Existence results for m-point boundary value problems of nonlinear fractional differential equations with p-Laplacian operator
© Lv; licensee Springer. 2014
Received: 21 November 2013
Accepted: 6 February 2014
Published: 18 February 2014
In this paper, we discuss the existence and multiplicity of positive solutions to m-point boundary value problems of nonlinear fractional differential equations with p-Laplacian operator
where , and are the standard Riemann-Liouville fractional derivatives with , , , , , , , , , and , , , . Our results are based on the monotone iterative technique and the theory of the fixed point index in a cone. Furthermore, two examples are also given to illustrate the results.
Fractional differential equations arise in various areas of science and engineering. Due to their applications, fractional differential equations have gained considerable attention (see, e.g., [1–26] and the references therein).
where , are the standard Riemann-Liouville fractional derivatives, , , , .
where , and are the standard Riemann-Liouville fractional derivatives with , , , , the constant σ is a positive number, .
where , , , are Caputo fractional derivatives, and is continuous.
where , , , are the standard Riemann-Liouville fractional derivatives, and .
where , is the standard Caputo fractional derivative, .
where , , for , and , , . is the standard Riemann-Liouville fractional derivative.
where is the standard Riemann-Liouville fractional derivative, , is continuous, , , , , , and .
where , and are the standard Riemann-Liouville fractional derivatives with , , , , , , , , , and , , , .
Our work presented in this paper has the following features. Firstly, to the best of the author’s knowledge, there are few results on the existence of solutions for nonlinear fractional p-Laplacian differential equations with m-point boundary value problems. Secondly, we transform (1.1) into an equivalent integral equation and discuss the eigenvalue interval for the existence of multiplicity of positive solutions. The paper is organized as follows. In Section 2, we present some background materials and preliminaries. Section 3 deals with some existence results. In Section 4, two examples are given to illustrate the results.
2 Background materials and preliminaries
where Γ is the gamma function.
Lemma 2.1 ()
where N is the smallest integer greater than or equal to α.
Lemma 2.2 ()
Lemma 2.3 ()
, for ,
, for ,
Lemma 2.6 ()
3 Main results
Then T has a solution if and only if the operator T has a fixed point.
Lemma 3.1 If , then the operator is completely continuous.
Proof From the continuity and non-negativeness of and , we know that is continuous.
This means that is uniformly bounded.
which implies that is equicontinuous. By the Arzela-Ascoli theorem, we obtain that is completely continuous. The proof is complete. □
Step 1: Problem (1.1) has at least one solution.
By Lemma 3.1, we can see that is completely continuous. Hence, by means of the Schauder fixed point theorem, the operator T has at least one fixed point, and BVP (1.1) has at least one solution in .
Step 2: BVP (1.1) has a positive solution in , which is a minimal positive solution.
Since T is compact, we obtain that is a sequentially compact set. Consequently, there exists such that ().
Let be any positive solution of BVP (1.1) in . It is obvious that .
Taking limits as in (3.5), we get for .
Step 3: BVP (1.1) has a positive solution in , which is a maximal positive solution.
Let be any positive solution of BVP (1.1) in .
Taking limits as in (3.6), we obtain for .
The proof is complete. □
Theorem 3.3 Assume that , and the following conditions hold:
(H2) There exists a constant such that for , .
Therefore, T has a fixed point and a fixed point . Clearly, , are both positive solutions of BVP (1.1) and . The proof of Theorem 3.3 is completed. □
In a similar way, we can obtain the following result.
Corollary 3.4 Assume that , and the following conditions hold:
(H2) There exists a constant such that for , .
Hence, by Theorem 3.2, BVP (4.1) has a minimal positive solution in and a maximal positive solution in .
Thus, condition (H2) is satisfied. It is obvious that condition (H1) holds.
This research is supported by Henan Province College Youth Backbone Teacher Funds (2011GGJS-213) and the National Natural Science Foundation of China (11271336).
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