A new class of fractional boundary value problems
© Ahmad et al.; licensee Springer. 2013
Received: 29 July 2013
Accepted: 1 December 2013
Published: 20 December 2013
In this paper, a fractional boundary value problem with a new boundary condition is studied. This new boundary condition relates the nonlocal value of the unknown function at ξ with its influence due to a sub-strip , where . The main results are obtained with the aid of some classical fixed point theorems and Leray-Schauder nonlinear alternative. A demonstration of applications of these results is also given.
Keywordsfractional differential equations nonlocal boundary conditions fixed point theorems
where denotes the Caputo fractional derivative of order q, f is a given continuous function, and a is a positive real constant.
In (1.1), the second condition may be interpreted as a more general variant of nonlocal integral boundary conditions, which states that the integral contribution due to a sub-strip for the unknown function is proportional to the value of the unknown function at a nonlocal point with . We emphasize that most of the work concerning nonlocal boundary value problems relates the contribution expressed in terms of the integral to the value of the unknown function at a fixed point (left/right end-point of the interval under consideration), for instance, see [1–3] and references therein.
The recent development in the theory, methods and applications of fractional calculus has contributed towards the popularity and importance of the subject. The tools of fractional calculus have been effectively applied in the modeling of many physical and engineering phenomena. Examples include physics, chemistry, biology, biophysics, blood flow phenomena, control theory, wave propagation, signal and image processing, viscoelasticity, percolation, identification, fitting of experimental data, economics, etc. [4–6]. For some recent work on the topic, we refer to [7–22] and the references therein.
Let us recall some basic definitions on fractional calculus.
provided the integral exists.
where denotes the integer part of the real number q.
where are arbitrary constants.
Substituting the values of , in (2.4), we get (2.2). This completes the proof. □
3 Existence results
Let denote the Banach space of all continuous functions from to ℝ endowed with the norm .
where A is given by (2.3). Observe that problem (1.1) has solutions if and only if the operator has fixed points.
Theorem 3.1 Let be a jointly continuous function satisfying the Lipschitz condition
(H1) , , , .
Then problem (1.1) has a unique solution if , where ω is given by (3.1).
which implies that .
Since , by the given assumption, therefore the operator is a contraction. Thus, by Banach’s contraction mapping principle, there exists a unique solution for problem (1.1). This completes the proof. □
The next result is based on Krasnoselskii’s fixed point theorem .
Theorem 3.2 Let be a continuous function satisfying (H1) and
(H2) , , and .
For , it is easy to show that , which implies that .
where . Hence, by the Arzelá-Ascoli theorem, Φ is compact on . Thus all the assumptions of Krasnoselskii’s fixed point theorem are satisfied. So problem (1.1) has at least one solution on . This completes the proof. □
Our next result is based on the following fixed point theorem .
Theorem 3.3 Let X be a Banach space. Assume that is a completely continuous operator and the set is bounded. Then T has a fixed point in X.
Theorem 3.4 Assume that there exists a positive constant such that for all , . Then there exists at least one solution for problem (1.1).
Therefore, is equicontinuous on . Thus, by the Arzelá-Ascoli theorem, the operator is completely continuous.
and . Hence, , . So V is bounded. Thus, Theorem 3.3 applies and, in consequence, problem (1.1) has at least one solution. This completes the proof. □
Our final result is based on Leray-Schauder nonlinear alternative.
Lemma 3.1 (Nonlinear alternative for single-valued maps )
has a fixed point in , or
there are (the boundary of V in ) and with .
Theorem 3.5 Let be a continuous function. Assume that
(H3) there exist a function and a nondecreasing function such that , ;
Then problem (1.1) has at least one solution on .
Clearly, the right-hand side tends to zero independently of as . Thus, by the Arzelá-Ascoli theorem, the operator is completely continuous.
In view of (H4), there exists M such that . Let us choose .
Observe that the operator is continuous and completely continuous. From the choice of , there is no such that for some . Consequently, by Lemma 3.1, we deduce that the operator has a fixed point which is a solution of problem (1.1). This completes the proof. □
Here, , , , and . With the given values, and as . Clearly, . Therefore, by Theorem 3.1, there exists a unique solution for problem (4.1).
Clearly, all the conditions of Theorem 3.2 are satisfied. Hence there exists a solution for problem (4.2).
it is found that with . Thus, Theorem 3.5 applies and there exists a solution for problem (4.3) on .
This research was partially supported by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, Saudi Arabia.
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