- Research Article
- Open Access
Unbounded solution for a fractional boundary value problem
© Guezane-Lakoud and Kılıçman; licensee Springer. 2014
- Received: 7 January 2014
- Accepted: 7 May 2014
- Published: 23 May 2014
This paper concerns the existence of unbounded positive solutions of a fractional boundary value problem on the half line. By means of the properties of the Green function and the compression and expansion fixed point theorem (Kwong in Fixed Point Theory Appl. 2008:164537, 2008), sufficient conditions are obtained to guarantee the existence of a solution to the posed problem.
MSC:26A33, 34B15, 34B27.
- integral condition
- Caputo derivative
- unbounded solution
- existence of solution
- Leray-Schauder nonlinear alternative
- Guo-Krasnosel’skii fixed point theorem
where is a given function, , , denotes the Caputo fractional derivative. Note that few papers in the literature dealing with fractional differential equations considered the nonlinearity f in (P) depending on the derivative of u.
Since many problems in the natural sciences require a notion of positivity (only non-negative densities, population sizes or probabilities make sense in real life), in the present study we discuss the existence of positive solutions for the problem (P). The proofs of the main results are based on the properties of the associated Green function, Leray-Schauder nonlinear alternative and Guo-Krasnosel’skii fixed point theorem on cone. Different methods are applied to investigate such boundary value problems, we can cite fixed point theory, topological degree methods, Mawhin theory, upper and lower solutions…; see [1–13].
Fractional boundary value problems on infinite intervals often appear in applied mathematics and physics. They can model some physical phenomena, such as the models of gas pressure in a semi-infinite porous medium; see . The population growth model can also be characterized by a nonlinear fractional Volterra integrodifferential equation on the half line . For more results on fractional differential equations in science and engineering and their applications we refer to [15–17].
by using Leray-Schauder nonlinear alternative. Here , and denotes the Riemann-Liouville fractional derivative.
In  by means of fixed point theorem on cone, the authors discussed the existence of multiple positive solutions for m-point fractional boundary value problems with p-Laplacian operator on infinite interval.
Applying a fixed point theorem and the monotone iterative technique, they proved the existence of positive solution.
The organization of this paper is as follows. In Section 2, we provide necessary background and properties of the Green function. The existence result is established under some sufficient conditions on the nonlinear term f. Section 3 is devoted to the existence of positive solutions on a cone. We conclude the paper with some examples.
where is the Gamma function, .
where ( is the entire part of q).
where , , and .
Lemma 4 Let , . Then and , for all .
To prove the main results of this paper we need the following lemma.
The proof is complete. □
Applying the same techniques to the other cases, the conclusion follows. □
and equipped with the norm , where and .
so we have transformed the problem (P) to a Hammerstein integral equation by using the Green function.
Lemma 8 The function is solution of the boundary value problem (P) if and only if , for all .
From this we see that to solve the problem (P) it remains to prove that the map T has a fixed point in E. Since the Arzela-Ascoli theorem cannot be applied in this situation, then, to prove that T is completely continuous, we need the following compactness criterion:
Lemma 9 
, (uniformly according to u).
We recall that a continuous mapping F from a subset M of a normed space X into another normed space Y is called completely continuous iff F maps bounded subset of M into relatively compact subset of Y.
then T is completely continuous. (Here .)
Proof The proof will be done in some steps.
thus is uniformly bounded.
which approaches zero uniformly when . Now we analyze the first integral on the right hand side of inequality (2.5) in different cases when the compact contains 1 or not.
which approaches zero uniformly when .
Thus T is equicontinuous on the compact .
consequently T is equiconvergent at ∞. The proof is complete. □
Now, we can give an existence result.
Then the fractional boundary value problem (P) has at least one nontrivial solution .
To prove this theorem, we apply the Leray-Schauder nonlinear alternative.
Lemma 12 
Let F be a Banach space and Ω a bounded open subset of F, . Let be a completely continuous operator. Then either there exist , , such that , or there exists a fixed point of T.
which contradicts the fact that . Lemma 12 allows one to conclude that the operator T has a fixed point and then the fractional boundary value problem (P) has a nontrivial solution . The proof is complete. □
To study the existence of positive solution of the problem (P), first, we will introduce a positive cone constituted of continuous positive functions or some suitable subset of it. Second, we will impose suitable assumptions on the nonlinear terms such that the hypotheses of the cone theorem are satisfied. Third, we will apply a fixed point theorem to conclude the existence of a positive solution in the annular region.
Definition 13 A function u is called positive solution of the problem (P) if , , and it satisfies the boundary conditions in (P).
for all and ;
Proof The proof is easy, we omit it. □
, , , where , and .
Lemma 16 We have .
Theorem 17 Under the hypothesis (H) and if , then the fractional boundary value problem (P) has at least one positive solution in the case and .
To prove Theorem 17, we apply the well-known Guo-Krasnosel’skii fixed point theorem on cone.
(Expansive form) , , and , ; or
(Compressive form) , , and , .
Then has a fixed point in .
Choosing , it yields , for any .
The first statement of Theorem 18 implies that T has a fixed point in . The proof is complete. □
It is proved in  that:
Lemma 19 If g is continuous then and .
Theorem 20 Under the hypothesis (H) and if and g is decreasing according to the both variables, then the problem (P) has at least one nontrivial positive solution in the cone K, in the case and .
then from the second statement of Theorem 18, T has a fixed point in . The proof is complete. □
therefore our conclusion follows.
by direct calculation we obtain , , and . Clearly hypothesis (H) is satisfied, so by Theorem 17 there exists at least one nontrivial positive solution in the cone K.
Easily we check the hypothesis (H) and find that g is decreasing with respect to u and v. Furthermore we have the case and . Thus by Theorem 20 there exists at least one nontrivial positive solution in the cone K.
The authors express their sincere thanks to the referees for the careful and details reading of the manuscript and very helpful suggestions. This joint work was done when the first author visited University Putra Malaysia as a visiting scientist during 21st December-30th December 2013. Thus the authors gratefully acknowledge that this research was partially supported by the University Putra Malaysia under the GP-IBT Grant Scheme having project number GP-IBT/2013/9420100.
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