AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 10.1155/2015/290674 290674 Research Article Existence and Uniqueness Results for Fractional Differential Equations with Riemann-Liouville Fractional Integral Boundary Conditions Abbas Mohamed I. 1 Banas Jozef Department of Mathematics and Computer Science Faculty of Science Alexandria University Alexandria 21511 Egypt alexu.edu.eg 2015 29122015 2015 12 09 2015 17 11 2015 29122015 2015 Copyright © 2015 Mohamed I. Abbas. 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.

We prove the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions. The first existence and uniqueness result is based on Banach’s contraction principle. Moreover, other existence results are also obtained by using the Krasnoselskii fixed point theorem. An example is given to illustrate the main results.

1. Introduction

Very recently, fractional differential equations have gained much attention due to extensive applications of these equations in the mathematical modelling of physical, engineering, biological phenomena and viscoelasticity (see, e.g., [1, 2]). There has been a significant development in fractional differential equations. One can see the monographs of Kilbas et al. , Miller and Ross , Lakshmikantham et al. , and Podlubny . In particular, Agarwal et al.  establish sufficient conditions for the existence and uniqueness of solutions for various classes of initial and boundary value problem for fractional differential equations and inclusions involving the Caputo fractional derivative in finite dimensional spaces.

Moreover, the theory of boundary value problems with integral boundary conditions for ordinary differential equations arises in different areas of applied mathematics and physics. For example, heat conduction, chemical engineering, underground water flow, thermoelasticity, population dynamics , and cellular systems  can be reduced to the nonlocal problems with integral boundary conditions. For boundary value problems with integral boundary conditions and comments on their importance, we refer the reader to the papers  and so forth.

In this paper, we study the existence and uniqueness of solution for fractional differential equations with Riemann-Liouville fractional integral boundary conditions of the following form:(1)DqCxt=ft,xt,tJ0,1,1<q2,α1x0+β1Ipxtt=0=γ1,α2x1+β2Ipxtt=1=γ2,0<p<1,where  CDq is the Caputo fractional derivative of order q, Ip is the Riemann-Liouville fractional integral of order p, and αi,βi,γi(i=1,2) are real constants.

Moreover, f:J×EE is a continuous function satisfying some assumptions that will be specified later and E is a Banach space with norm ·.

2. Preliminaries

We now gather some definitions and preliminary facts which will be used throughout this paper. Denote by C(J,E) the Banach space of continuous functions x:JE, with the usual supremum norm (2)x=suptJxt.

We need some basic definitions and properties [6, 14] of fractional calculus which are used in this paper.

Definition 1.

The Riemann-Liouville fractional integral of order q>0 of the function hL1([a,b]) is defined as(3)Iqht=1Γq0thst-s1-qds.

Definition 2.

For a function h defined on the interval [a,b], the Caputo fractional order derivative of h is defined by(4)DqCht=1Γn-q0thnst-sq-n+1ds, where n=[q]+1 and [q] denotes the integer part of q.

From the definition of the Caputo derivative, the following auxiliary results have been established in .

Lemma 3.

Let q>0; then the differential equation  CDtqh(t)=0 has solutions (5)ht=c0+c1t+c2t2++cm-1tm-1,for some ciR, i=0,1,,m-1.

Lemma 4.

Let q>0; then(6)IqDtqCht=ht+c0+c1t+c2t2++cm-1tm-1,for some ciR, i=0,1,,m-1, m=-[-q].

To study the boundary value problem (1), we first consider the associated linear problem and obtain its solution.

Lemma 5.

For a given σC(J,E), the unique mild solution of the fractional boundary value problem,(7)DtqCxt=σt,0<q<1,tJ,α1x0+β1Ipxtt=0=γ1,α2x1+β2Ipxtt=1=γ2,0<p<1,satisfies the following integral equation:(8)xt=γ1α1+ν1tγ2-γ1ν2α1-01α2Γq1-sq-1+β2Γp+q1-sp+q-1σsds+1Γq0tt-sq-1σsds,where(9)ν1=Γp+2α2Γp+2+β2,ν2=Γp+1α2Γp+1+β2.

Proof.

By Lemma 4, we reduce (7) to an equivalent integral equation(10)xt=Iqσt-c0-c1t=1Γq0tt-sq-1σsds-c0-c1t.Then we have(11)Ipxt=1Γp0tt-sp-1xsds=1Γp0tt-sp-11Γq0ss-τq-1στdτ-c0-c1sds=1ΓpΓq0tτtt-sp-1s-τq-1dsστdτ-c0tpΓp+1-c1Γp0tt-sp-1sds=Bp,qΓpΓq0tt-τp+q-1στdτ-c0tpΓp+1-c1Γp0tt-sp-1sds=1Γp+q0tt-τp+q-1στdτ-c0tpΓp+1-c1Γp0tt-sp-1sds,where the integral (12)τtt-sp-1s-τq-1ds=t-τp+q-1011-zp-1zq-1dz=t-τp+q-1Bp,qis evaluated with the help of the substitution s=τ+z(t-τ) and the definition of the beta function B(p,q)=01(t-s)p-1sq-1ds=Γ(p)Γ(q)/Γ(p+q).

Applying the boundary conditions in (7), we have(13)-c0α1+β1×0=γ1,whichimpliesthatc0=-γ1α1,(14)α2011-sq-1Γqσsds-c0-c1+β2011-sp+q-1Γp+qσsds-c0011-sp-1Γpds-c1011-sp-1sΓpds=γ2.Using (13) in (14) together with (9), we obtain (15)c1=ν101α21-sq-1Γq+β21-sp+q-1Γp+qσsds-ν1γ2-γ1ν2α1.Substituting the values of c0 and c1 in (10), we get (8). This completes the proof.

Lemma 6 (Arzelà-Ascoli, [<xref ref-type="bibr" rid="B15">15</xref>]).

If a sequence {xn} in a compact subset of X is uniformly bounded and equicontinuous, then it has a uniformly convergent subsequence.

Lemma 7 (Krasnoselskii, [<xref ref-type="bibr" rid="B15">15</xref>]).

Let Ω be a closed convex and nonempty subset of a Banach space X. Let A and B be two operators such that

Ax+ByΩ, wherever x,yΩ;

A is compact and continuous;

B is a contraction mapping.

Then there exists zΩ such that z=Az+Bz.

3. Main Results

In view of Lemma 5, we define the operator T:C(J,E)C(J,E) as follows: (16)Txt=γ1α1+ν1tγ2-γ1ν2α1-01α2Γq1-sq-1+β2Γp+q1-sp+q-1fs,xsds+1Γq0tt-sq-1fs,xsds,tJ.For the forthcoming analysis we impose suitable conditions on the functions involved in the boundary value problem (1). Namely, we assume the following:

The function f:J×EE is continuous and satisfies the following Lipschitz condition with constant k>0: (17)ft,x-ft,ykx-y,x,yE,tJ.

Let d and r be two nonnegative real numbers such that 0<d<1 and (18)k1+ν1α2Γq+1+ν1β2Γp+q+1<d,M1+ν1α2Γq+1+ν1β2Γp+q+1+ν1γ2+γ1ν2α1+γ1α11-dr,

where MsuptJft,0.

Our first result is based on Banach’s contraction principle.

Theorem 8.

Assume that conditions (H1) and (H2) are satisfied. Then the boundary value problem (1) has a unique solution in C(J,E).

Proof.

We need to prove that the operator T has a fixed point on the set Br={xE:xr}. For xBr, we have (19)Txtγ1α1+ν1γ2+γ1ν2α1+01α2Γq1-sq-1+β2Γp+q1-sp+q-1fs,xs-fs,0ds+01α2Γq1-sq-1+β2Γp+q1-sp+q-1fs,0ds+1Γq0tt-sq-1fs,xs-fs,0ds+1Γq0tt-sq-1fs,0dsγ1α1+ν1γ2+γ1ν2α1+kx01α2Γq1-sq-1+β2Γp+q1-sp+q-1ds+M01α2Γq1-sq-1+β2Γp+q1-sp+q-1ds+kxΓq0tt-sq-1ds+MΓq0tt-sq-1dsγ1α1+ν1γ2+γ1ν2α1+kα2Γq+1+β2Γp+q+1r+Mα2Γq+1+β2Γp+q+1+krΓq+1+MΓp+q+1=k1+ν1α2Γq+1+ν1β2Γp+q+1r+M1+ν1α2Γq+1+ν1β2Γp+q+1+ν1γ2+γ1ν2α1+γ1α1.Therefore, (20)Txdr+1-dr,which implies that TBrBr. Hence, T maps Br into itself.

Now, we will prove that T is a contraction mapping on Br. For x,yBr, we have(21)Txt-Tytν101α2Γq1-sq-1+β2Γp+q1-sp+q-1fs,xs-fs,ysds+1Γq0tt-sq-1fs,xs-fs,ysdsk1+ν1α2Γq+1+ν1β2Γp+q+1x-y.Therefore, (22)Tx-Tydx-y.Hence, the operator T is a contraction. Then T has a unique fixed point which is a solution of the boundary value problem (1).

The following result is based on the Krasnoselskii fixed point theorem. To apply this theorem, we need the following hypothesis:

There exist ϕ(t)L1(J,R+) such that (23)ft,xϕt,t,xJ×E.

Theorem 9.

Assume that conditions (H1) and (H3) are satisfied. Then the boundary value problem (1) has at least a solution in C(J,E), provided that(24)lkν1α2Γq+1+β2Γp+q+1<1.

Proof.

Letting ϕsuptJϕ(t), we fix (25)ργ1α1+ν1γ2+γ1ν2α1+1+ν1α2Γq+1+ν1β2Γp+q+1ϕ. On Bρ={xE:xρ}, we define the two operators R and S as follows: (26)Rxt=γ1α1+ν1tγ2-γ1ν2α1-01α21-sq-1Γq+β21-sp+q-1Γp+qfs,xsds,tJ,Sxt=1Γq0tt-sq-1fs,xsds,tJ.For x,yBρ, we have(27)Rx+Syγ1α1+ν1γ2+γ1ν2α1+01α2Γq1-sq-1+β2Γp+q1-sp+q-1fs,xsds+1Γq0tt-sq-1fs,ysdsγ1α1+ν1γ2+γ1ν2α1+01α2Γq1-sq-1+β2Γp+q1-sp+q-1ϕsds+1Γq0tt-sq-1ϕsdsγ1α1+ν1γ2+γ1ν2α1+ϕ01α2Γq1-sq-1+β2Γp+q1-sp+q-1ds+ϕ1Γq0tt-sq-1dsγ1α1+ν1γ2+γ1ν2α1+ϕα2Γq+1+β2Γp+q+1+ϕΓq+1=γ1α1+ν1γ2+γ1ν2α1+1+ν1α2Γq+1+ν1β2Γp+q+1ϕρ.Hence, Rx+SyBρ.

On the other hand, from (H1) together with (24), it is easy to see that (28)Rx-Rylx-y,and since l<1, then R is a contraction mapping.

Moreover, continuity of f implies that the operator S is continuous. Also, S is uniformly bounded on Bρ and (29)SxϕΓq+1.

Now we prove the compactness of the operator S.

In view of (H1) we define Lsup(t,x)J×Bρf(t,x) and consequently we have for t1,t2J, t1<t2, and xBρ(30)Sxt2-Sxt1=supt,xJ×Bρ1Γq0t2t2-sq-1-t1-sq-1fs,xsds+1Γqt1t2t1-sq-1fs,xsdsLΓqt2q-t1q,which is independent of x and tends to zero as t2-t10. Thus, S is equicontinuous. Hence, by the Arzelà-Ascoli Theorem (Lemma 6), S is compact on Bρ. Thus, all the assumptions of Lemma 7 are satisfied. So the conclusion of the Krasnoselskii fixed point theorem implies that the boundary value problem (1) has at least one solution on J. This completes the proof of Theorem 9.

4. An Example

Consider the following fractional boundary value problem:(31)Dt4/3Cxt=1et+2xtxt+1,t0,1,x0+I1/3xtt=0=12,12x1+13I1/3xtt=1=2,where q=4/3, p=1/3, α1=1, α2=1/2, β1=1, β2=1/3, γ1=1/2, γ2=2, and ν11.282114, ν21.145105.

Here, f:[0,1]×RR is given by (32)ft,x=1et+2xx+1,t,x0,1×R.Then we have (33)ft,x-ft,y1e2x-y. Therefore, (H1) is satisfied with k=1/e20.135335.

Moreover, (34)ft,x=1et+2xx+11et+2xx+11et+2. Therefore, ϕt=1/et+2, t[0,1], and ϕ=1/e20.135335.

Also, we get M=suptJ|f(t,0)|=1/e20.135335.

Finally, simple calculations give(35)k1+ν1α2Γq+1+ν1β2Γp+q+10.224974<1,l=kν1α2Γq+1+β2Γp+q+10.130704<1.Clearly, all the assumptions of Theorems 8 and 9 are satisfied. So there exists at least one solution of the boundary value problem (31) on [0,1].

5. Conclusion

In this paper, we study the existence and uniqueness for fractional order differential equations with Riemann-Liouville integral boundary conditions (1) in Banach spaces. Existence and uniqueness results of solutions are established by virtue of fractional calculus, Banach’s contraction principle, and the Krasnoselskii fixed point theorem. As applications, an example is presented to illustrate the main results.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Bagley R. L. Torvik P. J. A theoretical basis for the application of fractional calculus to viscoelasticity Journal of Rheology 1983 27 3 201 210 10.1122/1.549724 2-s2.0-0020765202 Diethelm K. Freed A. D. Keil F. Mackens W. Voss H. On the solution of nonlinear fractional order differential equations used in the modelling of viscoplasticity Scientific Computing in Chemical Engineering II-Computational Fluid Dynamics and Molecular Properties 1999 Heidelberg, Germany Springer 217 224 Kilbas A. A. Srivastava H. M. Trujillo J. J. Theory and Applications of Fractional Differential Equations 2006 204 Amsterdam, The Netherlands Elsevier North-Holland Mathematics Studies Miller K. S. Ross B. An Introduction to the Fractional Calculus and Differential Equations 1993 New York, NY, USA John Wiley & Sons MR1219954 Lakshmikantham V. Leela S. Devi J. V. Theory of Fractional Dynamic Systems 2009 Cambridge Scientific Publishers Podlubny I. Fractional Differential Equations 1999 San Diego, Calif, USA Academic Press MR1658022 Agarwal R. P. Benchohra M. Hamani S. A survey on existence results for boundary value problems of nonlinear fractional differential equations and inclusions Acta Applicandae Mathematicae 2010 109 3 973 1033 10.1007/s10440-008-9356-6 MR2596185 2-s2.0-77949264980 Blayneh K. W. Analysis of age structured host-parasitoid model Far East Journal of Dynamical Systems 2002 4 125 145 Zbl1059.92050 Adomian G. Adomian G. E. Cellular systems and aging models Computers & Mathematics with Applications 1985 11 1–3 283 291 10.1016/0898-1221(85)90153-1 MR787443 2-s2.0-46549103408 Ahmad B. Ntouyas S. K. Fractional differential inclusions with fractional separated boundary conditions Fractional Calculus and Applied Analysis 2012 15 3 362 382 10.2478/s13540-012-0027-y MR2944105 2-s2.0-84869162803 Ahmad B. Nieto J. J. Existence results for nonlinear boundary value problems of fractional integrodifferential equations with integral boundary conditions Boundary Value Problems 2009 2009 11 708576 10.1155/2009/708576 MR2525567 Ahmad B. Nieto J. J. Riemann-Liouville fractional integro-differential equations with frac tional nonlocal integral boundary conditions Boundary Value Problems 2011 2011, article 36 MR2851530 Benchohra M. Graef J. R. Hamani S. Existence results for boundary value problems with non-linear fractional differential equations Applicable Analysis 2008 87 7 851 863 10.1080/00036810802307579 MR2458962 Samko S. G. Kilbas A. A. Marichev O. I. Fractional Integrals and Derivatives: Theory and Applications 1993 Englewood Cliffs, NJ, USA Gordon and Breach Science Publishers Smart D. R. Fixed Point Theorems 1980 Cambridge University Press MR0467717