IJDE International Journal of Differential Equations 1687-9651 1687-9643 Hindawi 10.1155/2017/4683581 4683581 Research Article Existence and Uniqueness of Solutions for BVP of Nonlinear Fractional Differential Equation Su Cheng-Min 1 http://orcid.org/0000-0001-6533-3963 Sun Jian-Ping 1 Zhao Ya-Hong 1 Rossi Julio D. Department of Applied Mathematics Lanzhou University of Technology Lanzhou 730050 China lut.cn 2017 29 01 2017 2017 11 10 2016 14 12 2016 15 12 2016 29 01 2017 2017 Copyright © 2017 Cheng-Min Su et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

In this paper, we study the existence and uniqueness of solutions for the following boundary value problem of nonlinear fractional differential equation: D0+qCut=ft,ut,  t0,1, u0=u0=0,D0+σ1Cu1=λI0+σ2u1, where 2<q<3, 0<σ11, σ2>0, and λΓ2+σ2/Γ2-σ1. The main tools used are nonlinear alternative of Leray-Schauder type and Banach contraction principle.

National Natural Science Foundation of China 11661049
1. Introduction

Fractional calculus has wide applications in many fields of science and engineering, for example, fluid flow, biosciences, rheology, electrical networks, chemical physics, control theory of dynamical systems, and optics and signal processing .

Recently, nonlinear fractional differential equations have been discussed under the following boundary conditions (BCs for short):

Integer derivative BCs:

u0=u1=0,

u0+u0=0, u1+u1=0,

u0=u1=u0=0,

u0=0, u0+u0=0, u1+u1=0,

u0=u0, u0=u0, uT=uT,

u0=u1=u0==un-10=0;

see papers , respectively.

Integer derivative and integral BCs:

αu0-βu0=01gsusds, γu1+δu1=01hsusds,

u0=u0=u0=0, u1=λ0ηusds;

see papers [8, 9], respectively.

Integer and fractional derivative BCs:

u0=D0+σ1Cu1=0, u0=D0+σ2Cu1=0,

u0=u0=0, u1=D0+σCu1,

u0=u0=0, u1=D0+σCu1,

u0=0, D0+βu1=i=1m-2ξiD0+βuηi,

u0=0, u1+D0+βu1=kuξ+lD0+βuη,

u0=u0==un-20=0, D0+αu1=0;

see papers , respectively.

Integer derivative and fractional integral BCs:

u0=αI0+puη,

u0=0, u1=I0+σu1;

see papers [17, 18], respectively.

Besides, there are some other BCs involved in fractional differential equations, such as nonlinear BCs; refer to [19, 20].

Motivated greatly by the above-mentioned works, in this paper, we study the following boundary value problem (BVP for short) of nonlinear fractional differential equation with fractional integral BCs as well as integer and fractional derivative(1)D0+qCut=ft,ut,t0,1,u0=u0=0,D0+σ1Cu1=λI0+σ2u1,where D0+qC and D0+σ1C denote the standard Caputo fractional derivatives and I0+σ2 denotes the standard Riemann-Liouville fractional integral. Throughout this paper, we always assume that 2<q<3, 0<σ11, σ2>0, λΓ(2+σ2)/Γ(2-σ1), and f:[0,1]×RR is continuous.

In order to prove our main results, the following well-known fixed point theorems are needed.

Theorem 1 (nonlinear alternative of Leray-Schauder type [<xref ref-type="bibr" rid="B21">21</xref>]).

Let B be a Banach space with EB closed and convex. Assume Ω is a relatively open subset of E with θΩ and T:Ω¯E is a continuous and compact map. Then either

T has a fixed point in Ω¯ or

there exists uΩ and η(0,1) such that u=ηTu.

Theorem 2 (Banach contraction principle [<xref ref-type="bibr" rid="B22">22</xref>]).

Let (X,d) be a complete metric space and T:XX be contractive. Then T has a unique fixed point in X.

2. Preliminaries

In this section, we always assume that N={1,2,3,}, α,β>0, and [α] denotes the integer part of α. Now, for the convenience of the reader, we give the definitions of the Riemann-Liouville fractional integrals and fractional derivatives and the Caputo fractional derivatives on a finite interval of the real line, which may be found in .

Definition 3.

The Riemann-Liouville fractional integrals I0+αu and I1-αu of order α on [0,1] are defined by (2)I0+αut1Γα0tusdst-s1-α,I1-αut1Γαt1usdss-t1-α,respectively.

Definition 4.

The Riemann-Liouville fractional derivatives D0+αu and D1-αu of order α on [0,1] are defined by (3)D0+αutddtnI0+n-αut=1Γn-αddtn0tusdst-sα-n+1,D1-αut-ddtnI1-n-αut=1Γn-α-ddtnt1usdss-tα-n+1,respectively, where n=[α]+1.

Definition 5.

Let D0+α[u(s)](t)(D0+αu)(t) and D1-α[u(s)](t)(D1-αu)(t) be the Riemann-Liouville fractional derivatives of order α. Then the Caputo fractional derivatives   CD0+αu and   CD1-αu of order α on [0,1] are defined by (4)D0+αCutD0+αus-k=0n-1uk0k!skt,D1-αCutD1-αus-k=0n-1uk1k!1-skt, respectively, where(5)n=α+1,αN,α,αN.

Lemma 6 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

If α+β>1, then the equation (I0+αI0+βu)(t)=(I0+α+βu)(t), t[0,1], is satisfied for uL10,1.

Lemma 7 (see [<xref ref-type="bibr" rid="B23">23</xref>]).

Let β>α. Then the equation D0+αCI0+βut=I0+β-αut, t[0,1], is satisfied for uC0,1.

Lemma 8 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let n be given by (5). Then the following relations hold:

For k{0,1,2,,n-1}, D0+αCtk=0.

If β>n, then D0+αCtβ-1=Γβ/Γβ-αtβ-α-1.

Lemma 9 (see [<xref ref-type="bibr" rid="B1">1</xref>]).

Let n be given by (5) and uCn0,1. Then (6)I0+αD0+αCut=ut+c0+c1t+c2t2++cn-1tn-1,where ciR, i=0,1,,n-1.

For any xL10,1, we define (7)xL1=01xtdt.

Lemma 10.

Let uL1[0,1] be nonnegative. Then I0+α+1utI0+αuL1, t0,1.

Proof.

For any t0,1, we have (8)I0+α+1ut=1Γα+10tust-s-αds=1αΓα0tust-sαds=1Γα0tusstr-sα-1drds=1Γα0t0rusr-s1-αdsdr011Γα0rusr-s1-αdsdr=01I0+αurdr=I0+αuL1.

3. Main Results

In the remainder of this paper, for any nonnegative function gL1[0,1], we denote (9)Mg=I0+q-1gL1+Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2λI0+q+σ2g1+I0+q-σ1g1and for any yC[0,1], we use the norm(10)y=maxt0,1yt.

Lemma 11.

Let yC[0,1] be a given function. Then the BVP(11)D0+qCut=yt,t0,1,u0=u0=0,D0+σ1Cu1=λI0+σ2u1has a unique solution (12)ut=01Gt,sysds,t0,1,where(13)Gt,s=-Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλ1-sq+σ2-1Γq+σ2-1-sq-σ1-1Γq-σ1+t-sq-1Γq,0st1,0,0ts1.

Proof.

It follows from the equation in (11) and Lemma 9 that(14)ut=I0+qyt-c0-c1t-c2t2,t0,1.So, (15)ut=I0+q-1yt-c1-2c2t,t0,1,(16)ut=I0+q-2yt-2c2,t0,1.In view of (14), (16), and the BCs u(0)=u0=0, we get (17)c0=c2=0,and so, (18)ut=I0+qyt-c1t,t0,1. Then, by using Lemmas 6, 7, and 8, we may obtain(19)D0+σ1Cut=I0+q-σ1yt-c1Γ2Γ2-σ1t1-σ1,t0,1,I0+σ2ut=I0+q+σ2yt-c1Γ2Γ2+σ2t1+σ2,t0,1,which together with the BC (CD0+σ1u)1=λI0+σ2u1 implies that (20)c1=Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2λI0+q+σ2y1-I0+q-σ1y1.Therefore, the BVP (11) has a unique solution (21)ut=I0+qyt-Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2λI0+q+σ2y1-I0+q-σ1y1t=0t-Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλ1-sq+σ2-1Γq+σ2-1-sq-σ1-1Γq-σ1+t-sq-1Γqysds+t1-Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλ1-sq+σ2-1Γq+σ2-1-sq-σ1-1Γq-σ1ysds=01Gt,sysds,t0,1.

Lemma 12.

Let gL1[0,1] be nonnegative. Then (22)01Gt,sgsdsMg,t0,1.

Proof.

In view of Lemma 10, we have (23)01Gt,sgsds=0tGt,sgsds+t1Gt,sgsds0tΓ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλ1-sq+σ2-1Γq+σ2+1-sq-σ1-1Γq-σ1+t-sq-1Γqgsds+t1Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλ1-sq+σ2-1Γq+σ2+1-sq-σ1-1Γq-σ1gsds=1Γq0tgst-s1-qds+Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλΓq+σ201gs1-s1-q-σ2ds+1Γq-σ101gs1-s1-q+σ1ds=I0+qgt+Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2tλI0+q+σ2g1+I0+q-σ1g1I0+q-1gL1+Γ2+σ2Γ2-σ1λΓ2-σ1-Γ2+σ2λI0+q+σ2g1+I0+q-σ1g1=Mg,t0,1.

Now, we define an operator T:C[0,1]C[0,1] by (24)Tut=01Gt,sfs,usds,t0,1.Obviously, u is a solution of the BVP (1) if and only if u is a fixed point of T.

Theorem 13.

Assume that f(t,0)0, t(0,1), and there exist nonnegative functions g1,g2L1[0,1], nonnegative increasing continuous function ϕ defined on [0,+), and r>0 such that(25)ft,xg1t+g2tϕx,t,x0,1×R,(26)Mg1+ϕrMg2<r.Then the BVP (1) has one nontrivial solution.

Proof.

Let Ω=uC[0,1]:u<r. Since G(t,s) and f(t,x) are continuous on 0,1×0,1 and 0,1×R, respectively, we may denote(27)L=maxt,s0,1×0,1Gt,s,(28)H=maxt,x0,1×-r,rft,x.

First, we prove that T:Ω¯C[0,1] is continuous. Suppose that un (n=1,2,), u0Ω¯, and un-u00 (n). Then for any n and s[0,1], we have unsr. This together with (27) and (28) implies that, for any n and t[0,1], (29)Gt,sfs,unsLH,s0,1.By applying Lebesgue dominated convergence theorem, we get (30)limnTunt=limn01Gt,sfs,unsds=01Gt,sfs,u0sds=Tu0t,t0,1,which indicates that T:Ω¯C[0,1] is continuous.

Next, we show that T:Ω¯C[0,1] is compact. Assume that K is a subset of Ω¯. Then for any uK, we have(31)usr,s0,1.In what follows, we will prove that T(K) is relatively compact. On the one hand, for any yT(K), there exists uK such that y=Tu, and so, it follows from (27), (28), and (31) that (32)yt=Tut=01Gt,sfs,usds01Gt,sfs,usdsLH,t0,1,which shows that T(K) is uniformly bounded. On the other hand, for any ε>0, since G(t,s) is uniformly continuous on 0,1×0,1, there exists δ>0 such that, for any t1,t20,1 with t1-t2<δ,(33)Gt1,s-Gt2,s<εH,s0,1.For any yT(K), there exists uK such that y=Tu, and so, for any t1,t20,1 with t1-t2<δ, it follows from (28), (31), and (33) that (34)yt1-yt2=Tut1-Tut2=01Gt1,s-Gt2,sfs,usds01Gt1,s-Gt2,sfs,usdsH01Gt1,s-Gt2,sds<ε,which indicates that T(K) is equicontinuous. By Arzela-Ascoli theorem, we know that T(K) is relatively compact. Therefore, T:Ω¯C[0,1] is compact.

Now, we will prove that (a) of Theorem 1 is fulfilled. Suppose on the contrary that (b) of Theorem 1 is satisfied; that is, there exists uΩ and η(0,1) such that u=ηTu. Then, in view of (25), (26), and Lemma 12, we have (35)ut=ηTutTut=01Gt,sfs,usds01Gt,sfs,usds01Gt,sg1s+g2sϕusds01Gt,sg1sds+ϕr01Gt,sg2sdsMg1+ϕrMg2<r,t0,1,which shows that (36)u<r.This contradicts the fact uΩ.

So, it follows from Theorem 1 that T has a fixed point u, which is a desired solution of the BVP (1). At the same time, since f(t,0)0,t(0,1), we know that the zero function is not a solution of the BVP (1). Therefore, u is a nontrivial solution of the BVP (1).

Theorem 14.

Assume that there exists a nonnegative function g3L1[0,1] such that(37)ft,x-ft,yg3tx-y,t0,1,x,yR,(38)Mg3<1.Then the BVP (1) has a unique solution.

Proof.

For any u,vC[0,1], in view of (37) and Lemma 12, we have (39)Tut-Tvt=01Gt,sfs,us-fs,vsds01Gt,sfs,us-fs,vsds01Gt,sg3sus-vsdsu-v01Gt,sg3sdsMg3u-v,t0,1.This indicates that (40)Tu-TvMg3u-v,which together with (38) implies that T is contractive. So, it follows from Theorem 2 that T has a unique fixed point, and so, the BVP (1) has a unique solution.

Example 15.

We consider the BVP(41)D0+5/2Cut=t-t2ut,t0,1,u0=u0=0,D0+1/2Cu1=12I0+3/2u1.

Let f(t,x)=t-t/2x, t,x[0,1]×R. Then f:[0,1]×RR is continuous and ft,00, t0,1.

If we choose g1(t)=t, g2(t)=t/2, t[0,1], and ϕ(y)=y, y[0,+), then it is easy to verify that (25) is satisfied.

Since q=5/2, σ1=1/2, σ2=3/2, and λ=1/2, a direct calculation shows that (42)Γ2+σ2Γ2-σ1=154,Mg1=6656+4305π43680π,Mg2=6656+4305π87360π.If we choose r=1, then (26) is fulfilled.

Therefore, it follows from Theorem 13 that the BVP (41) has one nontrivial solution.

Example 16.

We consider the BVP(43)D0+5/2Cut=tπutarctanut-12ln1+u2t,t0,1,u0=u0=0,D0+1/2Cu1=12I0+3/2u1.

Let f(t,x)=t/πxarctanx-1/2ln1+x2, t,x0,1×R. Then f:[0,1]×RR is continuous.

If we choose g3t=t/2,t[0,1], then we may assert that (37) is satisfied. In fact, for any t[0,1], if x=y, then (37) is obvious. When xy, we may suppose that x<y. In this case, by Lagrange mean value theorem, there exists ξ(x,y) such that, for any t[0,1], (44)ft,x-ft,y=tπxarctanx-12ln1+x2-yarctany+12ln1+y2=tπarctanξx-yg3tx-y;that is, (37) is satisfied.

On the other hand, in view of Mg3=Mg2=6656+4305π/87360π, we know that (38) is fulfilled.

Therefore, it follows from Theorem 14 that the BVP (43) has a unique solution.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This paper is supported by the National Natural Science Foundation of China (11661049).

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