Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives
Advances in Difference Equations volume 2014, Article number: 136 (2014)
We investigate the uniqueness of solutions for fully anti-periodic fractional boundary value problems of order with nonlinearity depending on lower-order fractional derivatives. Our results are based on some standard fixed point theorems. The paper concludes with illustrative examples.
In this article, we show the existence of solutions for a fully fractional-order anti-periodic boundary value problem of the form
where denotes the Caputo fractional derivative of order q and f is a given continuous function.
As a second problem, we will discuss the existence of solutions for the following fractional differential equation with the boundary conditions (1.2):
In the last few decades, fractional calculus has evolved as an attractive field of research in view of its extensive applications in basic and technical sciences. Examples can be found in physics, chemistry, biology, economics, control theory, signal and image processing, biophysics, blood flow phenomena, aerodynamics, fitting of experimental data, etc. [2–5].
The subject of boundary value problems of differential equations, having an enriched history, has been progressing at the same pace as before. In the context of fractional boundary value problems, there has been a much development in the last ten years; for instance, see [6–25] and the references cited therein.
In view of the importance of anti-periodic boundary conditions in the mathematical modeling of a variety of physical processes [26–28], the study of anti-periodic boundary value problems has received considerable attention. Some recent work on anti-periodic boundary value problems of fractional order can be found in a series of papers [29–34] and the references therein.
Definition 2.1 The Riemann-Liouville fractional integral of order q for a continuous function g is defined as
provided the integral exists.
Definition 2.2 For a function , the Caputo derivative of fractional order q is defined as
where denotes the integer part of the real number q.
Lemma 2.1 
For any , the unique solution of the linear fractional equation , , with anti-periodic boundary conditions (1.2) is given by
where is the Green’s function (depending on q and p) given by
3 Uniqueness of solutions
This section is devoted to the uniqueness of solutions for the problems at hand by means of Banach’s contraction principle.
3.1 Uniqueness result for the problem (1.1)-(1.2)
For , let us define a space , where denotes the space of all continuous functions defined on . Note that the space endowed with the norm defined by is a Banach space.
In view of Lemma 2.1, let us define an operator associated with the problem (1.1)-(1.2) as
Observe that the problem (1.1)-(1.2) has a solution only if the operator has a fixed point.
Before proceeding further, let us introduce some notations:
Theorem 3.1 Assume that is a continuous function satisfying the condition
with , where N is given by (3.2). Then the anti-periodic boundary value problem (1.1)-(1.2) has a unique solution.
Proof Let us set and to show that , where . For , we have
Using the facts (b is a constant), , , , for , we get
As in the previous step, it can be shown that
Thus we get . Hence . Next, for and for each , we obtain
In a similar manner, we find that
By the given assumption, , it follows that the operator is a contraction. Thus, the conclusion of the theorem follows by the contraction mapping principle (Banach fixed point theorem). □
3.2 Uniqueness result for the problem (1.3)-(1.2)
Here, we study the uniqueness of solutions for the problem of (1.3)-(1.2). For that, let be a Banach space endowed with the norm , .
Relative to the problem (1.3)-(1.2), we define an operator as
In what follows, we set
where N is given by (3.2) and
Theorem 3.2 Let be a continuous function and there exists a positive number such that
Then the anti-periodic boundary value problem (1.3)-(1.2) has a unique solution on .
Proof We define , , and show that . In view of the given assumption, we have
Further, it can be shown in a similar way that
Next, for and for each , we obtain
Also, we have
Since , therefore, the operator is a contraction. Thus, it follows by the contraction mapping principle that the problem (1.3)-(1.2) has a unique solution on . □
Consider the anti-periodic fractional boundary value problem given by(4.1)
where , , , , and
With (), all the assumptions of Theorem 3.1 hold. Therefore, the problem (4.1) has a unique solution on .
Consider the following anti-periodic fractional boundary value problem:(4.2)
where , , , . With , , we can write
Furthermore, we have
Clearly all the assumptions of Theorem 3.2 are satisfied with (). Hence, the problem (4.2) has a unique solution on .
Sotiris K Ntouyas is a member of Nonlinear Analysis and Applied Mathematics (NAAM) - Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
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This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
The authors declare that they have no competing interests.
Each of the authors, AA, BA, NM, and SKN, contributed to each part of this work equally and read and approved the final version of the manuscript.
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Alsaedi, A., Ahmad, B., Mohamad, N. et al. Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives. Adv Differ Equ 2014, 136 (2014). https://doi.org/10.1186/1687-1847-2014-136