Abstract and Applied Analysis

Volume 2012 (2012), Article ID 294694, 21 pages

doi:10.1155/2012/294694

Research Article

Uniqueness and Asymptotic Behavior of Positive Solutions for a Fractional-Order Integral Boundary Value Problem

Min Jia, Xin Liu, and Xuemai Gu

Communication Research Center, Harbin Institute of Technology, Harbin 150080, China

Received 18 July 2012; Accepted 7 August 2012

Academic Editor: Xinguang Zhang

Copyright © 2012 Min Jia 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 study a model arising from porous media, electromagnetic, and signal processing of wireless communication system ,  ,   ,   , where ,   and , is the standard Riemann-Liouville derivative, is linear functionals given by Riemann-Stieltjes integrals, is a function of bounded variation, and can be a changing-sign measure. The existence, uniqueness, and asymptotic behavior of positive solutions to the singular nonlocal integral boundary value problem for fractional differential equation are obtained. Our analysis relies on Schauder's fixed-point theorem and upper and lower solution method.

1. Introduction

Recently, fractional-order models have proved to be more accurate than integer order models, that is, there are more degrees of freedom in the fractional-order models.Indeed, we can find numerous applications in viscoelasticity, electrochemistry control, porous media, electromagnetic, and signal processing of wireless communication system. Especially, in application of the digital signal processing, the fractional digital signal processing can greatly improve the high frequency components of signal, enhance an intermediate frequency component of signal, and reserve nonlinear low frequency signal. According to this analysis, fractional differential equation applied to the edge information extraction will get higher signal to noise ratio than that of the traditional method based on one- or-two order differential equation.

Many applications on digital signal processing were found recently, for example, Hartley et al. [1] derived a closed-loop control electromagnetic and signal processing system of equations with the Chua resistor

Caputo [2] presented a fractional model system to describe the relationship between the electric field and the current density in the study of a electrochemical polarization medium,

where , , , are constants and is a real number. Anastasio [3] believed that signal processing of communication system of vestibule visual reflex effect is fractional, which can be described by the following model

where is the vestibular reflex nerve discharge rate, is the head of the rotational angular velocity, is fraction.

Motivated by systems (1.1)–(1.3) and their application background in electromagnetic and signal processing of wireless communication system, in this paper, we consider the existence, uniqueness, and asymptotic behavior of positive solutions for the higher nonlocal fractional differential equation

where ,   and , is the standard Riemann-Liouville derivative, is linear functionals given by Riemann-Stieltjes integrals, is a function of bounded variation, and can be a changing-sign measure, and is continuous, may be singular at and .

The nonlocal integral-boundary value problems represent a class of interesting and important problems arising in physical, biological, and chemical processes and have attracted the attention of Khan [4], Gallardo [5], Karakostas and Tsamatos [6], Ahmad et al. [7], Feng et al. [8], Corduneanu [9], and Agarwal and O’Regan [10].

When is an integer, a lot of work has been done dealing with nonlocal 3-point boundary value problems, see [11, 12]. In [12], Eloe and Ahmad studied the following th-order nonlocal differential equation

where , . By applying the fixed point theorem in cones, the authors prove the existence of at least one positive solution when and is continuous and is either sublinear or superlinear. Recently, Hao et al. [13] studied the existence of positive solutions for the BVP (1.5) with integer order , and nonlinear term is replaced by , where can be singular at , can be singular at and no singularity at .

If is fractional, Yuan [14] studied the -type conjugate boundary value problem

where is continuous and semipositone, is the standard Riemann-Liouville derivative. By giving properties of Green’s function and using the Guo-Krasnosel’skii fixed-point theorem on cones, the existence of multiple positive solutions were obtained. More recently, Zhang [15] considered the following BVP whose nonlinear term and boundary condition contain derivatives of unknown functions

where is the standard Riemann-Liouville fractional derivative of order , may be singular at and may be singular at ,  , by using fixed-point theorem of the mixed monotone operator, the unique existence result of positive solution to problem (1.7) was established. Other some recent results, see [16–20]

Motivated by the results mentioned above, in this paper, we study the existence, uniqueness, and asymptotic behavior of positive solutions for the BVP (1.4) where the nonlinear terms and boundary conditions all involve derivatives of unknown functions and with Riemann-Stieltjes integral boundary condition, moreover may be singular at and . Our main tool relies on Schauder’s fixed-point theorem and upper and lower solution method.

2. Preliminaries and Lemmas

In this section, we present here the necessary definitions from fractional calculus theory. These definitions can be found in the recent literatures.

Definition 2.1 (see [21, 22]). The Riemann-Liouville fractional integral of order of a function is given by the following:

provided that the right-hand side is pointwise defined on .

Definition 2.2 (see [21, 22]). The Riemann-Liouville fractional derivative of order of a function is given by the following:

where , denotes the integer part of number , provided that the right-hand side is pointwise defined on .

Proposition 2.3 (see [21, 22]). (1) If , then

(2) If , then

Proposition 2.4 (see [21, 22]). Let , and is integrable, then

where , is the smallest integer greater than or equal to .

Our discussion is based on the assumption in this paper.

Let , by standard discuss, we easily reduce order the BVP (1.4) to the following equivalent BVP

In fact, let , then

By (2.7), we have and

Moreover . Thus, (1.4) is transformed into (2.6).

On the other hand, if is a solution for problem (2.6). Then, from Proposition 2.3 and (2.7), one has

Notice for any ,

which implies that . So , and from (2.7), for , we have , and

Consequently, the BVP (2.6) is transformed into the BVP (1.4).

Applying Propositions 2.3 and 2.4, by standard discuss, we have the following Lemma.

Lemma 2.5. Given , then the problem

has the unique solution

where is the Green function of the BVP (2.12) and is given by the following:

By Proposition 2.4, the unique solution of the problem

is . Let

and define

Then the Green function for the nonlocal BVP (2.6) is (the detail see [23] or [16])

Throughout paper, we always assume the following holds.

is a function of bounded variation such that for and , where is defined by (2.16).

Lemma 2.6. Suppose holds, then the Green function defined by (2.18) satisfies

(1), for all .(2)

where

Proof. (1) is obvious. For (2), by (2.14) and (2.18), we have

Definition 2.7. A continuous function is called a lower solution of the BVP (2.6), if it satisfies

Definition 2.8. A continuous function is called a upper solution of the BVP (2.6), if it satisfies

It follows from Lemma 2.5, we have the following maximum principle.

Lemma 2.9 (maximum principle). If satisfies

and for any . Then

3. Main Results

Let , and

Clearly, , so is nonempty. For any , define an operator by the following:

Let

then

The conditions imposed on are the following.

 (H1)  , and is decreasing in for ; (H2)  for any , ,  , and

Lemma 3.1. Suppose hold, then is well defined and .

Proof. For any , by the definition of , there exists two positive numbers such that for any . It follows from (2.19) and that

Now take , then for any ,

by . Thus by the continuity of , we have

This yields

By (2.19) and (3.6)–(3.9), we have

where

On the other hand, it follows from (2.19) that

where

It follows from (3.6)–(3.12) that is well defined and .

Take

By (H2) and Weierstrass distinguishing method, we know

is a continuous function on , that is, .

Theorem 3.2 (existence). Suppose hold, and if

Then the BVP (1.4) has at least a positive solution .

Proof. By (2.18) and (3.2), we have

It follows from (2.19) that

which implies that . Let

then we also have

Thus according to the fact that the operator is nonincreasing relative to , Lemma 3.1 and (3.20), we have

and . By (2.19), (3.12), and (3.20), one has

Thus, by (3.20), (3.21) and the operator is nonincreasing relative to , we have

However (3.17) implies that , satisfy boundary conditions of BVP (2.6). Let

then (3.21), (3.23) imply that are lower and upper solution of BVP (2.6), respectively, and .

Define the function and the operator in by the following:

Clearly, is continuous by (3.25). Consider the following boundary value problem

Obviously, a fixed point of the operator is a solution of the BVP (3.27).

For all , it follows from Lemma 2.6 and (3.22) that

So is bounded. It is easy to see is continuous from the continuity of and .

Let be bounded, that is, there exists a positive constant such that , for all . Let , and since is uniformly continuous on , then for any and , there exists such that

for . Then

This implies that is equicontinuous.

By the means of the Arzela-Ascoli theorem, we have is completely continuous. Thus, by using Schauder’s fixed-point theorem, has at least a fixed-point such that .

Now we prove

Let ,  . By is upper solution of BVP (2.6) and is fixed point of , we know

From the definition of and (3.22), we obtain

Thus (3.16) and (3.33) imply

By (3.32),  (3.34), and Lemma 2.9, we know which implies on . By the same way, it is easy to prove on . So we obtain

Consequently, ,  . Then is a positive solution of the BVP (2.6), which implies that is a positive solution of the BVP (1.4).

Theorem 3.3 (asymptotic behavior). Suppose the conditions of Theorem 3.2 are satisfied. Then there exist two constants such that the positive solution of the BVP (1.4) satisfies

where

Proof. By (3.22) and (3.35), we know

On the other hand, it follows from (3.38) and (2.19) that

Then

Since

so it follows from (3.40) that

That is,

In the end, from and L’Hospital Rule, we have

Theorem 3.4 (uniqueness). Suppose the conditions of Theorem 3.2 are satisfied and . Then the positive solution of the BVP (1.4) is unique.

Proof. We only need the BVP (2.6) has unique solution. Suppose that are positive solutions to the BVP (2.6). We may assume, without loss of generality, that there exists such that . Let

Evidently,

Thus

By the boundary conditions of the BVP (2.6), it is easy to check that there exist the following two possible cases:

Case 1. From and , we get that , which implies a contradiction with .

Case 2. In this case we have and . Since is increasing on , we also have is increasing on , that is, for any . Thus is nonincreasing on , which implies that , this is a contradiction with .

Therefore the BVP (2.6) has unique solution, and the BVP (1.4) also has unique solution.

Remark 3.5. We only get the uniqueness of positive solution for the BVP (1.4) as is integer, but it remains unknown as to whether an analogous unique result holds for the case . We believe the result should also hold and try to study this problem, but it is a pity that we failed to find the effective method to solve this problem.

Remark 3.6. The BVP (1.4) allows the nonlinearity has singularity at and ,  . If is continuous at ,  , that is

, and is decreasing in for ;

then the BVP (1.4) has at least a positive solution , and there exists a constant such that

If , the positive solution of the BVP (1.4) also is unique.

Proof. In fact, in Theorem 3.2 the set is replaced by the following:

and (3.21) can be replaced by the following:

Clearly , and

Thus the rest of proof is similar to those of Theorem 3.2.

On the other hand, by Lemma 2.6 and , the positive solution of the BVP (2.6) satisfies

and then

where

Moreover if , the positive solution of the BVP (1.4) also is unique.

Example 3.7. Consider the existence of positive solutions for the nonlinear fractional differential equation

where

Then the BVP (3.58) is equivalent to the following 4-point BVP with coefficients of both signs

Conclusion: The BVP (3.59) has at least a positive solution such that

Proof. Clearly,

Thus,

Clearly, and hold.

On the other hand, for any , , , , and

Thus is satisfied.

Now we compute . Since

Thus

So

which implies that (3.16) holds. Then the BVP (1.4) has at least a positive solution , and there exist two constants

such that

Acknowledgment

This work is supported by the National Natural Science Foundation of China (61201143) and the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2010091), the National Science Foundation for Postdoctoral Scientists of China (Grant no. 2012M510956), and the Postdoctoral Funds of Heilongjiang Province (Grant no. LBHZ11128).

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