Solvability for a coupled system of fractional differential equations at resonance

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Abstract

By using the coincidence degree theory due to Mawhin and constructing suitable operators, we study the existence of solutions for a coupled system of fractional differential equations at resonance. An example is given to illustrate our result.

Introduction

The existence of solutions for integer order differential equations at resonance has been studied by many authors (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and references cited therein). Since the extensive applicability of fractional differential equations (see [15], [16]), recently, more and more authors pay their close attention to the boundary value problems of fractional differential equations (see [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30]). In papers [27], [28], the existence of solutions to coupled systems of fractional differential equations at nonresonance has been given. In papers [29], [30], the solvability of fractional differential equations at resonance has been investigated. To the best of our knowledge, the existence of solutions to coupled systems of fractional differential equations at resonance has not been studied. We will fill this gap in the literature. In this paper, we discuss the existence of solutions to a coupled system of fractional differential equations at resonance: {D0+αu(t)=f(t,u(t),v(t)),u(0)=0,D0+γu(t)|t=1=i=1naiD0+γu(t)|t=ξi,D0+βv(t)=g(t,u(t),v(t)),v(0)=0,D0+δv(t)|t=1=i=1mbiD0+δv(t)|t=ηi, where t[0,1],1<α,β2,0<γα1,0<δβ1, 0<ξ1<ξ2<<ξn<1, 0<η1<η2<<ηm<1.

In this paper, we will always suppose that the following conditions hold.

(H1)i=1naiξiαγ1=1,i=1mbiηiβδ1=1,i=1naiξiαγ1,i=1mbiηiβδ1.

(H2)f,g:[0,1]×R×RR satisfy Carathéodory conditions, i.e., f(,u) and g(,u) are measurable for each fixed uR×R, f(t,) and g(t,) are continuous for a.e. t[0,1], and for each r>0, there exist Φr,ΨrL[0,1] such that |f(t,x,y)|Φr(t), |g(t,x,y)|Ψr(t) for all |x|,|y|[0,r], a.e. t[0,1].

Section snippets

Preliminaries

For abbreviation, we write D0+γu(ξ) instead of D0+γu(t)|t=ξ.

For convenience, we introduce some notations and a theorem. For more details, see [31].

Let X and Y be real Banach spaces and L:dom(L)XY be a Fredholm operator with index zero, P:XX,Q:YY be projectors such that ImP=KerL,KerQ=ImL,X=KerLKerP,Y=ImLImQ. It follows that L|domLKerP:domLKerPImL is invertible. We denote the inverse by KP.

If Ω is an open bounded subset of X,domLΩ¯, the map N:XY will be called L-compact on Ω¯ if QN(Ω

Main results

Define operators T1,T2:L[0,1]R as follows: T1u=01(1s)αγ1u(s)dsi=1nai0ξi(ξis)αγ1u(s)ds,T2u=01(1s)βδ1u(s)dsi=1mbi0ηi(ηis)βδ1u(s)ds.

In order to obtain our main results, we first present and prove the following lemmas.

Lemma 3.1

If (H1) holds, then L:domLXY is a Fredholm operator of index zero, the linear continuous projectors P:XX and Q:YY can be defined asP(u(t),v(t))=(D0+α1u(0)Γ(α)tα1,D0+β1v(0)Γ(β)tβ1),Q(x,y)=(Q1x,Q2y)(αγ1i=1naiξiαγT1x,βδ1i=1mbiηiβδT2y),respectively,

Example

Let us consider the following coupled system of fractional differential equations at resonance {D0+32u(t)=f(t,u(t),v(t)),u(0)=0,D0+12u(1)=D0+12u(14),a.e.t[0,1],D0+54v(t)=g(t,u(t),v(t)),v(0)=0,D0+14v(1)=D0+14v(116),a.e.t[0,1], where f(t,x,y)=t2sin(xy)+14e(1t)x+[Γ(32)3401(1s)12e(1s)ds]3sintsiny,g(t,x,y)=tcos(xy)+[Γ(54)3401(1s)14e(1s)ds]3costsinx+14e(1t)y.

Corresponding to the problem (1.1), we have that m=n=1,α=32,β=54,γ=12,δ=14,ξ1=14,η1=116, a1=b1=1. Take ψ1(t)=t2,ψ2(t)=t,h1(t)=[Γ(

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This work is supported by the Natural Science Foundation of China (11171088), the Natural Science Foundation of Hebei Province (A2009000664), the Doctoral Program Foundation of Hebei University of Science and Technology (QD201020) and the Foundation of Hebei University of Science and Technology (XL201136).

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