International Journal of Analysis and Applications
ISSN 2291-8639
Volume 15, Number 1 (2017), 75-85
http://www.etamaths.com

OPTIMALITY AND DUALITY DEFINED BY THE CONCEPT OF TEMPERED

FRACTIONAL UNIVEX FUNCTIONS IN MULTI-OBJECTIVE OPTIMIZATION

RABHA W. IBRAHIM

Abstract. In this paper, we purpose the concept of tempered Univex functions by utilizing a tem-

pered fractional difference-differential operator type Caputo. This instruction indicates a new class

of these functions in some optimal problems by exemplifying the settings on the modified formula.
We call it the class of tempered fractional Univex functions. Our study is based on the strong, weak,

converse, and strict converse duality propositions. A Multi-objective optimal problem includes the

new process is disentangled.

1. Introduction

In 1989, Dunkl imposed a difference-differential operator [1] setting on some Euclidean space and
realizing the commutative law for a differentiable function on Rn. This operator can be employed
in various parts in pure mathematics, such as Lee algebra, Clifford algebra and complex analysis.
In 1998, Rosler and Voit acquired into consideration this operator to adapt the tool of the Markov
Processes [2]. Unevenly, these operators can be expected as a simplification of the partial derivatives
and various constructions of operators like the Laplace operator, the Fourier transform, and the Hermite
polynomials. Also, these operators convoluted in famous processes such as the Brownian motion and
the Cauchy processes.

Recently, this operator and its some simplifications have improved significant care in many fields of
mathematics and physics. They shield a helpful method in the study of special functions and they
are closely related with definite demonstrations of degenerate affine Hecke algebras. Furthermore,
Dunkl operator is obviously convoluted in the algebraic explanation of definite devotedly resolvable
quantum multi-body systems. It can be used to identify the generalized method of the heat equation,
which is called the Dunkl heat equation. It can be recommended to adapt the idea of moments of
probability measures on Rn. Our goal is to generalize the Dunkl operator in view of the tempered
fractional calculus and propose it to simplify the class of non-linear Univex functions. This class
typically appears in many non-linear multi-objective problems. The benefit of exploiting the Dunkle
operator is that this operator deals with multi-dimensional spaces. Moreover, the author extend it
to the complex plane and provided a modified differential-difference Dunkl operator in the open unit
disk. The study was in the field of geometric function theory [3].

Fractional calculus is the most important branch of mathematical analysis, because it refers to the
non-linearity studies in all science. The most famous operators are the Riemann-Liouville, Caputo
(continuous operators) and Grunwald-Letnikov (discrete operator) (see [4], [5]). It has been presented
the fractional calculus discoveries usage in many categories of science and engineering, containing fluid
flow, diffusive transport theory, electrical networks, electromagnetic theory, probability and statistics.
The tempered fractional diffusion idea was established in statistics. This idea has demonstrated useful
applications in geophysics and finance [6]. Moreover, it applied to introduce a fractional multi-objective
function in optimal control [7].

Received 15th May, 2017; accepted 28th July, 2017; published 1st September, 2017.
2010 Mathematics Subject Classification. 34A08, 26A33, 49J35.
Key words and phrases. fractional calculus; fractional differential operator; fractional differential equation; univex

function.

c©2017 Authors retain the copyrights of
their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.

75



76 RABHA W. IBRAHIM

In this study, we aim to generalize the concept of tempered Univex functions by utilizing a tempered
fractional differential-difference operator (Dunkl operator), based on different types of fractional cal-
culus. This study gives us a new class of these functions in some optimal problems by illustrating
conditions on the generalized functions. We call it the class of tempered fractional Univex functions.
The strong, weak, converse, and strict converse duality theorems are proposed. The main tool employed
in the analysis is based on the tempered Caputo operator.

2. Handling

We need the following concepts in the sequel of the article:

2.1. Dunkl operator. Suppose the two column vectors x = (x1, ...,xn) and v = (v1, ...,vn) ∈ Rn
with their dot product x.v = xT .v =

∑n
i=1 vixi. The operator through the hyper-plane is defined by

σv x = x− 2
v.x

v.v
v.

In a matrix form, we have

σv = I − 2
vvT

vTv
.

Note that σv can be represented by a symmetric matrix. The Dunkl operator is formulated by:

Diφ(x) =
∂

∂xi
φ(x) +

∑
v∈R+

kv
(1 −σv)φ(x)

v.x
vi, i = 1, ...,n,

where vi is the i-th component of v, 1 ≤ i ≤ n, x ∈ Rn. , and φ smooth function on Rn. When kv = 0,
then we have

Diφ(x) =
∂

∂xi
φ(x).

One of the outcomes of these operators is satisfying

Di(Djφ(x)) = Dj(Diφ(x)). (2.1)

Moreover, the operator achieves the product

Di[φ(x)ψ(x)] = ψ(x)Diφ(x) + φ(x)Diψ(x).

Note that, if φ is a polynomial of degree n, then (1 − σv)φ(x)/v.x is a polynomial of degree n − 1.
Moreover, the path of the Dunkl process onto a subset of Rn is collected by the set

S = {x ∈ Rn : x.v > 0, ∀v ∈ R+}.

Finally, Dunkl processes are formulated as the Markov processes which achieve the Dunkl heat equation

∂

∂t
−

1

2

n∑
i=1

D2i = 0.

2.2. Fractional calculus. The Cauchy formula for frequent integration, to be specific as follows:

(Inφ)(χ) =
1

(n− 1)!

∫ χ
0

(χ− t)n−1φ(t) dt,

drives in an explicit way to a generalization for real n. Utilizing the gamma function to take off the
discrete nature of the factorial function allows us a natural candidate for fractional usage of the integral
operator.

(Iαφ)(χ) =
1

Γ(α)

∫ χ
0

(χ− t)α−1φ(t) dt

This operator is well-defined and it is represented to the classical fractional calculus, which is called the
Riemann-Liouville fractional integral operator. It is straightforward to show that the integral operator
achieves the semi-group property of fractional differ-integral operators

(Iα)(Iβφ)(χ) = (Iβ)(Iαφ)(χ) = (Iα+βφ)(χ) =
1

Γ(α + β)

∫ χ
0

(χ− t)α+β−1φ(t) dt.



OPTIMALITY AND DUALITY 77

Corresponding to the above fractional integral operator and for a general function φ(χ) and 0 < α < 1,
the complete fractional derivative is defined as follows:

Dαφ(χ) =
1

Γ(1 −α)
d

dχ

∫ χ
0

φ(t)

(χ− t)α
dt.

For the fractional power α < 1, since the gamma function is sloppy for values whose real part is a
negative integer with the imaginary part is equal to zero, it is important to employ the fractional
derivative after the integer derivative has been accurate. For example,

D5/4φ(χ) = D1/4
(
D1φ(χ)

)
= D1/4

( d
dχ
φ(χ)

)
.

In general, calculating n-th order derivative over the integral of order (n−α), is given by the formula

aD
α
χφ(χ) =

dn

dχn
aD
−(n−α)
χ φ(χ) =

dn

dχn
aI
n−α
χ φ(χ).

The Riemann-Liouville calculus admits a fast converge, historical property, natural generalization and
wide applications in almost all science. One of the most property of this calculus is as follows:

Dαχm =
Γ(m + 1)

Γ(m−α + 1)
χm−α, m ≥ 0.

There is another fractional differential operator called the Caputo operator, which is defined as follows:

C
a D

α
χφ(χ) =

1

Γ(n−α)

∫ χ
a

φ(n)(τ) dτ

(χ− τ)α+1−n
.

This type of fractional differential operator is applied to find the solutions of fractional differential
equations with initial conditions.

2.2.1. Tempered fractional calculus. The Riemann-Liouville tempered fractional derivative is formed
as follows [9]

D̂α,λφ(χ) = e−λχDα
(
eλχφ(χ)

)
−λαφ(χ), λ ≥ 0.

And the The Caputo tempered fractional derivative is formed as follows

C
a D̂

α,λφ(χ) = D̂α,λ[φ(χ) −
n−1∑
i=0

χi
i!
φ(i)(0)], λ ≥ 0,

where φ(i) is the derivative of order i, and n is the upper integer value less than α. When λ = 0, the
above equation reduces to the usual formula of the Caputo operator.

2.3. Tempered-Dunkl operator. Based on the Caputo tempered fractional calculus, assume that
the fractional partial derivative is denoted by Ca D̂

α,λ. Then for x ∈ Rnwe receive the tempered Dunkl
operator as follows:

D
α,λ
i φ(x) =

C
a D̂

α,λ
i φ(x) +

n∑
i=1,v∈R+

kv
(1 −σv)αφ(x)

v.x
vi, (2.2)

(
i = 1, ...,n, 0 < α < 1

)
,

where Ca D̂
α,λ
i is denoted the fractional derivative with respect the component xi, R+ is a positive

subsystem, satisfying for all u ∈ R+, u.v > 0. Dunkl operators in the direction of y ∈ Rn is defined as
follows:

Dα,λφ(x) =

n∑
i=1

yiD
α,λ
i φ(x).

Our aim is to include the generalized tempered Dunkl operator in a class of fractional stochastic
differential equations and to study the behavior of the solutions. We have the following properties of
the new operator:



78 RABHA W. IBRAHIM

Proposition 2.1 Let φ(x) be analytic function converging in the interval (0,ρ] with the approximate
form

φ(x) =

∞∑
m=0

am x
m+λ, λ > −1.

Then

D
α,λ
i

(
D
β,λ
j φ(x)

)
= D

β,λ
j

(
D
α,λ
i φ(x)

)
, α,β ∈ (0, 1), i = 1, ...,n.

Proof. In view of Theorem 3 in [3] and (2.1) , we have

D
α,λ
i

(
D
β,λ
j φ(x)

)
= D

α,λ
i

( ∂β
∂x

β
j

φ(x) +
∑
v∈R+

kv
(1 −σv)βφ(x)

v.x
vj

)
= D

β,λ
j

( ∂α
∂xαi

φ(x) +
∑
v∈R+

kv
(1 −σv)αφ(x)

v.x
vi

)
= D

β,λ
j

(
D
α,λ
i φ(x)

)
.

Proposition 2.2 Let φ and ψ be power functions in x. Then

Dαi [φ(x)ψ(x)] = ψ(x)D
α
i φ(x) + φ(x)D

α
i ψ(x).

Proof. By applying the fractional generalization of the Leibniz rule of the Caputo derivative [8]

∂α

∂xα
[φ(x)ψ(x)] =

∞∑
k=0

Γ(α + 1)

Γ(α−k + 1)Γ(k + 1)
∂α−kφ(x)∂kψ(x),

we conclude the desire result.

2.4. Tempered Univex function. In this subsection, we generalize the concept of tempered Univex
function, by using the fractional calculus. Let Ω be a nonempty subset of Rn,η : Ω × Ω → Rm, ξ be
an arbitrary point of Ω and h : Ω → Rm,φ : Rm → R.

Definition 2.1 A differential function h is said to be a tempered fractional univex function of order
α ∈ (0, 1) in the direction of ξ ∈ Ω if for all x ∈ Ω, we have

η(x,ξ).Dα,λ h(x) ≤ φ
(
h(x) −h(ξ)

)
,

where

Dα,λ h(x) =

n∑
i=1

ξiD
α,λ
i h(x), ξ = (ξ1, ...,ξn).

The advantage of using the tempered fractional Dunkl operator, is that can be acted on multi-
dimensional Euclidean spaces as well as it can be defined a parametric family of deformations of
the polynomial . Therefor, it can be employed in non-linear multi-objective problem

Minimize Ψ(x) =
(
ψ1(x), ...,ψm(x)

)
subjectto Θ(x) ≤ 0,

(2.3)

where Ψ : Ω → Rm and Θ : Ω → Rp and 0 is the zero vector in Rp. The function Ψ(x) can be applied in
various studies. It can be considered as a utility function over some set of needs (goods), cost function
of production presented a fixed quantity produced, growth function and others.

Definition 2.2 A point ξ ∈ Λ := {x ∈ Ω : Θ(x) ≤ 0} is said to be an efficient solution of (2.3), if
there exists no x ∈ Λ, such that Ψ(x) ≤ Ψ(ξ). And it is called a weak efficient solution if Ψ(x) < Ψ(ξ).

Next, we define a new class of fractional Univex function for the problem (2.3), we denote this class
by : (α,ρ,η,ϑ) as follows:



OPTIMALITY AND DUALITY 79

Definition 2.3 The couple (Ψ, Θ) is called (α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω if for all x ∈ Λ such
that

η1(x,ξ).D
α,λ Ψ(x) + ρ1‖ϑ(x,ξ)‖2 ≤ φ1

(
Ψ(x) − Ψ(ξ)

)
and

η2(x,ξ).D
α,λ Θ(x) + ρ2‖ϑ(x,ξ)‖2 ≤ −φ2

(
Θ(x) − Θ(ξ)

)
,

where η1 : Ω × Ω → Rm, η2 : Ω × Ω → Rp, ϑ : Ω × Λ → R,φ1 : Rm → R, φ2 : Rp → R and ρ1,ρ2 ∈ R.
We have the following facts:

Remark 2.1

• If φ1
(

Ψ(x) − Ψ(ξ)
)
≤ 0 ⇒ η1(x,ξ).Dα,λ Ψ(x) ≤ −ρ1‖ϑ(x,ξ)‖2 and φ2

(
Ψ(x) − Ψ(ξ)

)
≥

0 ⇒ η2(x,ξ).Dα,λ Ψ(x) ≤ −ρ2‖ϑ(x,ξ)‖2. Then the couple (Ψ, Θ) is called weak pseudo-quasi
(α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω.
• If φ1

(
Ψ(x) − Ψ(ξ)

)
≤ 0 ⇒ η1(x,ξ).Dα,λ Ψ(x) < −ρ1‖ϑ(x,ξ)‖2 and φ2

(
Ψ(x) − Ψ(ξ)

)
≥

0 ⇒ η2(x,ξ).Dα,λ Ψ(x) < −ρ2‖ϑ(x,ξ)‖2. Then the couple (Ψ, Θ) is called strong pseudo-quasi
(α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω.

3. Results

In this section, we investigate some sufficient optimality conditions for a point to be an efficient
solution of (2.3) under the tempered (α,λ,ρ,η,ϑ)-type Univex.

Theorem 3.1. Let ξ be an initial solution of the multi-objective problem (2.3) and c1 and c2 be two
non-negative constants such that

(A) Θ(ξ) = 0;

(B) c1
(
η1(x,ξ).D

α,λ Ψ(x)
)

+ c2
(
η2(x,ξ).D

α,λ Θ(x)
)
≥ 0;

(C) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω;
(D) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;
(E) c1ρ1 + c2ρ2 ≥ 0.
Then ξ is an efficient solution of (2.3).

Proof. Suppose that ξ is not an efficient solution of (2.3), then there exists x ∈ Λ such that Ψ(x) ≤
Ψ(ξ). By the assumptions (A) and (D), we have

φ1(Ψ(x) − Ψ(ξ)) ≤ 0, and φ2(Θ(ξ)) ≥ 0. (3.1)
In view of the assumption (C), we get

c1
(
η1(x,ξ).D

α,λ Ψ(x)
)
< −c1ρ1‖ϑ(x,ξ)‖2 (3.2)

and

c2
(
η2(x,ξ).D

α,λ Θ(x)
)
≤−c2ρ2‖ϑ(x,ξ)‖2. (3.3)

Summing the above inequalities and utilizing (E), we conclude that

c1
(
η1(x,ξ).D

α,λ Ψ(x)
)

+ c2
(
η2(x,ξ).D

α,λ Θ(x)
)
< −

(
c1ρ1 + c2ρ2

)
‖ϑ(x,ξ)‖2

≤ 0,

which contradicts the assumption (B). Hence, ξ is an efficient solution of (2.3). This completes the
proof.

Theorem 3.2. If the following conditions are satisfied:

(A) ξ is a weakly efficient solution of (2.3);

(B) Θ is continuous in ξ;

(C) The functions Ψ and Θ are fractional tempered Univex functions of order α ∈ (0, 1), λ ≥ 0 in the
direction of ξ ∈ Λ. Moreover, for some x̄ ∈ Λ, we have Θ(x̄) < 0.



80 RABHA W. IBRAHIM

Then there are two constants c1 ≥ 0 and c2 ≥ 0 such that
c1
(
η1(x,ξ).D

α,λ Ψ(x)
)

+ c2
(
η2(x,ξ).D

α,λ Θ(x)
)
≥ 0,(

x ∈ Ω, c2Θ(ξ) = 0, η1 : Ω × Ω → Rm, η2 : Ω × Ω → Rp
)
.

Proof. Our aim is to show that the system

η1(x,ξ).D
α,λ Ψ(x) < 0, η2(x,ξ).D

α,λ Θ(x) < 0,

has no solution for x ∈ Ω. Let the system has a solution y ∈ Ω. By the assumption (A), we have
Ψ(ξ + �1y) < Ψ(ξ) and Θ(ξ + �2y) < Θ(ξ),

for sufficient small arbitrary constants �1, �2 > 0. Now, we let x̄ := ξ + �2y; which implies that
x̄ ∈ Λ ∩N�2 (ξ) thus by (B) and (C), we have Θ(ξ + �2y) = Θ(x̄) < 0; which contradicts (A), where ξ
is a weak solution. Therefore, the above inequalities are non-negative. Hence, in view of (C) these are
two constants c1 and c2 satisfy the inequality

c1
(
η1(x,ξ).D

α,λ Ψ(x)
)

+ c2
(
η2(x,ξ).D

α,λ Θ(x)
)
≥ 0,

with the property c2Θ(ξ) = 0. This completes the proof.

Next, we consider the dual problem of (2.3) as follows:

Max Ψ(χ) =
(
ψ1(χ), ...,ψm(χ)

)
subjectto c1

(
η1(x,χ).D

α,λ Ψ(x)
)

+ c2
(
η2(x,χ).D

α,λ Θ(x)
)
≥ 0,

c2Θ(χ) ≥ 0,
(3.4)

where χ ∈ Ω, c1 and c2 be two non negative constants.

Theorem 3.3. Let x,χ be initial solutions of the multi-objective problems (2.3) and (3.4) respectively.
If

(A) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω;
(B) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;
(C) c1ρ1 + ρ2 ≥ 0;
then Ψ(x) � Ψ(χ).

Proof. Suppose that Ψ(x) ≤ Ψ(χ). Since c1ρ1 + ρ2 ≥ 0 then by (B), we obtain
φ1(Ψ(x) − Ψ(χ)) ≤ 0
φ2(Θ(χ)) ≥ 0.

In virtue of the assumption (A) the above inequalities yield(
η1(x,χ).D

α,λ Ψ(χ)
)
< −ρ1‖ϑ(x,χ)‖2(

η2(x,χ).D
α,λ Θ(χ)

)
≤−ρ2‖ϑ(x,χ)‖2,

consequently, we obtain
c1
(
η1(x,ξ).D

α,λ Ψ(x)
)
< −c1ρ1‖ϑ(x,χ)‖2

and
c2
(
η2(x,χ).D

α,λ Θ(x)
)
≤−ρ2‖ϑ(x,χ)‖2.

Summing the above inequalities and utilizing (C), we conclude that

c1
(
η1(x,χ).D

α,λ Ψ(χ)
)

+ c2
(
η2(x,χ).D

α,λ Θ(χ)
)
< −

(
c1ρ1 + ρ2

)
‖ϑ(x,χ)‖2

≤ 0,
which contradicts the assumption (C). This completes the proof.

Theorem 3.4. Let x0 and χ0 be initial solution for the problems (2.3) and (3.4) respectively. If
Ψ(x0) = Ψ(χ0) then the (weak or strong) duality problems (2.3) and (3.4) has efficient solutions x0
and χ0 respectively.



OPTIMALITY AND DUALITY 81

Proof. Suppose that x0 is not efficient for (2.3), then for some x ∈ Λ

Ψ(x) ≤ Ψ(x0) = Ψ(χ0),

which contradicts weak (strong) duality theorems as χ0 is initial solution for (3.4). Therefore, x0 is
efficient for (2.3). Similarly χ0 is efficient solution for (3.4). Hence the proof.

Theorem 3.5. Let χ0 be an initial solution of the multi-objective problem (3.4) and c1 and c2 be two
non negative constants such that

(A) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω;
(B) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;
(C) c1ρ1 + ρ2 ≥ 0.
Then χ0 is an efficient solution of (3.4).

Proof. Suppose that χ0 is not an efficient solution of (3.4), then there exists x0 ∈ Λ such that
Ψ(x0) ≤ Ψ(χ0). Now going on as in Theorem 3.3, we have a contradiction. Hence, χ0 is an efficient
solution of (3.4).

Theorem 3.6. Let x0,χ0 be initial solutions of the multi-objective problems (2.3) and (3.4) respec-
tively. If

(A) Ψ(x0) ≤ Ψ(χ0);
(B) The couple (Ψ, Θ) is a strong (or weak ) pseudo-quasi (α,λ,ρ,η,ϑ)-type univex at ξ ∈ Ω;
(C) u ≤ 0 ∈ Rm ⇒ φ1(u) ≤ 0 and v ≥ 0 ∈ Rp ⇒ φ2(v) ≥ 0;
(D) c1ρ1 + ρ2 ≥ 0;
then x0 = χ0.

Proof. Suppose that x0 6= χ0. Since χ0 is an initial solution for (3.4) then by (A) and (C), we have

φ1(Ψ(x0) − Ψ(χ0)) ≤ 0
φ2(Θ(χ0)) ≥ 0.

In virtue of the assumption (B) the above inequalities imply that(
η1(x0,χ0).D

α,λ Ψ(χ0)
)
< −ρ1‖ϑ(x0,χ0)‖2(

η2(x0,χ0).D
α,λ Θ(χ0)

)
≤−ρ2‖ϑ(x0,χ0)‖2,

which on summing yields

c1
(
η1(x0,χ0).D

α,λ Ψ(χ0)
)

+ c2
(
η2(x0,χ0).D

α,λ Θ(χ0)
)
< −

(
c1ρ1 + ρ2

)
‖ϑ(x0,χ0)‖2

≤ 0,

which contradicts to initially of χ0. Then we obtain x0 = χ0. This completes the proof.

4. Simulation

In this section, we illustrate a simulation to show how the tempered fractional calculus is effected
on the multi-objective functions.

Let Ψ, Θ : R → R2 such that

Ψ(x) =
(
x2,x3

)
; Θ(x) =

(
x,x2

)
.

Our aim is to show that the couple (Ψ, Θ) is (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. To determine the
fractional Dunkl operator on these functions, we shall introduce three cases depending on the value of
kv for v = 1.



82 RABHA W. IBRAHIM

4.1. Case (i) λ = 0, kv = 0. The tempered fractional Dunkl operator acts on the functions Ψ and Θ
as follows:

Dα,λΨ(x) =
( Γ(3)

Γ(3 −α)
x2−α,

Γ(4)

Γ(4 −α)
x3−α

)
; Dα,λΘ(x) =

( Γ(2)
Γ(2 −α)

x1−α,
Γ(3)

Γ(3 −α)
x2−α

)
.

Now, by letting

η1,2(x,ξ) =
(x− ξ

2
,
x− ξ

2

)
, ξ = 0,

we have

η1(x,ξ).D
α,λΨ(x) =

x3−α

Γ(3 −α)
+

3x4−α

Γ(4 −α)
; η2(x,ξ).D

α,λΘ(x) =
x2−α

2Γ(2 −α)
+

x3−α

Γ(3 −α)
.

Consider ρ1 = ρ2 = 1, x ∈ [0, 1] and ϑ(x,ξ) = x2 − ξ, therefore, we obtain

‖ϑ(x,ξ)‖2 = x4, ξ = 0.
It is clear that

Ψ(ξ) = Ψ(0) = (0, 0); Θ(ξ) = Θ(0) = (0, 0),

then by assuming

φ1

(
Ψ(x) − Ψ(ξ)

)
= 5x, φ2

(
Θ(x) − Θ(ξ)

)
= −5x, x ∈ [0, 1],

we conclude that

η1(x,ξ).D
α Ψ(x) + ρ1‖ϑ(x,ξ)‖2 =

x3−α

Γ(3 −α)
+

3x4−α

Γ(4 −α)
+ x4

< 5x, x ∈ [0, 1]

= φ1

(
Ψ(x) − Ψ(ξ)

) (4.1)
and

η2(x,ξ).D
α,λ Θ(x) + ρ2‖ϑ(x,ξ)‖2 =

x2−α

2Γ(2 −α)
+

x3−α

Γ(3 −α)
+ x4

< 5x, x ∈ [0, 1]

= −φ2
(

Θ(x) − Θ(ξ)
) (4.2)

Hence, the couple (Ψ, Θ) is (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. Table 1 shows that for various values
of α ∈ (0, 1), the outcomes yield the tempered fractional univexty of the couple (Ψ, Θ).

Table 1. Fractional multi-objective function, kv = 0
(α) Eq. (4.1) Eq. (4.2)

0.25 1.6 1.9

0.5 2.6 2.4

0.75 3.1 3.2

To apply the conditions of Theorem 3.1, we assume that c1 = c2 = 1; thus, we have c1ρ1 +c2ρ2 = 2 > 0
with the inequalities (4.1) and (4.2). This leads that all the conditions of Theorem 3.1 are achieved
and hence, ξ = 0 is an efficient solution. Note that if we let φ1(Y ) = 3Y and φ2(Y ) = −3Y, the couple
(Ψ, Θ) is not (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1].

4.2. Case (ii) λ = 0, kv = 1. To evaluate the tempered fractional Dunkl operator, a calculation
implies that

σx2 = x
2 − 2

v.x2

v.v
= −x2, σx3 = −x3.

Therefore, one can attain

η1(x,ξ).D
α,λΨ(x) =

x3−α

Γ(3 −α)
+
x(2x2)α

2
+

3x4−α

Γ(4 −α)
+
x(2x3)α

2



OPTIMALITY AND DUALITY 83

and

η2(x,ξ).D
αΘ(x) =

x2−α

2Γ(2 −α)
+
x(2x)α

2
+

x3−α

Γ(3 −α)
+
x(2x2)α

2
.

Table 2 shows the evaluation of the tempered fractional multi-objective functions for different values
of α.

Table 2. Fractional multi-objective function, kv = 1
(α) Eq. (4.1) Eq. (4.2)

0.25 2.7 2.9

0.5 5 3.8

0.75 4.7 4.8

Thus, we conclude that the conditions of Theorem 3.1 are satisfied when c1 = c2 = 1; such that
c1ρ1 + c2ρ2 = 2 > 0 with the inequalities (4.1) and (4.2). Consequently, we obtain ξ = 0 is an efficient
solution.

4.3. Case (iii) λ = 0, kv = 2. By applying (2.2), we have

η1(x,ξ).D
α,λΨ(x) =

x3−α

Γ(3 −α)
+ x(2x2)α +

3x4−α

Γ(4 −α)
+ x(2x3)α

and

η2(x,ξ).D
α,λΘ(x) =

x2−α

2Γ(2 −α)
+ x(2x)α +

x3−α

Γ(3 −α)
+ x(2x2)α.

Table 3 shows the evaluation of the tempered fractional multi-objective functions for different values of
α. It is clear that the couple (Ψ, Θ) is not (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. It is of (α,λ,ρ,η,ϑ)-
type univex at ξ ∈ [0, 1], when α ∈ (0, 0.25]. Hence, Theorem 3.1 can be applied only for this value of
α.

Table 3. Fractional multi-objective function, kv = 2
(α) Eq. (4.1) Eq. (4.2)

0.25 3.5 4.1

0.5 5.4 5.2

0.75 6.4 5.5

4.4. Case (iv) λ = 1, kv = 0. The tempered fractional Dunkl operator acts on the functions Ψ and
Θ as follows:

Dα,λΨ(x) =
( Γ(3)

Γ(3 −α)
x2−α + x2ex,

Γ(4)

Γ(4 −α)
x3−α + x3ex

)
;

Dα,λΘ(x) =
( Γ(2)

Γ(2 −α)
x1−α + xex,

Γ(3)

Γ(3 −α)
x2−α + x2ex

)
.

Now, by letting

η1,2(x,ξ) =
(x− ξ

2
,
x− ξ

2

)
, ξ = 0,

we have

η1(x,ξ).D
α,λΨ(x) =

x3−α

Γ(3 −α)
+
x3ex

2
+

3x4−α

Γ(4 −α)
+
x4ex

2
;

η2(x,ξ).D
α,λΘ(x) =

x2−α

2Γ(2 −α)
+
x2ex

2
+

x3−α

Γ(3 −α)
+
x3ex

2
.

Consider ρ1 = ρ2 = 1, x ∈ [0, 1] and ϑ(x,ξ) = x2 − ξ, therefore, we obtain

‖ϑ(x,ξ)‖2 = x4, ξ = 0.
It is clear that

Ψ(ξ) = Ψ(0) = (0, 0); Θ(ξ) = Θ(0) = (0, 0),



84 RABHA W. IBRAHIM

then by assuming

φ1

(
Ψ(x) − Ψ(ξ)

)
= 7x, φ2

(
Θ(x) − Θ(ξ)

)
= −7x, x ∈ [0, 1],

we conclude that

η1(x,ξ).D
α Ψ(x) + ρ1‖ϑ(x,ξ)‖2 =

x3−α

Γ(3 −α)
+
x3ex

2
+

3x4−α

Γ(4 −α)
+
x4ex

2
+ x4

< 7x, x ∈ [0, 1]

= φ1

(
Ψ(x) − Ψ(ξ)

) (4.3)
and

η2(x,ξ).D
α,λ Θ(x) + ρ2‖ϑ(x,ξ)‖2 =

x2−α

2Γ(2 −α)
+
x2ex

2
+

x3−α

Γ(3 −α)
+
x3ex

2
+ x4

< 7x, x ∈ [0, 1]

= −φ2
(

Θ(x) − Θ(ξ)
) (4.4)

Hence, the couple (Ψ, Θ) is (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. Table 1 shows that for various values
of α ∈ (0, 1), the outcomes yield the tempered fractional univexty of the couple (Ψ, Θ).

Table 4. , kv = 0,λ = 1
(α, 1) Eq. (4.3) Eq. (4.4)

0.25 4.3 4.6

0.5 5.4 5.1

0.75 5.8 5.9

To apply the conditions of Theorem 3.1, we assume that c1 = c2 = 1; thus, we have c1ρ1 +c2ρ2 = 2 > 0
with the inequalities (4.3) and (4.4). This leads that all the conditions of Theorem 3.1 are achieved
and hence, ξ = 0 is an efficient solution. Note that if we let φ1(Y ) = 5Y and φ2(Y ) = −5Y, the couple
(Ψ, Θ) is not (α,λ,ρ,η,ϑ)-type univex at ξ ∈ [0, 1]. Also, the case λ = 1 and kv = 1 does not imply
the univex function when φ1(Y ) = 7Y and φ2(Y ) = −7Y.

5. Conclusion

This effort is generalized, for the first time, two important concepts in science. The Dunkl tempered
fractional operator and the tempered Univex function, by utilizing the Caputo tempered fractional
differential operator. These two generalizations are combined to deliver the fractional multi-objective
problems. We studied the duality cases by minimize and maximize the desired function in the Rn. Sim-
ulation is provided to apply the existing solutions. It has been found that the fractional case converges
to the ordinary case. These problems can be employed in many studies not only in mathematics, but
also in the economy; such as the utility function the cost function and the entropy function.

References

[1] C.F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1) (1989),
167–183.

[2] M. Rosler and M. Voit, Markov processes related with Dunkl operators. Adv. Appl. Math. 21(4)(1998) 575–643.

[3] R. W. Ibrahim, New classes of analytic functions determined by a modified differential-difference operator in a

complex domain, Karbala Int. J. Modern Sci. 3 (2017) 53–58.
[4] K.S. Miller, B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wily&

Sons (1993).
[5] A.A. Kilbas, H.M. Srivastava, J.J. Trujiilo, Theory and Applications of Fractional Differential Equations. Amster-

dam, Netherlands: Elsevier (2006).

[6] B. Baeumer, M.M. Meerschaert, Tempered stable Levy motion and transient super-diffusion, J. Comput. Appl.
Math. 233 (2010) 243–2448.

[7] R. W. Ibrahim, Fractional calculus of Multi-objective functions & Multi-agent systems. LAMBERT Aca- demic

Publishing, Saarbrcken, Germany 2017.
[8] V.E. Tarasov, Leibniz rule and fractional derivatives of power functions. J. Comput. Nonlinear Dyn. 11 (3) (2016),

Art. ID 031014–4.



OPTIMALITY AND DUALITY 85

[9] M.M. Meerschaert, A. Sikorskii, Stochastic Models for Fractional Calculus, De Gruyter, Berlin, 2012.

Faculty of Computer Science and Information Technology, University of Malaya, Malaysia

Corresponding author: rabhaibrahim@yahoo.com


	1. Introduction
	2. Handling
	2.1. Dunkl operator
	2.2.  Fractional calculus
	2.3. Tempered-Dunkl operator
	2.4. Tempered Univex function

	3. Results
	4. Simulation
	4.1. Case (i) =0, kv=0.
	4.2. Case (ii) =0, kv=1.
	4.3. Case (iii) =0, kv=2.
	4.4. Case (iv) =1, kv=0.

	5. Conclusion
	References