Elsevier

Applied Mathematics and Computation

Volume 226, 1 January 2014, Pages 708-718
Applied Mathematics and Computation

Existence of positive solutions for a class of nonlinear fractional differential equations with integral boundary conditions and a parameter

https://doi.org/10.1016/j.amc.2013.10.089Get rights and content

Abstract

In this paper, we study the existence of positive solutions for the following nonlinear fractional differential equations with integral boundary conditions:D0+αu(t)+h(t)f(t,u(t))=0,0<t<1,u(0)=u(0)=u(0)=0,u(1)=λ0ηu(s)ds,where 3<α4,0<η1,0ληαα<1,D0+α is the standard Riemann–Liouville derivative. h(t) is allowed to be singular at t=0 and t=1. By using the properties of the Green function, u0-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator, we obtain some existence results of positive solution.

Introduction

Fractional differential equations have been of great interest recently. It is caused both by the intensive development of the theory of fractional calculus itself and by the applications of such constructions in various fields of sciences and engineering such as control, porous media, electromagnetic, and other fields (see [1], [2], [3], [4], [5], [6], [7], [8], [10], [11], [12] and the references therein).

In present, most papers are devoted to the sublinear problem, see [2], [3]. However, there are few papers consider the superlinear problem, see [1]. In this paper, we consider the following nonlinear fractional differential equations with integral boundary conditionsD0+αu(t)+h(t)f(t,u(t))=0,0<t<1,u(0)=u(0)=u(0)=0,u(1)=λ0ηu(s)ds,under sublinear and superlinear cases, where 3<α4,0<η1,0ληαα<1,D0+α is the standard Riemann–Liouville derivative. The existence and multiplicity of positive solutions are obtained by means of the properties of the Green function, u0-bounded function and the fixed point index theory under some conditions concerning the first eigenvalue with respect to the relevant linear operator. The methods are different from those in previous works. A function uC3([0,1],R+)C4((0,1),R+) is called a positive solution of FBVP (1) if it satisfies (1).

The paper is organized as follows. Firstly, we derive the corresponding Green’s function known as fractional Green’s function and argue its positivity. This is our main work in this paper. Consequently, BVP (1) is reduced to an equivalent Fredholm integral equation. Finally, the existence results of positive solutions are obtained by the use of fixed point index and spectral radii of some related linear integral operators.

Section snippets

Background materials and Green’s function

For the convenience of the reader, we present here the necessary definitions from fractional calculus theory to facilitate analysis of BVP (1). These definitions can be found in the recent literature, see [1], [2], [3], [4]. Let E=C[0,1] be the Banach space with the maximum norm u=maxt[0,1]|u(t)|.

Definition 2.1

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

Definition 2.2

The

Main results and proof

Theorem 3.1

Suppose that conditions (H1)–(H4) are satisfied, then the BVP (1) has at least one positive solution.

Proof

By (H3), there exist r>0 and ε>0 such thatf(t,u)(λ1+ε)u,t[0,1],u[0,r].

Let φ be the positive eigenfunction of L corresponding to λ1, i.e. φ=λ1Lφ. For all uBrP, one has by (6),(Au)(t)(λ1+ε)01G(t,s)h(s)u(s)ds=(λ1+ε)(Lu)(t),t[0,1].

Without loss of generality, we may assume that A has no fixed points on BrP. Now, we show thatu-Auμφ,uBrP,μ0.Otherwise, there exist u1BrP and μ0

Acknowledgements

The authors were supported financially by the National Natural Science Foundation of China (11371221, 11071141), the Foundation for Outstanding Middle-Aged and Young Scientists of Shandong Province (BS2010SF004), a Project of Shandong Province Higher Educational Science and Technology Program (No. J10LA53, J11LA02), the Specialized Research Foundation for the Doctoral Program of Higher Education of China (20123705110001) and the Program for Scientific Research Innovation Team in Colleges and

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