- Open Access
Positive solutions to boundary value problems of a high-order fractional differential equation in a Banach space
© Zhao et al.; licensee Springer. 2013
- Received: 9 July 2013
- Accepted: 29 October 2013
- Published: 27 November 2013
In this paper, by using the fixed-point theorem in the cone of strict-set-contraction operators, we study a class of higher-order boundary value problems of nonlinear fractional differential equation in a Banach space. The sufficient conditions for the existence of at least two positive solutions is obtained. In addition, an example to illustrate the main results is given.
- fractional differential equation
- boundary value problem
- measure of noncompactness
- strict-set-contraction operators
Fractional calculus is a generalization of ordinary differentiation and integration to arbitrary noninteger order. The fractional differential equations have been of great interest recently. This is because of both the intensive development of the theory of fractional calculus itself and its numerous applications in various fields of science and engineering including fluid flow, rheology, control, electrochemistry, electromagnetic, porous media and probability, etc. (see [1–4]).
in a reflexive Banach space E, where is the pseudo fractional differential operator of order , .
where is the Caputo derivative of order .
in a Banach space E. is the Riemann-Liouville fractional derivative.
where θ is the zero element of E, , α, β, γ and δ are nonnegative constants satisfying , and is the Caputo fractional derivative. Note that the nonlinear term f depends on u and its derivatives .
The paper is organized as follows. In Section 2 we give some basic definitions in Riemann-Liouville fractional calculus and the Kuratowski noncompactness. In Section 3 we present the expression and properties of Green’s function associated with BVP (1.1), and by using the fixed-point theorem for strict-set-contraction operators and introducing a new cone Ω, we obtain the existence of at least two positive solutions for BVP (1.1) under certain conditions on the nonlinearity. Moreover, an example illustrating our main result is given in Section 4.
For convenience of the reader, we present here some definitions and preliminaries which are used throughout the paper. These definitions and lemmas can be found in the recent literature such as [1, 5].
Definition 2.1 ()
provided that the right-hand side is defined pointwise.
Definition 2.2 ()
where , denotes the integer part of number q, provided that the right-hand side is defined pointwise. In particular, for , .
Lemma 2.3 ()
has the unique solution , , , here .
Lemma 2.4 ()
for some , , here N is the smallest integer greater than or equal to q.
Let the real Banach space E with the norm be partially ordered by a cone P of E, i.e., if and only if , and P is said to be normal if there exists a positive constant N such that implies , where the smallest N is called the normal constant of P. For details on cone theory, see .
The basic space used in this paper is . For any , evidently, is a Banach space with the norm , and is a cone of the Banach space . We use α, to denote the Kuratowski noncompactness measure of bounded sets in the spaces E, , respectively. As for the definition of the Kuratowski noncompactness measure, we refer to Ref. .
Definition 2.5 (, Strict-set contraction operator)
Let , be real Banach spaces, . is a continuous and bounded operator. If there exists a constant k such that , then T is called a k-set contraction operator. When , T is called a strict-set contraction operator.
Lemma 2.6 ()
Lemma 2.7 ()
, and , , or
, and , ,
then T has a fixed point in .
(H0) , where is defined as
(H1) There exist and such that(2.1)
(H2) for any , f is uniformly continuous on and there exist nonnegative constants , , with(2.2)such that(2.3)
where is Green’s function defined by (3.3). This completes the proof. □
Moreover, there is one paper  in which the following statement was shown.
Lemma 3.2 ()
is continuous on ;
if , then for any .
where , , . Moreover, if is a solution of problem (3.3) and , , then the function is a positive solution of (1.1).
Lemma 3.5 Assume that (H0)-(H2) hold. Then is a strict-set contraction operator.
Then , which implies , i.e., .
Next we prove that T is continuous on Ω. Let and (). Hence is a bounded subset of Ω. Thus, there exists such that and .
for , .
This implies T is continuous on Ω.
By the properties of continuity of , it is easy to see that T is equicontinuous on I.
Noticing that (2.3), we obtain that T is a strict-set contraction operator. This completes the proof. □
Now we are in a position to give the main result of this work.
(H3) There exist , and such thatand
(H4) There exist , and such thatand
(H5) There exists such that
Then problem (1.1) has at least two different positive solutions.
where satisfies .
where satisfies .
Applying Lemma 2.7 to (3.12), (3.13) and (3.14) yields that T has a fixed point , , and a fixed point , . Noticing (3.13), we get and . This and Lemma 3.4 complete the proof. □
Theorem 3.7 Let the cone P be normal and conditions (H0) ∼ (H3) hold. In addition, assume that the following condition is satisfied:
Then problem (1.1) has at least one positive solution.
Since , applying Lemma 2.7 to (3.11) and (3.17) yields that T has a fixed point , . This and Lemma 3.5 complete the proof. □
Conclusion Problem (4.1) has at least two positive solutions.
i.e., condition (H2) holds for , .
From Theorem 3.6, the conclusion follows and the proof is complete. □
The authors are highly grateful for the referees’s careful reading and comments on this paper. The first author is supported financially by Hunan Provincial Natural Science Foundation of China (Grant No: 13JJ3106).
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