Abstract

In this paper, we discussed the existence of a four point boundary value problem for q-fractional differential equation in a Banach space, particulary, using the Banach contraction principle for certain conditions on f.

1. Introduction

Fractional calculus has been always an attractive field for many researchers due to its very wide applications in the study of modeling for various fields in science and engineering, such as physics, biophysics, chemistry, controlling engineering, visco-elasticity and many more. For further details, see ^{1, 2, 3, 4, 5} and the references therein. Years ago, the fractional equation has a been a significant progress (see ^{6, 7} and the references therein).

The q-calculus or known as "quantum"-calculus was originally developed by Jackson ⁸ and some of its basic definitions and properties can be found in ⁹.

This study focuses on showing the existence of the solution for the following problem:

(1)

where and and are the q-Caputo fractional Derivatives, are real constants and is a continuous function on This paper is organized as follows: in section 2, we introduce the prelimimaries of a q-analogue and definitions of q-fractional integral and differential. In section 3, we give some hypotheses to prove the existence of solution and also using the Banach contraction priciple.

2. Mathematical Preliminaries

The following notations, properties, definitions, and preliminary facts will be utlized throughout this study. Let then we have the q-analogue of a real number defined by ¹⁰

The q-analogue of the power function with n is a nonnegative integer is

in general, if

The q-derivative of a function is defined by

and the q-integral of a function in the interval is given by

The following an properties is useful in studying q-fractional calculus

Remark 2.1. It is to be noted that if and then

Definition 2.2. The Reimann-Liouville q-fractional integral type for is defined by

(2)

where f is continuous on and

And the q-Gamma function is defined as

Definition 2.3. The Reimann-Liouville q-fractional derivative type for is defined by

(3)

where is the smallest possible integer greater than or equal to

Definition 2.4. The Caputo q-fractional derivative type for is defined by

(4)

where is the smallest possible integer greater than or equal to

Theorem 2.5. Let t > 0 and Then the following equality holds

(5)

For further details regarding these properties, we refer the reader to ¹¹.

3. Mathematical Result

Let us consider a Banach space given by the norms and Then we have the following Lemmas:

Lemma 3.1. The general solution of q-fractional differential equation for is given by

(6)

where

Lemma 3.2. Let Then

Lemma 3.3. Let then the solution for the equation

(7)

subject to the boundary conditions

is given by

(8)

where

Proof. By lemmas 3.1 and 3.2, general solution of (7) is given by

(9)

and using the boundary condition, we can get

and

Substituting these obtained constant to equation (9) and making use of the lemma (3.3), we can have (8).

In this section, we will show that there exist a solution of the problem (1) and show the contribution of the term on the solution. Suppose the following quantities:

(10)

(11)

Following the hypotheses of Houas and Dahmani ³, we can consider the following the hypothesis:

Hypothesis 3.4. The function is continous.

In accordance also to the work of El-Shahed and Al-Yami ¹², we have

Hypothesis 3.5. Assume that there exist a nonegative constant M > 0 such that

Hypothesis 3.6. There exist a nonnegative function u, v on such that for

where and

Theorem 3.7. Suppose hypothesis (3.6) holds if

then the solution for the problem (1) exist on [0, 1]^{, 1}.

Proof. Consider a q-operator

such that, it is defined as

In which we must show that is a contraction

For any using Hypothesis 3.6 we can obtain

To reduce the expression, we use Hypothesis (3.5) and (3.6) and simplifying further we can obtain the relation

Furthermore, to finally show the contraction of We can use the definition of a q-Caputo Fractional Derivative ^{3, 11}, by lemma (3.3) and lastly the hypotheses (3.4-3.6). we can finally obtain

4. Conclusion

Hence, by the hypothesis (3.6), we deduced that is a consequence of Banach contraction principle and the solution of equation (1) exist for For a Generic case, we can say that the contraction obtained in ¹², which is a three point q-fractional boundary problem, coincides with the obtained result in this study.

Acknowledgements

The authors would like to thank the Department of Physics, College of Natural Sciences and Mathematics, Minadanao State University-Main Campus, Marawi City.

References