Abstract and Applied Analysis

Volume 2012 (2012), Article ID 740760, 14 pages

doi:10.1155/2012/740760

Research Article

Eigenvalue Problem of Nonlinear Semipositone Higher Order Fractional Differential Equations

Jing Wu¹ and Xinguang Zhang²

¹School of Economic Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, Sichuan, China

²School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, Shandong, China

Received 22 October 2012; Accepted 19 November 2012

Academic Editor: Dragoş-Pătru Covei

Copyright © 2012 Jing Wu and Xinguang Zhang. 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 the eigenvalue interval for the existence of positive solutions to a semipositone higher order fractional differential equation = =   where ,  , , , satisfying , is the standard Riemann-Liouville derivative, , and is allowed to be changing-sign. By using reducing order method, the eigenvalue interval of existence for positive solutions is obtained.

1. Introduction

In this paper, we consider the eigenvalue interval for existence of positive solutions to the following semipositone higher order fractional differential equation:

where ,   ,  ,  , and , ,   satisfying , is the standard Riemann-Liouville derivative, is continuous.

Recently, one has found that fractional models can sufficiently describe the operation of variety of computational, economic mathematics, physical, and biological processes and systems, see [1–9]. Accordingly, considerable attention has been paid to the solution of fractional differential equations, integral equations, and fractional partial differential equations of physical phenomena [10–24]. One of the most frequently used tools in the theory of fractional calculus is furnished by the Riemann-Liouville operators. It possesses advantages of fast convergence, higher stability and higher accuracy to derive the solution of different types of fractional equations.

In this work, we will deal with the eigenvalue interval for existence of positive solutions to the higher order fractional differential equation when may be negative. This type of differential equation is called semipositone problem which arises in many interesting applications as pointed out by Lions in [25]. For example, the semipositone differential equation which can be derived from chemical reactor theory, design of suspension bridges, combustion, and management of natural resources, see [26–28]. To our knowledge, few results were established, especially for higher order multipoint boundary value problems with the fractional derivatives.

2. Preliminaries and Lemmas

We use the following assumptions in this paper:

  is continuous, and there exist functions and continuous function such that

Now we begin this section with some preliminaries of fractional calculus. Let 0 and , where is the smallest integer greater than or equal to . For a function , we define the fractional integral of order of as

provided the integral exists. The fractional derivative of order of a continuous function is defined by

provided the right side is pointwise defined on . We recall the following properties [8, 9] which are useful for the sequel.

Lemma 2.1 (see [8, 9]).

(1) If ,  , and , then

(2) If ,  , then

Lemma 2.2 (see [8]). Assume that and . Then

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

Let , and consider the following modified integro-differential equation:

The following Lemmas 2.3–2.5 are obtained by Zhang et al. [10].

Lemma 2.3. The higher order multipoint boundary value problem (1.1) has a positive solution if and only if nonlinear integro-differential equation (2.8) has a positive solution. Moreover, if is a positive solution of (2.8), then is positive solution of the higher order multipoint boundary value problem (1.1).

Lemma 2.4. If and , then the boundary value problem

has the unique solution

where

is the Green function of the boundary value problem (2.9), and

Lemma 2.5. The Green function of the boundary value problem (2.9) satisfies

where

Define a modified function for any by

and consider the following boundary value problem

Lemma 2.6. Suppose ,   is a solution of the problem (2.16), then is a positive solution of the problem (2.8), consequently, is also a positive solution of the semipositone higher differential equation (1.1).

Proof. Since is a solution of the BVP (2.16) and for any , then we have

Let , then we have

and , which implies that

Substituting the above into (2.17), then solves the (2.8), that is, is a positive solution of the semipositone differential equation (2.8). By Lemma 2.3, is a positive solution of the singular semipositone differential equation (1.1). This completes the proof of Lemma 2.5.

Let

Lemma 2.7 (see [10]). The solution of (2.9) satisfies

where

It is well known that the BVP (2.17) is equivalent to the fixed points for the mapping by

The basic space used in this paper is , where is the set of real numbers. Obviously, the space is a Banach space if it is endowed with the norm as follows:

for any . Let

then is a cone of .

Lemma 2.8. Assume that holds. Then is a completely continuous operator.

Proof. By using similar method to [10] and standard arguments, according to the Ascoli-Arzela Theorem, one can show that is a completely continuous operator.

Lemma 2.9 (see [29]). Let be a real Banach space, be a cone. Assume are two bounded open subsets of with ,  , and let be a completely continuous operator such that either

(1) and , or (2) and .

Then has a fixed point in ).

3. Main Results

Theorem 3.1. Suppose that holds, and

Then there exists some constant such that the higher order multipoint boundary value problem (1.1) has at least one positive solution for any .

Proof. By Lemma 2.8, we know is a completely continuous operator. Take

where is defined by Lemma 2.5. Let . Then, for any ,  , we have

Choose

where

and is defined by Lemma 2.7. Thus, for any ,  , and , by (3.3), we have

Therefore,

On the other hand, choose a real number such that

By (3.1), for any , there exists a constant such that

Take

let and . Then for any ,  , by Lemma 2.7, we have

And then, for any ,  , one gets

It follows from (3.12) that, for any ,

So, we have

By Lemma 2.9, has at least a fixed point such that .

It follows from and that

Let , then

By Lemma 2.6, we know that the differential equation (1.1) has at least a positive solutions .

Theorem 3.2. Suppose holds and

Then there exists such that the higher order multipoint boundary value problem (1.1) has at least one positive solution for any .

Proof. By (3.17), there exists such that for any we have

Let

In the following of the proof, we suppose . Take

and let and . Then for any ,  , by Lemma 2.7, we have

So for any ,  , one gets

Thus, by (3.22), for any , we have

So, we have

Next, take

Let us choose such that

Then for the above , by (3.17), there exists such that, for any ,

Thus, by (3.3) and (3.27), if

we have

Take

where

Then .

Now let and . Then, for any , we have

which implies that

By Lemma 2.9, has at least a fixed point such that .

It follows from that

Let , then

By Lemma 2.6, we know that the differential equation (1.1) has at least a positive solutions .

Example 3.3. Consider the existence of positive solutions for the nonlinear higher order fractional differential equation with four-point boundary condition

Then there exists such that the higher order four-point boundary value problem (1.1) has at least one positive solution for any .

Proof. Let

then

Clearly, , and

By Theorem 3.2, there exists such that the higher order multipoint boundary value problem (3.36) has at least one positive solution for any .

References

H. M. Srivastava and S. Owa, Univalent Functions, Fractional Calculus, and Their Applications, Halsted Press, New York, NY, USA, John Wiley and Sons, Chichester, UK, 1989.

L. Gaul, P. Klein, and S. Kemple, “Damping description involving fractional operators,” Mechanical Systems and Signal Processing, vol. 5, no. 2, pp. 81–88, 1991.

W. G. Glockle and T. F. Nonnenmacher, “A fractional calculus approach to self-similar protein dynamics,” Biophysical Journal, vol. 68, no. 1, pp. 46–53, 1995.

K. Diethelm, The Analysis of Fractional Differential Equations, Springer-Verlag, Berlin, Germany, 2010.

R. Metzler, W. Schick, H. G. Kilian, and T. F. Nonnenmacher, “Relaxation in filled polymers: a fractional calculus approach,” The Journal of Chemical Physics, vol. 103, no. 16, pp. 7180–7186, 1995.

K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York, NY, USA, 1974.

A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, “Theory and applications of fractional differential equations,” in North-Holland Mathematics Studies, vol. 204, Elsevier, Amsterdam, The Netherlands, 2006.

K. S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Differential Equations, John Wiley & Sons, New York, NY, USA, 1993.

I. Podlubny, Fractional Differential Equations, Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1999.

X. Zhang, L. Liu, and Y. Wu, “Existence results for multiple positive solutions of nonlinear higher order perturbed fractional differential equations with derivatives,” Applied Mathematics and Computation, vol. 219, no. 4, pp. 1420–1433, 2012.

B. Ahmad, A. Alsaedi, and B. S. Alghamdi, “Analytic approximation of solutions of the forced Duffing equation with integral boundary conditions,” Nonlinear Analysis: Real World Applications, vol. 9, no. 4, pp. 1727–1740, 2008.

X. Zhang and Y. Han, “Existence and uniqueness of positive solutions for higher order nonlocal fractional differential equations,” Applied Mathematics Letters, vol. 25, no. 3, pp. 555–560, 2012.

X. Zhang, L. Liu, and Y. Wu, “The eigenvalue problem for a singular higher order fractional differential equation involving fractional derivatives,” Applied Mathematics and Computation, vol. 218, no. 17, pp. 8526–8536, 2012.

X. Zhang, L. Liu, B. Wiwatanapataphee, and Y. Wu, “Positive solutions of eigenvalue problems for a class of fractional differential equations with derivatives,” Abstract and Applied Analysis, vol. 2012, Article ID 512127, 16 pages, 2012.

X. Zhang, L. Liu, and Y. Wu, “Multiple positive solutions of a singular fractional differential equation with negatively perturbed term,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1263–1274, 2012.

H. A. H. Salem, “On the fractional order m-point boundary value problem in reflexive Banach spaces and weak topologies,” Journal of Computational and Applied Mathematics, vol. 224, no. 2, pp. 565–572, 2009.

C. S. Goodrich, “Existence of a positive solution to a class of fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1050–1055, 2010.

M. Rehman and R. A. Khan, “Existence and uniqueness of solutions for multi-point boundary value problems for fractional differential equations,” Applied Mathematics Letters, vol. 23, no. 9, pp. 1038–1044, 2010.

B. Ahmad and J. J. Nieto, “Riemann-Liouville fractional integro-differential equations with fractional nonlocal integral boundary conditions,” Boundary Value Problems, vol. 2011, 2011.

C. S. Goodrich, “Existence of a positive solution to systems of differential equations of fractional order,” Computers and Mathematics with Applications, vol. 62, no. 3, pp. 1251–1268, 2011.

C. S. Goodrich, “Existence and uniqueness of solutions to a fractional difference equation with nonlocal conditions,” Computers and Mathematics with Applications, vol. 61, no. 2, pp. 191–202, 2011.

C. S. Goodrich, “Positive solutions to boundary value problems with nonlinear boundary conditions,” Nonlinear Analysis. Theory, Methods & Applications, vol. 75, no. 1, pp. 417–432, 2012.

M. Jia, X. Zhang, and X. Gu, “Nontrivial solutions for a higher fractional differential equation with fractional multi-point boundary conditions,” Boundary Value Problems, vol. 2012, 70 pages, 2012.

M. Jia, X. Liu, and X. Gu, “Uniqueness and asymptotic behavior of positive solutions for a fractional-order integral boundary value problem,” Abstract and Applied Analysis, vol. 2012, Article ID 294694, 21 pages, 2012.

P.-L. Lions, “On the existence of positive solutions of semilinear elliptic equations,” SIAM Review, vol. 24, no. 4, pp. 441–467, 1982.

R. Aris, Introduction to the Analysis of Chemical Reactors, Prentice-Hall, Englewood Cliffs, NJ, USA, 1965.

A. Castro, C. Maya, and R. Shivaji, “Nonlinear eigenvalue problems with semipositone,” Electronic Journal of Differential Equations, vol. 5, pp. 33–49, 2000.

V. Anuradha, D. D. Hai, and R. Shivaji, “Existence results for superlinear semipositone BVP'S,” Proceedings of the American Mathematical Society, vol. 124, no. 3, pp. 757–763, 1996.

D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cone, Academic Press, New York, NY, USA, 1988.