Abstract and Applied Analysis

Volume 2013, Article ID 579740, 20 pages

http://dx.doi.org/10.1155/2013/579740

## Positive Solutions to Fractional Boundary Value Problems with Nonlinear Boundary Conditions

^{1}Department of Mathematics, Faculty of Sciences, Razi University, Kermanshah 67149, Iran^{2}Department of Mathematics and Computer Sciences, Faculty of Art and Sciences, Cankaya University, 06530 Ankara, Turkey^{3}Institute of Space Sciences, P.O. BOX MG-23, Magurele, 76900 Bucharest, Romania^{4}Department 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.

#### Abstract

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.

#### 1. Introduction

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 [10]; electrochemistry and material science are also described by differential equations of fractional order [11–15]. In [16], 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 [17], 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 [18] introduced the -Laplacian equation as follows: where , . Obviously, is invertible and its inverse operator is , where is a constant such that .

Ahmad et al. [19] 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 [20] 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 .

In the past few decades, many important results relative to (6) with certain boundary value conditions have been obtained; we refer the reader to [21–25] and the references therein.

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.

#### 2. Preliminaries

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 .

*Definition 2 (see [26, 27]). *The Riemann-Liouville fractional integral operator of order of function is defined as
where is the Euler gamma function.

*Definition 3 (see [26, 27]). *The Riemann-Liouville fractional derivative of order of a continuous function is defined as
where .

Lemma 4 (see [28]). *The equality , , holds for .*

Lemma 5 (see [28]). *Let . Then the differential equation
**
has a unique solution , , , where .*

Lemma 6 (see [28]). *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 [16], so we omit it here.

Lemma 8 (see [16]). *For and ,
*

Lemma 9 (see [29]). *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:

We next recall the Krasnoselskii's fixed point theorem (see [30]). This lemma will be of use in Sections 3 and 4 of this paper.

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 .

Theorem 15. *Assume that (L1), (L2), and (L3) are satisfied. Then, system (6) and (7) has at least one positive solution. *

*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 , .

*Remark 16. *Condition (67) in (L4) is true only if for each there is such that whenever , it follows that . The same is true for condition (68) involving .

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: