Existence of the solutions for a class of nonlinear fractional order three-point boundary value problems with resonance
© Ouyang and Li; licensee Springer 2012
Received: 4 December 2011
Accepted: 9 May 2012
Published: 1 July 2012
A class of nonlinear fractional order differential equation
is investigated in this paper, where is the standard Riemann-Liouville fractional derivative of order , , . Using intermediate value theorem, we obtain a sufficient condition for the existence of the solutions for the above fractional order differential equations.
where is the standard Riemann-Liouville fractional derivative of order , and .
In the last few decades, many authors have investigated fractional differential equations which have been applied in many fields such as physics, mechanics, chemistry, engineering etc. (see references [1, 6, 10, 21–23]). Especially, many works have been devoted to the study of initial value problems and bounded value problems for fractional order differential equations [12, 13, 15, 24].
Recently, the existence of positive solutions of fractional differential equations has attracted many authors’ attention [2–5, 8, 9, 12, 14, 17–20, 25, 26]. Using some fixed point theorems, they obtained some nice existence conditions for positive solutions.
where is the standard Riemann-Liouville fractional derivative of order and is continuous. Using some properties of the Green function , they obtain some new sufficient conditions for the existence of positive solutions for the above problem.
where is the standard Riemann-Liouville fractional derivative of order , and is continuous.
In this paper, we discuss the boundary value problem (1.1)-(1.2). Using some properties of the Green function and intermediate value theorem, we establish some sufficient conditions for the existence of the positive solutions of the problem (1.1)-(1.2).
The paper is arranged as follows: In Section 2, we introduce some definitions for fractional order differential equations and give our main results for the boundary value problem (1.1)-(1.2). We give some lemmas for our results in Section 3. In Section 4, we prove our main result; and finally, we give an example to illustrate our results.
2 Main results
In this section, we introduce some definitions and preliminary facts which are used in this paper.
provided that the integral on the right-hand side is point-wise defined on , where Γ is the Gamma function.
where n is a positive integer.
We call the function a solution of (1.1)-(1.2) if with a fractional derivative of order α belongs to and satisfies Equation (1.1) and the boundary condition (1.2).
We also need to introduce some lemmas as follows, which will be used in the proof of our main theorems.
Lemma 2.1 ()
Assume thatwith a fractional derivative of orderbelongs to. Then, the fractional equation
Lemma 2.2 ()
for some, , .
Lemma 2.3 ()
T has a fixed point in , or
there exist and with .
Throughout this paper, we assume that satisfies the following:
We have our main results:
3 Some lemmas
then Ω is a Banach space.
We first give some lemmas as follows:
Lemma 3.1 Problem (1.1)-(1.2) is equivalent to the following integral equation
Proof The sufficiency is obvious, we only need to prove the necessity.
According to (3.3), it is easy to show that (3.2) holds. The proof is completed. □
Lemma 3.2 For any, is continuous, andfor any.
Proof The continuity of for is obvious.
for . The proof is completed. □
The new Green’s function has the following properties:
Lemma 3.4 For any, is nonincreasing with respect to. Especially, for any, for, andfor. That is, where
We can divide our proof into the following two steps:
It suffices to show that for any given real number μ, (3.13) has a solution , which implies that Equation (1.1) has a solution which satisfies the first boundary value condition .
Second, we show that there exists a μ such that the solution of (3.13) satisfies , which implies that the solution of (1.1) also satisfies the boundary value condition .
Lemma 3.5 Suppose that, and (2.4) hold, then the operator T is completely continuous in Ω.
Proof It is easy to show that the operator T maps Ω into itself. We divide the proof into the following three steps.
Step 1. is continuous with respect to .
Thus, the operator T is continuous in Ω.
Step 2. T maps bounded set in Ω into bounded set.
This gives that the operator T maps bounded set into bounded set in Ω.
Step 3. T is equicontinuous in Ω.
According to Step 1-Step 3, the operator T is completely continuous in Ω. The proof is completed. □
Further, we have
Lemma 3.6 Suppose that, and (2.4) and (2.6) holds, then, for any real number μ, the integral Equation (3.13) has at least one solution.
To show the existence of a fixed point of T by Lemma 2.3, we need to verify that the second possibility in Lemma 2.3 cannot happen.
It is obvious that (3.18) contradicts our assumption that . Therefore, by Lemma 2.3, it follows that T has a fixed point . Hence, the integral Equation (3.14) has at least a solution . The proof is completed. □
4 The proof of the main results
Now, we prove Theorem 2.1 by Lemma 3.4-3.5 and the intermediate value theorem.
Proof of Theorem 2.1 It is obvious that the right-hand side of (3.14) is continuously dependent on the parameter μ, so we need to find a μ such that , which implies that .
It is obvious that is continuously dependent on the parameter μ. In order to prove that there exists a such that , we only need to show that , and .
which contradicts our assumption.
Then, is not empty.
It is easy to know that , and , and we have from (H) that is not empty.
This contradicts (4.9).
Now, we have proved that . By a similar method, we can prove that . The detail is omitted.
Notice that is continuous with respect to . It follows from the intermediate value theorem  that there exists a such that , that is , which satisfies the second boundary value condition of (1.2). The proof is completed. □
Thus, the conditions of Theorem 2.1 are satisfied. Therefore, the problem (5.1) has at least a nontrivial solution.
Each of the authors, ZO and GL contributed to each part of this study equally and read and approved the final version of the mnanuscript.
Supported partially by China Postdoctoral Science Foundation under Grant No.20110491280 and the Subject Lead Foundation of University of South China No. 2007XQD13.
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