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Abstract and Applied Analysis
Volumeย 2010, Article IDย 501230, 15 pages
http://dx.doi.org/10.1155/2010/501230
Research Article

Positive Solutions to Nonlinear Higher-Order Nonlocal Boundary Value Problems for Fractional Differential Equations

1Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Sector H-12, Islamabad 46000, Pakistan
2University of Malakand, Chakdara Dir(L), Khyber Pakhutoonkhwa, Pakistan

Received 28 June 2010; Revised 21 September 2010; Accepted 28 October 2010

Academic Editor: Paulย Eloe

Copyright ยฉ 2010 Mujeeb Ur Rehman and Rahmat Ali Khan. 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 study existence of positive solutions to nonlinear higher-order nonlocal boundary value problems corresponding to fractional differential equation of the type , , . , , , , where, , , , the boundary parameters and is the Caputo fractional derivative. We use the classical tools from functional analysis to obtain sufficient conditions for the existence and uniqueness of positive solutions to the boundary value problems. We also obtain conditions for the nonexistence of positive solutions to the problem. We include examples to show the applicability of our results.

1. Introduction

Fractional calculus goes back to the beginning of the theory of differential calculus and is developing since the 17th century through the pioneering work of Leibniz, Euler, Abel, Liouville, Riemann, Letnikov, Weyl, and many others. Fractional calculus is the generalization of ordinary integration and differentiation to an arbitrary order. For almost 300 years, it was seen as interesting but abstract mathematical concept. Nevertheless the applications of fractional calculus just emerged in the last few decades in various areas of physics, chemistry, engineering, biosciences, electrochemistry, and diffusion processes. For details, we refer the readers to [1โ€“5].

The existence and uniqueness of solutions for fractional differential equations is well studied in [6โ€“10] and references therein. It should be noted that most of the papers and books on fractional calculus are devoted to the solvability of initial value problems for fractional differential equations. In contrast, the theory of boundary value problems for nonlinear fractional differential equations has received attention quiet recently, and many aspects of the theory need to be further investigated.

There are some recent development dealing with the existence and multiplicity of positive solutions to nonlinear boundary value problems for fractional differential equations, see, for example, [11โ€“18] and the reference therein. However, few results can be found in the literature concerning the existence of positive solutions to nonlinear three-point boundary value problems for fractional differential equations. For example, Li and coauthors [19] obtained sufficient conditions for the existence and multiplicity results to the following three point fractional boundary value problem where is standard Riemann-Liouville fractional order derivative.

Bai [20] studied the existence and uniqueness of positive solutions to the following three-point boundary value problem for fractional differential equations where , , is standard Riemann-Liouville fractional order derivative. The function is assumed to be continuous on .

The purpose of the present work is to investigate sufficient conditions for the existence, uniqueness, and nonexistence of positive solutions to more general boundary value problems for higher-order nonlinear fractional differential equations where, , ; , the boundary parameters and , is the Caputo fractional derivative. The function is assumed to be continuous and nonnegative on . To the best of our knowledge, existence and uniqueness of positive solution to the boundary value problem (1.3) have never been studied previously.

This paper is organized as follows: in Section 2, we recall some basic definitions and preliminary results which are needed for our main results. In Section 3, we study existence and uniqueness and nonexistence of positive solutions to the boundary value problem (1.3) under certain assumptions on the function . Moreover, examples are provided to illustrate the applicability of main results.

2. Background Materials and Lemmas

For the convenience of the readers, in this section, we provide definitions of Riemann-Liouville fractional integral and fractional derivative and some of their basic properties which will be helpful in the forth coming investigations.

Definition 2.1 (see [2, 5]). For a function , the Riemann-Liouville fractional integral of order is defined by
The provided that the integral on the right hand side exists. For , the fractional integral satisfies the semigroup property In addition, if or if , then the identity is true for every .

Definition 2.2 (see [2, 5]). The standard Riemann-Liouville fractional derivative of order of a function is given by , where , , provided that the right hand side is pointwise defined on .

Definition 2.3. For a given function , the Caputo fractional derivative of order is defined by , where , .

Lemma 2.4 (see [2]). If , then . In particular, if is positive integer and , then .

The following two lemmas play a fundamental role to obtain an equivalent integral representation to the boundary value problem (1.3).

Lemma 2.5 (see [2]). Let , then where , and .

Lemma 2.6 (see [2]). For , the fractional differential equation has a general solution , where , and .

For the existence of positive solutions to the boundary value problem (1.3), we use the following fixed point theorem due to Krasnoselโ€™skii.

Theorem 2.7 (see [21]). Let be a Banach space, and let be a cone. Assume that , are open subsets of such that . Let be completely continuous such that either (i), for and , for , or (ii), for and , for . Then, has a fixed point in .

3. Main Results

Lemma 3.1. Let , then the unique solution of the linear problem is given by where, and .

Proof. In view of Lemma 2.5, (3.1) is equivalent to the integral equation Using Lemma 2.4, we obtain where . The boundary conditions, , , lead to , . Using the boundary conditions , and the fact that , we obtain which implies that and Hence, the unique solution of the linear fractional boundary value problem (3.1), (3.2) is given by

Lemma 3.2. The functions and satisfy the following properties: (i), and for all , ,(ii)For , , , (iii), (iv)for , .

Proof. (i) For , in view of the expression for , it follows that and for all , .
For and , we have . Hence, . Also, we note that for .
(ii) Now, Since which implies that Now, for , we have For , obviously, . From the expression of , it clearly follows that
(iii) From the definition of , we have Therefore, is nonincreasing in , so its minimum value occurs at for , and its maximum value occurs at for . That is, Since and , therefore Also, as , therefore Subletting (3.18) and (3.19) in (3.16), we have

Remark 3.3. For , .

Proof. As , therefore is a decreasing function. Hence, Thus, we have .
Let be endowed with the norm . Define a cone by and an operator by By Lemma 3.1, the boundary value problem (1.3) has a solution if and only if has a fixed point.

Lemma 3.4. The operator is completely continuous.

Proof. Firstly, we show that . From (3.23), Lemma 3.2, and Remark 3.3, we have Hence, we have .
Next, we show that is uniformly bounded. For fixed , consider a bounded subset of defined by and define , then for , we have Hence, is bounded.
Finally, we show that is equicontinuous. Define and choose such that . Then, for all and , we have Using the mean value theorem, we obtain . Hence, it follows that By means of Arzela-Ascoli theorem is completely continuous operator.

For convenience, we introduce following notations:

Theorem 3.5. If there exist constants such that and functions such that (i), ,(ii), , then the boundary value problem (1.3) has at least one positive solution for , small enough and has no positive solution for , large enough.

Proof. By Lemma 3.4, the operator is completely continuous. The proof is divided into two steps.Step 1. We prove that the boundary value problem (1.3) has at least one positive solution.
Define an open subset of and choose , such that and . Then, for any , we have and in view of Lemma 3.2 and (3.23), it follows that which implies that , for .
Define . For any and , using Lemma 3.2, we have Therefore, by (3.3) and Remark 3.3, we have the following estimate: which implies that for
Hence, by Theorem 2.7, it follows that has a fixed point in .
Step 2. Now, we prove that for large values of , , the boundary value problem (1.3) has no positive solution. Otherwise, for , there exists , with , such that for any positive integer , the boundary value problem has a positive solution given by If , then and if , then . Therefore, as .
Let and define . For , using (3.3) and Remark 3.3, we have which is a contradiction. Hence, the boundary value problem (1.3) have no positive solution for , large enough.

Theorem 3.6. Assume that is satisfied and there exists real-valued function such that then the boundary value problem (1.3) has a unique positive solution.

Proof. For , using (3.23) and Lemma 3.2, we obtain where we have taken into account , we obtain the relation . Hence, it follows by Banach contraction principle that the boundary value problem (1.3) has a unique positive solution.

Example 3.7. Consider the boundary value problem where, , , , and . Let , . For , we observe that Taking and . For , , by computations, we find that , , , also . Therefore, and . Hence, Assumptions (i) and (ii) of the Theorem 3.5 are satisfied. Therefore, the boundary value problem (3.38) has a positive solution for , and no positive solution for , .

Example 3.8. Consider the boundary value problem where , , , , and . Let , . For , , we have the following estimate: Let , then by computations, we have , . All the conditions of Theorem 3.6 are satisfied. Therefore, by Theorem 3.6, the boundary value problem (3.41), (3.41) has a unique positive solution.

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