Singular Positone and Semipositone Boundary Value Problems of Nonlinear Fractional Differential Equations
Chengjun Yuan,1,2Daqing Jiang,1and Xiaojie Xu1
Academic Editor: Victoria Vampa
Received08 Jan 2009
Accepted15 Apr 2009
Published22 Jul 2009
We present some new existence results for singular positone and semipositone nonlinear fractional boundary value problem , , , where , and are continuous, is a real number, and is Riemann-Liouville fractional derivative. Throughout our nonlinearity may be singular
in its dependent variable. Two examples are also given to illustrate the main results.
Fractional calculus has played a significant role in engineering, science, economy, and other fields. Many papers and books on fractional calculus, and fractional differential equations have appeared recently, (see [1–9]). It should be noted that most of papers and books on fractional calculus, are devoted to the solvability of initial fractional differential equations (see [3, 4]). Here, we consider positive solutions of nonlinear fractional differential equation conjugate boundary value problem involving Riemann-Liouville derivative:
where , , and are continuous. is a real number, and is the Riemann-Liouville fractional derivative.
It is well known that in mechanics the boundary value problem (1.1) where describes the deflection of an elastic beam rigidly fixed at both ends. The integer order boundary value problem (1.2) has been studied extensively. For details, see for instance, the papers [10–13] and the references therein. In [10, 12], Yao considered
and using a Krasnosel'skii fixed-point theorem, derived a -interval such that, for any lying in this interval, the beam equation has existence and multiplicity on positive solution. In this paper, we will consider a more general situation, namely, the boundary value problem (1.1). To the best of our knowledge, there have been few papers which deal with the boundary value problem (1.1) for nonlinear fractional differential equation.
In this paper, in analogy with boundary value problems for differential equations of integer order, we firstly derive the corresponding Green's function named the fractional Green' function. Consequently problem (1.1) is reduced to an equivalent Fredholm integral equation of the second kind. Finally, using Krasnosel'skii's fixed-point theorems, the existence of positive solutions are obtained.
For completeness, in this section, we will demonstrate and study the definitions and some fundamental facts of Riemann-Liouville derivatives of fractional order which can be found in .
Definition 2.1 (see [5, Definition 2.1]). The integral
where , is called the Riemann-Liouville fractional integral of order .
Definition 2.2 (see [5, page 36-37]). For a function given in the interval , the expression
where denotes the integer part of number , is called the Riemann-Liouville fractional derivative of order . From the definition of Riemann-Liouville derivative, for , we have
giving in particular , where is the smallest integer greater than or equal to .
Lemma 2.3. Let , then the differential equation
has solutions , , , as unique solutions, where is the smallest integer greater than or equal to . As , from Lemma 2.3, we deduce the following statement.
Lemma 2.4. Let , then
for some , , is the smallest integer greater than or equal to . The following Krasnosel'skii's fixed-point theorem will play a major role in our next analysis.
Theorem 2.5 (see ). Let be a Banach space, and let be a cone in . Assume that are open subsets of with , and let be a completely continuous operator such that, either (1), , , , or(2), , .Then has a fixed-point in ∖ .
3. Green's Function and Its Properties
In this section, we derive the corresponding Green's function for boundary-value problem (1.1), and obtain some properties of Green's function.
Lemma 3.1. Let be a given function, then the boundary-value problem,
has a unique solution
Here is called Green's function of boundary-value problem (3.1).
Proof. By means of the Lemma 2.4, we can reduce (3.1) to an equivalent integral equation
From , we have and
Therefore, the unique solution of (3.1) is
The proof is finished.
Lemma 3.2. The function defined by (3.3) has the following properties: (1);(2) and for where .
Proof. Observing the expression of , it is clear that for . In the following, we consider . For , we have
For , since , we have
Thus , for . Combining , we have
This completes the proof.
We note that is a solution of (1.1) if and only if
For our constructions, we will consider the Banach space equipped with standard norm . We define a cone by
Define an integral operator by
Notice from (3.13) and Lemma 3.2 that, for , on and
On the other hand, we have
Thus, . In addition, standard arguments show that is completely continuous.
4. Singular Positone Problems
In this section we present some new result for the singular problem
where and nonlinearity may be singular at .
Using Theorem 2.5 we establish the following main result.
Theorem 4.1. Suppose that the following conditions are satisfied.
here is Green's function and
Then (4.1) has two nonnegative solutions with and for .
Proof. First we will show that there exists a solution to (4.1) with for and Let
We now show
To see this, let . Then and for So for , we have
This together with (4.7) yields
so (4.12) is satisfied. Next we show
To see this, let so , and let for . We have
This together with (4.9) yields
Thus so (4.15) is held. Now Theorem 2.5 implies that has a fixed-point , that is, and for . It follows from (4.12) and (4.15) that so we have Similarly, if we put
we can show that there exists a solution to (4.1) with for and This completes the proof of Theorem 4.1.
Similar to the proof of Theorem 4.1, we have the following result.
Theorem 4.2. Suppose that (4.2)–(4.8) hold. In addition suppose
Then (4.1) has a nonnegative solution with and for .
Remark 4.3. If in (4.19) we have , then (4.1) a nonnegative solution with and for . It is easy to use Theorem 4.2 and Remark 4.3 to write theorems which guarantee the existence of more than two solutions to (4.1). We state one such result.
Theorem 4.4. Suppose that (4.2)–(4.6) and (4.8) hold. Assume that and constants with , and
In addition suppose for each that
hold. Then (4.1) has nonnegative solutions with for .
Example 4.5. Consider the boundary value problem
where is such that
Then (4.23) has two solutions with for To see this we will apply Theorem 4.1 with (here will be chosen below)
Clearly (4.2)–(4.6) and (4.8) hold, and . Now (4.7) holds with since
Finally notice (4.9) is satisfied for small and large since
as , since . Thus all the conditions of Theorem 4.1 are satisfied so existence is guaranteed.
5. Singular Semipositone Problems
In this section we present a new result for the singular semipositone problem:
where and nonlinearity may be singular at .
Before we prove our main result, we first state a result.
Lemma 5.1. Suppose with on . Then the boundary value problem,
has a solution with
In fact, from Lemma 3.1, (5.2) has solution
According to Lemma 3.2, we have
The above Lemma together with Theorem 2.5 establish our main result.
Theorem 5.2. Suppose that the following conditions are satisfied.
here is any constant (choose and fix it) so that (note exists since in fact we can have ) ) and is Green's function and
Then (5.1) has a solution with for .
Proof. To show that (5.1) has a nonnegative solution we will look at the boundary value problem
( is as in Lemma 5.1). We will show, using Theorem 2.5, that there exists a solution to (5.16) with for . If this is true then is a nonnegative solution (positive on ) of (5.1), since
As a result, we will concentrate our study on (5.16). Let as in Section 2, and let
Next let be defined by
In addition, standard argument shows that and is completely continuous. We now show
To see this, let . Then and for Now for , the Lemma 5.1 implies
so for we have
This together with (5.12) yields
so (5.21) is satisfied. Next we show
To see this let so and for . Also for we have
As a result
This together with (5.14) yields
Thus so (5.25) is held. Now Theorem 2.5 implies that has a fixed-point , that is, and for . Thus is a solution of (5.16) with for . Thus (5.1) has a positive solution for .
Example 5.3. Consider the boundary value problem
where is such that
Then (5.30) has a solution with for To see this we will apply Theorem 5.2 with (here will be chosen later, in fact here we choose so that works, i.e., we choose so that ),
Clearly (5.7)–(5.11) and (5.13) hold. Now (5.12) holds with since
from (5.31). Finally notice (5.14) is satisfied for large since
as , since . Thus all the conditions of Theorem 5.2 are satisfied so existence is guaranteed.
This paper is supported by Key Subject of Chinese Ministry of Education (no. 109051) and Scientific Research Fund of Heilongjiang Provincial Education Department (no. 11544032).
O. P. Agrawal, “Formulation of Euler-Lagrange equations for fractional variational problems,” Journal of Mathematical Analysis and Applications, vol. 272, no. 1, pp. 368–379, 2002.
Q. Yao, “Existence and multiplicity of positive solutions to a singular elastic beam equation rigidly fixed at both ends,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 8, pp. 2683–2694, 2008.