# Existence Results for a Class of Fractional Differential Equations with Periodic Boundary Value Conditions and with Delay

Authors: Hadi Karami; Azizollah Babakhani; Dumitru Baleanu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We discuss the existence and uniqueness of solution for two types of fractional order ordinary and delay differential equations. Fixed point theorems are the main tool used here to establish the existence and uniqueness results.
First we use Banach contraction principle to prove the uniqueness of solution and then Krasnoselskii's fixed point theorem to show the existence of the solution under certain conditions in a Banach space.


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## Body

## 1. Introduction

In mathematics delay differential equations are a type of differential equation in which the derivative of unknown function at a certain time is given in terms of the values of the function at previous times.While physical events such as acceleration and deceleration take little time compared to the times needed to travel most distances, times involved in biological processes such as gestation and maturation can be substantial when compared to the data-collection times in most population studies. Therefore, it is often imperative to explicitly incorporate these process times into mathematical models of population dynamics. These process times are often called delay times, and the models that incorporate such delay times are referred as delay differential equation models [1, 2].Recently theory of fractional differential equations attracted many scientists and mathematicians to work on them [3–12]. For the existence of solutions for fractional differential equations, one can see [13–30] and references therein. The results have been obtained by using fixed point theorems like Picard’s, Schauder fixed-point theorem, and Banach contraction mapping principle. About the development of existence theorems for fractional functional differential equations, many publications exist [31–35]. Many applications of fractional calculus amount to replacing the time derivative in a given evolution equation by a derivative of fractional order. The results of several studies clearly stated that the fractional derivatives seem to arise generally and universally from important mathematical reasons. Recently, interesting attempts have been made to give physical meaning to the initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives which could be found in [8, 9, 36, 37].Recently Benchoohra et al. [28] studied existence of solutions for a class of fractional differential equations with infinite delay; namely,
(1)Dαu(t)=u(t)+f(t,ut),foreacht∈[0,b],0<α<1,u(t)=ϕ(t),t∈[-∞,0],
where Dα is the standard Riemann-Liouville fractional derivative and f satisfies some assumptions.First, in this paper we consider nonlinear delayed fractional differential equations:(2)Dαu(t)=u(t)+f(t,ut),t∈(0,1),0<α<1,u(t)=ϕ(t),t∈[-τ,0],
associated with boundary conditions
(3)u(0)=limt→0+t1-αu(t)=u(1),
where Dα is the standard Riemann-Liouville fractional derivative and f is a continuous function. Here ut(·) represents the properitoneal state from time -τ up to time t which is defined by ut(θ)=u(t+θ),  -τ≤θ≤0. We proved the uniqueness of existence solutions for (2) with periodic boundary condition (3) under some further conditions.For investigating to establish an existence theorem, we also consider a class of nonlinear delayed fractional differential equations of the form(4)Dαu(t)=u(t)+f(t,u(t),u(t-τ)),t∈(0,1),u(t)=ϕ(t),t∈[-τ,0],0<α<1,
with periodic boundary condition
(5)u(0)=limt→0+t1-αu(t)=c=u(1),D-(1-α)u(t)|t=0=cΓ(α).
The paper has been organized as follows. In Section 2 we give basic definitions and preliminary. Unique solution of (2)-(3) under some conditions is proved in Section 3. The existence solution of (4)-(5) under some assumptions has been presented in Section 4.


## 2. Preliminaries

For the convenience of the readers, we firstly present the necessary definitions from the fractional calculus theory and functional analysis. These definitions and results can be found in the literature [3, 7, 38].LetC[0,1] be the Banach space of all continuous real functions defined on [0,1] with the norm
(6)∥u∥=:max{|u(t)|:t∈[0,1]}.
Let Cr[0,1],r≥0, be the space of all functions f such that tru(t)∈C[0,1] which is a Banach space when endowed with the norm
(7)∥u∥r=:max{tr|u(t)|:t∈[0,1]}.Definition 1.
For a functionu defined on an interval [a,b], the Riemann-Liouville fractional integral of u of order α>0 is defined by
(8)Da+-αu(t)=1Γ(α)∫at‍(t-s)α-1u(s)ds,t>a,
and Riemann-Liouville fractional derivative of u(t) of order α>0 defined by
(9)Da+αu(t)=dndtn{Da+-(α-n)u(t)},n-1<α≤n,
provided that the right-hand side of the pervious equation is pointwise defined on (a,+∞).We denoteDa+αu(t) as Daαy(t) and Da+-αu(t) as Da-αu(t). Further D0+αu(t) and D0+-αu(t) are referred as Dαu(t) and D-αu(t), respectively.Definition 2.
A two-parameter function of the Mittag-Leffler type is defined by(10)Eα,β(z)=∑k=0∞‍zkΓ(αk+β),(α>0,β>0).Definition 3.
The beta function is usually defined by(11)B(z,w)=∫01‍τz-1(1-τ)w-1dτ,(Re(z)>0,Re(w)>0).
and we have also the following expression for the beta function:
(12)B(z,w)=Γ(z)Γ(w)Γ(z+w).Theorem 4 (Arzela-Ascoli’s theorem).
A subset ofC[a,b] is compact if and only if it is closed, bounded, and equicontinuous.Theorem 5 (Banach’s fixed point theorem).
Consider a metric spaceX=(X,d), where X≠∅. Suppose that X is complete and T:X→X is a contraction on X. Then T has precisely one fixed point.Theorem 6 (Krasnoselskii’s fixed point theorem).
LetB be a nonempty closed convex subset of a Banach space (X,∥·∥). Suppose that T1, and T2 map B into X such that  (1)
for anyx,y∈B we have T1x+T2y∈B,(2)
T1 is a contraction,(3)
T2 is continuous and T2(B) is contained in a compact set. Then there exists z∈B such that z=T1z+T2z.


## 3. Uniqueness of Solution

In this section we prove (2) with boundary condition (3) and another condition on f has a unique solution. Before proving, we need to introduce some notations that will be provided in the following.LetΩ={u:[-τ,1]→ℝ,u∈C1-α[0,1]}. Consider the operator N:Ω→Ω defined by
(13)Nu(t)={ϕ(t),t∈[-τ,0],∫01‍G1,α(t,s)f(s,us)ds,t∈(0,1),
where
(14)G1,α(t,s)={Γ(α)Eα,α(tα)Eα,α((1-s)α)tα-1(1-s)α-11-Γ(α)Eα,α(1)+(t-s)α-1Eα,α((t-s)α),0≤s≤t≤1,Γ(α)Eα,α(tα)Eα,α((1-s)α)tα-1(1-s)α-11-Γ(α)Eα,α(1),0≤t≤s≤1,
is given in Section 4. Let x(·):[-τ,1]→ℝ be the function defined by
(15)x(t)={0,t∈(0,1),ϕ(t),t∈[-τ,0].
For each z∈C1-α([0,1],ℝ) with limt→0+t1-αz(t)=c we denote z¯ the function defined by
(16)z¯(t)={z(t),t∈(0,1),0,t∈[-τ,0].
If u(·) satisfies the integral equation,
(17)u(t)=∫01‍G1,α(t,s)f(s,us)ds,
we can decompose u(·) as u(t)=z¯(t)+x(t),  0<t<1, which implies ut=z¯t+xt for every 0<t<1, and function z(·) satisfies
(18)z(t)=∫01‍G1,α(t,s)f(s,z¯s+xs)ds.
Set C0={z∈(C1-α[0,1],ℝ):limt→0+t1-αz(t)=c}. C0 is Banach space with the norm ∥·∥r. Let T:C0→C0 be defined by
(19)Tz(t)=∫01‍G1,α(t,s)f(s,z¯s+xs)ds,0<t<1.
Note that as operator N has a fixed point, equivalently T has a fixed point and so instead we try to prove that T has a fixed point.Theorem 7.
Assume that there exists a constantK>0 such that |f(t,u)-f(t,v)|≤K|u-v| for each t∈[0,1] and all u,v∈ℝ. Then the problem (2)-(3) has a unique solution in C0⊆C1-α[0,1] provided that
(20)K(Γ(α)Eα,α(1)|1-Γ(α)Eα,α(1)|+1)Eα,α(1)(Γ(α))2Γ(2α)<1.Proof.
We prove thatT:C0→C0 is a contraction map. For each t∈[0,1] and for z1,z2∈C0 we have
(21)t1-α|Tz1(t)-Tz2(t)|0000≤t1-α∫01‍|G1,α(t,s)|000000000000×|f(s,z¯1s+xs)-f(s,z¯2s+xs)|ds,0000≤Kt1-α∫01‍|G1,α(t,s)||z¯1s-z¯2s|ds;
using the definition of ∥·∥1-α we get
(22)∥Tz1(t)-Tz2(t)∥1-α0000≤Kmaxt∈[0,1]{t1-α∫01‍|G1,α(t,s)|sα-1ds}000000×∥z1-z2∥1-α.
Moreover,
(23)t1-α∫01‍|G1,α(t,s)|sα-1ds0000≤(Γ(α)Eα,α(1)|1-Γ(α)Eα,α(1)|+1)Eα,α(1)Γ(α)2Γ(2α).
Indeed we have
(24)t1-α∫01‍|G1,α(t,s)|sα-1ds00≤Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|0000×∑i=0∞‍1Γ(αi+α)∫01‍(1-s)α-1sα-1ds0000+tα-1∑i=0∞‍1Γ(αi+α)0000×∫0t‍(t-s)α-1sα-1dsΓ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|0000×Eα,α(1)∫01‍(1-s)α-1sα-1ds0000+t1-αEα,α(1)∫0t‍(t-s)α-1sα-1ds.
Note that
(25)∫0t‍sα-1(t-s)α-1ds=t2α-1Γ(α)2Γ(2α).
Hence we have
(26)t1-α∫01‍|G1,α(t,s)|sα-1ds00≤Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|Eα,α(1)Γ(α)2Γ(2α)0000+t1-αEα,α(1)t2α-1Γ(α)2Γ(2α)00=(Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|+tα)Eα,α(1)(Γ(α))2Γ(2α).
Using (22) and (23) we get
(27)∥Tz1-Tz2∥1-α≤K(Γ(α)Eα,α(1)|1-Γ(α)Eα,α(1)|+1)×Eα,α(1)Γ(α)2Γ(2α)∥z1-z2∥1-α.
This completes the proof.


## 4. Existence of Solution

In this section, by using Krasnoselskii’s theorem, we discuss the existence solution of (4) under some assumptions on f and further conditions. Before proving this theorem, we prove the following lemma which will be used in the next theorem.Lemma 8.
Consider the following nonlinear fractional differential equation of the form(28)Dαu(t)=u(t)+h(t),0<t<1,0<α<1,
with periodic boundary conditions
(29)u(0)=limt→0+t1-αu(t)=c=u(1),D-(1-α)u(t)|t=0=cΓ(α),
where h is a continuous function. Then the periodic boundary value problem (28)-(29) is equivalent to an integral equation given by u(t)=∫01‍G1,α(t,s)h(s)ds∈C1-α[0,1], where
(30)G1,α(t,s)={Γ(α)Eα,α(tα)Eα,α((1-s)α)tα-1(1-s)α-11-Γ(α)Eα,α(1)+(t-s)α-1Eα,α((t-s)α),0≤s≤t≤1,Γ(α)Eα,α(tα)Eα,α((1-s)α)tα-1(1-s)α-11-Γ(α)Eα,α(1),0≤t≤s≤1.Proof.
We consider the following fractional differential equation:(31)Dαy(t)-y(t)=h(t),(0<t<1,0<α<1),
with
(32)[Dα-1y(t)]t=0=b1=cΓ(α),
where limt→0+t1-αy(t)=c. Laplace transform of (31) yields
(33)sαY(s)-Y(s)=H(s)+b1
from which
(34)Y(s)=H(s)sα-1+b1sα-1
and the inverse Laplace transform gives the solution
(35)y(t)=b1tα-1Eα,α(tα)+∫0t‍(t-s)α-1Eα,α(t-s)αh(s)ds.
Hence we have
(36)y(t)=cΓ(α)tα-1Eα,α(tα)+∫0t‍(t-s)α-1Eα,α(t-s)αh(s)ds.
Therefore,
(37)y(1)=cΓ(α)+Eα,α(1)+∫01‍(1-s)α-1Eα,α((1-s)α)h(s)ds,
which leads to
(38)c(1-Γ(α)Eα,α(1))0000=∫01‍(1-s)α-1Eα,α((1-s)α)h(s)ds;
since Γ(α)Eα,α(1)≠1 we have
(39)c=11-Γ(α)Eα,α(1)∫01‍(1-s)α-1Eα,α((1-s)α)h(s)ds.
Then the solution of the problem (28)-(29) is given by
(40)y(t)=Γ(α)1-Γ(α)Eα,α(1)tα-1Eα,α(tα)×∫01‍(1-s)α-1Eα,α((1-s)α)h(s)ds+∫0t‍(t-s)α-1Eα,α((t-s)α)h(s)ds.
This completes the proof.Now we prove our main result using Lemma8 and two more assumptions which follow next.(H1)
We assume thatf(t,u,v) can be written as f1(t,u)+f2(t,u,v), where f1, f2 are Lipschitz continuous. Moreover assume that the function f1, f2 satisfies the following relations:
(41)|f1(t,u(t))|≤M1∥u(t)∥1-α,|f2(t,u(t),v(t))|≤M2∥u(t)∥1-α∥v(t)∥1-α.(H2)
LetΩ=C1-α([-τ,1],ℝ) denote collection of the space of all function u such that t1-αu(t)∈C[0,1]. Define the set B={u∈Ω:t1-α|u|≤r}, where r satisfies
(42)(M1∥u∥1-α+M2∥u∥1-α2)0000×(|Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|Eα,α+1(1)+tEα,α+1(tα))≤r.Furthermore we assume that
(43)Lf1(Γ(α)Eα,α(1)|1-Γ(α)Eα,α(1)|+1)Eα,α(1)(Γ(α))2Γ(2α)<1,
where Lf1 is a Lipschitz constant of f.Theorem 9.
If the assumptions (H1) and (H2) satisfied, then the problem (4) with periodic boundary value condition (5) has at least one solution.Proof.
(i) Note that by Lemma8, (4)-(5) is equivalent to integral equation (17). Define T1,T2:Ω→Ω by
(44)T1u(t)=∫01‍G1,α(t,s)f1(s,u(s))ds,T2u(t)=∫01‍G1,α(t,s)f2(s,u(s),u(s-τ))ds.
For u∈B we have,
(45)t1-α|T1u(t)+T2u(t)|00≤t1-α∫01‍|G1,α(t,s)|0000000000×|f1(s,u(s))0000000000000+f2(s,u(s),u(s-τ))|ds00≤(M1∥u∥1-α+M2∥u∥1-α2)t1-α0000×∫01‍|G1,α(t,s)|ds00≤(M1∥u∥1-α+M2∥u∥1-α2)0000×(Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|Eα,α+1(1)0000000+tEα,α+1(tα)Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|)≤r.
(ii) We will prove thatT1 is a contraction:
(46)|T1u-T1v||1-α00=t1-α|T1u(t)-T1v(t)|,00≤t1-α∫01‍|G1,α(t,s)|0000000000×|f1(s,u(s))-f1(s,v(s))|ds,00≤Lf1t1-α∫01‍|G1,α(t,s)|u-v|ds,00≤Lf1t1-α∫01‍|G1,α(t,s)sα-1s1-α|u-v|ds00≤Lf1(Γ(α)Eα,α(1)|1-Γ(α)Eα,α(1)|+1)0000×Eα,α(1)(Γ(α))2Γ(2α)∥u-v∥1-α.
Then  T1 is a contraction.
(iii) Finally we prove thatT2 is continuous and T2(B) is contained in a compact set. To prove the continuity of T2 let us consider a sequence un converging to u. Taking the norm of T2un(t)-T2u(t) we have
(47)∥T2un(t)-T2u(t)∥1-α00=t1-α|T2un(t)-T2u(t)|00≤t1-α∫01‍|G1,α(t,s)|0000000000×|f2(s,un(s),un(s-τ))0000000000000-f2(s,u(s),u(s-τ))|ds00≤t1-αLf2∫01‍|G1,α(t,s)|0000000000000×(|un(s)-u(s)|0000000000000000+|un(s-τ)-u(s-τ)|)ds00≤2Lf2t1-α∫01‍|G1,α(t,s)|sα-10000000000000×ds∥un-u∥1-α.
Hence whenever un→u we have T2un→T2u. This proves the continuity of T2.
On the other hand for0≤t1<t2≤1 and u∈B we have
(48)|t11-αT2u(t1)-t21-αT2u(t2)|00=|t11-α∫01‍G1,α(t1,s)f2(s,u(s),u(s-τ))ds00000-t21-α∫01‍G1,α(t2,s)f2(s,u(s),u(s-τ))ds|00≤|Γ(α)Eα,α(t1α)1-Γ(α)Eα,α(1)00000×∫0t1‍Eα,α((1-s)α)(1-s)α-1000000000×f2(s,u(s),u(s-τ))ds00000+t11-α∫0t1‍Eα,α((t1-s)α)(t1-s)α-10000000000000×f2(s,u(s),u(s-τ))ds00000-Γ(α)Eα,α(t2α)1-Γ(α)Eα,α(1)00000×∫0t1‍Eα,α((1-s)α)(1-s)α-1000000000×f2(s,u(s),u(s-τ))00000-t21-α∫0t1‍Eα,α((t2-s)α)(t2-s)α-10000000000000×f2(s,u(s),u(s-τ))00000+Γ(α)Eα,α(t1α)1-Γ(α)Eα,α(1)00000×∫t1t2‍Eα,α((1-s)α)(1-s)α-1000000000×f(s,u(s),u(s-τ))ds00000-Γ(α)Eα,α(t2α)1-Γ(α)Eα,α(1)00000×∫t1t2‍Eα,α((1-s)α)(1-s)α-1000000000×f2(s,u(s),u(s-τ))ds00000-t21-α∫t1t2‍Eα,α(t2-s)α(t2-s)α-1000000000000×f2(s,u(s),u(s-τ))ds00000+Γ(α)Eα,α(t1α)1-Γ(α)Eα,α(1)00000×∫t21‍Eα,α((1-s)α)(1-s)α-1000000000×f2(s,u(s),u(s-τ))ds00000-Γ(α)Eα,α(t2α)1-Γ(α)Eα,α(1)00000×∫t21‍Eα,α((1-s)α-1)(1-s)α-1000000000×f2(s,u(s),u(s-τ))dsΓ(α)Eα,α(t1α)1-Γ(α)Eα,α(1)|.
Then we have
(49)|t11-αT2u(t1)-t21-αT2u(t2)|00≤M2∥u∥1-α20000×(Γ(α)1-Γ(α)Eα,α(1)|Eα,α(t1α)-Eα,α(t2α)|0000000×∫0t1‍|Eα,α((1-s)α)|(1-s)α-1ds0000000+∫0t1‍|t11-αEα,α((t1-s)α)(t1-s)α-10000000000000-t21-α(t2-s)α-1Eα,α((t2-s)α)|ds0000000+Γ(α)|1-Γ(α)Eα,α(1)||Eα,α(t1α)-Eα,α(t2α)|0000000×∫t1t2‍|Eα,α((1-s)α)|(1-s)α-1ds0000000-t21-α∫t1t2‍|Eα,α((t2-s)α)|(t2-s)α-1ds0000000+Γ(α)|1-Γ(α)Eα,α(1)|0000000×|Eα,α(t1α)-|Eα,α(t2α)|0000000×∫t21‍|Eα,α((1-s)α)|(1-s)α-1ds).
Hence we deduce that if |t1-t2|→0 then |t11-αT2(t2)-t21-αT2(t2)|→0.
Then{t1-αT2u:u∈B} is equicontinuous. Moreover we show that {t1-αT2u:u∈B} is a bounded set in C[0,1]. Indeed we have
(50)t1-α|T2u(t)|00≤M2∥u∥1-α2t1-α∫01‍|G1,α(t,s)|ds00≤M2∥u∥1-α20000×(Γ(α)|Eα,α(tα)||1-Γ(α)Eα,α(1)|Eα,α+1(1)+tEα,α+1(tα)).
Then by Arzela-Ascoli’s theorem we conclude that T2 is compact. By using Krasnoselskii’s theorem there exists z∈B such that z is a fixed point of T. This completes the proof.


## 5. Conclusions

We considered two types of nonlinear delay fractional differential equations (FDE) with periodic boundary conditions involving Remann-Liouville fractional derivative possessing with a lower terminal at 0. In order to obtain the results in this paper, we have shown the existence and the uniqueness of solution for a class of nonlinear delayed FDE by Banach contraction principle. Then using Krasnoselskii's fixed point theorem we established an existence theorem for a different type of the equation that we have proven its uniqueness theorem.



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