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: mesamei@basu.ac.ir, mesamei@gmail.com

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)

International Journal of Statistical Analysis 3

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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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