex mapping under the condition of lower or upper semicontinuity. In 2012, Rufian-

Lizana et al. [29] reviewed the existing literature and made appropriate modifications to the 

results obtained by Wu and Xu [32] regarding invex-FMs. For differentiable and twice 

differentiable preinvex-FMs, Rufian-Lizana et al. [30] gave the required and sufficient 

conditions. They demonstrated the validity of characterizations with the help of examples and 

improved the previous results provided by Li and Noor [19]. We refer to the readers for further 

analysis of literature on the applications and properties of variational-like inequalities and 

generalized convex-FMs, see ([8], [13], [15], [24], [32]) and the references therein. 

Motivated and inspired by the ongoing research work, we note that convex and 

generalized convex-FMs play an important role in fuzzy optimization. The paper is organized 

as follows. Section 2 recalls some basic and new definitions, preliminary notations, and new 

results which will be helpful for further study. Section 3 introduces and considers a family of 

classes of nonconvex-FMs is called exponentially preinvex-FMs and investigates some 



Int. J. Anal. Appl. 19 (4) (2021) 521 

 

properties. Section 4 derives some relations among the exponentially preinvex-FMs, 

exponentially-IFMs, and exponentially monotonicity under some mild conditions.  Section 5 

introduces the new classes of fuzzy variational-like inequality which is known as exponentially 

fuzzy variational-like inequality and exponentially fuzzy mixed variational-like inequality. 

Several special cases are also discussed. These inequalities give an interesting outcome of our 

main results. 

 

2. PRELIMINARIES 

Let ℝ be the set of real numbers. A fuzzy subset set 𝒜 of ℝ is distinguished by a function 

𝜑:ℝ → [0,1] called the membership function. In this study this depiction is approved. Moreover, 

the collection of all fuzzy subsets of  ℝ is denoted by �̃�(ℝ). 

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| 𝜑(𝓊) ≥ 𝛾}.    (2.1) 

Definition 2.2. A fuzzy number (FN)  𝜑 is a fuzzy set in ℝ with the following properties:  

(1) 𝜑 is normal i.e. there exists 𝑥 ∈ ℝ such that 𝜑(𝑥) = 1; 

(2) 𝜑 is upper semi continuous i.e., for given 𝑥 ∈ ℝ, for every 𝑥 ∈ ℝ there exist > 0 there exist 

𝛿 > 0 such that 𝜑(𝑥) − 𝜑(𝑦) <  for all 𝑦 ∈ ℝ with |𝑥 − 𝑦| < 𝛿. 

(3) 𝜑 is fuzzy convex i.e., 𝜑((1 − 𝜏)𝑥 + 𝜏𝑦) ≥ 𝑚𝑖𝑛(𝜑(𝑥),𝜑(𝑦)), ∀ 𝑥,𝑦 ∈ ℝ and 𝜏 ∈ [0,1]; 

(4)  𝜑 is compactly supported i.e., 𝑐𝑙{𝑥 ∈ ℝ| 𝜑(𝑥) > 0} is compact. 

The LR-FNs first introduced by Dubois and Prade [13] are defined as follows: 

Definition 2.3. Let 𝐿,𝑅:[0,1] → [0,1] be two decreasing and upper semicontinuous functions 

with 𝐿(0) =  𝑅(0) = 1 and 𝐿(1) = 𝑅(1) = 0. Then the FN is defined as 

  𝜑(𝓊) =

{
 
 

 
 𝐿(

𝑐−𝓊

𝜌
)                  𝑐 −  𝜌 ≤ 𝓊 < 𝑑,

1                          𝑐 ≤ 𝓊 < 𝑑,

𝐿(
𝓊−𝑑

𝑟
)                 𝑐 ≤ 𝓊 < 𝑑 + 𝑟,

0                        otherwise,

                                                  

where 𝑟,𝜌 > 0 and 𝑑 ≥ 𝑐.  



Int. J. Anal. Appl. 19 (4) (2021) 522 

 

 Let 𝔽(ℝ) denotes the set of all FNs and let 𝜑 ∈ 𝔽(ℝ) be FN, 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) = 𝑖𝑛𝑓{𝓊 ∈ ℝ| 𝜑(𝓊) ≥ 𝛾}, 𝜑
∗(𝛾) = 𝑠𝑢𝑝{𝓊 ∈ ℝ| 𝜑(𝓊) ≥ 𝛾}. 

Since each 𝜌 ∈ ℝ is also a FN, defined as 

�̃�(𝓊) = {
1  if 𝓊 = 𝜌
0 if 𝓊 ≠ 𝜌 

.  

It is well known that a FN is a real fuzzy interval and 𝜑 can also be identified by a triplet: 

{(𝜑∗(𝛾),𝜑
∗(𝛾),𝛾):𝛾 ∈ [0,1]}. 

This notion help us to characterize a FN in terms of the two end point functions 𝜑∗(𝛾) and 

𝜑∗(𝛾). 

Theorem 2.1 ([13]). Suppose that 𝜑∗(𝛾):[0,1] → ℝ and 𝜑
∗(𝛾):[0,1] → ℝ satisfy the following 

assertions: 

• 𝜑∗(𝛾) is a non-decreasing function. 

• 𝜑∗ (𝛾) is a non-increasing 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  

𝜑(𝓊) = 𝑠𝑢𝑝{𝛾:𝜑∗(𝛾) ≤ 𝓊 ≤ 𝜑
∗(𝛾)},  

is a FN, parameterization is given by {(𝜑∗(𝛾),𝜑
∗(𝛾),𝛾):𝛾 ∈ [0,1]}. Moreover, if 𝜑:ℝ → [0,1] is a 

FN 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 ∈ 𝔽(ℝ)  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 ∈ 𝔽(ℝ), we have 𝜑 ⪯ 𝜙 or 𝜑 ⪰ 𝜙 otherwise they are 

non-comparable. Some time we may write 𝜑 ⪯ 𝜙 instead of 𝜙 ⪰ 𝜑 and note that, we may say 

that 𝔽(ℝ) is a partial ordered set under the relation ⪯.  



Int. J. Anal. Appl. 19 (4) (2021) 523 

 

If 𝜑,𝜙 ∈ 𝔽(ℝ), there exist 𝜓 ∈ 𝔽(ℝ)  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 [32]. 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 FNs under addition and scalar multiplication, if 𝜑,𝜙 ∈ 𝔽(ℝ) 

and 𝜌 ∈ ℝ then 𝜑+̃𝜙 and 𝜌𝜑 define as 

  𝜑+̃𝜙 = {(𝜑∗(𝛾) + 𝜙∗(𝛾),𝜑
∗(𝛾) + 𝜙∗(𝛾),𝛾):𝛾 ∈ [0,1]},  (2.2) 

     𝜌𝜑 = {(𝜌𝜑∗(𝛾),𝜌𝜑
∗(𝛾),𝛾):𝛾 ∈ [0,1]}.    (2.3) 

Remark 2.1. Obviously, 𝔽(ℝ) is closed under addition and nonnegative scaler multiplication 

and above defined properties on 𝔽(ℝ) 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]}.     (2.4) 

Definition 2.4. A mapping 𝒯:𝐾 ⊂ ℝ → 𝔽(ℝ) is called fuzzy mapping (FM). For each 𝛾 ∈ [0,1], 

denote [𝒯(𝓊)]𝛾 = [𝒯∗(𝓊,𝛾),𝒯
∗(𝓊,𝛾)]. Thus a FM 𝒯 can be identified by a parametrized triples 

𝒯(𝓊) = {(𝒯∗(𝓊,𝛾),𝒯
∗(𝓊,𝛾),𝛾):𝛾 ∈ [0,1]}. 

Definition 2.5. Let 𝒯:𝐾 ⊂ ℝ → 𝔽(ℝ) be a FM. Then 𝒯(𝓊) is said to be continuous at 𝓊 ∈ 𝐾, if for 

each 𝛾 ∈ [0,1], both end point functions 𝒯∗(𝓊,𝛾) and 𝒯
∗(𝓊,𝛾) are continuous at 𝓊 ∈ 𝐾. 

Definition 2.6 ([8]). Let 𝐿 = (𝑚,𝑛)  and 𝓊 ∈ 𝐿 . Then FM 𝒯:(𝑚,𝑛) → 𝔽(ℝ)  is said to be a 

generalized differentiable (in short, G-differentiable) at 𝓊 if there exists an element 𝒯 ,(𝓊) ∈

𝔽(ℝ) such that 

for all 0 < 𝜏, sufficiently small, there exist 𝒯(𝓊 + 𝜏)−̃𝒯(𝓊), 𝒯(𝓊)−̃𝒯(𝓊 − 𝜏) and the limits 

 lim
𝜏→0+

𝒯(𝓊+𝜏)−̃𝒯(𝓊)

𝜏
= lim

𝜏→0+

𝒯(𝓊)−̃𝒯(𝓊−𝜏)

𝜏
=  𝒯 ,(𝓊) or lim

𝜏→0+

𝒯(𝓊)−̃𝒯(𝓊+𝜏)

−𝜏
= lim

𝜏→0+

𝒯(𝓊−𝜏)−̃𝒯(𝓊)

−𝜏
=  𝒯 ,(𝓊) or 

 lim
𝜏→0+

𝒯(𝓊+𝜏)−̃𝒯(𝓊)

𝜏
= lim

𝜏→0+

𝒯(𝓊−𝜏)−̃𝒯(𝓊)

−𝜏
=  𝒯 ,(𝓊) or lim

𝜏→0+

𝒯(𝓊)−̃𝒯(𝓊+𝜏)

−𝜏
= lim

𝜏→0+

𝒯(𝓊)−̃𝒯(𝓊−𝜏)

𝜏
=  𝒯 ,(𝓊), 

where the limits are taken in the metric space (𝔽(ℝ),𝐷), for 𝜑,𝜙 ∈ 𝔽(ℝ) 

𝐷(𝜑,𝜙) = sup
0≤𝛾≤1

𝐻(𝜑𝛾,𝜙𝛾), 

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

Definition 𝟐.𝟕 ([21]). A FM 𝒯:𝐾 → 𝔽(ℝ) is called convex on the convex set 𝐾 if  

                              𝒯((1 − 𝜏)𝓊 + 𝜏𝜗) ⪯ (1 − 𝜏)𝒯(𝓊)+̃𝜏𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾,  𝜏 ∈ [0,1]. 

Definition 𝟐.𝟖 ([21]). A FM 𝒯:𝐾 → 𝔽(ℝ) is called quasi-convex on the convex set 𝐾 if  



Int. J. Anal. Appl. 19 (4) (2021) 524 

 

                               𝒯((1 − 𝜏)𝓊 + 𝜏𝜗) ⪯ 𝑚𝑎𝑥(𝒯(𝓊),𝒯(𝜗)),∀ 𝓊,𝜗 ∈ 𝐾,  𝜏 ∈ [0,1]. 

Definition 𝟐.𝟗 ([22]). A FM 𝒯:𝐾𝜕 → 𝔽(ℝ) is called preinvex-FM on the invex set 𝐾𝜕 w.r.t. bi-

function  𝜕 if  

                              𝒯(𝓊 + 𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝒯(𝓊)+̃𝜏𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾𝜕,  𝜏 ∈ [0,1], 

where 𝜕:𝐾𝜕 × 𝐾𝜕 → ℝ. 

Lemma 2.1 ([19]). Let 𝐾𝜕 be an invex set w.r.t. 𝜕 and let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a FM parametrized by  

𝒯(𝓊) = {(𝒯∗(𝓊,𝛾),𝒯
∗(𝓊,𝛾),𝛾):𝛾 ∈ [0,1]},∀ 𝓊 ∈ 𝐾𝜕.  

Then 𝒯 is preinvex-FM on 𝐾𝜕 if and only if, for all 𝛾 ∈ [0,1],𝒯∗(𝓊,𝛾) and 𝒯
∗(𝓊,𝛾) are preinvex 

functions w.r.t. 𝜕 on 𝐾𝜕. 

Definition 𝟐.𝟏𝟎 ([22]). A comparable FM 𝒯:𝐾𝜕 → 𝔽(ℝ) is called quasi-preinvex-FM on the 

invex set 𝐾𝜕 w.r.t. 𝜕 if  

                               𝒯(𝓊 + 𝜏𝜕(𝜗,𝓊)) ⪯ 𝑚𝑎𝑥(𝒯(𝓊),𝒯(𝜗)), ∀ 𝓊,𝜗 ∈ 𝐾𝜕,  𝜏 ∈ [0,1]. 

Definition 2.11 ([28]). A function  𝒯:𝐾 → ℝ is called exponentially convex on 𝐾, if 

𝑒𝒯((1−𝜏)𝓊+𝜏𝜗) ⪯ (1 − 𝜏)𝑒𝒯(𝓊) + 𝜏𝑒𝒯(𝜗),∀ 𝓊,𝜗 ∈ 𝐾,𝜏 ∈ [0,1]. 

The Definition 2.11, can also be written in the following equivalent way, which is due to 

Antczak [4]. 

Definition 2.12.  A function  𝒯:𝐾 → ℝ is called exponentially convex on 𝐾, if 

𝒯((1 − 𝜏)𝓊 + 𝜏𝜗) ⪯ 𝑙𝑜𝑔[(1 − 𝜏)𝑒𝒯(𝓊) + 𝜏𝑒𝒯(𝜗)], ∀ 𝓊,𝜗 ∈ 𝐾,𝜏 ∈ [0,1]. 

Strictly exponentially convex function if strict inequality holds for 𝒯(𝓊) ≠ 𝒯(𝜗). 𝒯:𝐾 → 𝔽0 is 

called exponentially concave function if −𝒯 is exponentially convex on 𝐾. Strictly exponentially 

concave function if strict inequality holds for 𝒯(𝓊) ≠ 𝒯(𝜗).  

Antczak [4] and Alirezaei and Mathar [3 have discussed the applications of exponentially 

convex function in the mathematical programing and information theory. Similarly, Alirezaei 

and Mazhar [3] explore the applications of exponentially concave function with following 

example in communication and information theory. 

Example. The error function 

𝑒𝑟𝑓(𝓊) =
2

√𝛱
∫ 𝑒−𝑡

2
𝓊

0

𝑑𝓊, 

becomes an exponentially concave function in the form 𝑒𝑟𝑓(√𝓊), 𝓊 ≥ 0, which defines the 

bit/symbol error probability of communication system depending on the square root of the 



Int. J. Anal. Appl. 19 (4) (2021) 525 

 

underlying signal-to-noise ratio. This demonstrates that the exponentially concave functions in 

the theory communication and information theory may play an important role. 

Definition 2.13 ([28]). A function  𝒯:𝐾 → ℝ is called exponentially quasi convex on 𝐾, if 

𝑒𝒯((1−𝜏)𝓊+𝜏𝜗) ⪯ max (𝑒𝒯(𝓊),𝑒𝒯(𝜗)),∀ 𝓊,𝜗 ∈ 𝐾,𝜏 ∈ [0,1]. 

From above definitions we can easily prove that each exponentially convex function is 

exponentially quasi convex function but the converse is not true such that 

       𝑒𝒯((1−𝜏)𝓊+𝜏𝜗) ⪯ (1 − 𝜏)𝑒𝒯(𝓊) + 𝜏𝑒𝒯(𝜗), 

      ⪯ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)). 

Definition 2.14. Let 𝐾𝜕 be an invex set. Then FM 𝒯:𝐾𝜕 → 𝔽0 is said to be: 

• exponentially preinvex-FM (exponentially preinvex-FM) on 𝐾𝜕 w.r.t. bi-function  𝜕(. , . ) if  

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1].   (2.5) 

• exponentially preconcave-FM on 𝐾𝜕 if inequality (2.5) is reversed. 

• strictly exponentially preinvex-FM on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ) if  

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1].   (2.6) 

• strictly exponentially preconcave-FM on 𝐾𝜕 if inequality (2.6) is reversed. 

Remark 2.2. 

(i) If 𝒯 is exponentially preinvex-FM, then 𝜆𝒯 is also exponentially preinvex-FM for 𝜆 ≥

0. 

(ii) If 𝒯(𝓊) and 𝐺(𝓊) both are exponentially preinvex-FMs, then max(𝒯(𝓊),𝐺(𝓊)) is also 

exponentially preinvex-FMs. 

Special cases 

(iii) The exponentially preinvexity of 𝒯 on 𝐾𝜕 is equivalent to preinvexity of 𝑒
𝒯(𝓊). 

(iv) The exponentially prconcavity of 𝒯 on 𝐾𝜕 is equivalent to preconcavety of  𝑒
𝒯(𝓊). 

(v) By using remark (iii), if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊,  then exponentially preinvex-FM becomes 

exponentially convex-FM, that is 

𝑒𝒯(𝓊+𝜏(𝜗−𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),∀  𝓊,𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1]. 

(vi) If 𝜏 =
1

2
, then (2.5) becomes 

                                             𝑒
𝒯(

2𝓊+𝜕(𝜗,𝓊)

𝟐
)
⪯
𝑒𝒯(𝓊)+̃𝑒𝒯(𝜗)

2
,∀ 𝓊,𝜗 ∈ 𝐾𝜕.                                               (2.7) 



Int. J. Anal. Appl. 19 (4) (2021) 526 

 

The inequality (2.7) is known as exponentially Jenson preinvex-FMs. 

We can easily discuss the next result with the help of Remark 2.2 (iii). 

Theorem 2.2. Let K∂ be an invex set w.r.t. ∂ and let 𝒯:K∂ → 𝔽0 be a FM parameterized by  

𝒯(𝓊) = {(𝒯∗(𝓊,𝛾),𝒯
∗(𝓊,𝛾),𝛾):𝛾 ∈ [0,1]},∀ 𝓊 ∈ K∂.  

Then 𝒯 is exponentially preinvex-FM on K∂ if and only if, for all 𝛾 ∈ [0,1], 

 𝒯∗(𝓊,𝛾) and 𝒯
∗(𝓊,𝛾) are exponentially preinvex function w.r.t. ∂.  

Proof. The demonstration of proof is similar to Lemma 2.1, by Remark 2.2 (iii).   

Example 2.1. We consider the FMs 𝒯:(−1,1) → 𝔽0 defined by, 

𝒯(𝓊)(σ) = {
σ − 𝓊2

1 − 𝓊2
    σ ∈ [𝓊2,1)

0            otherwise.

, 

Then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [𝛾 + (1 − 𝛾)𝓊
2,1 ]. Since end point functions 𝒯∗(𝛾), 

𝒯∗(𝛾) are exponentially preinvex functions for each 𝛾 ∈ [0,1], then 𝒯 is exponentially preinvex-

FM w.r.t. bi-function  

∂(ϑ,𝓊) = {
ϑ2 − 𝓊2

2𝓊
     ϑ2 − 𝓊2 < 0,

0             otherwise.

 

Example 2.2. We consider the FMs 𝒯:(0,∞) → 𝔽0 defined by, 

𝒯(𝓊)(σ) =

{
 
 

 
  

σ

𝓊2
               σ ∈ [0,𝓊2],

2𝓊2 − σ

𝓊2
           σ ∈ (𝓊2,2𝓊2],

0                    otherwise.

 

Then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [𝛾𝓊
2, (2 − 𝛾)𝓊2]. Since end point functions 𝒯∗(𝛾), 

𝒯∗(𝛾) are exponentially preinvex functions for each 𝛾 ∈ [0,1], then 𝒯 is exponentially preinvex-

FM w.r.t.  

∂(ϑ,𝓊) = ϑ − 𝓊. 

Definition 2.15. A comparable FM 𝒯:𝐾𝜕 → 𝔽0 is said to be: 

• exponentially quasi-preinvex-FM on 𝐾𝜕 w.r.t. bi-function  𝜕(. , . ), if  

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1].   (2.8) 

• exponentially quasi-preconcave-FM on 𝐾𝜕 if inequality (2.8) is reversed. 

• strictly exponentially quasi-preinvex-FM on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ), if  

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)),∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1].   (2.9) 



Int. J. Anal. Appl. 19 (4) (2021) 527 

 

• strictly exponentially quasi-preconcave-FM on 𝐾𝜕 if inequality (2.9) is reversed. 

Proposition 2.1. Let comparable FM 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-FM, such that 

𝒯(𝜗) ≺ 𝒯(𝓊). Then 𝒯 is strictly exponentially quasi-preinvex-FM. 

Proof. Let 𝑒𝒯(𝜗) ≺ 𝑒𝒯(𝓊) and 𝒯 be exponentially preinvex-FM. Then, for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈

[0,1] we have 

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), 

Since 𝒯(𝜗) ≺ 𝒯(𝓊), we have 

           𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ 𝑒𝒯(𝓊) = max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)). 

Hence, 𝒯 is strictly exponentially quasi-preinvex-FM. 

Definition 2.16. A FM 𝒯:𝐾𝜕 → 𝔽0 is said to be: 

• exponentially logarithmic preinvex-FM on 𝐾𝜕 w.r.t. bi-function  𝜕(. , . ), if  

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (𝑒𝒯(𝓊))
1−𝜏
(𝑒𝒯(𝜗))

𝜏
,∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1].   (2.10) 

• exponentially logarithmic preconcave-FM on 𝐾𝜕 if inequality (2.10) is reversed. 

• strictly exponentially logarithmic preinvex-FM on 𝐾𝜕 w.r.t. bi-function 𝜕(. , . ), if  

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ (𝑒𝒯(𝓊))
1−𝜏
(𝑒𝒯(𝜗))

𝜏
,∀ 𝜗 ∈ 𝐾𝜕,𝜏 ∈ [0,1].   (2.11) 

• strictly exponentially logarithmic preconcave-FM on 𝐾𝜕 if inequality (2.11) is reversed, 

where 𝒯(.) ≻ 0̃. 

For further study, let 𝐾𝜕 be a nonempty closed invex set in ℝ. Let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a FM and 

𝜕:𝐾𝜕 × 𝐾𝜕 → ℝ be an arbitrary bifunction. We denote  〈. , . 〉 be the inner product. 

Next, we will discuss exponentially preinvex-FMs by using Remark 2.2, (iii). 

 

3. BASIC RESULTS 

In this section, we investigate some basic properties of exponentially preinvex-FMs and obtain 

some characterization for exponentially preinvex-FMs by using the concepts of level sets and 

epigraph of FMs.   

Definition 3.1. Let 𝒯:𝒟 → 𝔽0 and 𝐺:𝒟 → 𝔽0 be two FMs on 𝒟 ⊆ ℝ . Then 𝒯(𝓊) and 𝐺(𝓊) are 

said to be fuzzy comonotonic on 𝒟 if, for all 𝓊,𝜗 ∈ 𝒟, 

[𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗)][𝑒𝐺(𝓊)−̃𝑒𝐺(𝜗)] ⪰ 0̃. 



Int. J. Anal. Appl. 19 (4) (2021) 528 

 

Theorem 3.1. Let 𝒯(𝓊) and 𝐺(𝓊) be two exponentially preinvex-FMs on 𝐾𝜕 . Then 𝒯(𝓊) =

𝒯(𝓊)+̃𝐺(𝓊)  is exponentially preinvex-FM on 𝐾𝜕,  provide that 𝒯(𝓊)  and 𝐺(𝓊)  are fuzzy 

comonotonic on 𝐾𝜕. 

Proof. Let 𝒯(𝓊) and 𝐺(𝓊) be two exponentially preinvex-FMs on 𝐾𝜕. Then, for all  𝓊,𝜗 ∈ 𝐾𝜕 and 

𝜏 ∈ [0,1], we have  

                    𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗),     (3.1) 

     𝑒𝐺(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝐺(𝓊)+̃𝜏𝑒𝐺(𝜗).    (3.2) 

Now, 

    𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))+̃𝐺(𝓊+𝜏𝜕(𝜗,𝓊)) = 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝑒𝐺(𝓊+𝜏𝜕(𝜗,𝓊)), 

⪯ [(1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗)][(1 − 𝜏)𝑒𝐺(𝓊)+̃𝜏𝑒𝐺(𝜗)], 

               = (1 − 𝜏)2𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃(1 − 𝜏)𝜏[𝑒𝒯(𝓊)+̃𝐺(𝜗)+̃𝑒𝒯(𝜗)+̃𝐺(𝓊)] 

     +̃𝜏2𝑒𝒯(𝜗)+̃𝐺(𝜗).       (3.3) 

Since 𝒯(𝓊) and 𝐺(𝓊) are fuzzy comonotonic on 𝐾𝜕, such that 

[𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗)][𝑒𝐺(𝓊)−̃𝑒𝐺(𝜗)] ⪰ 0̃, 

which implies that 

   𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃𝑒𝒯(𝜗)+̃𝐺(𝜗) ⪰ 𝑒𝒯(𝓊)+̃𝐺(𝜗)+̃𝑒𝒯(𝜗)+̃𝐺(𝓊).    (3.4) 

It follows by substituting (3.4) into (3.3) that 

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))+̃𝐺(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)2𝑒𝒯(𝓊)+̃𝐺(𝓊) + (1 − 𝜏)𝜏[𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃𝑒𝒯(𝜗)+̃𝐺(𝜗)] 

     +̃𝜏2𝑒𝒯(𝜗)+̃𝐺(𝜗), 

           = (1 − 𝜏)𝑒𝒯(𝓊)+̃𝐺(𝓊)+̃𝜏𝑒𝒯(𝜗)+̃𝐺(𝜗). 

Hence, 𝒯(𝓊)+̃𝐺(𝓊) is exponentially preinvex-FM on 𝐾𝜕. 

Theorem 3.2. Let 𝒯:𝐾𝜕 → 𝔽0 be a FM, with 𝜃 = inf𝓊∈𝐾𝜕(𝑒
𝒯(𝓊)) exists in 𝔽0. 

(1) If 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-FM on 𝐾𝜕, then set 𝛺 = {𝓊:𝓊 ∈ 𝐾𝜕, 𝑒
𝒯(𝓊) = 𝜃} 

is an invex set. 

(2) If 𝒯:𝐾𝜕 → 𝔽0 is a strictly exponentially preinvex-FM on 𝐾𝜕, then 𝛺 is a singleton set or 

empty. That is, if  𝒯 is strictly exponentially preinvex-FM, then 𝒯 has at least one global 

minimum. 



Int. J. Anal. Appl. 19 (4) (2021) 529 

 

Proof. (1). Let 𝒯 be exponentially preinvex-FM. If 𝛺 is an empty set, then 𝛺 is an invex set. 

Assume that, 𝓊,𝜗 ∈ 𝛺 , that is  𝓊,𝜗 ∈ 𝐾𝜕  and 𝑒
𝒯(𝓊) = 𝜃 = 𝑒𝒯(𝜗) . Since 𝒯  is exponentially 

preinvex-FM,  

    𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), 

              = (1 − 𝜏)𝜃+̃𝜏𝜃 = 𝜃, 

for all 𝜏 ∈ (0,1). Hence, all points of the form 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝛺, 𝜏 ∈ [0,1], which implies that 𝛺 is 

an invex set. 

(2). Let 𝒯 be a strictly exponentially preinvex-FM on 𝐾𝜕. Contrary suppose that there exist 

distinct points 𝓊,𝜗 ∈ 𝛺 such that  𝑒𝒯(𝓊) = 𝜃 = 𝑒𝒯(𝜗). Since 𝒯 is strictly exponentially preinvex-

FM then, for all 𝓊,𝜗 ∈ 𝐾𝜕  and 𝜏 ∈ (0,1), we have  

    𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ≺ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), 

              = (1 − 𝜏)𝜃+̃𝜏𝜃 = 𝜃, 

which implies that  𝜃 ≠ inf𝓊∈𝐾𝜕(𝑒
𝒯(𝓊)), so this contradict the fact. Hence, the result follows. 

Theorem 3.3. Let 𝐾𝜕 be a nonempty set. Suppose that 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-

FM on 𝐾𝜕, with 𝜃 = inf𝓊∈𝐾𝜕(𝑒
𝒯(𝓊)) exists in [0,1], and that the set 

𝛺 = {𝓊:𝓊 ∈ 𝐾𝜕,𝑒
𝒯(𝓊) = 𝜃} ≠ ∅. 

If 𝓊 is local minimum of 𝒯, then it is also a global minimum of 𝒯 on 𝐾𝜕.  

Proof. Let 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-FM and  𝓊 is local minimum of  𝒯. If  𝓊 is 

not global minimum of 𝒯, then 𝓊 ∉ 𝛺. By hypothesis, 𝛺 ≠ ∅. If 𝜗 ∈ 𝛺, then we must have 

𝒯(𝜗) ≺ 𝒯(𝓊). Since 𝒯 is exponentially preinvex-FM then, for 𝜏 ∈ (0,1),  we have  

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ≺ 𝑒𝒯(𝓊), because 𝒯(𝜗) ≺ 𝒯(𝓊), 

for any small positive number 𝜏, and this contradiction proves the required result. 

Theorem 3.4. Let 𝒯:𝐾𝜕 → 𝔽0 be an exponentially quasi-preinvex-FM on 𝐾𝜕. If 𝓊 ∈ 𝐾𝜕 is a strict 

local minimum of 𝒯, then it is also a strict global minimum of 𝒯 on 𝐾𝜕. 

Proof. Let 𝓊 ∈ 𝐾𝜕 is a strict local minimum of 𝒯. Then by definition of strict local minimum, 

there exist a 𝛿 > 0 such that 𝒯(𝓊) ≺ 𝒯(𝜗) for all 𝜗 ∈ 𝐾𝜕 ∩ 𝐵𝛿(𝓊). Assume contrary, that is, for 

some 𝜗 ∈ 𝐾𝜕, we have 𝒯(𝓊) ⪰ 𝒯(𝜗). Since 𝒯 is an exponentially quasi-preinvex-FM then, for 𝜏 ∈

[0,1], we have 

        𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ max(𝑒𝒯(𝓊),𝑒𝒯(𝜗)) = 𝑒𝒯(𝓊), 



Int. J. Anal. Appl. 19 (4) (2021) 530 

 

since 𝒯(𝓊) ⪰ 𝒯(𝜗), we have  

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑒𝒯(𝓊) 

This contradicts that 𝓊 ∈ 𝐾𝜕 is a strict local minimum of 𝒯, and hence the results follow. 

Definition 3.2. A set 𝐾𝜕 ⊆ ℝ
2 is said to an invex set w.r.t. 𝜕, if  

    (𝓊 + 𝜏𝜕(𝜗,𝓊), (1 − 𝜏)𝛽1+̃𝜏𝛽2) ∈ 𝐾𝜕, 

 (𝓊,𝛽1),(𝜗,𝛽2) ∈ 𝐾𝜕, 𝓊,𝜗 ∈ ℝ, 𝜏 ∈ [0,1]. 

The epigraph of exponentially preinvex-FM 𝒯:𝐾𝜕 → 𝔽0 can be given as 

𝐸(𝒯) = {(𝓊,𝛽):𝓊 ∈ 𝐾𝜕,𝛽 ∈ 𝔽0,𝑒
𝒯(𝓊) ⪯ 𝛽}. 

We now give a characterization of exponentially preinvex-FMs in terms of its epigraph 𝐸(𝒯).  

Theorem 3.5. A FM 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-FM w.r.t. 𝜕, if and only if, its 

epigraph is an invex set w.r.t. �̃�, where �̃�: 𝐸(𝒯) × 𝐸(𝒯) ⟶ ℝ × 𝛺 with 

 �̃�((𝜗,𝛽2),(𝓊,𝛽1)) = (𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1),  

for (𝓊,𝛽1),(𝜗,𝛽2) ∈ 𝐸(𝒯). 

Proof. Let (𝓊,𝛽1) , (𝜗,𝛽2) ∈ 𝐸(𝒯) . Then  𝑒
𝒯(𝓊) ⪯ 𝛽1  and 𝑒

𝒯(𝜗) ⪯ 𝛽2 . Since FM 𝒯:𝐾𝜕 → 𝔽0  is 

exponentially preinvex-FM w.r.t. 𝜕, so for all 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1], we have 

     𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ⪯ (1 − 𝜏)𝛽1+̃𝜏𝛽2, 

from which it follows that 

(𝓊 + 𝜏𝜕(𝜗,𝓊),(1 − 𝜏)𝛽1+̃𝜏𝛽2) ∈ 𝐸(𝒯), 

which implies that   

  (𝓊 + 𝜏𝜕(𝜗,𝓊),(1 − 𝜏)𝛽1+̃𝜏𝛽2) = (𝓊,𝛽1)+ 𝜏(𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1), 

      = (𝓊,𝛽1)+ 𝜏�̃�((𝜗,𝛽2),(𝓊,𝛽1)) ∈ 𝐸(𝒯). 

 Hence, 𝐸(𝒯)  is an invex set w.r.t. �̃�:𝐸(𝒯) × 𝐸(𝒯) ⟶ ℝ × 𝛺, with �̃�((𝜗,𝛽2),(𝓊,𝛽1)) =

(𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1) for (𝓊,𝛽1),(𝜗,𝛽2) ∈ 𝐸(𝒯). 

Conversely, 𝐸(𝒯)  is an invex set w.r.t. �̃�:𝐸(𝒯) × 𝐸(𝒯) ⟶ ℝ × 𝛺,  with �̃�((𝜗,𝛽2),(𝓊,𝛽1)) =

(𝜕(𝜗,𝓊),𝛽2+̃(−1)𝛽1) for  (𝜗,𝛽2),(𝓊,𝛽1) ∈ 𝐸(𝒯). Since  (𝓊,𝑒
𝒯(𝓊)),(𝜗,𝑒𝒯(𝜗)) ∈ 𝐸(𝒯), we have for 

𝜏 ∈ [0,1]  

(𝓊,𝑒𝒯(𝓊))+ 𝜏�̃�((𝜗,𝑒𝒯(𝜗)),(𝓊,𝑒𝒯(𝓊))) ∈ 𝐸(𝒯), 

since �̃�((𝜗,𝛽2),(𝓊,𝛽1)) = (𝜕(𝜗,𝓊),𝛽2+̃(−1) 𝛽1), then 

(𝓊,𝑒𝒯(𝓊)) + 𝜏(𝜕(𝜗,𝓊),𝑒𝒯(𝜗)+̃(−1)𝑒𝒯(𝓊)) ∈ 𝐸(𝒯), 



Int. J. Anal. Appl. 19 (4) (2021) 531 

 

which implies that 

(𝓊 + 𝜏𝜕(𝜗,𝓊),(1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗)) ∈ 𝐸(𝒯), 

so we have 

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗). 

Hence, 𝒯:𝐾𝜕 → 𝔽0 is an exponentially preinvex-FM w.r.t. 𝜕. 

Theorem 3.6. Let {𝒯𝑗}𝒋∈𝐼
:𝐾𝜕 → 𝔽0 be a family of exponentially preinvex-FMs and bounded above 

on an invex set 𝐾𝜕. Then the mapping 𝒯(𝓊) = 𝑠𝑢𝑝𝒋∈𝐼𝒯𝑗(𝓊) is an exponentially preinvex-FM on 

𝐾𝜕.  

Proof. Since each 𝒯𝑗 is an exponentially preinvex-FM on the invex set 𝐾𝜕, its epigraph 𝐸(𝒯) =

{(𝓊,𝛽):𝓊 ∈ 𝐾𝜕,𝛽 ∈ 𝔽0,𝑒
𝒯𝑗(𝓊) ⪯ 𝛽,𝑗 ∈ 𝐼}  is an invex set w.r.t. �̃�,  by Theorem 3.5. Then their 

intersection 

    ⋂ 𝐸(𝒯) = {(𝓊,𝛽):𝓊 ∈ 𝐾𝜕,𝛽 ∈ 𝔽0,𝑒
𝒯𝑗(𝓊) ⪯ 𝛽,𝑗 ∈ 𝐼}𝒋∈𝐼 , 

is also an invex set w.r.t. 𝜕. Hence, 𝒯(𝓊) = 𝑠𝑢𝑝𝒋∈𝐼𝒯𝑗(𝓊) is an exponentially preinvex-FM. 

Definition 3.3. Let 𝒯:𝐾𝜕 → 𝔽0 be an FM then the level set of 𝒯 is denoted by 𝒯𝛽 and defined as,  

𝒯𝛽 = {𝓊:𝓊 ∈ 𝐾𝜕,𝑒
𝒯(𝓊) ⪯ 𝛽,𝛽 ∈ 𝔽0}. 

Note that the level set is also called 𝛽-cut of 𝒯 and the set 𝒯𝛽 generalizes the standard form of 𝛽-

level of 𝒯. The set of all level sets of 𝒯 is represented as 𝐿(𝒯𝛽). 

Theorem 3.7.  Let 𝒯:𝐾𝜕 → 𝔽0 be an exponentially preinvex-FM. Then 𝒯𝛽 is an invex set. 

Proof.  Let 𝓊,𝜗 ∈ 𝒯𝛽 . Then 𝑒
𝒯(𝓊) ⪯ 𝛽 and 𝑒𝒯(𝜗) ⪯ 𝛽 . Since 𝒯:𝐾𝜕 → 𝔽0  is an exponentially 

preinvex-FM, 

                                  𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) ⪯ (1 − 𝜏)𝛽 + 𝜏𝛽 = 𝛽, 

hence 𝒯𝛽 is an invex set. 

Theorem 3.8. The comparable FM 𝒯:𝐾𝜕 → 𝔽0 is said to be exponentially quasi-preinvex-FM if 

and only if, the level set 𝒯𝛽 is an invex set. 

Proof. Let 𝓊,𝜗 ∈ 𝒯𝛽. Then by definition of 𝒯𝛽, we have 𝑒
𝒯(𝓊) ⪯ 𝛽 and 𝑒𝒯(𝜗) ⪯ 𝛽. Since 𝓊,𝜗 ∈ 𝐾𝜕 

and 𝐾𝜕 is invex set then, 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝐾𝜕. Since 𝒯 is exponentially quasi-preinvex-FM then, 

                   𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑚𝑎𝑥(𝑒𝒯(𝓊), 𝑒𝒯(𝜗)) ⪯ 𝑚𝑎𝑥(𝛽,𝛽) = 𝛽, ∀ 𝓊,𝜗 ∈ 𝑅,  𝜏 ∈ [0,1], 

from which we can note that 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝒯𝛽 is an invex set. 



Int. J. Anal. Appl. 19 (4) (2021) 532 

 

Conversely, assume that 𝒯𝛽 be an invex set to prove 𝒯 is exponentially quasi-preinvex-FM. Since 

𝒯𝛽 is an invex set then, for any 𝓊,𝜗 ∈ 𝒯𝛽 such that 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝒯𝛽 with 𝜏 ∈ [0,1]. Now we take  

𝑚𝑎𝑥(𝑒𝒯(𝓊),𝑒𝒯(𝜗)) = 𝛽 and 𝑒𝒯(𝓊) ⪰ 𝑒𝒯(𝜗). 

Now by the definition of invex set 𝒯𝛽, we have 

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑚𝑎𝑥(𝑒𝒯(𝓊),𝑒𝒯(𝜗)) ⪯ 𝛽. 

Hence, 𝒯 is exponentially quasi-preinvex-FM. 

Lemma 3.1. For a comparable FM 𝒯:𝐾𝜕 → 𝔽0: 

Exponentially logarithmic preinvexity ⇒  exponentially preinvexity ⇒  exponentially quasi-

preinvexity. 

Proof. For 𝜏 ∈ [0,1], and 𝓊,𝜗 ∈ 𝐾𝜕, we have 

 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (𝑒𝒯(𝓊))
1−𝜏
(𝑒𝒯(𝜗))

𝜏
 

       ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗) 

       ⪯ 𝑚𝑎𝑥(𝑒𝒯(𝓊),𝑒𝒯(𝜗)). 

Theorem 3.9.  Let FM 𝒯:𝐾𝜕 → 𝔽0 be a exponentially logarithmic preinvex-FM. Then its epigraph 

𝐸(𝒯) is an invex set. 

Proof. From Lemma 3.1, it follows that exponentially logarithmic preinvex-FM 𝒯  is 

exponentially preinvex-FM. Hence by Theorem 3.5, epigraph 𝐸(𝒯) is an invex set.  

Theorem 3.10. Let {𝒯𝑗}𝒋∈𝐼
:𝐾𝜕 → 𝔽0 be a family of exponentially logarithmic preinvex-FMs and 

bounded above on an invex set 𝐾𝜕. Then the mapping 𝒯(𝓊) = 𝑠𝑢𝑝𝒋∈𝐼𝒯𝑗(𝓊) is an exponentially 

logarithmic preinvex-FM on 𝐾𝜕.  

Proof. By using Lemma 3.1 and Theorem 3.6, the demonstration of proof is analogous to 

Theorem 3.9. 

 

4. EXPONENTIALLY INVEX FUZZY MAPPINGS 

In this section, we propose the concept of exponentially-IFMs and some relations are 

established between exponentially preinvex-FM, exponentially-IFM and exponentially invex 

fuzzy monotone mapping.   

Definition 4.1. A FM 𝒯:𝐾𝜕 → 𝔽(ℝ) is called sharply exponentially pseudo invex w.r.t. 𝜕 on 𝐾𝜕, 

if  



Int. J. Anal. Appl. 19 (4) (2021) 533 

 

〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉 ⪰ 0̃ ⟹ 𝑒𝒯(𝜗+𝜏𝜕(𝜗,𝓊)) ⪯ 𝑒𝒯(𝜗),   

∀ 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈ [0,1], where 𝒯
,, is G-differentiable of 𝒯 at 𝓊. 

Theorem 4.1. Let 𝒯 be a sharply exponentially pseudo preinvex-FM on 𝐾𝜕. Then 

−̃〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝜗,𝓊)〉 ⪰ 0̃, ∀ 𝓊,𝜗 ∈ 𝐾𝜕. 

Proof. Let 𝒯 be a sharply exponentially pseudo preinvex-FM on 𝐾𝜕. Then for all 𝓊,𝜗 ∈ 𝐾𝜕, 𝜏 ∈

[0,1], we have 

𝑒𝒯(𝜗) ⪰ 𝑒𝒯(𝜗+𝜏𝜕(𝜗,𝓊)), 

from which we get 

𝑒𝒯(𝜗+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝜗)

𝜏
⪯ 0̃. 

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

    −̃〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝜗,𝓊)〉 ⪰ 0̃, ∀ 𝓊,𝜗 ∈ 𝐾𝜕. 

Hence, the result follows. 

Definition 4.2. A G-differentiable FM 𝒯:𝐾𝜕 → 𝔽(ℝ) is said to be: 

• invex w.r.t. 𝜕, if 

𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉,     (4.1) 

• strictly incave w.r.t. 𝜕, if inequality (4.1) is reversed. 

• invex w.r.t. 𝜕, if 

𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉,     (4.2) 

• strictly incave w.r.t. 𝜕, if inequality (4.2) is reversed. 

∀ 𝓊,𝜗 ∈ 𝐾𝜕. 

Example 4.1. We consider the FMs 𝒯:(0,∞) → 𝔽(ℝ) defined by, 

𝒯(𝓊)(σ) =

{
 
 

 
  
4σ − 𝓊

𝓊
     σ ∈ [

1

4
𝓊,𝓊],

3𝓊 − 2σ

𝓊
   σ ∈ (𝓊,

3

2
𝓊],

0                  otherwise.

 

Then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [(
1

4
+
3

4
𝛾)𝓊,(

3

2
−
1

2
𝛾)𝓊]. Since end point functions 

𝒯∗(𝛾), 𝒯
∗(𝛾) are exponentially invex functions for each 𝛾 ∈ [0,1], then 𝒯 is exponentially-IFM 

w.r.t.  

∂(ϑ,𝓊) = ϑ. 



Int. J. Anal. Appl. 19 (4) (2021) 534 

 

Example 4.2. We consider the FMs 𝒯:(0,∞) → 𝔽(ℝ) defined by, 

𝒯(𝓊)(σ) =

{
 
 

 
  
−𝓊 + σ

𝓊
    σ ∈ [𝓊,2𝓊],

4𝓊 − σ

2𝓊
   σ ∈ (2𝓊,4𝓊],

0                 otherwise.

 

Then, for each 𝛾 ∈ [0,1], we have 𝒯𝛾(𝓊) = [(1 + 𝛾)𝓊,2(2 − 𝛾)𝓊]. Since end point functions 𝒯∗(𝛾), 

𝒯∗(𝛾) are exponentially invex functions for each 𝛾 ∈ [0,1], then 𝒯 is strictly exponentially-IFM 

w.r.t.  

∂(ϑ,𝓊) = ϑ + 𝓊. 

Theorem 4.2.  Let  𝒯:𝐾𝜕 → 𝔽(ℝ) be a G-differentiable FM and let 

(i) 𝜕(𝜗,𝓊 + 𝜏𝜕(𝜗,𝓊)) = (1 − 𝜏)𝜕(𝜗,𝓊),   (ii) 𝜕(𝓊,𝓊 + 𝜏𝜕(𝜗,𝓊)) = −𝜏𝜕(𝜗,𝓊). 

Then 𝒯 is an exponentially preinvex-FM, if and only if 𝒯 is an exponentially-IFM. 

Proof. Let  𝒯:𝐾𝜕 → 𝔽(ℝ)  be G-differentiable exponentially preinvex-FM. Since 𝒯  is 

exponentially preinvex-FM then, for each 𝓊,𝜗 ∈ 𝐾𝜕 and 𝜏 ∈ [0,1],  therefore  

         𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), 

       = 𝑒𝒯(𝓊)+̃𝜏(𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊)), 

which implies that 

     𝜏(𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊)) ⪰ 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊), 

𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰
𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊)

𝜏
. 

On taking limit in the above inequality as 𝜏 → 0, we obtain 

        𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. 

Conversely, let 𝒯  be an exponentially-IFM. 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 (4.1), we get   

𝑒𝒯(𝜗𝜏)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗𝜏,𝓊)〉. 

From (i) and (ii), we have 

𝑒𝒯(𝜗𝜏)−̃𝑒𝒯(𝓊) ⪰ (1 − 𝜏)〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉.    (4.3) 

In a similar way, we get 

𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗𝜏) ⪰ 〈𝑒𝒯(𝓊) 𝒯,(𝓊),𝜕(𝓊,𝜗𝜏)〉  

         = −𝜏〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝜗𝜏)〉,    (4.4) 

Multiplying (4.3) by 𝜏 and (4.4) by (1 − 𝜏), and adding the resultant, we have 



Int. J. Anal. Appl. 19 (4) (2021) 535 

 

𝑒𝒯(𝜗𝜏) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), 

which implies that 

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗). 

Hence, 𝒯 is exponentially preinvex-FM w.r.t. 𝜕. 

Theorem 4.3. Let 𝒯 be a G-differentiable exponentially preinvex-FM on the invex set 𝐾𝜕 and let 

 (i) 𝜕(𝜗,𝓊 + 𝜏𝜕(𝜗,𝓊)) = (1 − 𝜏)𝜕(𝜗,𝓊),   (ii) 𝜕(𝓊,𝓊 + 𝜏𝜕(𝜗,𝓊)) = −𝜏𝜕(𝜗,𝓊). 

 Then the followings are equivalent 

(1) 𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉,        (4.5) 

(2) 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃〈𝑒𝒯(𝜗)𝒯 ,(𝜗),𝜕(𝓊,𝜗)〉 ⪯ 0̃,     (4.6) 

for all 𝓊,𝜗 ∈ 𝐾𝜕. 

Proof. (1) implies (2), let (1) holds. Then, by replacing 𝜗 by 𝓊 and 𝓊 by 𝜗 in (4.5), we get 

𝑒𝒯(𝓊)−̃𝑒𝒯(𝜗) ⪰ 〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝓊,𝜗)〉,          (4.7) 

adding (4.5) and (4.7), we have  

〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃〈𝑒𝒯(𝜗)𝒯 ,(𝜗),𝜕(𝓊,𝜗)〉 ⪯ 0̃, 

(2) implies (1), assume that (2) holds. Then  

〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉 ⪯ −̃〈𝑒𝒯(𝜗)𝒯,(𝜗),𝜕(𝓊,𝜗)〉,    (4.8)  

Since 𝐾𝜕  is an invex set so we have, 𝜗𝜏 = 𝓊 + 𝜏𝜕(𝜗,𝓊) ∈ 𝐾𝜕  for all 𝓊,𝜗 ∈ 𝐾𝜕  and 𝑡 ∈ [0,1]. 

Taking 𝜗=𝜗𝜏 in (4.8), we  obtain 

〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜕(𝓊,𝓊 + 𝜏𝜕(𝜗,𝓊))〉 ⪯ −̃〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝓊 + 𝜏𝜕(𝜗,𝓊),𝓊)〉,    

From (i) and (ii), we have 

 〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜏𝜕(𝜗,𝓊)〉 ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜏𝜕(𝜗,𝓊)〉, 

  〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜕(𝜗,𝓊)〉 ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉,    (4.9) 

Let 

ℋ(𝜏) = 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)). 

Taking G-derivative w.r.t. 𝜏, we get 

ℋ ,(𝜏) = 𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)).𝜕(𝜗,𝓊) = 〈𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))𝒯,(𝓊 + 𝜏𝜕(𝜗,𝓊)),𝜕(𝜗,𝓊)〉. 

The above inequality along with (4.9) gives 

    ℋ ,(𝜏) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉,        (4.10) 

By integrating (4.10) between 0 to 1 w.r.t. 𝜏, we get 



Int. J. Anal. Appl. 19 (4) (2021) 536 

 

    ℋ(1)−̃ℋ(0) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, 

              𝑒𝒯(𝓊+𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. 

Since 𝒯 is strongly exponentially preinvex-FM, then for 𝜏 = 1, we have 

𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, for all 𝓊,𝜗 ∈ 𝐾𝜕. 

Note that, if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊 in (4.5), then we obtain the G-differentiable exponentially convex-

FM. Consequently, the exponentially-IFM is more general than G-differentiable convex-FM.  

The exponentially-IFM type is equivalent to the type of FM whose stationary points are global 

minima. 

 

5. APPLICATIONS 

In this section, by using the familiar fact that fuzzy variational inequality problem has deep 

relationship with the fuzzy optimization problems, we try to explore some applications of 

exponentially preinvex-FMs in fuzzy optimization and prove that the minimum of PFMs can be 

distinguish with variational-like inequality.  

Theorem 5.1.  Let 𝒯 be an exponentially preinvex-FM on 𝐾𝜕. Then 𝓊 ∈ 𝐾𝜕 is the minimum of the 

FM 𝒯, if and only if, for 𝓊 ∈ 𝐾𝜕 such that  

〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉 ⪰ 0̃,       (5.1) 

for all 𝜗 ∈ 𝐾𝜕.  

Proof. Let 𝓊 ∈ 𝐾𝜕 be a minimum of 𝒯. Then 

                                                    𝒯(𝓊) ⪯ 𝒯(𝜗),   ∀ 𝜗 ∈ 𝐾𝜕,     

From which, we have 

                                            𝑒𝒯(𝓊) ⪯ 𝑒𝒯(𝜗), ∀ 𝜗 ∈ 𝐾𝜕.     (5.2) 

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

(5.2), we obtain 

 0̃ ⪯
𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊)

𝜏
, 

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

      0̃ ⪯ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉. 

Conversely, since 𝒯:𝐾𝜕 → 𝔽(ℝ) is an exponentially preinvex-FM, so 

𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊)) ⪯ (1 − 𝜏)𝑒𝒯(𝓊)+̃𝜏𝑒𝒯(𝜗), 



Int. J. Anal. Appl. 19 (4) (2021) 537 

 

𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰
𝑒𝒯(𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊)

𝜏
, 

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

𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉, 

The above inequality and  (5.1) yield  

               𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊) ⪰ 0̃. 

Hence, the result follows. 

Remark 5.1. The inequality of the type (5.1) is called exponentially fuzzy variational-like 

inequality. We  emphasize that if 𝜕(𝜗,𝓊) = 𝜗 − 𝓊, the optimality conditions of the exponentially 

convex-FM can be characterized by the following inequality 

〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜗 − 𝓊〉 ⪰ 0̃, ∀ 𝓊,𝜗 ∈ 𝐾𝜕,     

which is called exponentially fuzzy variational inequality.  

We consider the functional 𝐼(𝜗), defined as 

     𝐼(𝜗) = 𝒯(𝜗)+̃𝒥(𝜗), ∀ 𝜗 ∈ ℝ,     (5.3) 

where 𝒯  is a G-differentiable exponentially preinvex-FM and 𝒥  is a non G-differentiable 

exponentially preinvex-FM. 

Since both 𝒯(𝜗) and 𝒥(𝜗) both are exponentially preinvex-FMs, then by Remark 2.2 (iii), (5.3) 

can be written as, 

𝐼(𝜗) = 𝑒𝒯(𝜗)+̃𝑒𝒥(𝜗),∀ 𝜗 ∈ ℝ,     (5.4) 

where 𝑒𝒯(𝜗) and 𝑒𝒥(𝜗) both are preinvex-FM w.r.t. same 𝜕. 

We know that the minimum of the functional 𝐼(𝜗), can be characterized by a class of variational 

like-inequalities. 

Theorem 5.2.  Let 𝒯:𝐾𝜕 → 𝔽(ℝ) be a G-differentiable exponentially preinvex-FM and 𝒥:𝐾𝜕 →

𝔽(ℝ)  be a non G-differentiable exponentially preinvex-FM. Then the functional 𝐼(𝜗)  has 

minimum 𝓊 ∈ 𝐾𝜕, if and only if, 𝓊 ∈ 𝐾𝜕 satisfies  

〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊) ⪰ 0̃, ∀ 𝜗 ∈ 𝐾𝜕.      (5.5)                           

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

𝐼(𝓊) ⪯ 𝐼(𝜗).      (5.6) 

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 

(5.6), we get 



Int. J. Anal. Appl. 19 (4) (2021) 538 

 

𝐼(𝓊) ⪯ 𝐼(𝓊 + 𝜏𝜕(𝜗,𝓊)), 

which implies that, using (5.4), we have 

                                          𝑒𝒯(𝓊)+̃𝑒𝒥(𝓊) ⪯ 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))+̃𝑒𝒥((𝓊+𝜏𝜕(𝜗,𝓊)) . 

Since 𝒥 is exponentially preinvex-FM then, 

                                      𝑒𝒯(𝓊)+̃𝑒𝒥(𝓊) ⪯ 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))+̃(1 − 𝜏)𝑒𝒥(𝓊)+̃𝜏𝑒𝒥(𝜗), 

that is 

                                     0̃ ⪯ 𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊)+̃𝜏(𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊)). 

On dividing by “𝜏” and taking lim
𝜏→0

, we get 

0̃ ⪯ lim
𝜏→0

𝑒𝒯((𝓊+𝜏𝜕(𝜗,𝓊))−̃𝑒𝒯(𝓊)

𝜏
+̃𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊). 

Conversely, let (5.5) be satisfy to prove 𝓊 ∈ 𝐾𝜕 is a minimum of 𝐼. Assume that 

0̃ ⪯ 〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉+̃𝑒𝒥(𝜗)−̃𝑒𝒥(𝓊).                         

 for all 𝜗 ∈ 𝐾𝜕, we have 

                        𝐼(𝓊)−̃𝐼(𝜗) = 𝑒𝒯(𝓊)+̃𝑒𝒥(𝓊)−̃𝑒𝒯(𝜗)−̃𝑒𝒥(𝜗),                    

                                = −̃(𝑒𝒯(𝜗)−̃𝑒𝒯(𝓊))+̃𝑒𝒥(𝓊)−̃𝑒𝒥(𝜗), 

Using Theorem 4.2, we obtain 

𝐼(𝓊)−̃𝐼(𝜗) ⪯ −̃[〈𝑒𝒯(𝓊)𝒯,(𝓊),𝜕(𝜗,𝓊)〉−̃𝑒𝒥(𝜗)+̃𝑒𝒥(𝓊)] ⪯ 0̃, 

From (5.5), we have  

             𝐼(𝓊)−̃𝐼(𝜗) ⪯ 0̃, 

 hence, 𝐼(𝓊) ⪯ 𝐼(𝜗). 

Note that the (5.5) is called exponentially fuzzy mixed variational like-inequality. This result 

shows that the fuzzy optimality conditions of fuzzy functional 𝐼(𝜗) can be characterized by 

exponentially fuzzy mixed variational-like inequalities. If 𝜕(𝜗,𝓊) = 𝜗 − 𝓊, then from (5.1) and 

(5.2), we can obtain the family of some new classes. 

 

𝟔. CONCLUSION 

Convex and generalized convex-FMs play an important role in fuzzy optimization. 

Therefore, by the importance of nonconvex-FMs, we have introduced and considered a class of 

nonconvex-FMs is called exponentially preinvex-FMs. It is illustrated that classical convexity 



Int. J. Anal. Appl. 19 (4) (2021) 539 

 

and nonconvexity are special cases of exponentially preinvex-FMs. We have also introduced the 

notions of quasi-preinvex and log-preinvex-FMs and investigated some properties.  Some 

relations among the exponentially preinvex-FMs, exponentially invex-FMs, and exponentially 

monotonicities are derived under some mild conditions. We have proved that optimality 

conditions of G-differentiable exponentially preinvex-FMs and for the sum of G-differentiable 

preinvex-FMs and non G-differentiable exponentially preinvex-FMs can be characterized by 

exponentially fuzzy variational-like inequalities and exponentially fuzzy mixed variational-like 

inequalities, respectively. The inequalities (5.1) and (5.2) are the interesting outcome of our main 

results. It is itself an engaging problem to flourish some well-organized some numerical 

methods for solving exponentially fuzzy variational-like inequalities and exponentially fuzzy 

mixed variational-like inequalities together with applications in applied and pure sciences. In 

the future, we will try to investigate the applications of exponentially fuzzy variational-like 

inequalities and exponentially 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 there are no conflicts of interest regarding the 

publication of this paper. 

 

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