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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 579740, 20 pages
Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions
1Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran
2Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey
3Institute of Space Sciences, P.O. BOX MG-23, Magurele, 76900 Bucharest, Romania
4Department of Chemical and Materials Engineering, Faculty of Engineering, King Abdulaziz University, P.O. Box 80204, Jeddah 21589, Saudi Arabia
Received 20 March 2013; Accepted 5 April 2013
Academic Editor: Juan J. Trujillo
Copyright © 2013 Nemat Nyamoradi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We consider a system of boundary value problems for fractional differential equation given by , , , , where , , , , are eigenvalues, subject either to the boundary conditions , , , , , or , , , , , , where , and , , are continuous functions. The Krasnoselskiis fixed point theorem is applied to prove the existence of at least one positive solution for both fractional boundary value problems. As an application, an example is given to demonstrate some of main results.
Fractional calculus is the field of mathematical analysis which deals with the investigation and applications of integrals and derivatives of arbitrary order; the fractional calculus may be considered an old and yet novel topic.
Recently, fractional differential equations have found numerous applications in various fields of physics and engineering [1, 2]. It should be noted that most of the books and papers on fractional calculus are devoted to the solvability of initial value problems for differential equations of fractional order. In contrast, the theory of boundary value problems for nonlinear fractional differential equations has received attention quite recently and many aspects of this theory need to be explored. For more details and examples, see [3–9] and the references therein; moreover, fractional derivative arises from many physical processes, such as a charge transport in amorphous semiconductors ; electrochemistry and material science are also described by differential equations of fractional order [11–15]. In , Bai and Lü considered the boundary value problem of fractional order differential equation where is the standard Riemann-Liouville fractional derivative of order and is continuous.
In , Salem considered the following nonlinear -point boundary value problem of fractional type: where , with , is a real valued continuous function, and is a nonlinear Pettis integrable function.
The turbulent flow in a porous medium is a fundamental mechanics problem. For studying this type of problems, Leibenson  introduced the -Laplacian equation as follows: where , . Obviously, is invertible and its inverse operator is , where is a constant such that .
Ahmad et al.  also considered the existence of solutions for the following three-point boundary value problem of Langevin equation with two different fractional orders: where is the Caputo fractional derivative, is a given continuous function, and is a real number.
Dai  considered the following problem of ordinary differential equations: By means of global bifurcation techniques and the approximation of connected components, existence and multiplicity results for positive solutions were obtained.
Motivated by the works above, our purpose in this paper is to show the existence of at least one positive solution for the following fractional -Laplacian system: where , , , is the Riemann-Liouville fractional derivative of order , , and is integer.
We first consider the problem (6) with following boundary condition: We then consider the case in which the boundary conditions are changed to where are continuous functions, where means the set of continuous, real valued functions on the unit interval .
In the cases, we assume that .
The following conditions will be used in this paper:(H1), is a -laplacian operator. Obviously, is invertible and , where is a constant such that ;(H2) for and ;(H3) is a given continuous function and is a positive real valued continuous function, .
The rest of the paper is organized as follows: in Section 2, we will recall certain results from the theory of the continuous fractional calculus; in Section 3, we will provide some conditions under which the problem (6) and (7) has at least one positive solution; in Section 4, by suitable conditions, we will prove that the problem (6) and (8) has at least one positive solution; finally, in Section 4, we will provide some numerical examples, which will explicate the applicability of our results.
In this section, we present some notations and preliminary lemmas that will be used in the proofs of the main results.
Definition 1. Let be a real Banach space. A nonempty closed set is called a cone of if it satisfies the following conditions:(1), , implies (2), , implies .
Lemma 4 (see ). The equality , , holds for .
Lemma 5 (see ). Let . Then the differential equation has a unique solution , , , where .
Lemma 6 (see ). Let . Then the following equality holds for , , , , where .
In the following, we present the Green function of fractional differential equation boundary value problem.
Let then, the problem where , is turned into problem
Lemma 7. Suppose that , then the boundary value problem (15) has a unique solution where
Proof. The proof is similar to that of Lemma 2.3 in , so we omit it here.
Lemma 8 (see ). For and ,
Lemma 9 (see ). Suppose that and , are two constants such that ; then,
Lemma 10. Suppose that (H1) and (H2) hold. Then, for , the boundary value problem has a unique solution where , for ,
Proof. By applying Lemma 6, (20) is equivalent to the following integral equation:
for some arbitrary constants .
By the boundary condition , we conclude that ; then we have It follows from Lemmas 8 and 9 that So, by the boundary condition , we obtain that Then, the unique solution of (20) is given by the formula Then, the proof is completed.
Lemma 11. Assume , then; for all , we have(i), , for any ;(ii)there exists a positive function such that , ,where with .
Proof. (i) If , we have
If , we get
Thus, , for any . It is obvious that .
Now, we show that for any . We define One can get on the other hand, it is obvious that , .
Thus For any , then, is nonincreasing with respect to on ; hence, we obtain that Also, we have then, is increasing with respect to on . Then, by the fact that , we have
(ii) Since is nonincreasing and is nondecreasing, for all , we have where is the solution of so, we get where is given in (28). This completes the proof.
Remark 12. If and , then Lemma 11 satisfies.
In this paper, we assume that and .
Now, we consider system (6). Assume that (H1), (H2), and (H3) hold; then, by applying Lemmas 7 and 10, is a solution of system (6) if and only if is a solution of the following nonlinear integral system:
Theorem 13. Let be a Banach space and let be a cone. Assume that and are open sets contained in such that and . Assume, further, that is a completely continuous operator. If eigher (1) for and for or(2) for and for ,
then, has at least one fixed point in .
3. Existence of a Positive Solution: Case I
Let and , and define In this section, we consider the following assumption:(L1)There exist numbers and , with , such that (L2)There exist numbers and , with , such that (L3)There are numbers and , where such that , .
The basic space used in this paper is a real Banach space with the norm , where .
Then, choose a cone , by and define an operator by where
Lemma 14. Suppose that (H1), (H2), and (H3) hold. Then, the operator is well defined, that is, .
Proof. For any , by (H1), (H2), (H3) and Lemma 11, , , , and it follows from (47) that
Thus, for any , it follows from Lemma 11 and (49) that
In the same way, for any , we have
Therefore, From the above, we conclude that , that is, . This completes the proof.
It is clear that the existence of a positive solution for system (6) is equivalent to the existence of a nontrivial fixed point of in .
Proof. It follows from Lemma 14 that . Furthermore, by the application of the Ascoli-Arzela theorem, which we omit, is a completely continuous operator.
Condition (L3) implies that there is sufficiently small such that
Now given this , it follows from condition (L1) that there exists some number such that whenever . Similarly, by condition (L1), for the same , there exists some number such that whenever . In particular, by putting , we conclude that both (54) and (55) hold whenever . So, define by Then, for , we have Also, by similarly argument, we get for . Thus, for , we have
On the other hand, letting be the same number selected at the beginning this proof, it follows from condition (L2) that there exists number such that whenever . Let Moreover, let Then, .
If , then it follows that for any ,
Thus, (63) shows that for , (60) holds, whenever .
So, for each , we have Similarly, we obtain that for . Thus, for , we have Thus, all conditions of Theorem 13 are satisfied. Consequently, we conclude that has a fixed point on . This is a positive solution of systems (6) and (7). The proof is completed.
4. Existence of a Positive Solution: Case II
In this section, we assume that and . We now provide a set of conditions under which the problem (6) and (8) will have at least one positive solution. we need conditions (L1) and (L2) in this section. furthermore, we use notations , , and which were defined in Section 3. we will introduce new conditions.
(L4) The functionals and are continuous in and and nonnegative for and satisfy
(L5) There are numbers and , where such that , .
By repeating the way that we used in Section 3, with a minor modification, we can get that is a solution of systems (6) and (8) if and only if is a solution of the following nonlinear integral system: