Functional fractional boundary value problems with singular -Laplacian
Introduction
Bereanu and Mawhin [1] studied the functional differential equationand the differential equationsubject to the nonhomogeneous Dirichlet boundary conditionsHere satisfies the condition
, is increasing and such that and ,
The purpose of this paper is to show that the procedures given in [1] can be applied to fractional functional differential equations with singular -Laplacian and functional boundary conditions.
We discuss the fractional functional differential equationand the fractional differential equationwhere and denotes the Caputo fractional derivative of order .
The Caputo fractional derivative of order of a function is defined as [4], [5]where for and means the integral part of the number . is the Euler gamma function.
Fractional differential equations have received a lot of attention recently. In the literature there is only a few papers dealing with boundary value problems for fractional differential equations with -Laplacian. In [8], [9], [10] fractional boundary value problems are investigated for equations only with the p-Laplacian operator , . Wang et al. [8] considered the equationand in [9] the equation
Paper [10] deals with the equation
In these equations denotes the Riemann–Liouville fractional derivative.
Let and let be the space equipped with the norm , where is the norm in . As we will prove in Lemma 6, X is a Banach space. We denote by the set of functionals which are bounded (i.e., ) and continuous. If , we write .
We investigate Eqs. (1.4), (1.5) subject to the functional boundary conditions
Note that if are constant functionals, then (1.6) reduces to (1.3).
We work with the following condition on the operator F in (1.4).
is continuous and takes bounded sets into bounded sets.
We say that a function is a solution of problem (1.4) and (1.6) if , , u satisfies the boundary conditions (1.6) and equality (1.4) holds for .
A function is called a solution of problem (1.5), (1.6) if , u satisfies (1.6), (1.5) holds for .
Our paper is organized as follows. Section 2 contains the definition of Riemann–Liouville fractional integral and preliminary facts from the fractional calculus, which are used throughout this paper. In Section 3 we introduce an operator and give its properties. In particular, it is proved that u is a solution of problem (1.4) and (1.6) if and only if it is a fixed point of . In Section 4 our main results are stated and proved. We present an example to illustrate our results.
Section snippets
Preliminaries
The Riemann–Liouville fractional integral of order of a function is given by Diethelm [4], Kiblas et al. [5] and Podlubny [6] Lemma 1 Let . Then . Lemma 2 Let . Then for and . Lemma 3 Let . Then for and . Lemma 4 Let , and let . Then the equality is fulfilled for . Remark 1 The equality in Lemma 4 can be written in the form[3], [4, Theorem 2.5], [5, Lemma 2.8]
[5, Lemma 2.21]
[5, Lemma 2.22]
[4, Theorem 2.2], [5, Lemma 2.3]
Operator and its properties
In order to introduce an operator , we need the following technical result. Letwhere a is given in . Lemma 7 Let conditions and be satisfied. Then for each there exists a unique such that Moreover, the function is continuous. Proof Let us choose . It is not difficult to verify that the functionis well defined, continuous and decreasing on . Since
Existence results
Theorem 1 Let and hold. Let be such that . Then problem (1.4), (1.6) has a solution. Proof By Lemma 9 we need to prove that has a fixed point. Let W be a positive constant given in Lemma 10 and let To simplify notation, we use the same letter for the restriction of on . Then, by Lemma 8, Lemma 10, is a compact operator and for . Hence, by the homotopy property (see, e.g., [7])
Acknowledgements
The research of the first author was supported by FEDER and Ministerio de Educación y Ciencia, Spain, project MTM2010–15314.
The research of the second author was supported by the Grant PrF-2012–017.
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