International Journal of Statistical Analysis
Existence of Solution for a Class of Fuzzy Fractional Q-Integral Equation
Samei ME*
Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Iran
*Corresponding Author: Mohammad Esmael Samei, Department of Mathematics, Faculty of Basic Science, Bu-Ali Sina University, Hamedan 65178, Iran, Tel: +98 (918) 8121728, E-mail: [email protected], [email protected]
Citation: Samei ME (2019) Existence of Solution for a Class of Fuzzy Fractional Q-Integral Equation. International Jour- nal of Statistical Analysis. V1(1): 1-9.
Received Date: Nov 29, 2019 Accepted Date: Dec 20, 2019 Published Date: Dec 23, 2019
1. Abstract
We study solution for a fuzzy fractional q-integral equation. We
considered the fractional derivative in the sense of Riemann-
Liouville and we investigate existence of the solution of fuzzy
fractional type q-integral equations using the Hausdorff
measure of non-compactness.
2. Keywords: Fractional calculus; existence of solution;
measure of compactness; fuzzy fractional q-integral equation
3. Introduction
Fractional calculus is an important branch in mathematical
analysis. However, after Leibniz and Newton invented
differential calculus, it has been a subject of interest among
mathematicians, physicists, and engineers. It is known that
fractional calculus has numerous applications in different
sciences such as mechanics, electricity, biology, control theory,
signal and image processing [1,28]. During the last decade the
fractional differential equations were developed intensively
[4,8,10,30,35-37]. On the other hand, a great attention was
devoted to the fuzzy fractional equations [12, 34,38]. Studies
on q-difference equations appeared already at the beginning of
the last century in intensive works especially by [20,11,27,2,39].
The goal of the manuscript is to analyze the existence of
solutions for the fuzzy fractional q-integral equation
presentation here can be found in, for example, [5,22,24,31]
4. Basic Concept
Let as the Euclidean space wia a norm and ), the
set of all nonempty, convex and compact subset of . By the
operations defined as follows, is a semilinear space:
for all ), . Define
the Hausdorff metric on →[0,∞) by
(2.1)
where and
is a complete separable
metric space with respect to the topology generated by the
Hausdorff metric [3].
4.1. The Fractional Q-Integral and Fractional Q-Derivative
Let and be a non-zero real number. Define
[20]. In fact, q-analogue of the power function
with is and
for and [7,33]. If , then we define power
function for by
where
,
and . Here is the space of normal, fuzzy upper
semi-continuous, compactly supported and convex fuzzy sets
on .
We recall some known facts on fractional q-calculus and
fundamental results of the q-calculus and fuzzy set theory. The
where . It can be seen, if , then . In
[20], Jackson has been defined the q-Gamma function by
where [20]. Note
that, .
A simplified analysis can be performed to estimate the value
of q-Gamma function, , for input values qand xby
Copyright: © 2019 Samei ME, et al. Volume 1 | Issue 1
Research Article Open Access
International Journal of Statistical Analysis 2
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Obviously, we have . we can
counting the number of sentences nin summation. To this
aim, we consider a pseudo-code description of the method for
calculating q-Gamma function of order nin [Algorithm 1].
According to the base definitions, the q-derivative of a function
is given by
where [17]. Also
where and [18]. Consider
then the following is valid:
and [1]. Also, the
q-derivative of higher order of a function is defined by
for all and
for [1]. Also, the q-integralis
definedof a function defined in the interval is given by
for , provided that the sum converges absolutely [1].
If , then
whenever the series exists. The operator is defined by
and for all
[1]. It has been proved that and
whenever is continuous at
[1]. The fractional Riemann-Liouville type q-integral
of the function f on [0,1] with order is defined by
and
for and [6,16].
Also, the fractional Caputo type q-derivative of the function is defined by
for and [6,16]. It has been proved that
where [33]. Particularly, for and ,
we obtain
calculate which is shown in (Algorithm 4).
4.2. Fuzzy Set Theory
We denote by the space of all fuzzy sets , and
we consider the following properties [21]:
I. is normal, in the other words, there exists an
such that ,
II. set is compact,
III. is upper semi-continuous,
IV. we have ,
for all , and for all ( is fuzzy convex).
For is called -level
set where . The fuzzy zero is define
by whenever and otherwise. The
following operations define a semilinear structure on ,
where and [21]. Lakshmikantham and
Mohapatra showed the α-level set of fuzzy sets satisfies
, and , for all ,
and [24]. We define metric on by
where is the Hausdorff metric. Then is complete
metric space [32]. Also the metric satisfies the following
properties:
(i)
(ii)
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For measurable function define the norm
(iii)
for all and . We denote the space of all
continuous fuzzy functions and the space of all the absolutely
continuous functions on interval by and
, respectively. Let and for
, define a metric D by
Thus, is a complete metric space [32].
Lemma4.1.[29] Suppose that be a nonempty set and let
be a family of subsets of M such that
(i)
(ii)
(iii)
Then there exists a unique function
define by such that
for all .
Let denote the space of Lebesgue
integrable functions from to . Suppose that
.
We denote by the set of all Lebsgue integrable selections of
, that is
The Aumann integral is defined by [9]
A function is called measurable, if
for all closed set , where denotes the
Borel algebra of [19]. Also, F is called integrably bounded
if there exists a function such that
for almost . If such an has mesurable selectors, then
they are also integrable and is nonempty. If is measurable
and integrably bounded, then [19]. Also, If is
integrably bounded, then in [19].
Theorem 4.2.[19] If are measurable and
if there exists such that
A fuzzy function is measurable if, the set-valued
function , defined by
is measurable, for all . Moreover, the fuzzy function
u is itegrably bounded, if there exists a function
such that for all . A measurable and
integrably bounded fuzzy function is said
to be integrable on if there exists such that
for all .
Kaleva showed can be embedded as a closed convex cone
in a Banach space [21]. The embedded map is
an isometry and an isomorphism. Moreover, if
is integrable then is strongly measurable,
Bochner integrable and
for all [21]. The fractional integral of the function
of order is defined by [15]
Agarwal, Arshad, O’Regan, Lupulescu investigated that
if be an integrably fuzzy function , then
for each fixed and for each fixed
, there exists a unique fuzzy set such that
for all [3]. Now,
Let be an integrable fuzzy function. The fuzzy
fractional q-integral of order of the function ,
is defined by . Also, its
level sets are given by
for all and . So, we have
where is the Lebsque measure on and . Let
be the Banach space of all measurable functions
with . We denote the space of all functions
such that the function belong to
, by for [13]. Therefore, for
if , then
, and
.
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is a metric on , for , and
is a metric on
[13]. Lakshmikantham and Vatsala introduce
Kamke function [25]. By using the notion, we use Kamke
function for fraction q-integral equation as follows.
Definition 4.3. A function is called a
Kamke function if is a measurable function for each
fixed is continuous for each fixed
and there exists a function
such that for and for all
with for , such
that is the only solution of
5. Main Results
For complete metric space and nonempty bounded subset
in , the Hausdorff measure defined by
such that where is ball with .
Suppose that be two bounded. Then
(i) if and only if is compact,
(ii)
(iii) ,
(iv) ,
(v) .
Furthermore, if is a complete semilinear space, then
(vi)
(vii)
Lemma 5.1.[23] If be a separable Banach space and
be an integrally bounded sequence of measurable functions
from into , then the function is
measurable and
Here, we analyze the fuzzy fractional q-integral equation
(1.1) that presented in introduction. A continuous
fuzzy function u on is called a solution for (1.1) if
, for almost all . Now,
we present the main theorem of the paper.
Theorem 5.4. Consider the following conditions:
(A1) Suppose that a fuzzy function
such that:
(i) is measurable for all ,
(ii) is continuous for ,
(iii) there exists and such
that for .
(A2) Let be a
continuous operator such that for each and for all
, with for every , we
have for .
(A3) For all nonempty bounded we have
for all , where is Kamke
function and
Then (1.1) has at least one solution on an interval .
Proof. Put and .
First, we show that a solution u of (1.1) is bounded. Note that
and
for .
Lemma 5.2.[3] If is a bounded subset in then
is a bounded subset in Banch space and
, where is an
embedding map. Using the description of the first and the
Lemma 5.2, we can prove the following lemma.
Lemma 5.3.[14] Let be a set of integrable fuzzy
functions from into . Then the function
is measurable and for we have
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We put and define the set
Also, we define the operator by
So, for , we obtain
As a result, . Let
and
we obtain
By assumption (A1) and the continuity of function G that P is a
continuous operator. Define the sequence whenever
and
whenever , for each . Put . This
implies that A is uniformly bounded on . Now, we prove
that the set A is equicontinuous on . We show that
. For this purpose, we consider three
so
Case III. if , then
so when , we obtain . Thus, A is
uniformly equicontinuous on . For each fixed and
we obtain
cases.
Case I. If , then .
Case II. If , then
International Journal of Statistical Analysis 6
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Hence
For any given , we can find such that
. By using property (vii), we have
Thus,
for each . In addition to, 1 exists, such that
for . So, use again the property (vii) of the measure of
noncompactness, we obtain
For Therefore, using the property (i) and (ii), we
obtain
Hence, we have
By assumption (A3) and Lemma 5.3, we have
Note that, we can do this levels for each . However, we
have for , because, and
is a Kamke function. Therefore, is a relatively subset of
. Then, there exists a subsequence
which converges uniformly on I to a continuous function
[24]. By selecting two cases for , we have
Case I. If , then
Case II. If , then
as . Since is continuous, we have
. So, for .
for , is a solution of (1.1).
Remark 5.5.[26] If we replace the condition (A3) with the
following condition
for almost all and a Kamke function
, the conclusion of Theorem 5.4 is also
true.
Remark 5.6. If the fuzzy function u is a solution of the fuzzy
fractional q-integral equation (1.1), then by (2.3), it follows that
u is a solution of the fuzzy fractional q-diferential equation
for
6. Algorithms
In this part, we give a simplified analysis can be executed to
calculate the value of q-Gamma function, , for input
values and by counting the number of sentences in
summation. To this aim, we consider a pseudo-code description
of the method for calculated q-Gamma function of order in
(Algorithm 2) (for more details, see the following link https://
en.wikipedia.org/ wiki/Q-gamma_function). (Table 1) shows
that when q is constant, the q-Gamma function is an increasing
function. Also, for smaller values of x, an approximate result is
obtained with less values of n. It has been shown by underlined
rows. (Table 2)shows that the q-Gamma function for values
near to one is obtained with more values of in comparison
with other columns. They have been underlined in line 8 of the
first column, line 17 of the second column and line 29 of third
columns of (Table 2). Also, (Table 3) is the same as (Table 2),
but x values increase in 3. Similarly, the q-Gamma function
for values q near to one is obtained with more values of n in
comparison with other columns.
International Journal of Statistical Analysis 7
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Table 1: Some numerical results for calculation of with which is constant, for
in Algorithm 2.
n
1 2.679786 4432.545834 1804225.635 1.29090809480473E + 45
2 2.674552 4423.888518 1800701.757 1.28838678993206E + 45
3 2.673899 4422.808467 1800262.132 1.28807224237593E + 45
4 2.673818 4422.673494 1800207.192 1.28803293353064E + 45
5 2.673808 4422.656623 1800200.325 1.28802802007493E + 45
6 2.673806 4422.654514 1800199.467 1.28802740589531E + 45
7 2.673806 4422.654250 1800199.369 1.28802732912289E + 45
8 2.673806 4422.654217 1800199.346 1.28802731952634E + 45
9 2.673806 4422.654213 1800199.344 1.28802731832677E + 45
10 2.673806 4422.654213 1800199.344 1.28802731817683E + 45
11 2.673806 4422.654212 1800199.344 1.28802731815808E + 45
12 2.673806 4422.654212 1800199.344 1.28802731815574E + 45
13 2.673806 4422.654212 1800199.344 1.28802731815545E + 45
14 2.673806 4422.654212 1800199.344 1.28802731815541E + 45
15 2.673806 4422.654212 1800199.344 1.28802731815541E + 45
16 2.673806 4422.654212 1800199.344 1.28802731815541E + 45
17 2.673806 4422.654212 1800199.344 1.28802731815541E + 45
18 2.673806 4422.654212 1800199.344 1.28802731815541E + 45
19 2.673806 4422.654212 1800199.344 1.28802731815541E + 45
Table 2: Some numerical results for calculation of with for of Algorithm 2.
1 2.679786 136.0462 79062.14 6301918
2 2.674552 119.0815 41793.34 2528395
3 2.673899 111.6582 26290.73 1232716
4 2.673818 108.1782 18589.88 689176.8
5 2.673808 106.4926 14278.33 426538.4
6 2.673806 105.6629 11650.59 285518.7
7 2.673806 105.2513 9946.351 203363.8
. . . . .
. . . . .
. . . . .
26 2.673806 104.8418 5522.284 25842.86
27 2.673806 104.8418 5513.202 25230.37
28 2.673806 104.8418 5505.95 24699.65
29 2.673806 104.8418 5500.155 24238.45
. . . . .
. . . . .
. . . . .
106 2.673806 104.8418 5477.048 20879.61
107 2.673806 104.8418 5477.048 20879.57
108 2.673806 104.8418 5477.048 20879.53
. . . . .
. . . . .
. . . . .
118 2.673806 104.8418 5477.048 20879.34
119 2.673806 104.8418 5477.048 20879.33
120 2.673806 104.8418 5477.048 20879.32
Table 3: Some numerical results for calculation of with for
of Algorithm 2.
1 1804225.635 2.43388915243820E + 32 1.10933564801075E
+ 75
2.3996994906237E
+ 102
2 1800701.757 2.12965300838343E + 32 5.41355796236824E
+ 74
7.1431517307455E
+ 101
3 1800262.132 1.99654969535946E + 32 3.19616462101800E
+ 74
2.6837217226512E
+ 101
4 1800207.192 1.93415751737948E + 32 2.14884539802207E
+ 74
1.1944485864825E
+ 101
5 1800200.325 1.90393630617042E + 32 1.58553847001434E
+ 74
6.0526350536381E
+ 100
6 1800199.467 1.88906180377847E + 32 1.25302695267477E
+ 74
3.3987862057282E
+ 100
7 1800199.36 1.88168265610746E + 32 1.04280391429109E
+ 74
2.0741306563269E
+ 100
8 1800199.346 1.87800749466975E + 32 9.02841142168746E
+ 73
1.3555712905453E
+ 100
9 1800199.344 1.87617350297573E + 32 8.05899312693661E
+ 73
9.38129101307050E
+ 99
10 1800199.344 1.87525740263248E + 32 7.36673088857628E
+ 73
6.81335603265770E
+ 99
11 1800199.344 1.87479957611817E + 32 6.86049299667128E
+ 73
5.15556440821410E
+ 99
12 1800199.344 1.87457071874804E + 32 6.48333340557523E
+ 73
4.04051908444650E
+ 99
. . . . .
. . . . .
. . . . .
48 1800199.344 1.87434189862553E + 32 5.18960499065178E
+ 73
6.66324790738213E
+ 98
. . . . .
. . . . .
. . . . .
90 1800199.344 1.87434189862553E + 32 5.18923469131315E
+ 73
6.50025876524830E
+ 98
91 1800199.344 1.87434189862553E + 32 5.18923468501255E
+ 73
6.50013085733126E
+ 98
92 1800199.344 1.87434189862553E + 32 5.18923467997207E
+ 73
6.50001716364224E
+ 98
93 1800199.344 1.87434189862553E + 32 5.18923467593968E
+ 73
6.49991610435300E
+ 98
. . . . .
. . . . .
. . . . .
118 1800199.344 1.87434189862553E + 32 5.18923465987107E
+ 73
6.49915022957670E
+ 98
119 1800199.344 1.87434189862553E + 32 5.18923465985889E
+ 73
6.49914550293450E
+ 98
120 1800199.344 1.87434189862553E + 32 5.18923465984914E
+ 73
6.49914130147782E
+ 98
Input:
1:
2: if then
3:
4: else
5: for to do
6:
7: end for
8:
9: end if
Output
Algorithm 1 The proposed method for calculated
:
International Journal of Statistical Analysis 8
Volume 1 | Issue 1
4: end for
Input:
1:
2: if then
3:
4: else
5:
6: end if
Output:
Algorithm 4 The proposed method for calculated
which id defined by Riemann-Liouville
Input:
1:
2: for to do
3:
4:
5: end for
6:
Output
Input:
1:
2: for do
3:
4: end for
5:
Output:
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Algorithm 5 The proposed method for calculated
Output:
5:
3:
2: for to do
1:
Input:
Algorithm 2 The proposed method for calculated
Algorithm 3 The proposed method for calculated
that is the standard -defivative of function
:
International Journal of Statistical Analysis 9
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