Existence of positive solution for singular fractional differential equation

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Abstract

In this paper, we establish the existence of a positive solution to a singular boundary value problem of nonlinear fractional differential equation. Our analysis rely on nonlinear alternative of Leray–Schauder type and Krasnoselskii’s fixed point theorem in a cone.

Introduction

Many papers and books on fractional calculus differential equation have appeared recently [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11]. Most of them are devoted to the solvability of the linear fractional equation in terms of a special function and to problems of analyticity in the complex domain (see, for example [3], [7]). Moreover, Delbosco and Rodino [2] considered the existence of a solution for the nonlinear fractional differential equation D0+αu=f(t,u), where 0<α<1, and f:[0,a]×RR,0<a+ is a given function, continuous in (0,a)×R. They obtained results for solutions by using the Schauder fixed point theorem and the Banach contraction principle. Very recently, Zhang [9] considered the existence of positive solution for equation D0+αu=f(t,u), where 0<α<1, and f:[0,1]×[0,+)[0,+) is a given continuous function, by using the sub- and super-solution method.

In this paper, we discuss the existence of a positive solution to boundary value problem of nonlinear fractional differential equation:D0+αu(t)+f(t,u(t))=0,0<t<1,u(0)=u(1)=u(0)=0,where 2<α3 is a real number, D0+α is the Caputo’s differentiation, and f:(0,1]×[0,)[0,),limt0+f(t,·)=+ (that is f is singular at t=0).

About the development of the existence theorems for fractional differential equation and also many references with real applications, we refer to the Survey paper by Kilbas and Trujillo [5], [6]. Concerning the definitions of a fractional integral and derivative and related basic properties used in the text, the readers can refer to Samko et al. [8] or Delbosco and Rodino [2].

In conclusion we obtain two results about this boundary value problem, by using Krasnoselskii’s fixed point theorem in a cone and nonlinear alternative of Leray–Schauder, respectively.

Section snippets

Background materials and preliminaries

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory. These definitions and properties can be found in the recent literature.

Definition 2.1

The Riemann–Liouville fractional integral of order α>0 of a function f:(0,)R is given byI0+αf(t)=1Γ(α)0t(t-s)α-1f(s)ds,provided that the right side is pointwise defined on (0,).

Definition 2.2

The Caputo’s fractional derivative of order α>0 of a continuous function f:(0,)R is given byD0+αf(t)=1Γ(n-α)0tf(n)(s)(t-s)α-n+1ds,

Main results

For our construction, we let E=C[0,1] with u=max0t1|u(t)| be Banach space.We seek solutions of (1.1) that lie in a cone P, defined byP={uE|u(t)0,0t1}.Define operator T:PPTu(t)=01G(t,s)f(s,u(s))ds.

Lemma 3.1

Let 0<σ<1,2<α3,F:(0,1]R is continuous, and limt0+F(t)=. Suppose that tσF(t) is continuous function on [0, 1]. Then the functionH(t)=0tG(t,s)F(s)ds,is continuous on [0, 1].

Proof

By the continuity of tσF(t) and H(t)=0tG(t,s)s-σsσF(s)ds It is easily to know that H(0)=0. The proof is divided into

Acknowledgements

This work is supported by the National Nature Science Foundation of China (10371006) and the Mathematics Tianyuan Foundation of China (10626033).

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