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A new class of fractional boundary value problems
Advances in Difference Equations volume 2013, Article number: 373 (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.
We study a boundary value problem of Caputo-type fractional differential equations with new boundary conditions given by
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.
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 at least n th continuously differentiable 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 boundary value problem
Proof It is well known that the general solution of the fractional differential equation in (2.1) can be written as
where are arbitrary constants.
Applying the given boundary conditions, we find that , and
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 .
In relation to the given problem, we define an operator as
where A is given by (2.3). Observe that problem (1.1) has solutions if and only if the operator has fixed points.
For the sake of computational convenience, we set
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).
Proof Let us denote and show that , where with . For , , we have
which implies that .
Now, for and for each , we obtain
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 .
Then problem (1.1) has at least one solution on if
Proof Fixing , we consider and define the operators Φ and Ψ on as
For , it is easy to show that , which implies that .
In view of the assumption , the operator Ψ is a contraction. The continuity of f implies that the operator Φ is continuous. Also, Φ is uniformly bounded on as . Moreover, Φ is relatively compact on as
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).
Proof As a first step, we show that the operator is completely continuous. Clearly, the continuity of follows from the continuity of f. Let be bounded. Then, , it is easy to establish that . Furthermore, we find that
Hence, for , it follows that
Therefore, is equicontinuous on . Thus, by the Arzelá-Ascoli theorem, the operator is completely continuous.
Next, we consider the set . To show that V is bounded, let , . Then
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 )
Let E be a Banach space, be a closed, convex subset of E, V be an open subset of and . Suppose that is a continuous, compact (that is, is a relatively compact subset of ) map. Then either
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 , ;
(H4) there exists a constant such that
Then problem (1.1) has at least one solution on .
Proof Let us consider the operator defined by (3.1) and show that maps bounded sets into bounded sets in . For a positive number r, let be a bounded set in . Then, for together with (H3), we obtain
Next, it will be shown that maps bounded sets into equicontinuous sets of . Let with and . Then
Clearly, the right-hand side tends to zero independently of as . Thus, by the Arzelá-Ascoli theorem, the operator is completely continuous.
Let x be a solution for the given problem. Then, for , following the method of computation used in proving that is bounded, we have
which implies that
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. □
Example 4.1 Consider a fractional boundary value problem given by
Here, , , , and . With the given values, and as . Clearly, . Therefore, by Theorem 3.1, there exists a unique solution for problem (4.1).
Example 4.2 Consider a fractional boundary value problem given by
Observe that implies that and . With the given data, it is found that and
Clearly, all the conditions of Theorem 3.2 are satisfied. Hence there exists a solution for problem (4.2).
Example 4.3 Consider the problem
Here, , , , , and . Let us fix , . Further, by the condition
it is found that with . Thus, Theorem 3.5 applies and there exists a solution for problem (4.3) on .
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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, BA, AA, AAS, and RPA, contributed to each part of this work equally and read and approved the final version of the manuscript.
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Ahmad, B., Alsaedi, A., Assolami, A. et al. A new class of fractional boundary value problems. Adv Differ Equ 2013, 373 (2013). https://doi.org/10.1186/1687-1847-2013-373
- fractional differential equations
- nonlocal boundary conditions
- fixed point theorems