- Open Access
Uniqueness results for fully anti-periodic fractional boundary value problems with nonlinearity depending on lower-order derivatives
© Alsaedi et al.; licensee Springer. 2014
- Received: 20 February 2014
- Accepted: 2 April 2014
- Published: 7 May 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.
- differential equations of fractional order
- anti-periodic fractional boundary conditions
- fixed point
where denotes the Caputo fractional derivative of order q and f is a given continuous function.
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.
provided the integral exists.
where denotes the integer part of the real number q.
Lemma 2.1 
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.
Observe that the problem (1.1)-(1.2) has a solution only if the operator has a fixed point.
with , where N is given by (3.2). Then the anti-periodic boundary value problem (1.1)-(1.2) has a unique solution.
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 , .
Then the anti-periodic boundary value problem (1.3)-(1.2) has a unique solution on .
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 . □
- (a)Consider the anti-periodic fractional boundary value problem given by(4.1)
- (b)Consider the following anti-periodic fractional boundary value problem:(4.2)
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.
This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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