# Existence and Controllability Results for Fractional Impulsive Integrodifferential Systems in Banach Spaces

Authors: Haiyong Qin; Xin Zuo; Jianwei Liu
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We firstly study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equations and then present controllability results  for fractional impulsive integrodifferential systems in Banach spaces. The method we adopt is
based on fixed point theorem, semigroup theory, and generalized Bellman inequality. The results obtained in this paper improve and
extend some known results. At last, an example is presented to demonstrate the applications of our main results.


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## Body

## 1. Introduction

Fractional calculus is an area having a long history whose infancy dates back to three hundred years. However, at the beginning of fractional calculus, it develops slowly due to the disadvantage of technology. In recent decades, as the ancient mathematicians expected, fractional differential equations have been found to be a powerful tool in many fields, such as viscoelasticity, electrochemistry, control, porous media, and electromagnetic. For basic facts about fractional derivative and fractional calculus, one can refer to the books [1–4]. Since the fractional theory has played a very significant role in engineering, science, economy, and many other fields, during the past decades, fractional differential equations have attracted many authors, and there has been a great deal of interest in the solutions of fractional differential equations in analytical and numerical sense (see, e.g., [5–10] and references therein).On the other hand, the impulsive differential systems are used to describe processes which are subjected to abrupt changes at certain moments [11–13]. The study of dynamical systems with impulsive effects has been an object of intensive investigations. It is well known that controllability is a key topic for control theory. Controllability means that it is possible to steer any initial state of the system to any final state in some finite time using an admissible control. We refer the readers to the survey [14] and the reference therein for controllability of nonlinear systems in Banach spaces. The sufficient controllability conditions for fractional impulsive integrodifferential systems in Banach spaces have already been obtained in [15–18].Balachandran and Park [17] studied the controllability of fractional integrodifferential systems in Banach spaces without impulse
(1)dqx(t)dtq=Ax(t)+f(t,x(t),∫0t‍h(t,s,x(s))ds)==+Bu(t),t∈J=[0,b],x(0)=x0∈𝕏,
where 0<q<1, the state x(·) takes values in the Banach space 𝕏, f:J×𝕏×𝕏→𝕏, h:Δ×𝕏→𝕏 are continuous functions, and here Δ={(t,s):0≤s≤t≤b}. The control function u∈L2[J,U], a Banach space of admissible control functions with U as a Banach space, and B:U→𝕏 is a bounded linear operator.In [19], Mophou considered the existence and uniqueness of a mild solution for impulsive fractional semilinear differential equation
(2)Dtαx(t)=Ax(t)+f(t,x(t)),t∈I=[0,T],t≠tk,x(0)=x0∈𝕏,Δx|t=tk=Ik(x(tk-)),k=1,2,…,m,
where Dtα is the Caputo fractional derivative, and 0<α<1. The operator A:D(A)⊂𝕏→𝕏 is a generator of 𝒞0-semigroup (T(t))t≥0 on a Banach space 𝕏, and Ik:𝕏→𝕏 are impulsive functions.To consider fractional systems in the infinite dimensional space, the first important step is to define a new concept of the mild solution. Unfortunately, By Hernández et al. [20], we know that the concept of mild solutions used in [15–17, 19], inspired by Jaradat et al. [21], was not suitable for fractional evolution systems at all. Therefore, it is necessary to restudy this interesting and hot topic again.Recently, in Wang and Zhou [18], a suitable concept of mild solutions was introduced, using Krasnoselskii’s fixed point theorem and Sadovskii’s fixed point theorem, investigating complete controllability of fractional evolution systems in the infinite dimensional spaces
(3)Dtqcx(t)=Ax(t)+f(t,x(t))+Bu(t),Ax(t)+f(tx(t))t∈J=[0,b],x(0)=x0∈𝕏,
where Dtqc is the Caputo fractional derivative of the order 0<q≤1 with the lower limit zero, the state x(·) takes values in Banach space 𝕏, and the control function u(·) is given in L2[J,U], with U as a Banach space. A:D(A)⊂𝕏→𝕏 is the infinitesimal generator of a strongly continuous semigroup (T(t))t≥0 in 𝕏, B is a bounded linear operator from U to 𝕏, and f:J×𝕏→𝕏 is given 𝕏-value functions. Some sufficient conditions for complete controllability of the previous system were obtained.Inspired by the work of the previous papers and many known results in [22–24], we study the existence of mild solutions for impulsive fractional semilinear integrodifferential equation
(4)Dtqx(t)=Ax(t)+f(t,x(t),(Hx)(t)),Dtqx(t)=Ax(t)(t)t∈I=[0,b],t≠tk,x(0)=x0∈𝕏,Δx|t=tk=Ik(x(tk-)),k=1,2,…,m,
where Dtq is the Caputo fractional derivative, 0<q<1, the state x(·) takes values in Banach space 𝕏. A:D(A)⊂𝕏→𝕏 is the infinitesimal generator of a strongly continuous semigroup (T(t))t≥0 of a uniformly bounded operator on 𝕏, and A is a bounded linear operator. f:J×𝕏×𝕏→𝕏 is given 𝕏-value functions, H is defined as
(5)(Hx)(t)=∫0th(t,s,x(s))ds,
where h:Δ×𝕏→𝕏 are continuous, here Δ={(t,s):0≤s≤t≤b}, Ik:𝕏→𝕏 are impulsive functions, 0=t0<t1<⋯<tm<tm+1=b, Δx|t=tk=x(tk+)-x(tk-), and x(tk+)=limh→0+x(tk+h) and x(tk-)=limh→0-x(tk+h) represent the right and left limits of x(t) at t=tk, respectively.We also define a controlu and present controllability results for fractional integrodifferential systems in Banach spaces
(6)Dtqx(t)=Ax(t)+f(t,x(t),(Hx)(t))+Bu(t),f(t,x(t),(Hx)(t))t∈I=[0,b],t≠tk,x(0)=x0∈𝕏,Δx|t=tk=Ik(x(tk-)),k=1,2,…,m,
where B is a bounded linear operator from U to 𝕏, and the control function u(·) is given in L2[J,U], with U as a Banach space. The method we adopt is based on the ideas in [17–19, 22–24]. Compared with the previous results, this paper has three advantages. Firstly, we add operator H in the nonlinear term f and introduce a suitable concept of mild solutions of (4) and (6). Secondly, we not only study the existence of PC-mild solutions for impulsive fractional semilinear integrodifferential equation (4) but also present controllability results for fractional impulsive integrodifferential systems (6), and the results in [17, 19] could be seen as the special cases. Thirdly, our method avoids the compactness conditions on the semigroup (T(t))t≥0, and some other hypotheses are more general compared with the previous research (see the conditions (H1)–(H3) and (H5)–(H8)).The rest of the paper is organized as follows. In Section2, we present some preliminaries and lemmas that are to be used later to prove our main results. In Section 3, the existence of PC-mild solutions for (4) is discussed. In Section 4, by introducing a class of controls, we present the controllability results for fractional impulsive integrodifferential systems (6). In Section 5, an example is given to illustrate the theory.


## 2. Preliminaries and Lemmas

Let us consider the set of functionsPC[I,𝕏]={x:I→𝕏:x∈C[(tk,tk+1),𝕏], and there exist x(tk-) and x(tk+), k=0,1,2,…,m with x(tk-)=x(tk)}. Endowed with the norm ∥x∥PC=supt∈I∥x(t)∥, it is easy to know that (PC[I,𝕏],∥·∥PC) is a Banach space. Throughout this paper, let A be the infinitesimal generator of a C0-semigroup (T(t))t≥0 of a uniformly bounded operators on 𝕏. Let LB(𝕏) be the Banach space of all linear and bounded operator on 𝕏. For a C0-semigroup (T(t))t≥0, we set M1=supt∈I∥T(t)∥LB(𝕏). For each positive constant r, set Br={x∈PC[I,𝕏]:∥x∥≤r}.Definition 1.
The fractional integral of orderγ with the lower limit zero for a function f is defined as
(7)Iγf(t)=1Γ(γ)∫0t‍f(s)(t-s)1-γds,t>0,γ>0,
provided that the right side is point-wise defined on [0,+∞), where Γ(·) is the gamma function.Definition 2.
The Riemann-Liouville derivative of the orderγ with the lower limit zero for a function f:[0,∞]→R can be written as
(8)DγLf(t)=1Γ(n-γ)dndtn==×∫0tf(s)(t-s)1-n+γds,t>0,n-1<γ<n.Definition 3.
The Caputo derivative of the orderγ for a function f:[0,∞]→R can be written as
(9)Dγf(t)=DLγ(f(t)-∑k=0n-1tkk!f(k)(0)),=DLDLγt>0,n-1<γ<n.Remark 4.
(1) Iff(t)∈Cn[0,∞), then
(10)Dγf(t)=1Γ(n-γ)∫0tf(n)(s)(t-s)1-n+γds=In-rf(n)(t),t>0,n-1<γ<n.
(2) The Caputo derivative of a constant is equal to zero.
(3) Iff is an abstract function with values in 𝕏, then integrals which appear in Definitions 1, 2, and 3 are taken in Bochner’s sense.Definition 5 (see [22]).
A mild solution of the following nonhomogeneous impulsive linear fractional equation of the form(11)Dtqx(t)=Ax(t)+h(t),t∈I=[0,b],0<q<1,t≠tk,x(0)=x0∈𝕏,Δx|t=tk=Ik(x(tk-)),k=1,2,…,m,
is given by
(12)x(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)h(s)ds,t∈[0,t1],𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)h(s)ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)h(s)ds,t∈(tm,b],
where 𝒯(·) and 𝒮(·) are called characteristic solution operators and given by
(13)𝒯(t)=∫0∞ξq(θ)T(tqθ)dθ,𝒮(t)=q∫0∞θξq(θ)T(tqθ)dθ,
and for θ∈(0,∞),
(14)ξq(θ)=1qθ-1-(1/q)ϖq(θ-1/q)≥0,ϖq(θ)=1π∑n=1∞(-1)n-1θ-qn-1Γ(nq+1)n!sin(nπq),
where ξq is a probability density function defined on (0,∞); that is,
(15)ξq(θ)≥0,θ∈(0,∞),∫0∞ξq(θ)dθ=1.Definition 6.
By a PC-mild solution of (4), we mean that a function x∈PC[I,𝕏], which satisfies the following integral equation:
(16)x(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)(t)x0+++×f(s,x(s),(Hx)(s))ds,t∈[0,t1],𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯++×f(s,x(s),(Hx)(s))ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯++×f(s,x(s),(Hx)(s))ds,t∈(tm,b].Definition 7.
By a PC-mild solution of the system (6), we mean that a function x∈PC[I,𝕏], which satisfies the following integral equation:
(17)x(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)(t)x0+++×[f(s,x(s),(Hx)(s))(t)x0+++++Bu(s)f]ds,t∈[0,t1],𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯+∫0t‍(t-s)q-1𝒮(t-s)+++×[f(s,x(s),(Hx)(s))++++++Bu(s)f]ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯+∫0t(t-s)q-1𝒮(t-s)+++×[f(s,x(s),(Hx)(s))++++++Bu(s)f(s,x(s),(Hx)(s))]ds,t∈(tm,b].Definition 8.
The system (6) is said to be controllable on the interval J if, for every x0,x1∈𝕏, there exists a control u∈L2(J,U) such that a mild solution x of (6) satisfies x(b)=x1.Definition 9 (see [25]).
Let𝕏 be a Banach space, and a one parameter family T(t), 0≤t<+∞, of bounded linear operators from 𝕏 to 𝕏 is a semigroup of bounded linear operators on 𝕏 if(1)
T(0)=I (here, I is the identity operator on 𝕏);(2)
T(t+s)=T(t)T(s) for every t,s≥0 (the semigroup property).A semigroup of bounded linear operator, T(t), is uniformly continuous if limt↓0∥T(t)-I∥=0.Lemma 10 (see [25]).
Linear operatorA is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator.Lemma 11 (see [19]).
LetT be a continuous and compact mapping of a Banach space 𝕏 into itself, such that
(18){x∈𝕏:x=λTxforsome0≤λ≤1}
is bounded. Then, T has a fixed point.Lemma 12.
The operators𝒯(t) and 𝒮(t) have the following properties. (i)
For any fixedt≥0, 𝒯(t) and 𝒮(t) are linear and bounded operators; that is, for any x∈𝕏,
(19)∥𝒯(t)x∥≤M1∥x∥,∥𝒮(t)x∥≤qM1Γ(1+q)∥x∥.(ii)
{𝒯(t),t≥0} and {𝒮(t),t≥0} are strongly continuous.(iii)
{𝒯(t),t≥0} and {𝒮(t),t≥0} are uniformly continuous; that is, for each fixed t>0, and ϵ>0, there exists h>0 such that
(20)∥𝒯(t+ϵ)-𝒯(t)∥≤ε,fort+ϵ≥0,|ϵ|<h,∥𝒮(t+ϵ)-𝒮(t)∥≤ε,fort+ϵ≥0,|ϵ|<h.Proof.
For the proof of (i) and (ii), the reader can refer to [23, Lemma 2.9] and [24, Lemmas 3.2–3.5]. For each fixed t>0, and h>ϵ>0, one can obtain
(21)∥𝒯(t+ϵ)-𝒯(t)∥≤∫0∞ξq(θ)∥T((t+ϵ)qθ)-T(tqθ)∥dθ≤M1∫0∞ξq(θ)∥T((t+ϵ)qθ-tqθ)-I∥dθ,∥𝒮(t+ϵ)-𝒮(t)∥≤qM1∫0∞‍θξq(θ)∥T((t+ϵ)qθ-tqθ)-I∥dθ.
Because A is a bounded linear operator, from Lemma 10 and Definition 9, we know that A is the infinitesimal generator of a uniformly continuous semigroup. Thus, by the properties of uniformly continuous semigroup (T(t))t≥0, we get
(22)∥𝒯(t+ϵ)-𝒯(t)∥≤ε,∥𝒮(t+ϵ)-𝒮(t)∥≤ε;
that is, {𝒯(t),t≥0} and {𝒮(t),t≥0} are uniformly continuous.We list here the hypotheses to be used later.(H1)
f:I×𝕏×𝕏→𝕏 is continuous and there exist functions μ1,μ2∈L[I,ℝ+] such that
(23)∥f(t,x1,y1)-f(t,x2,y2)∥≤μ1(t)∥x1-x2∥+μ2(t)∥y1-y2∥,x1,x2,y1,y2∈𝕏.(H2)
h:Δ×𝕏×𝕏→𝕏 is continuous and there exist function ν1∈C[I,ℝ+] such that
(24)∥h(t,s,x1)-h(t,s,x2)∥≤ν1(t)∥x1-x2∥,x1,x2∈𝕏.(H3)
There existωk∈C[I,ℝ+] such that
(25)∥Ik(x1)-Ik(x2)∥≤ωk(t)∥x1-x2∥,ωk∥x1x∥x1,x2∈𝕏,k=1,2,…,m.(H4)
The functionΩm(t):I→ℝ+ is defined by
(26)Ωm(t)=qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))ds+mω0M1,whereν10=max{ν1(t)∣t∈I}, ω0=max{ωk(t)t∈I,k=1,2,…,m}, and 0<Ωm(t)<1, t∈I.(H4′)
The constantsΩu and Ωm′(t):I→ℝ+ are defined by
(27)Ωu=qM1KΓ(1+q)∫0b(b-s)q-1(μ1(s)+ν10bμ2(s))ds+ω0mM1,Ωm′(t)=qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))ds=+qM1ΩuΓ(1+q)∫0t(t-s)q-1ds+ω0mM1,and0<Ωm′(t)<1, t∈I.


## 3. Existence of Mild Solutions

Theorem 13.
If the hypotheses(H1)–(H4) are satisfied, then the fractional impulsive integrodifferential equation (4) has a unique mild solution x∈PC[I,𝕏].Proof.
Define an operatorQ on PC[I,𝕏] by
(28)(Qx)(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯𝒯𝒯𝒯×f(s,x(s),𝒯𝒯𝒯𝒯𝒯𝒯𝒯(Hx)(s))ds,t∈[0,t1],𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯++×f(s,x(s),(Hx)(s))ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯++×f(s,x(s),(Hx)(s))ds,t∈(tm,b].
We will show that Q is well defined on PC[I,𝕏]. For 0≤τ<t≤t1, applying (28), we obtain
(29)∥(Qx)(t)-(Qx)(τ)∥≤≤≤∥𝒯(t)-𝒯(τ)∥∥x0∥≤≤≤+∥∫τt(t-s)q-1𝒮(t-s)f(s,x(s),(Hx)(s))ds≤≤≤≤≤+∫0τ(t-s)q-1𝒮(t-s)f(s,x(s),(Hx)(s))ds≤≤≤≤≤+∫0τ(t-s)q-1𝒮(τ-s)f(s,x(s),(Hx)(s))ds≤≤≤≤≤-∫0τ(t-s)q-1𝒮(τ-s)f(s,x(s),(Hx)(s))ds≤≤≤≤≤-∫0τ(τ-s)q-1𝒮(τ-s)f(s,x(s),(Hx)(s))ds∥≤≤≤∥𝒯(t)-𝒯(τ)∥∥x0∥≤≤≤+∥∫τt(t-s)q-1𝒮(t-s)f(s,x(s),(Hx)(s))ds∥≤≤≤+∥∫0τ(t-s)q-1[𝒮(t-s)-𝒮(τ-s)]≤≤≤≤≤≤×f(s,x(s),(Hx)(s))ds∫0τ(t-s)q-1∥≤≤≤+∥∫0τ‍[(t-s)q-1-(τ-s)q-1]≤≤≤≤≤≤×𝒮(τ-s)f(s,x(s),(Hx)(s))ds∫0τ‍∥.
From the well-known inequality |tσ-τσ|≤(t-τ)σ for σ∈(0,1] and 0<τ≤t and Lemma 12, it is obvious that ∥(Qx)(t)-(Qx)(τ)∥→0 as t→τ. Thus, we deduce that Qx∈C[[0,t1],𝕏].
Fort1<τ<t≤t2, we have
(30)∥(Qx)(t)-(Qx)(τ)∥≤≤≤≤∥𝒯(t)-𝒯(τ)∥∥x0∥+≤≤≤+∥𝒯(t-t1)-𝒯(τ-t1)∥∥I1(x(t1-))∥+≤≤≤+∥∫τt(t-s)q-1𝒮(t-s)f(s,x(s),(Hx)(s))ds∥+≤≤≤+∥∫0τ(t-s)q-1[𝒮(t-s)-𝒮(τ-s)]++≤≤≤≤≤×f(s,x(s),(Hx)(s))ds∫0τ(t-s)q-1∥+≤≤≤+∥∫0τ[(t-s)q-1-(τ-s)q-1]+≤≤≤≤≤≤×𝒮(τ-s)f(s,x(s),(Hx)(s))ds∫0τ[(t-s)q-1-(τ-s)q-1]∥.
It is easy to get that, as t→τ, the right-hand side of the previous inequality tends to zero. Thus, we can deduce that Qx∈C[(t1,t2],𝕏]. By repeating the same procedure, we can also obtain that Qx∈C[(t2,t3],𝕏],…,Qx∈C[(tm,b],𝕏]. That is, Qx∈PC[I,𝕏].
Taket∈[0,t1]; then,
(31)∥(Qx)(t)-(Qy)(t)∥≤∫0t(t-s)q-1∥-f(s,y(s),(Hy)(s))𝒮(t-s)×(f(s,x(s),(Hx)(s))-f(s,y(s),(Hy)(s)))∥ds≤qM1Γ(1+q)∫0t(t-s)q-1×(μ1(s)∥x(s)-y(s)∥+μ2(s)∥(Hx)(s)-(Hy)(s)∥)ds.
From (H2) and (H4), we obtain
(32)∥(Qx)(t)-(Qy)(t)∥≤qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))×∥x(s)-y(s)∥ds.
So we deduce that
(33)∥(Qx)(t)-(Qy)(t)∥PC≤qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))ds×∥x-y∥PC.
In general, for each t∈(ti,ti+1], 1≤i≤m, using the assumptions,
(34)∥(Qx)(t)-(Qy)(t)∥PC≤qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))ds∥x-y∥PC+∥∑k=1i𝒯(t-tk)Ik(x(tk-))-∑k=1i𝒯(t-tk)Ik(y(tk-))∥≤(qM1Γ(1+q)∫0t(t-s)q-1(μ1(t)+ν10bμ2(t))ds+iω0M1qM1Γ(1+q)∫0t(t-s)q-1)∥x-y∥PC≤Ωi(t)∥x-y∥PC;
when i=m, obviously
(35)∥(Qx)(t)-(Qy)(t)∥PC≤(qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))ds+mω0M1qM1Γ(1+q)∫0t(t-s)q-1)∥x-y∥PC≤Ωm(t)∥x-y∥PC.
Noting that Ωi(t)≤Ωm(t), with assumption (H4) and in the view of the contraction mapping principle, we know that Q has a unique fixed point x∈PC[I,𝕏]; that is,
(36)x(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)(t)x0+++×f(s,x(s),(Hx)(s))ds,t∈[0,t1],𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)(tt)x0+×f(s,x(s),(Hx)(s))ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)(tt)x0+×f(s,x(s),(Hx)(s))ds,t∈(tm,b],
is a PC-mild solution of (4).In order to obtain results by the Schaefer fixed point theorem, let us list the following hypotheses.(H5)
f:I×𝕏×𝕏→𝕏 is continuous and there exist functions μ3,μ4,μ5∈L[I,ℝ+] such that
(37)∥f(t,x,y)∥≤μ3(t)+μ4(t)∥x∥+μ5(t)∥y∥,t∈I,x,y∈𝕏.(H6)
h:Δ×𝕏×𝕏→𝕏 is continuous and there exist functions ν2,ν3∈C[I,ℝ+] such that
(38)∥h(t,s,x)∥≤ν2(s)+ν3(s)∥x∥,x∈𝕏.(H7)
There existψk∈C[I,ℝ+] such that
(39)∥Ik(x)∥≤ψk(t)∥x∥,x∈𝕏.(H8)
For all bounded subsetsBr, the set
(40)Πh,δ(t)={∫0t-h(t-s)q-1𝒯δ(t)x0+∫0t-h(t-s)q-1𝒮δ(t-s)F(s)ds+∑k=1m𝒯δ(t-tk)Ik(x(tk-)):x∈Br}is relatively compact in𝕏 for arbitrary h∈(0,t) and δ>0, where
(41)𝒯δ(t)=∫δ∞ξq(θ)T(tqθ)dθ,𝒮δ(t)=q∫δ∞θξq(θ)T(tqθ)dθ.
(H8′)
For all bounded subsetsBr, the set
(42)Πh,δ′(t)={∫0t-h(t-s)q-1𝒯δ(t)x0+∫0t-h(t-s)q-1𝒮δ(t-s)[F(s)+Bu(s)]ds+∑k=1m𝒯δ(t-tk)Ik(x(tk-)):x∈Br}
is relatively compact in𝕏 for arbitrary h∈(0,t) and δ>0.Theorem 14.
If the hypotheses(H5)–(H8) are satisfied, the fractional impulsive integrodifferential equation (4) has at least one mild solution x∈PC[I,𝕏].Proof.
From Theorem13, we know that operator Q is defined as follows:
(43)(Qx)(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯𝒯𝒯𝒯𝒯×f(s,x(s),𝒯𝒯𝒯𝒯𝒯𝒯𝒯𝒯(Hx)(s))ds,t∈[0,t1],𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯𝒯𝒯×f(s,x(s),(Hx)(s))ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯𝒯+∫0t(t-s)q-1𝒮(t-s)𝒯𝒯𝒯𝒯×f(s,x(s),(Hx)(s))ds,t∈(tm,b].
We will prove the results in five steps.
Step 1 (continuity of Q on (ti,ti+1](i=0,1,2,…,m)). Let xn,x∈PC[I,𝕏] such that ∥xn-x*∥PC→0(n→+∞), and then r=supn∥xn∥PC<∞ and ∥x*∥PC<r; for every t∈(ti,ti+1](i=0,1,2,…,m), we have
(44)∥(Qxn)(t)-(Qx)(t)∥≤qM1Γ(1+q)∫0t(t-s)q-1×∥f(s,xn(s),(Hxn)(s))-f(s,x(s),(Hx)(s))∥ds+∥∑k=1m‍𝒯(t-tk)Ik(xn(tk-))-∑k=1m‍𝒯(t-tk)Ik(x(tk-))∥.
Since the functions f and Ik are continuous,
(45)f(s,xn(s),(Hxn)(s))⟶f(s,x(s),(Hx)(s)),Ik(xn(tk-))⟶Ik(x(tk-))n⟶∞.
By conditions (H5) and (H6), we know that
(46)∥f(s,xn(s),(Hxn)(s))-f(s,x(s),(Hx)(s))∥≤2μ3(s)+μ4(s)(∥x∥+∥xn∥)+μ5(s)(∥Hx∥+∥Hxn∥)≤2μ3(s)+2μ5(s)∫0sν2(θ)dθ+(μ4(s)+μ5(s)∫0sν3(θ)dθ)(∥x∥+∥xn∥)≤2μ3(s)+2μ5(s)∫0sν2(θ)dθ+(2μ4(s)+2μ5(s)∫0s‍ν3(θ)dθ)r.
Hence,
(47)(t-s)q-1∥f(s,xn(s),(Hxn)(s))(t-sss)q-1-f(s,x(s),(Hx)(s))∥∈L1[I,ℝ+].
By the Lebesgue dominated convergence theorem, we get
(48)∫0t(t-s)q-1∥f(s,xn(s),(Hxn)(s))∫0t(t-sss)q-1-f(s,x(s),(Hx)(s))∥ds⟶0.
It is easy to obtain that
(49)limn→∞∥(Qxn)(t)-(Qx)(t)∥PC=0.
Thus, Q is continuous on (ti,ti+1](i=0,1,2,…,m).
Step 2 (Q maps bounded sets into bounded sets in PC[I,𝕏]). From (43), we get
(50)∥(Qx)(t)∥≤∥𝒯(t)x0∥+qM1Γ(1+q)×∫0t(t-s)q-1∥f(s,x(s),(Hx)(s))∥ds+m∥𝒯(t-tk)Ik(x(tk-))∥,
and we know that
(51)∥f(s,x(s),(Hx)(s))∥≤μ3(s)+μ5(s)∫0sν2(θ)dθ+(μ4(s)+μ5(s)∫0sν3(θ)dθ)∥x∥≤φ1(s)+φ2(s)∥x∥.
From (50) and (51), we obtain
(52)∥(Qx)(t)∥≤M1∥x0∥+mM1ψ0∥x∥∥(Qx)(t)(t)∥+qbqM1Γ(1+q)∫0t(φ1(s)+φ2(s)∥x∥)ds,
where ψ0=max{ψk(t)∣t∈I,k=1,2,…,m}. Thus, for any x∈Br={x∈PC[I,𝕏]:∥x∥PC≤r},
(53)∥(Qx)(t)∥≤M1∥x0∥+qbqM1Γ(1+q)∫0bφ1(s)ds+(qbqM1Γ(1+q)∫0t‍φ2(s)ds+mMψ0)r=γ1.
Hence, we deduce that ∥(Qx)(t)∥≤γ1; that is, Q maps bounded sets to bounded sets in PC[I,𝕏].
Step 3. (Q(Br) is equicontinuous with Br on (ti,ti+1](i=0,1,2,…,m)). For any x∈Br, t′,t′′∈(ti,ti+1](i=0,1,2,…,m), we obtain
(54)∥(Qx)(t′′)-(Qx)(t′)∥≤∥𝒯(t′′)x0-𝒯(t′)x0∥+∥∫0t′′(t′′-s)q-1𝒮(t′′-s)F(s)ds-∫0t′‍(t′-s)q-1𝒮(t′-s)F(s)ds∥+∥∑k=1m𝒯(t′′-tk)Ik(x(tk-))-∑k=1m𝒯(t′-tk)Ik(x(tk-))∥;
after some elementary computation, we have
(55)∥(Qx)(t′′)-(Qx)(t′)∥≤∥𝒯(t′′)-𝒯(t′)∥∥x0∥+∥∫t′t′′(t′′-s)q-1𝒮(t′′-s)F(s)ds∥+∥∫0t′‍[(t′′-s)q-1-(t′-s)q-1]𝒮(t′′-s)F(s)ds∥+∥∫0t′(t′-s)q-1[𝒮(t′′-s)-𝒮(t′-s)]F(s)ds∥+m∥𝒯(t′′-t′)∥∥Ik(x(tk-))∥.
Using the fact that 𝒯(t) and 𝒮(t) are uniformly continuous, and the well-known inequality |t′σ-t′′σ|≤(t′′-t′)σ for σ∈(0,1] and 0<t′≤t′′, we can conclude that limt′′→t′∥(Qx)(t′′)-(Qx)(t′)∥=0. Thus Q(Br) is equicontinuous with Br on (ti,ti+1](i=0,1,2,…,m).
Step 4  (Q maps Br into a precompact set in 𝕏). We define Π=QBr and Π(t)={(Qx)(t):x∈Br} for t∈I. Set
(56)Πh,δ(t)={(Qh,δx)(t):x∈Br},
where
(57)Πh,δ(t)={𝒯δ(t)x0+∫0t-h(t-s)q-1𝒮δ(t-s)F(s)dsΠh,δ(t)==+∑k=1m𝒯δ(t-tk)Ik(x(tk-)):x∈Br}.
From Lemma 12(ii)-(iii), (H8), and the same method used in Theorem 3.2 of [18], we can verify that the set Π(t) can be arbitrary approximated by the relatively compact set Πh,δ(t). Thus, Q(Br)(t) is relatively compact in 𝕏.
Step5 (the set E={x∈PC[I,𝕏]:x=λQx for some 0<λ<1} is bounded). Let x∈E, and then(58)x(t)={λ𝒯(t)x0+λ∫0t(t-s)q-1𝒮(t-s)++++++×f(s,x(s),(Hx)(s))ds,t∈[0,t1],λ𝒯(t)x0+λ𝒯(t-t1)I1(x(t1-))+++λ∫0t(t-s)q-1𝒮(t-s)++++×f(s,x(s),(Hx)(s))ds,t∈(t1,t2],𝒯(t)x0+++++⋮λ𝒯(t)x0+λ∑k=1m𝒯(t-tk)Ik(x(tk-))+++λ∫0t(t-s)q-1𝒮(t-s)++++×f(s,x(s),(Hx)(s))ds,t∈(tm,b].Similar to the results of (53), we know that
(59)∥x(t)∥≤λM1∥x0∥+λqbqM1Γ(1+q)∫0bφ1(s)ds+λ(qbqM1Γ(1+q)∫0t‍φ2(s)ds+mM1ψ0)∥x(t)∥.
Obviously there exists λ sufficiently small such that ρ=1-λmM1ψ0>0, and then we get
(60)∥x(t)∥≤λM1ρ∥x0∥+λqbqM1ρΓ(1+q)∫0bφ1(s)ds+λqbqM1ρΓ(1+q)∫0tφ2(s)∥x(s)∥ds.
Let
(61)N3=λM1ρ∥x0∥+λqbqM1ρΓ(1+q)∫0bφ1(s)ds,f(t)=λqbqM1ρΓ(1+q)∫0tφ2(s)ds.
It is clear that f(t) is nonnegative continuous function on [0,+∞), and generalized Bellman inequality implies that
(62)∥x(t)∥≤N3e∫0t‍f(s)ds≤N3e∫0b‍f(s)ds=C0,
where C0 is a constant. Obviously, the set E is bounded on (ti,ti+1](i=0,1,2,…,m). Since Q is continuous and compact, thanks to Schaefer’s fixed point Theorem, Q has a fixed point (36) which is a PC-mild solution of (4).


## 4. Controllability Results

By introducing a class of controls, we present the controllability results for fractional impulsive integrodifferential systems (6).(H9)
The linear operatorWi from L2[(ti-1,ti],U] into 𝕏 defined by
(63)Wiu=∫0ti(ti-s)q-1𝒮(ti-s)Bu(s)ds,𝒮(ti-s)(s)Budsi=1,2,…,m,m+1,induces an invertible operatorW~i- defined on L2[(ti-1,ti],U]/KerWi, and there exists a positive constant K>0 such that ∥BW~i-∥≤K.Theorem 15.
If the hypotheses(H1)–(H3), (H4′), and (H9) are satisfied, then the fractional impulsive integrodifferential system (6) is controllable on I.Proof.
Using the condition(H9), for an arbitrary function x(·), define the control(64)u(t)={W~1-[∫0t1x1-x0m+1(t1-s)q-1x0+x1-x0m+1-𝒯(t1)x0-∫0t1(t1-s)q-1𝒮(t1-s)×f(s,x(s),(Hx)(s))ds∫0t1x1-x0m+1(t1-s)q-1](t),t∈[0,t1],W~2-[∫0t2x1-x0m+1(t2-s)q-1x0+2(x1-x0)m+1-𝒯(t2)x0-∫0t2(t2-s)q-1𝒮(t2-s)×f(s,x(s),(Hx)(s))ds-𝒯(t2-t1)I1(x(t1-))∫0t2x1-x0m+1(t2-s)q-1](t),t∈(t1,t2],𝒯(t)x0+++++⋮W~m+1-[∑k=1m𝒯(b-tk)x1-𝒯(b)x0-∫0b‍(b-s)q-1𝒮(b-s)×f(s,x(s),(Hx)(s))ds-∑k=1m𝒯(b-tk)Ik(x(tk-))](t),t∈(tm,b].Define the operator Q:PC[I,𝕏]→PC[I,𝕏], where(65)(Qx)(t)={𝒯(t)x0+∫0t(t-s)q-1𝒮(t-s)𝒯(t)x0++×[f(s,x(s),(Hx)(s))𝒯(t)x0+++++Bu(s)f(s,x(s),(Hx)(s))]ds,t∈[0,t1]𝒯(t)x0+𝒯(t-t1)I1(x(t1-))𝒯(t)+∫0t(t-s)q-1𝒮(t-s)𝒯(t)x0+×[f(s,x(s),(Hx)(s))𝒯(t)x0++++Bu(s)f(s,x(s),(Hx)(s))]ds,t∈(t1,t2],𝒯(t)x0+++++⋮𝒯(t)x0+∑k=1m𝒯(t-tk)Ik(x(tk-))𝒯(t)+∫0t(t-s)q-1𝒮(t-s)𝒯(t)x0++×[f(s,x(s),(Hx)(s))(t)x0++++++Bu(s)f(s,x(s),(Hx)(s))]ds,t∈(tm,b].By Theorem 13, we know that Q is well defined, and we will prove that when using the previous control, operator Q has a fixed point. Clearly, this fixed point is a PC-mild solution of the control problem (6) and x(b)=x1; that is, the control we defined steers the system (6) from initial x0 to x1 in the time b.
For anyx1,x2∈C[(ti,ti+1],𝕏](i=0,1,2,…,m), by conditions (H1)–(H3), (H4′), and (H9), we get
(66)∥Bu1(t)-Bu2(t)∥≤(qM1KΓ(1+q)∫0b(b-s)q-1×(μ1(s)+ν10bμ2(s))ds+ω0mM1qM1KΓ(1+q)∫0b(b-s)q-1)×∥x1(s)-x2(s)∥PC≤Ωu∥x1(s)-x2(s)∥PC,(67)∥(Qx1)(t)-(Qx2)(t)∥≤∫0t(t-s)q-1∥f(s,x1(s),(Hx1)(s))𝒮(t-s)×[f(s,x1(s),(Hx1)(s))𝒮(t-s)𝒮(tt---s)-f(s,x2(s),(Hx2)(s))]∥ds+∫0t(t-s)q-1×∥𝒮(t-s)[Bu1(s)-Bu2(s)]∥ds+∑k=1m∥𝒯(t-tk)(Ik(x1(tk-))-Ik(x2(tk-)))∥.
Therefore,
(68)∥(Qx1)(t)-(Qx2)(t)∥≤(qM1Γ(1+q)∫0t(t-s)q-1(μ1(s)+ν10bμ2(s))ds+qM1ΩuΓ(1+q)∫0t(t-s)q-1ds+ω0mM1)×∥x1(s)-x2(s)∥PC≤Ωm′(t)∥x1(s)-x2(s)∥PC.
Since 0<Ωm′(t)<1, then Q is contraction mapping. Any fixed point of Q is a PC-mild solution of (6) which satisfies x(b)=x1. Thus, the system (6) is controllable on I.Theorem 16.
If the hypotheses(H5)–(H7), (H8′), and (H9) are satisfied, the fractional impulsive integrodifferential system (6) is controllable on I.Proof.
Using the condition(H9), for an arbitrary function x(·), define the control(69)u(t)={W~1-[∫0t1‍x0+x1-x0m+1-𝒯(t1)x0WW~1--∫0t1‍(t1-s)q-1𝒮(t1-s)W~1-W~1-W~1-×f(s,x(s),(Hx)(s))ds∫x1-x0m+1](t),t∈[0,t1],W~2-[x0+2(x1-x0)m+1-𝒯(t2)x0WW~1--∫0t2(t2-s)q-1𝒮(t2-s)W~1-W~1-W~1-×f(s,x(s),(Hx)(s))dsWW~1--𝒯(t2-t1)I1(x(t1-))2(x1-x0)m+1](t),t∈(t1,t2],𝒯(t)x0+++++⋮W~m+1-[∑k=1m𝒯(b-tk)Ik(x(tk-))x1-𝒯(b)x0WW~m+1--∫0b(b-s)q-1𝒮(b-s)W~1-W~1-W~1-×f(s,x(s),(Hx)(s))dsW~m+1-W~--∑k=1m𝒯(b-tk)Ik(x(tk-))](t),t∈(tm,b].We will prove that when using the previous control, operator Q defined in (65) has a fixed point.
We discuss that in five steps.
Step 1 (continuity of Q on (ti,ti+1](i=0,1,2,…,m)). Let xn,x∈PC[I,𝕏] such that ∥xn-x*∥PC→0(n→+∞), and then r=supn∥xn∥PC<∞ and ∥x*∥PC<r. For every t∈(ti,ti+1](i=0,1,2,…,m), we have
(70)∥Qxn(t)-Qx(t)∥≤qM1Γ(1+q)∫0t(t-s)q-1×∥f(s,xn(s),(Hxn)(s))-f(s,x(s),(Hx)(s))∥ds+qM1Γ(1+q)∫0t(t-s)q-1∥Bxn(s)-Bx(s)∥ds+ψ0M1∑k=1m∥Ik(xn(tk-))-Ik(x(tk-))∥.
Since
(71)∥Bxn(s)-Bx(s)∥≤(qM1KΓ(1+q)∫0b(b-s)q-1×∥f(s,xn(s),(Hxn)(s))-f(s,x(s),(Hx)(s))∥ds∫0bqM1KΓ(1+q)(b-s)q-1)+ψ0M∑k=1m∥xn(tk-)-x(tk-)∥,
by (47), (71), and the Lebesgue dominated convergence theorem, it is easy to know that
(72)limn→∞∥(Qxn)(t)-(Qx)(t)∥PC=0.
Consequently, Q is continuous on (ti,ti+1](i=0,1,2,…,m).
Step 2. (Q maps bounded sets into bounded sets in PC[I,𝕏]). Since
(73)∥Bu(s)∥≤∥BW~i-∥×(+qM1Γ(1+q)∫0b(b-s)q-1∥x0∥+2∥x1∥+M1∥x0∥+qM1Γ(1+q)aaaaa×∫0b(b-s)q-1∥f(s,x(s),(Hx)(s))∥dsaaaaa+ψ0M∑k=1m∥x(tk-)∥)≤∥BW~i-∥×(qM1Γ(1+q)∫0b(b-s)q-1φ1(s)ds∥x0∥aaaaaaaaaaaa+2∥x1∥+M1∥x0∥+qM1∥x∥Γ(1+q)aaaaaaaaaaaa×∫0b(b-s)q-1φ2(s)dsaaaaaaaaaaaa+ψ0M∑k=1m∥x(tk-)∥)≤N1+N2∥x∥,
thus, from (65), we get, for any x∈Br={x∈PC[I,𝕏]:∥x∥PC≤r},
(74)∥(Qx)(t)∥≤M1∥x0∥+qbqM1Γ(1+q)∫0b(φ1(s)+N1)ds+(qbqM1Γ(1+q)∫0b(φ2(s)+N2)ds+mMψ0)r=γ2.
Hence, we deduce that ∥(Qx)(t)∥≤γ2; that is, Q maps bounded sets to bounded sets in PC[I,𝕏]. Using the same method used in Theorem 14, we can verify that Q(Br) is equicontinuous with Br on (ti,ti+1](i=0,1,2,…,m), Q maps Br into a precompact set in 𝕏, and Q(Br)(t) is relatively compact in 𝕏. Steps 3 and 4 are omitted.
Step 5 (the set E={x∈PC[I,𝕏]:x=λQxforsome0<λ<1} is bounded). Let x∈E, and similar to the results (74) we know that
(75)∥x(t)∥≤λM1∥x0∥+λqbqM1Γ(1+q)∫0t(φ1(s)+N1)ds+(λqbqM1Γ(1+q)∫0t(φ2(s)+N2)ds+λmMψ0)×∥x(t)∥.
There exists a λ  sufficiently small such that ρ2=1-λmMψ0>0, and then
(76)∥x(t)∥≤λM1ρ2∥x0∥+λqbqM1ρ2Γ(1+q)∫0b(φ1(s)+N1)ds+λqbqM1ρ2Γ(1+q)∫0t(φ1(s)+N1)∥x(s)∥ds.
Let
(77)N4=λM1ρ2∥x0∥+λqbqM1ρ2Γ(1+q)∫0b(φ1(s)+N1)ds,f(s)=λqbqM1ρ2Γ(1+q)∫0t(φ2(s)+N2)ds.
It is clear that f(s) is nonnegative continuous function on [0,+∞), and generalized Bellman inequality implies that
(78)∥x(t)∥≤N4e∫0t‍f(s)ds≤N4e∫0b‍f(s)ds=C1,
where C1 is a constant. Thus the set E is bounded. Since Q is continuous and compact, thanks to Schaefer’s fixed point Theorem, Q has a fixed point (36), and this fixed point is a PC-mild solution of (6) which satisfies x(b)=x1. Hence, the system (6) is controllable on I.


## 5. An Example

Consider the following nonlinear partial integrodifferential equation of the form(79)∂2/3∂t2/3z(t,y)=∫01‍(y-s)z(s,y)ds+f(t,z(t,y),Hz(t,y))+μ(t,y),t∈J=[0,1],z(t,0)=z(t,1)=0,z(0,y)=0,0<y<1,Δz|t=1/2=I1(z(12-,y)),
where 0<q<1, μ:J×(0,1)→(0,1) is continuous. Let us take 𝕏=C([0,1]). Consider the operator A:D(A)⊂𝕏→𝕏 defined by
(80)(Aw)(t)=∫01(y-s)w(s)ds.
It is easy to get
(81)∥Aw∥=∥w∥∫01|y-s|ds=(12-y(1-y))∥w∥≤12∥w∥;
clearly A is the infinitesimal generator of a uniformly continuous semigroup (T(t))t≥0 on 𝕏. Put x(t)=z(t,·) and u(t)=μ(t,·), and take
(82)f(t,x,Hx)=et+a(t)(∥x∥1+∥x∥)+∫0t‍k(t,s)(∥x∥1+∥x∥)ds,I1(x)=∥x∥,
where a(t)∈C[0,1], k(t,s)∈C([0,1]×[0,1]). Then clearly, f:[0,1]×ℝ×ℝ→ℝ and I1:ℝ→ℝ are continuous functions. f, I1, and h satisfy (H5)–(H8′), respectively. Equations (79) are an abstract formulation of (6). For y∈(0,1), we define
(83)W1u=∫01/2(12-s)-1/3𝒮(12-s)Bu(s)ds,W2u=∫01(1-s)-1/3𝒮(1-s)Bu(s)ds,
where
(84)𝒯(t)w(s)=∫0∞ξ2/3(θ)w(t2/3θ+s)dθ,𝒮(t)w(s)=23∫0∞θξ2/3(θ)w(t2/3θ+s)dθ,
and for θ∈(0,∞),
(85)ξ2/3(θ)=32θ-5/2ϖ2/3(θ-3/2),ϖ2/3(θ)=1π∑n=1∞(-1)n-1θ-((2n+3)/3)Γ((2n+3)/3)n!sin(2nπ3).
Assume that the linear operator Wi from L2[(ti-1,ti],U](i=1,2) into 𝕏 induces an invertible operator W~i- defined on L2[(ti-1,ti],U]/KerWi and there exists a positive constant K>0 such that ∥BW~i-∥≤K. Moreover, (H9) is satisfied. All conditions of Theorem 16 are now fulfilled, so we deduce that (79) is controllable on I. On the other hand, we have
(86)∥f(t,x,Hx)-f(t,y,Hy)∥≤a(t)∥x-y∥+∫0tk(t,s)∥x-y∥ds,k(t,s)∥x∥-k(t,s)∥y∥≤k0(∥x∥-∥y∥),k0=max{k(t,s)∣(t,s)∈I×I},∥I1(x)-I1(y)∥≤∥x-y∥.
Further, other conditions (H1)–(H3) are satisfied and it is possible to choose a(t), k(t,s) in such a way that condition (H4′) is satisfied. Hence, by Theorem 15, the system (79) is controllable on I.



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*Source: 295837-2013-06-10.xml*