AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 592964 10.1155/2013/592964 592964 Research Article The Existence and Uniqueness of Solutions for a Class of Nonlinear Fractional Differential Equations with Infinite Delay 0000-0002-5342-1322 Babakhani Azizollah 1 0000-0002-0286-7244 Baleanu Dumitru 2, 3, 4 Agarwal Ravi P. 5, 6 Ahmad Bashir 1 Department of Mathematics Faculty of Basic Science Babol University of Technology Babol 47148-71167 Iran nit.ac.ir 2 Department of Mathematics and Computer Science Cankaya University Turkey cankaya.edu.tr 3 Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz University P.O. Box 80204 Jeddah 21589 Saudi Arabia kau.edu.sa 4 Institute of Space Sciences P.O. Box MG-23 r 76900 Magurele-Bucharest Romania spacescience.ro 5 Department of Mathematics Texas A & M University-Kingsville 700 University Boulevard Kingsville USA tamu.edu 6 Department of Mathematics King Abdulaziz University P.O. Box 80204 Jeddah 21589 Saudi Arabia kau.edu.sa 2013 19 3 2013 2013 28 10 2012 02 02 2013 2013 Copyright © 2013 Azizollah Babakhani 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.

We prove the existence and uniqueness of solutions for two classes of infinite delay nonlinear fractional order differential equations involving Riemann-Liouville fractional derivatives. The analysis is based on the alternative of the Leray-Schauder fixed-point theorem, the Banach fixed-point theorem, and the Arzela-Ascoli theorem in ={y:(,b]:y|(,0]} such that y|[0,b] is continuous and is a phase space.

1. Introduction

Fractional derivatives and integrals have been vastly used in different fields, facing a huge development especially during the last few decades (see, e.g.,  and the references therein). The approaches based on fractional calculus establish models of engineering systems better than the ordinary derivatives approaches .

In particular, fractional differential equations as an important research branch of fractional calculus attracted much more attention (see, e.g.,  and the references therein). Also varieties of schemes for numerical solutions of fractional differential equations are reported (see, e.g., [6, 2123] and the references therein). We notice that some investigations have been done on the existence and uniqueness of solutions for fractional differential equations with delay (see, e.g., [24, 25] and the references therein).

Having all the aforementioned facts in mind, in this paper we study the existence and uniqueness of solutions for a class of delayed fractional differential equations, namely, (1)(𝒟)y(t)=f(t,yt),tJ=[0,b],y(t)=ϕ(t),t(-,0], where (𝒟)=D0+α-tnD0+β,0<β<α<1,n is a positive integer, f:J× is a given function satisfying some assumptions that will be specified later, ϕ with ϕ(0)=0, and   is called a phase space that will be defined later. D0+α and D0+β are the standard Riemann-Liouville fractional derivatives. yt, which is an element , is defined as any function y on (-,b] as follows: (2)yt(s)=y(t+s),s(-,0],tJ.

Here yt(·) represents the preoperational state from time - up to time t. We also consider the following nonlinear fractional differential equation: (3)(𝒟){y(t)-g(t,yt)}=f(t,yt),tJ,y(t)=ϕ(t),t(-,0], where α, β, f, ϕ, and (𝒟) are as (1) and g:J× is a given function which satisfies g(0,ϕ)=0.

The notion of the phase space plays an important role in the study of both qualitative and quantitative theories for functional differential equations. A common choice is a seminormed space satisfying suitable axioms, which was introduced by Hale and Kato .

Our approach is based on the Banach fixed-point theorem and on the nonlinear alternative of Leray-Schauder type [27, 28]. The organization of the paper is as follows.

In Section 2, we present some basic mathematical tools used in the paper. The main results are presented in Section 3. Section 4 is dedicated to our conclusions.

2. Preliminaries

In this section, we present some basic notations and properties which are used throughout this paper. First of all, we will explain the phase space introduced by Hale and Kato . Let 0=(-,0],0=[0,+),=(-,+), and let E be a Banach space with norm |·|E. Further, let be a linear space of functions mapping - into E with seminorm |·| having the following axioms,

If y:(-,σ+b)E,b>0 is continuous on [σ,σ+b) and yσ, then yt and yt are continuous for any t[σ,σ+b).

There exist functions k(t)>0 and m(t)0 with the following properties.  (i) k(t) is continuous for t0.  (ii) m(t) is locally bounded for t0.  (iii) For every function, y has the properties of (B1) and t[σ,σ+b), holds that |yt|k(t-σ)sup{|y(s)|E:σst}+m(t-σ)|yσ|.

There exists a positive constant L such that |ϕ(0)|EL  |ϕ| for all ϕ.

The quotient space ^:=/|·| is a Banach space.

We notice that in this paper, we select σ=0 and E=; thus (iii) can be converted to |yt|k(t)sup{|y(s)|E:0st}+m(t)|y0|,  for all t[0,b).

See  for examples of the phase space satisfying all axioms (B1)(B4).

Let +=(0,+) and C0(+) be the space of all continuous real function on +. Consider also the space C0()0 of all continuous real functions on 0 which later identifies with the class of all fC0(+) such that limt0+f(t)=f(0+). By C(J,), we denote the Banach space of all continuous functions from J into with the norm y:=sup{|y(t)|:tJ}, where |·| is a suitable complete norm on .

The most common notation for αth order derivative of a real-valued function y(t), which is defined in an interval denoted by (a,b), is Daαy(t). Here, the negative value of α corresponds to the fractional integral.

Definition 1.

For a function y defined on an interval [a,b], the Riemann-Liouville fractional integral of y of order α>0 is defined by [1, 6] (4)Ia+αy(t)=1Γ(α)at(t-s)α-1y(s)ds,t>a, and the Riemann-Liouville fractional derivative of y(t) of order α>0 reads as (5)Da+αy(t)=dndtn{Ia+n-αy(t)},  n-1<αn. Also, we denote Da+αy(t) as Daαy(t) and Ia+αy(t) as Iaαy(t). Further, D0+αy(t) and I0+αy(t) are referred to as Dαy(t) and Iαy(t), respectively. If the fractional derivative Daαy(t) is integrable, then we have [4, page 71] (6)Iaα(Daβy(t))=Iaα-βy(t)-[Ia1-βy(t)]t=a(t-a)α-1Γ(α),0<βα<1. If y is continuous on [a,b], then Daαy(t) is integrable, I1-βy(t)|t=a=0, and (7)Iaα(Daβy(t))=Iaα-βy(t),0<βα<1.

Proposition 2.

Let y be continuous on [0,b] and n a nonnegative integer, then (8)(i)Iα(tny(t))=k=0n(-αk)[Dktn][Iα+ky(t)]=k=0n(-αk)n!tn-k(n-k)!Iα+ky(t),(9)(ii)Iα(tnDβy(t))=k=0n(-αk)n!tn-k(n-k)!Iα-β+ky(t), where (10)(-αk)=(-1)kΓ(α+1)k!Γ(α)=(-1)k(α+k-1k)=Γ(1-α)Γ(k+1)Γ(1-α-k).

Proof.

( i ) can be found in [6, page 53], and (ii) is an immediate consequence of (7), and (i).

Lemma 3 (see [<xref ref-type="bibr" rid="B29">29</xref>]).

Let v:[0,b][0,) be a real function and w(·) a nonnegative, locally integrable function on [0,b]. If there exist positive constants a and α(0,1) such that v(t)w(t)+a0t(t-s)-αv(s)ds, then there exists a constant K=K(α) such that v(t)w(t)+Ka0tw(s)(t-s)-αds, for all t[0,b].

In this paper we use the alternative Leray-Schauder’s theorem and Banach’s contraction principle for getting the main results. These theorems can be found in [27, 28].

3. Existence and Uniqueness

In this section, we prove the existence results for (1) and (3) by using the alternative of Leray-Schauder’s theorem. Further, our results for the unique solution is based on the Banach contraction principle. Let us start by defining what we mean by a solution of (1). Let the space (11)Ω={y:(-,b]:y(-,0]  and  y|[0,b]  is  continuous}. A function yΩ is said to be a solution of (1) if y satisfies (1).

For the existence results on (1), we need the following lemma.

Lemma 4.

Equation (1) is equivalent to the Volterra integral equation (12)y(t)=k=0n(-αk)n!tn-k(n-k)!Iα-β+ky(t)+Iαf(t,yt),tJ.

Proof.

The proof is an immediate consequence of Proposition 2.

To study the existence and uniqueness of solutions for (1), we transform (1) into a fixed-point problem. Consider the operator P:ΩΩ defined by (13)Py(t)={(I)y(t)+Iαf(t,yt),t[0,b],ϕ(t),t(-,0], where, (14)(I)=k=0n(-αk)n!tn-k(n-k)!Iα-β+k. Let x(·):(-,b] be the function defined as (15)x(t)={0,if  t[0,b],ϕ(t),if  t(-,0]. Then, we get x0=ϕ. For each zC([0,b],) with z(0)=0, we denote by z- the function defined as follows: (16)z-(t)={z(t),if  t[0,b],0,if  t(-,0]. If y(·) satisfies the integral equation y(t)=(I)y(t)+Iαf(t,yt), then we can decompose y(·) as y(t)=z-(t)+x(t), -<tb, which implies yt=z-t+xt for every 0tb, and the function z(·) satisfies (17)z(t)=(I)z(t)+Iαf(t,z-t+xt), set C0={zC([0,b],):z(0)=0}, and let ·b be the seminorm in C0 defined by zb=z0+sup{|z(t)|:0tb}=sup{|z(t)|:0tb},zC0. C0 is a Banach space with norm ·b. Let the operator F:C0C0 be defined by (18)Fz(t)=(I)z(t)+Iαf(t,z-t+xt), where t[0,b]. The operator P has a fixed point equivalent to F that has a fixed point too.

Theorem 5.

Assume that f is a continuous function, and there exist p,qC(J,+) such that |f(t,u)|p(t)+q(t)u,tJ,u. Then, (1) has at least one solution on (-,b].

Proof.

It is enough to show that the operator F:C0C0 defined as (18) satisfies the following: (i) F is continuous, (ii) F maps bounded sets into bounded sets in C0, (iii) F maps bounded sets into equicontinuous sets of C0, and (iv) F is completely continuous.

Let {zn} converges to z in C0, then (19)Fzn(t)-Fz(t)k=0n|(-αk)|n!tn-k(n-k)!Iα-β+k|zn(t)-z(t)|+Iα|f(t,(z-n)t+xt)-f(t,z-t+xt)|k=0n|(-αk)|n!  bn-kzn-z(n-k)!Γ(α-β+k+1)+bαf(t,(z-n)t+xt)-f(t,z-t+xt)Γ(α+1). Hence, Fzn(t)-Fz(t)0 as znz, and thus f is continuous.

For any λ>0, let λ={zC0:zbλ} be a bounded set. We show that there exists a positive constant μ such that Fzμ. Let zλ, since f is a continuous function, we have for each t[0,b], (20)|Fz(t)|k=0n|(-αk)|n!tn-k(n-k)!Γ(α-β+k)×0b(t-s)α-β+k-1z(s)+1Γ(α)0t(t-s)α-1f(s,z-s+xs)dsk=0n|(-αk)|n!bn+α-β(n-k)!Γ(α-β+k+1)zb+1Γ(α)×0t(t-s)α-1[p(s)+q(s)z-s+xs]dsk=0n|(-αk)|n!bn+α-β(n-k)!Γ(α-β+k+1)zb+bαpΓ(α+1)+bαqΓ(α+1){z-s+xs}k=0n|(-αk)|n!bn+α-β(n-k)!Γ(α-β+k+1)zb+bαpΓ(α+1)+kbλ+mbϕ:=μ, where mb=sup{|m(t)|:t[0,b]}, and kb=sup{|k(t)|:t[0,b]}. Hence, we obtain Fzμ.

Let t1,t2[0,b] and t1<t2. Let λ be a bounded set of C0 as in (ii) and zλ, then given ϵ>0 choose (21)δ=min{12Λ1ϵ1/α,12(n+1)Λ2ϵ1/(α-β+k):k=0,1,,n12  (n+1)Λ2}, where (22)Λ2=Λ1=2p+ΛqΓ(α+1),Λ2=k=0n2|(-αnk)|k!bn-kzb(n-k)!Γ(α-β+k+1), and Λ=kbλ+mbϕ. If |t2-t1|<δ, then (23)|Fz(t2)-Fz(t1)|k=0n|(-αnk)|k!bn-k(n-k)!Γ(α-β+k)zb×|0t1{(t2-s)α-β+k-1-(t1-s)α-β+k-1}ds  +t1t2(t2-s)α-β+k-1ds|+1Γ(α)|0t1{(t2-s)α-1-(t1-s)α-1}f(s,z-s+xx)ds+t1t2(t2-s)α-1f(s,z-s+xx)ds|k=0n2|(-αnk)|k!bn-k(n-k)!Γ(α-β+k+1)zb(t2-t1)α-β+k+p+ΛqΓ(α+1){0t1{(t2-s)α-1-(t1-s)α-1}ds+p+  ΛqΓ(α+1)+t1t2(t2-s)α-1ds}k=0n2|(-αnk)|k!bn-k(n-k)!Γ(α-β+k+1)zb(t2-t1)α-β+k+2p+ΛqΓ(α+1)(t2-t1)α=Λ2δα-β+k+Λ1δα<ϵ2+ϵ2=ϵ, where z-s+xsz-s+xskbλ+mbϕ:=Λ. Hence, F(λ) is equicontinuous.

It is an immediate consequence from (i)–(iii), together with the Arzela-Ascoli theorem.

We show in the following that there exists an open set UC0 with zγF(z) for γ(0,1) and zU. Let zC0 and z=γF(z) for some 0<γ<1. Then, for each t[0,b], we have z(t)=λ{(I)z(t)+Iαf(t,z-t+xt)}. It follows by assumption of the theorem (24)|z(t)|k=0n|(-αnk)|k!bn-k(n-k)!Γ(α-β+k)0t(t-s)α-β+k-1|z(s)|ds+1Γ(α)0t(t-s)α-1|f(s,z-s+xx)|dsk=0n|(-αnk)|k!bn-kzb(n-k)!Γ(α-β+k+1)+1Γ(α)0t(t-s)α-1q(s)z-s+xsds+bαpΓ(α+1). On other hand, we have (25)z-s+xsBz-s+xsk(t)sup{|z(s)|:0st}+m(t)z0+k(t)sup{|x(s)|:0st}+m(t)x0kbsup{|z(s)|:0tt}+mbϕ. If we let δ(t) the right-hand side of (25), then z-s+xsδ(t) and, therefore, (26)|z(t)|k=0n|(-αnk)|k!bn-kzb(n-k)!Γ(α-β+k+1)+1Γ(α)0t(t-s)α-1q(s)δ(s)ds+bαpΓ(α+1). Using the aforementioned inequality and the definition of δ, we get (27)δ(t)k=0n|(-αnk)|k!bn-kzbkb(n-k)!Γ(α-β+k+1)+mbϕ+kbbαpΓ(α+1)+kbqΓ(α)×0t(t-s)α-1δ(s)ds. Then, using Lemma 3, there exists a constant Δ such that (28)|δ(t)|12kbΛ2+mbϕ+kbbαpΓ(α+1)+ΔkbqΓ(α)0t(t-s)α-1Rds, where Λ2 is mentioned in (22), and (29)R=12kbΛ2+mbϕ+kbbαpΓ(α+1). Hence, (30)δR+RΔbαkbqΓ(α+1):=M~, and then zΛ2+M~Iαq+bαp/Γ(α+1). Therefore, (31)zM~Iαq+bαp/Γ(α+1)1-Λ2:=Δ*. Set U={zC0:zb<Δ*+1}. Then, F:U¯C0 is continuous and completely continuous. From the choice of U, there is no zU such that z=γF(z), for γ(0,1); therefore, by the nonlinear alternative of the Leray-Schauder theorem, the proof is complete.

Theorem 6.

Let f:J×B be a continuous function. If there exists a positive constant l such that |f(t,u)-f(t,v)|lu-v,tJ,u,v, and 0<T+lkbbα/Γ(α+1):=L<1 then (1) has a unique solution in the interval (-,b], where, (32)T=k=0n|(-αnk)|k!bn-k(n-k)!Γ(α-β+k+1).

Proof.

The solution of (1) is equivalent to the solution of the integral equation (17). Hence, it is enough to show that the operator F:C0C0, satisfies the Banach fixed-point theorem. Consider u,vC0 and for each t[0,b], we have (33)|F(z)(t)-F(u)(t)|Tu-vb+1Γ(α)0t(t-s)α-1lu-s-v-sdsTu-vb+lΓ(α)0t(t-s)α-1u-s-v-sdsTu-vb+lΓ(α)0t(t-s)α-1×kbsupu(s)-v(s)ds{T+lkbΓ(α)0t(t-s)α-1lds}|u-v|b{T+lkbbαΓ(α+1)}u-vb=Lu-vb. Hence, F(z)-F(v)bL  z-z*b, and then F is a contraction. Therefore, F has a unique fixed point by Banach’s contraction principle.

Theorem 7.

Let f:J× be a continuous function, and let the following assumptions hold.

There exist p,qC(J,0) such that |f(t,u)|p(t)+q(t)u for each tJ,u and and Iαp<+.

The function g is continuous and completely continuous. For any bounded set 𝒟 in Ω, the set {tg(t,yt):y𝒟} is equicontinuous in C([0,b],). There exist positive constants d1 and d2 such that |g(t,u)|d1u+d2 for each t[0,b] and u.

If kbd1(0,1), then (3) has at least one solution on (-,b], where kb=sup{|k(t)|:t[0,b]}.

Proof.

Consider the operator P*:ΩΩ defined by (34)P*(y)(t)={(I)y(t)+Iαf(t,yt)+g(t,yt),t[0,b],ϕ(t),t(-,0], where (35)(I)=k=0n(-αk)n!tn-k(n-k)!Iα-β+k. In analog to Theorem 5, we consider the operator F*:C0C0 defined by (36)F*z(t)=(I)z(t)+Iαf(t,z-t+xt)+g(t,z-t+xt). By using (H2) and Theorem 5, the operator F* is continuous and completely continuous. Now, it is sufficient to show that there exists an open set U*C0 with zλF*(z) for γ(0,1) and zU*.

Let zC0 and z=γF*(z) for some γ(0,1). Then, for each t[0,b],z(t)=γ[g(t,z-t+xt)+(I)z(t)+Iαf(t,z-t+xt)]. Hence, (37)|z(t)|d1z-t+xt+d2+k=0n|(-αk)|k!bn-k(n-k)!Γ(α-β+k+1)zb+bαnpΓ(α+1)+1Γ(α)0t(t-s)αn-1q(s)z-s+xsds,d1δ(t)+d2+bαpΓ(α+1)+1Γ(α)0t(t-s)α-1q(s)δ(s)ds+k=0n|(-αk)|k!bn-k(n-k)!Γ(α-β+k+1)zb, where δ(t) is named the in right-hand side of (25) such that z-s-xsδ(t). Since 0<kbd1<1, if we let T*=k=0n(|(-αk)|k!bn-kzb  kb/(n-k)!Γ(α-β+k+1)), then (38)δ(t)kbd1δ(t)+kbd2+mbϕ+T*+mbϕ+kbbαpΓ(α+1)+kbqΓ(α)0t(t-s)α-1δ(s)dsδ(t)11-kbd1{0tkbd2+mbϕ+T*+mbϕ2+kbbαpΓ(α+1)+kbqΓ(α)2×0t(t-s)α-1δ(s)ds}. Then, using Lemma 3, there exists a constant Δ* such that (39)δ(t)kbd1δ(t)+kbd2+mbϕδ(t)+T*+mbϕ+kbbαpΓ(α+1)δ(t)+kbqΓ(α)0t(t-s)α-1δ(s)dsδ(t)11-kbd1δ(t)×{kbbαpΓ(α+1)kb  d2+mbϕ+T*+mbϕ+kbbαpΓ(α+1)δ(t)+Δ*kbqΓ(α)0t(t-s)α-1δ(s)dskbbαpΓ(α+1)}, and, therefore, wR+RΔ*kbq*/Γ(α+1):=L*, where q*=q/(1-kbd1) and R=1/(1-kbd1)[kbd2+mbϕ+(kbbαp)/Γ(α+1)+T*]. Then, (40)zd1L*+d2+bαpΓ(α+1)+LIαq+T*, and, hence, (41)zd1L*+d2+bαp/Γ(α+1)+L*Iαq1-zT*:=M*. Set U*={zC0:zb<M*+1}. From the choice of U*, there is no zU* such that z=γF*(z) for γ(0,1). As a consequence of the nonlinear alternative of the Leray-Schauder theorem, we deduce that F* has a fixed-point z* in U*, which is a solution of (3).

The unique solution of (3), under some conditions, is studied in the following theorem which is the result of the Banach contraction mapping.

Theorem 8.

Let f:J× be a continuous function, and there exist positive constants l,μ, such that (42)|f(t,u)-f(t,v)|lu-v,|g(t,u)-g(t,v)|μu-v, where tJ and u,v. Then, (3) with the following conditions has a unique solution in the interval (-,b](43)0<T+lkbbαΓ(α+1)<1,0<kbμ+T+kblbαΓ(α+1)<1, such that T is defined in Theorem 6.

Proof.

The proof is a similar process Theorem 6.

4. Conclusions

In this paper, the existence and the uniqueness of solutions for the nonlinear fractional differential equations with infinite delay comprising standard Riemann-Liouville derivatives have been discussed in the phase space. Leray-Schauder’s alternative theorem and the Banach contraction principle were used to prove the obtained results. Further generalizations can be developed to some other class of fractional differential equations such as (D)y(t)=f(t,yt), where (D)=Dαn-j=1n-1pj(t)Dαn-j,  0<α1<<αn<1,pj(t)=k=0Njajktk, and Nj is nonnegative integer.

Acknowledgments

The authors would like to thank the referee for helpful comments and suggestions. This paper was funded by King Abdulaziz University, under Grant no. (130-1-1433/HiCi). The authors, therefore, acknowledge technical and financial support of KAU.

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