Functional fractional boundary value problems with singular ϕ-Laplacian

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Abstract

This paper discusses the existence of solutions of the fractional differential equations cDμ(ϕ(cDαu))=Fu,cDμ(ϕ(cDαu))=f(t,u,cDνu) satisfying the boundary conditions u(0)=A(u),u(T)=B(u). Here μ,α(0,1],ν(0,α],cD is the Caputo fractional derivative, ϕC(-a,a) (a>0), F is a continuous operator, A,B are bounded and continuous functionals and fC([0,T]×R2). The existence results are proved by the Leray–Schauder degree theory.

Introduction

Bereanu and Mawhin [1] studied the functional differential equation(ϕ(u(t)))=(Hu)(t)and the differential equation(ϕ(u(t)))=h(t,u(t),u(t))subject to the nonhomogeneous Dirichlet boundary conditionsu(0)=A,u(T)=B(A,BR).Here ϕ satisfies the condition

  • (ϕ)

    ϕC(-a,a) (a(0,)), ϕ is increasing and such that ϕ(0)=0 and limx±aϕ(x)=±,

(they call it singular). By the Leray–Schauder degree method, they proved that if H:C1[0,T]C[0,T] is continuous and takes bounded sets into bounded sets and if |B-A|<aT, then problem (1.1), (1.3) has a solution. Hence an immediately consequence is that if hC([0,T]×R2) and if |B-A|<aT, then problem (1.2), (1.3) has a solution. The case A=B=0 was discussed in [2]

The purpose of this paper is to show that the procedures given in [1] can be applied to fractional functional differential equations with singular ϕ-Laplacian and functional boundary conditions.

We discuss the fractional functional differential equationcDμ(ϕ(cDαu(t)))=(Fu)(t)and the fractional differential equationcDμ(ϕ(cDαu(t)))=f(t,u(t),cDνu(t)),where μ,α(0,1],ν(0,α] and cDγ denotes the Caputo fractional derivative of order γ.

The Caputo fractional derivative cDγ of order γ>0 of a function x:[0,T]R is defined as [4], [5]cDγu(t)=1Γ(n-γ)dndtn0t(t-s)n-γ-1x(s)-k=0n-1x(k)(0)k!skds,ifγN,x(γ)(t),ifγN,where n=[γ]+1 for γN and [γ] means the integral part of the number γ. Γ 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 ϕ-Laplacian. In [8], [9], [10] fractional boundary value problems are investigated for equations only with the p-Laplacian operator ϕp,ϕp(s)=|s|p-2s, p>1. Wang et al. [8] considered the equationDγ(ϕp(Dδu(t)))+f(t,u(t))=0,0<γ1,1<δ2and in [9] the equationcDγ(ϕp(cDδu(t)))+f(t,u(t),cDρu(t))=0,0<γ<1,2<δ<3,0<ρ1.

Paper [10] deals with the equationDγ(ϕp(Dδu(t)))=f(t,u(t)),γ,δ(1,2).

In these equations Dγ denotes the Riemann–Liouville fractional derivative.

Let α(0,1] and let X={uC[0,T]:cDαuC[0,T]} be the space equipped with the norm u+cDαu, where u=maxt[0,T]|u(t)| is the norm in C[0,T]. As we will prove in Lemma 6, X is a Banach space. We denote by A the set of functionals A:XR which are bounded (i.e., sup{|A(u)|:uX}<) and continuous. If AA, we write As=sup{|A(u)|:uX}.

We investigate Eqs. (1.4), (1.5) subject to the functional boundary conditionsu(0)=A(u),u(T)=B(u)(A,BA).

Note that if A,BA are constant functionals, then (1.6) reduces to (1.3).

We work with the following condition on the operator F in (1.4).

  • (F)

    F:XC[0,T] is continuous and takes bounded sets into bounded sets.

We say that a function u:[0,T]R is a solution of problem (1.4) and (1.6) if uX, cDμ(ϕ(cDαu))C[0,T], u satisfies the boundary conditions (1.6) and equality (1.4) holds for t[0,T].

A function uX is called a solution of problem (1.5), (1.6) if cDμ(ϕ(cDαu))C[0,T], u satisfies (1.6), (1.5) holds for t[0,T].

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 QA,B:[0,1]×XX 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 QA,B(1,·). 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 Iγx of order γ>0 of a function x:[0,T]R is given by Diethelm [4], Kiblas et al. [5] and Podlubny [6]Iγx(t)=1Γ(γ)0t(t-s)γ-1x(s)ds.

Lemma 1

[3], [4, Theorem 2.5], [5, Lemma 2.8]

Let γ(0,1]. Then Iγ:C[0,T]C[0,T].

Lemma 2

[5, Lemma 2.21]

Let γ(0,1]. Then cDγIγx(t)=x(t) for t[0,T] and xC[0,T].

Lemma 3

[5, Lemma 2.22]

Let γ(0,1]. Then IγcDγx(t)=x(t)-x(0) for t[0,T] and xC[0,T].

Lemma 4

[4, Theorem 2.2], [5, Lemma 2.3]

Let β,τ(0,), β+τ1 and let xL1[0,T]. Then the equality IβIτx(t)=Iβ+τx(t) is fulfilled for t[0,T].

Remark 1

The equality IβIτx(t)=Iβ+τx(t) in Lemma 4 can be written in the form0

Operator QA,B and its properties

In order to introduce an operator QA,B, we need the following technical result. LetZ=[0,1]×X×-aTαα,aTαα,where a is given in (ϕ).

Lemma 7

Let conditions (ϕ) and (F) be satisfied. Then for each (λ,x,d)Z there exists a unique mΛϕ(λ,x,d) such that0T(T-t)α-1ϕ-1[λIμ(Fx)(t)-m]dt=d.

Moreover, the function Λϕ:ZR is continuous.

Proof

Let us choose (λ,x,d)Z. It is not difficult to verify that the functionρ(c)=0T(T-t)α-1ϕ-1[λIμ(Fx)(t)-c]dtis well defined, continuous and decreasing on R.

Sincelimc±ρ(c)=a0T(T-t)α-1dt

Existence results

Theorem 1

Let (ϕ) and (F) hold. Let A,BA be such that (B-A)s<aTαΓ(α+1). Then problem (1.4), (1.6) has a solution.

Proof

By Lemma 9 we need to prove that QA,B(1,·) has a fixed point. Let W be a positive constant given in Lemma 10 and letΩ={xX:x<W,cDαx<W}.

To simplify notation, we use the same letter QA,B for the restriction of QA,B on [0,1]×Ω¯. Then, by Lemma 8, Lemma 10, QA,B:[0,1]×Ω¯X is a compact operator and QA,B(λ,x)qx for (λ,x)[0,1]×Ω. Hence, by the homotopy property (see, e.g., [7])deg(I-QA,B(1,

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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