- Research Article
- Open Access
Unbounded solution for a fractional boundary value problem
Advances in Difference Equations volume 2014, Article number: 154 (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.
In this paper, we investigate the existence of solutions for a fractional boundary value problem (P) on the half line:
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].
Fractional boundary value problems on infinite intervals have been investigated by many authors; see [1, 2, 4, 5, 8, 11–13]. In , the authors proved the existence of unbounded solutions for the following nonlinear fractional boundary value problem:
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.
Further, in , the authors studied the following second order nonlinear differential equation on the half line:
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.
2 Existence results
Definition 1 The Riemann-Liouville fractional integral of order α of a function g is defined by
where is the Gamma function, .
Definition 2 The Caputo fractional derivative of order q of a function g is defined by
where ( is the entire part of q).
Lemma 3 For , , the homogeneous fractional differential equation has a solution
where , , and .
Lemma 4 Let , . Then and , for all .
Lemma 5 Let and , then the following relations hold:
To prove the main results of this paper we need the following lemma.
Lemma 6 Let with , the linear nonhomogeneous boundary value problem
has a unique solution
Proof By Lemmas 3 and 4, we obtain
The boundary conditions , imply that
applying Lemma 5 and the condition , we obtain
substituting b by its value, it yields
The proof is complete. □
Lemma 7 Assume that , then for all we have
Proof Simple computations give
Let us consider the case , then we get
Firstly if , then
Secondly if , then
Applying the same techniques to the other cases, the conclusion follows. □
In this paper, we will use the Banach space E defined by
and equipped with the norm , where and .
Define the integral operator by
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 
Let , , . V is relatively compact in E, if and are both equicontinuous on any finite subinterval of and equiconvergent at ∞, that is for any , there exists such that
, (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.
Lemma 10 Assume that , on any subinterval of and there exist non-negative functions and nondecreasing on , such that
then T is completely continuous. (Here .)
Proof The proof will be done in some steps.
Step 1: T is continuous. Let be a convergent sequence to u in E. Let and , then we obtain with the help of Lemma 7, hypothesis (2.1) and some elementary inequalities
Using similar techniques we prove that
hence the integrals are convergent. With the help of Lebesgue dominated convergence theorem and the fact that f is continuous we get
Step 2: T is relatively compact. Let , first let us show that is uniformly bounded. Let , taking (2.1) into account and the fact that and are nondecreasing on , it yields
Similarly, we prove that
this along with (2.3) yields
thus is uniformly bounded.
Next, we show that is equicontinuous on any compact interval of . Let , , , we have
which approaches zero uniformly when . On the other hand we have
Let us estimate the second integral on the right hand side of the inequality (2.5):
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.
If , then
which approaches zero uniformly when .
If , then
If , then
Thus T is equicontinuous on the compact .
Step 3: T is equiconvergent at ∞. Since
consequently T is equiconvergent at ∞. The proof is complete. □
Now, we can give an existence result.
Theorem 11 Assume that the hypotheses of Lemma 10 hold and that there exists , such that
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.
Proof of Theorem 11 From the proof of Lemma 10, we know that T is a completely continuous operator. Now we apply the nonlinear alternative of Leray-Schauder to prove that T has at least one nontrivial solution in E. Let , such that , ; we get with the help of (2.4):
This together with (2.6) implies
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. □
3 Positive solutions
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).
Definition 14 A nonempty subset P of a Banach space E is called a cone if P is convex, closed, and satisfies the conditions
for all and ;
Lemma 15 Assume that , then for and we have
Proof The proof is easy, we omit it. □
Let us make the following hypotheses on the nonlinear term f:
, , , where , and .
Define the cone K by
Lemma 16 We have .
Proof Taking Lemma 7 into account, we get
Lemma 15 implies for all
Let us introduce the following notation:
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.
Let E be a Banach space, and let , be a cone. Assume and are open subsets of E with , and let
be a completely continuous operator such that
(Expansive form) , , and , ; or
(Compressive form) , , and , .
Then has a fixed point in .
Proof of Theorem 17 From , we deduce that for any , there exists , such that if , then , . Let and , by Lemma 7 we get
Similarly we obtain
Choosing , it yields , for any .
Now, since , then for any , there exists , such that
for . Let and denote by . For and , we obtain
Using Lemma 15 and the fact that , we obtain for all
similarly, we get
Let us choose M such that
then we obtain
The first statement of Theorem 18 implies that T has a fixed point in . The proof is complete. □
Now define the function
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 .
Proof Since , then for , there exists , such that if , then for all , we have
Let , we should prove the second statement of Theorem 18. Suppose , then
Now from the fact that g is decreasing, we get
From and Lemma 19, we get , so for , there exists , such that if , then . Let , where , then . Suppose that , then it yields
On the other hand we get
then from the second statement of Theorem 18, T has a fixed point in . The proof is complete. □
Remark If , then every positive solution of the problem (P) is unbounded. Indeed
therefore our conclusion follows.
Example 21 Let us consider the problem (P) with
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.
Example 22 Let us reconsider the above example with
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.
Agarwal RP, O’Regan D: Infinite Interval Problems for Differential, Difference and Integral Equations. Kluwer Academic, Dordrecht; 2001.
Agarwal RP, O’Regan D: Boundary value problems of nonsingular type on the semi-infinity interval. Tohoku Math. J. 1999, 51: 391–397. 10.2748/tmj/1178224769
Guezane-Lakoud A, Khaldi R: Solvability of a three-point fractional nonlinear boundary value problem. Differ. Equ. Dyn. Syst. 2012, 20(4):395–403. 10.1007/s12591-012-0125-7
Liang S, Zhang J: Existence of three positive solutions of m -point boundary value problems for some nonlinear fractional differential equations on an infinite interval. Comput. Math. Appl. 2011, 61: 3343–3354. 10.1016/j.camwa.2011.04.018
Liang S, Shi S: Existence of multiple positive solutions for m -point fractional boundary value problems with p -Laplacian operator on infinite interval. J. Appl. Math. Comput. 2012, 38: 687–707. 10.1007/s12190-011-0505-0
Liu Y, Ahmad B, Agarwal RP: Existence of solutions for a coupled system of nonlinear fractional differential equations with fractional boundary conditions on the half-line. Adv. Differ. Equ. 2013., 2013: Article ID 46
Nyamoradi N, Baleanu D, Agarwal RP: On a multipoint boundary value problem for a fractional order differential inclusion on an infinite interval. Adv. Math. Phys. 2013., 2013: Article ID 823961
Su X, Zhang S: Unbounded solutions to a boundary value problem of fractional order on the half-line. Comput. Math. Appl. 2011, 61: 1079–1087. 10.1016/j.camwa.2010.12.058
Wang H: On the numbers of positive solutions of a nonlinear systems. J. Math. Anal. Appl. 2003, 281: 287–306. 10.1016/S0022-247X(03)00100-8
Webb JRL, Infante G: Positive solutions of nonlocal boundary value problems involving integral conditions. NoDEA Nonlinear Differ. Equ. Appl. 2008, 15(1–2):45–67. 10.1007/s00030-007-4067-7
Yan BQ, O’Regan D, Agarwal RP: Unbounded solutions for singular boundary value problems on the semi-infinite interval: upper and lower solutions and multiplicity. J. Comput. Appl. Math. 2006, 197: 365–386. 10.1016/j.cam.2005.11.010
Zhao X, Ge W: Unbounded solutions for a fractional boundary value problems on the infinite interval. Acta Appl. Math. 2010, 109: 495–505. 10.1007/s10440-008-9329-9
Zhang X: Existence of positive solutions for boundary value problems for second order nonlinear differential equations on the half line. Electron. J. Differ. Equ. 2009, 141: 1–10.
Xu H: Analytical approximations for a population growth model with fractional order. Commun. Nonlinear Sci. Numer. Simul. 2009, 14(5):1978–1983. 10.1016/j.cnsns.2008.07.006
Kilbas AA, Srivastava HM, Trujillo JJ North-Holland Mathematics Studies 204. In Theory and Applications of Fractional Differential Equations. Elsevier, Amsterdam; 2006.
Mainardi F: Fractals and Fractional Calculus in Continuum Mechanics. Springer, New York; 1997.
Podlubny I: Fractional Differential Equations Mathematics in Sciences and Engineering. Academic Press, New York; 1999.
Lakshmikantham V, Vatsala AS: Basic theory of fractional differential equations. Nonlinear Anal. 2008, 69(8):2677–2682. 10.1016/j.na.2007.08.042
Corduneanu C: Integral Equations and Stability of Feedback Systems. Academic Press, New York; 1973.
Deimling K: Nonlinear Functional Analysis. Springer, Berlin; 1985.
Guo DJ, Lakshmikantham V Notes and Reports in Mathematics in Science and Engineering 5. In Nonlinear Problems in Abstract Cones. Academic Press, Boston; 1988.
Kwong MK: On Krasnoselskii’s cone fixed point theorem. Fixed Point Theory Appl. 2008., 2008: Article ID 164537
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.
The authors declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final draft.
About this article
Cite this article
Guezane-Lakoud, A., Kılıçman, A. Unbounded solution for a fractional boundary value problem. Adv Differ Equ 2014, 154 (2014). https://doi.org/10.1186/1687-1847-2014-154
- integral condition
- Caputo derivative
- unbounded solution
- existence of solution
- Leray-Schauder nonlinear alternative
- Guo-Krasnosel’skii fixed point theorem