# Existence of Mild Solutions and Controllability of Fractional Impulsive Integrodifferential Systems with Nonlocal Conditions

Authors: Haiyong Qin; Zhenyun Gu; Youliang Fu; Tongxing Li
Journal: Journal of Function Spaces
 (2017)

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## Abstract

This paper is concerned with the existence results of nonlocal problems for a class of fractional impulsive integrodifferential equations in Banach spaces. We define a piecewise continuous control function to obtain the results on controllability of the corresponding fractional impulsive integrodifferential control systems. The results are obtained by means of fixed point methods. An example to illustrate the applications of our main results is given.


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## Body

## 1. Introduction

In recent decades, existence of mild solutions of nonlocal Cauchy problems has been investigated extensively by many researchers (see [1–15] and the references cited therein). The study of abstract nonlocal semilinear initial value problems was initiated by Byszewski and Lakshmikantham [11] and Byszewski [12]. Byszewski [12] considered the existence and uniqueness of mild, strong, and classical solutions of nonlocal Cauchy problems. Lin and Liu [8] studied the existence and uniqueness of mild and classical solutions of semilinear integrodifferential equations with nonlocal Cauchy problems. Using Krasnoselskii’s fixed point theorem, Schauder’s fixed point theorem, and Banach contraction principle, Zhou and Jiao [13] obtained several criteria on the existence and uniqueness of mild solutions of nonlocal Cauchy problems for fractional evolution equations without impulse.Such analysis on nonlocal Cauchy problems is important from an applied viewpoint, since the nonlocal condition has a better effect in applications than a classical initial one. For instance, the diffusion phenomenon of a small amount of gas in a transparent tube can be given a better description than using the usual local Cauchy problem. On the other hand, controllability of nonlocal problems in Banach spaces has become an active area of investigation; we refer the reader to, for example, the papers [16–29]. The most common method is to transform the controllability problem into a fixed-point problem of solutions for an appropriate operator in a function space, that is, the existence problem of differential and integrodifferential equations. Unfortunately, by [16], we know that the concept of mild solutions used in [14, 15, 17] was not suitable for fractional evolution systems.Chang et al. [18] investigated the controllability of a class of first-order semilinear differential systems with nonlocal initial conditions in a Banach space: (1)x′t=Axt+ft,xt+But,t∈J=0,b,x0+gx=x0∈X, where A generates a strongly continuous, not necessarily compact, semigroup (T(t))t≥0 in the Banach space X. Sufficient conditions for the controllability of the first-order semilinear differential system with nonlocal initial conditions were established. The approach used is Sadovskii’s fixed point theorem.Balachandran et al. [19] discussed the controllability of a class of fractional integrodifferential systems with nonlocal conditions in a Banach space: (2)dqxtdtq=Axt+ft,xt,Hxt+But,t∈J=0,b,x0+gx=x0∈X.Motivated by the work of the above papers and wide applications of nonlocal Cauchy problems in various fields of natural sciences and engineering, in this paper, we study the existence of nonlocal problems for a class of fractional impulsive integrodifferential systems in a Banach space of the following type:(3)Dtqxt=Axt+ft,xt,Hxt,t∈I=0,b,t≠tk,Δxt=tk=Ikxtk-,k=1,2,…,m,x0+gx=x0∈X,where (Hx)(t)=∫0th(t,s,x(s))ds and Dtq is the Caputo fractional derivative (0<q<1); the state x(·) takes values in the Banach space X. A:D(A)⊆X→X is the infinitesimal generator of a strongly continuous semigroup (T(t))t≥0 of uniformly bounded operators in X, and A is a bounded linear operator. f:I×X×X→X is a given X-value function; h:Δ×X→X is continuous; here Δ={(t,s):0≤s≤t≤b}, Ik:X→X, 0=t0<t1<⋯<tm<tm+1=b, Δx|t=tk=x(tk+)-x(tk-), 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. Using the similar method and a piecewise continuous control function, we consider the controllability of a class of fractional impulsive integrodifferential systems with nonlocal initial conditions: (4)Dtqxt=Axt+ft,xt,Hxt+But,t∈I=0,b,t≠tk,Δxt=tk=Ikxtk-,k=1,2,…,m,x0+gx=x0∈X,where B is a bounded linear operator from U to X and the control function u(·) is given in L2[I,U], with U as a Banach space.We study the nonlocal initial problem (3) that describes a more general form than the previous ones reported in [18, 19]. We introduce a suitable concept of PC-mild solutions for nonlocal initial problem (3). We not only study the existence and uniqueness of a mild solution for impulsive fractional semilinear integrodifferential equation (3) but also define a piecewise continuous control function and present the results on the controllability of the corresponding fractional impulsive integrodifferential system (4) which include some known results obtained in [14, 17]. Assumptions in our results are less restrictive.


## 2. Preliminaries and Lemmas

Throughout this paper, let us consider the set of functionsPC[I,X]={x:I→X∣x∈Ctk,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 xPC=supt∈Ix(t), it is easy to verify that (PC[I,X],·PC) is a Banach space. Let LB(X) be the Banach space of all linear and bounded operators on X. For a C0-semigroup (T(t))t≥0, we set M1=supt∈ITtLB(X). For each positive constant r, we set Br={x∈PC[I,X]:x≤r}. Obviously, Br is a bounded closed and convex subset.Definition 1.
The fractional integral of orderγ with the lower limit zero for a function f is defined as (5)Iγft=1Γγ∫0tfst-s1-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 orderγ with the lower limit zero for a function f:[0,∞)→R can be written as (6)LDγft=1Γn-γdndtn∫0tfst-s1-n+γds,t>0,n-1<γ<n.Definition 3.
The Caputo derivative of orderγ for a function f:[0,∞)→R can be written as (7)Dtγft=DLγft-∑k=0n-1tkk!fk0,t>0,n-1<γ<n.Remark 4.
Iff is an abstract function with values in X, then integrals that appear in Definitions 1–3 are taken in Bochner’s sense.Definition 5 (see [20]).
LetX be a Banach space; 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; 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 operators, T(t), is uniformly continuous if limt↓0⁡Tt-I=0.Definition 6 (see [21]).
By a PC-mild solution of system (3), we mean a function x∈PC[I,X] that satisfies the following integral equation: (8)xt=Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds,t∈0,t1,Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+Tt-t1I1xt1-,t∈t1,t2,⋮Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+∑k=1mTt-tkIkxtk-,t∈tm,b,where T(·) and S(·) are called characteristic solution operators and are given by (9)Tt=∫0∞ξqθTtqθdθ,St=q∫0∞θξqθTtqθdθ,and, for θ∈(0,∞),(10)ξqθ=1qθ-1-1/qϖqθ-1/q≥0,ϖqθ=1π∑n=1∞-1n-1θ-qn-1Γnq+1n!sin⁡qnπ,where ξq is a probability density function defined on (0,∞); that is, (11)ξqθ≥0,θ∈0,∞,∫0∞ξqθdθ=1.Definition 7 (see [21]).
By a PC-mild solution of system (4), we mean a function x∈PC[I,X] that satisfies the following integral equation: (12)xt=Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxs+Busds,t∈0,t1,Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxs+Busds+Tt-t1I1xt1-,t∈t1,t2,⋮Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxs+Busds+∑k=1mTt-tkIkxtk-,t∈tm,b.Definition 8.
System (4) is said to be controllable on the interval I if, for every x0,x1∈X, there exists a control u∈L2[I,U] such that a mild solution x of (4) satisfies x(b)+g(x)=x1.Lemma 9 (see [20]).
Linear operatorA is the infinitesimal generator of a uniformly continuous semigroup if and only if A is a bounded linear operator.Lemma 10 (see [13] Krasnoselskii's fixed point theorem).
LetX be a Banach space, let B be a bounded closed and convex subset of X, and let F1,F2 be maps of B into X such that F1x+F2y∈B for every pair x,y∈B. If F1 is a contraction and F2 is completely continuous, then the equation F1x+F2x=x has a solution in B.Lemma 11 (see [22, 23]).
The operatorsT(t) and S(t) defined by (9) 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, (13)Ttx≤M1x,Stx≤qM1Γ1+qx.(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.Remark 12.
Since the infinitesimal generatorA is a linear bounded operator and thanks to Definition 5 and Lemma 9, we can get that (iii) is satisfied.Lemma 13 (see [21]).
Forσ∈(0,1] and 0<a≤b, aσ-bσ≤(b-a)σ.


## 3. Existence and Uniqueness of PC-Mild Solutions

In order to prove the existence and uniqueness of mild solutions of (3), we have the following assumptions:(H1)
f:I×X×X→X is continuous and there exist two functions μ1,μ2∈L[I,R+] such that (14)ft,x1,y1-ft,x2,y2≤μ1tx1-x2+μ2ty1-y2,x1,x2,y1,y2∈X.(H2)
h:∆×X→X is continuous and there exists a function ν1∈C[I,R+] such that (15)ht,s,x1-ht,s,x2≤ν1tx1-x2,x1,x2∈X.(H3)
Ik:X→X are continuous and there exist ωk∈C[I,R+] such that (16)Ikx1-Ikx2≤ωktx1-x2,x1,x2∈X,k=1,2,…,m.H4
g is continuous and there exists a function ϕ∈C[I,R+] such that (17)gx1-gx2≤ϕtx1-x2.(H5)
The functionΩm(t):I→R+ is defined by (18)Ωmt=mω0M1+M1ϕt+qM1Γ1+q×∫0tt-sq-1μ1s+ν10bμ2sds,whereν10=max⁡{ν1(t)∣t∈I}, ω0=max⁡{ωk(t)∣t∈I,k=1,2,…,m}, and 0<Ωm(t)<1, t∈I.(H5′)
The constantΩu and function Ωm′(t):I→R+ are defined by (19)Ωu=ω0mM1+ϕ0M1+qKM1Γ1+q×∫0bb-sq-1μ1s+ν10bμ2sds,Ωm′t=ω0mM1+ϕ0M1+qM1Γ1+q×∫0tt-sq-1μ1t+ν10bμ2tds+qM1ΩuΓ1+q∫0tt-sq-1ds,whereϕ0=max⁡{ϕ(t)∣t∈I} and 0<Ωm′(t)<1, t∈I.Theorem 14.
If hypotheses(H1)–(H5) are satisfied, then (3) has a unique PC-mild solution.Proof.
Define the operatorQ on PC[I,X] by (20)Qxt=Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds,t∈0,t1,Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+Tt-t1I1xt1-,t∈t1,t2,⋮Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+∑k=1mTt-tkIkxtk-,t∈tm,b.
For0≤τ<t≤t1, by virtue of (20), we conclude that(21)Qxt-Qxτ≤Tt-Tτx0-gx+∫τtt-sq-1St-sfs,xs,Hxsds+∫0τt-sq-1St-s-Sτ-s×fs,xs,Hxsds+∫0τt-sq-1-τ-sq-1Sτ-s×fs,xs,Hxsds.It follows from Lemma 11, part (iii) and Lemma 13 that(22)Qxt-Qxτ⟶0ast⟶τ.Thus, we deduce that Qx∈C[[0,t1],X]. For t1<τ<t≤t2, we have (23)Qxt-Qxτ≤Tt-Tτx0-gx+Tt-t1-Tτ-t1I1xt1-+∫τtt-sq-1St-s×fs,xs,Hxsds+∫0τt-sq-1St-s-Sτ-s×fs,xs,Hxsds+∫0τt-sq-1-τ-sq-1Sτ-s×fs,xs,Hxsds.From (23), we know that Qx∈C[(t1,t2],X]. Using the same method, we obtain Qx∈C[(t2,t3],X],…, Qx∈C[(tm,b],X], and therefore Qx∈PC[I,X]. For each t∈(ti,ti+1],1≤i≤m, x,y∈PC[I,X], (24)Qxt-Qyt≤M1ϕt+qM1Γ1+q∫0tt-sq-1×μ1s+ν10bμ2sdsx-yPC+∑k=1iTt-tiIkxtk--∑k=1iTt-tiIkytk-≤Ωitx-yPC.When i=m, we get(25)Qxt-Qyt≤Ωmtx-yPC.It follows now from Ωi(t)≤Ωm(t), (H5), and the contraction mapping principle that Q has a unique fixed point x∈PC[I,X]; that is, (26)xt=Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds,t∈0,t1,Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+Tt-t1I1xt1-,t∈t1,t2,⋮Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+∑k=1mTt-tkIkxtk-,t∈tm,b,is a unique PC-mild solution of (3). The proof is complete.In order to obtain more existence results, we have the following assumptions:H6
f:I×X×X→X is continuous and there exist three functions μ3,μ4,μ5∈L[I,R+] such that (27)ft,x,y≤μ3t+μ4tx+μ5ty,t∈I,x,y∈X.(H7)
h:∆×X→X is continuous and there exist two functions ν2,ν3∈C[I,R+] such that (28)ht,s,x≤ν2s+ν3sx,x∈X.(H8)
Ik:X→X are continuous and there exist ψk∈C[I,R+] such that (29)Ikx≤ψktx,x∈X.Defineψ0=max⁡{ψkt∣t∈I,k=1,2,…,m}.(H9)
There exists a functionκ∈C[I,R+] such that (30)gx≤κtx,x∈X.Defineκ0=max⁡κt∣t∈I.(H10)
For all bounded subsetsBr, the set (31)Πm,h,δt=∫0t-ht-sq-1Sδt-sFsds+∑k=1mTδt-tkIkxtk-:x∈Bris relatively compact inX for arbitrary h∈(0,t) and δ>0, where Tδ(t) and Sδ(t) are defined by (32)Tδt=∫δ∞ξqθTtqθdθ,Sδt=q∫δ∞θξqθTtqθdθ.(H10′)
For all bounded subsetsBr, the set (33)Πm,h,δ′t=∫0t-ht-sq-1Sδt-sFs+Busds+∑k=1mTδt-tkIkxtk-:x∈Bris relatively compact inX for arbitrary h∈(0,t) and δ>0.Theorem 15.
Let hypotheses(H4) and (H6)–(H10) be satisfied. If the inequalities (34)qbqM1Γ1+q∫0bφ2sds+mM1ψ0+M1κ0<1,ϕ0M1<1hold, where φ2(s)=μ4(s)+μ5(s)∫0s‍ν3(θ)dθ and ϕ0 is as in (H5′), then (3) has at least one PC-mild solution.Proof.
We shall present the results in six steps.
Step  1 (Continuity of Q defined by (20) on ti,ti+1  (i=0,1,2,…,m)). Let xn,x∈PC[I,X] and xn-x∗PC→0  (n→∞). Then r=supn⁡xnPC<∞ and x∗PC<r. For t∈ti,ti+1  (i=0,1,2,…,m), we have (35)Qxnt-Qxt≤qM1Γ1+q∫0tt-sq-1×fs,xns,Hxns-fs,xs,Hxsds+ψ0M1∑k=1mIkxntk--Ikxtk-+M1gxn-gx.Since the functions f, Ik, and g are continuous, we conclude that (36)fs,xns,Hxns⟶fs,xs,Hxs,gxn⟶gx,Ikxntk-⟶Ikxtk-,n⟶∞.Applications of (H6) and (H7) yield (37)fs,xns,Hxns-fs,xs,Hxs≤2μ3s+2μ5s∫0sν2θdθ+2μ4s+2μ5s∫0sν3θdθr,which implies that (38)t-sq-1fs,xns,Hxns-fs,xs,Hxs∈L1I,R+.By Lebesgue’s dominated convergence theorem, we get (39)∫0tt-sq-1×fs,xns,Hxns-fs,xs,Hxsds⟶0, and so (40)limn→∞⁡Qxnt-QxtPC=0.Step  2 (Q maps bounded sets into bounded sets in PC[I,X]). From (20), we get (41)Qxt=Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxsds+∑k=1mTt-tkIkxtk-,where (42)fs,xs,Hxs≤μ3s+μ5s∫0sν2θdθ+μ4s+μ5s∫0sν3θdθx≤φ1s+φ2sx.By Lemma 11 and (42), we obtain (43)Qxt≤qbqM1Γ1+q∫0tφ1s+φ2sxds+M1x0+M1κ0x+mM1ψ0x.Thus, for any x∈Br={x∈PC[I,X]:xPC≤r}, we have (44)Qxt≤M1x0+qbqM1Γ1+q∫0bφ1sds+qbqM1Γ1+q∫0bφ2sds+mM1ψ0+M1κ0r=γ1.Hence, we deduce that (Qx)(t)≤γ1, that is, Q maps bounded sets into 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 (45)Qxt′′-Qxt′≤Tt′′x0-Tt′x0+Tt′′gx-Tt′gx+∫0t′′t′′-sq-1St′′-sFsds-∫0t′t′-sq-1St′-sFsds+∑k=1mTt′′-tkIkxtk--∑k=1mTt′-tkIkxtk-.Based on a straightforward computation, we have (46)Qxt′′-Qxt′≤Tt′′-Tt′x0+Tt′′gx-Tt′gx+∫t′t′′t′′-sq-1St′′-sFsds+∫0t′t′′-sq-1-t′-sq-1St′′-sFsds+∫0t′t′-sq-1St′′-s-St′-sFsds+mM1Tt′′-t′-IIkxtk-.It follows from Lemma 11, part (iii) and Lemma 13 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 (Pi map Br into a precompact set in X(i=1,…,m)). We define the operator (47)Qxt=Pixt+Lxt,where (48)Lxt=Ttx0-gx,Pixt=∫0tt-sq-1St-sfs,xs,Hxsds+∑k=1iTt-tkIkxtk-,i=1,…,m.
DefineΠ=PiBr and Π(t)={(Pix)(t):x∈Br} for t∈I. Set (49)Πi,h,δt=Pi,h,δxt:x∈Br,where (50)Πi,h,δt=∫0t-ht-sq-1Sδt-sFsds+∑k=1iTδt-tkIkxtk-:x∈Br.
From hypotheses we imposed and the same method used in [16, Theorem3.2], it is not difficult to verify that the set Π(t) can be arbitrary approximated by the relatively compact set Πi,h,δ(t). Thus, Pi(Br)(t) are relatively compact in X.
Step  5 (Lx+Piy∈Br for x,y∈Br(i=1,…,m)). Note that (51)qbqM1Γ1+q∫0bφ2sds+mM1ψ0+M1κ0<1.
Choose(52)M1x0+qbqM1/Γ1+q∫0bφ1sds1-qbqM1/Γ1+q∫0bφ2sds-mM1ψ0-M1κ0≤rand define operators L and Pi on Br by (53)Lxt=Ttx0-gx,Pixt=∫0tt-sq-1St-sfs,xs,Hxsds+∑k=1iTt-tkIkxtk-,i=1,…,m.
It is sufficient to proceed exactly as in step 1 to step 4 of the proof to deduce thatPi are continuous and compact. Thus, to complete this proof, it suffices to show that L is a contraction mapping and that Lx+Piy∈Br for x,y∈Br. Indeed, for any x∈Br, by virtue of (43) and (51), we have (54)Qxt≤M1x0+qbqM1Γ1+q∫0bφ1sds+qbqM1Γ1+q∫0bφ2sds+mM1ψ0+M1κ0r≤r.Consequently, if x,y∈Br, then Lx+Piy∈Br.
Step  6(L is a contraction mapping). For any t′,t′′∈ti,ti+1(i=0,1,2,…,m) and x,y∈PC[I,X], we have (55)Lxt-Lyt≤Ttgx-gy≤Ttgx-gy≤ϕ0M1x-yPC.Since ϕ0M1<1, L is a contraction mapping. Hence, by Lemma 10, we conclude that (3) has at least one PC-mild solution on I. This completes the proof.


## 4. Controllability Results

In this section, we impose the following conditions to prove the results.(H11)
DefineIi=ti-1,ti  (i=1,2,…,m+1). The linear operator Wi from L2[Ii,U] into X defined by (56)Wiu=∫0titi-sq-1Sti-sBusdsinduces an invertible operatorW~i- defined on L2[Ii,U]/KerWi and there exists a positive constant K>0 such that BW~i-≤K.Theorem 16.
If hypotheses(H1)–(H4), (H5′), and (H11) are satisfied, then system (4) is controllable on I.Proof.
Using(H11), for an arbitrary function x(·), we define the piecewise continuous control u by (57)ut=W~1-x0+x1-x0m+1-Tt1x0-gx-∫0t1t1-sq-1St1-sfs,xs,Hxsdst,t∈0,t1,W~2-x0+2x1-x0m+1-Tt2x0-gx-∫0t2t2-sq-1St2-sfs,xs,Hxsds-Tt2-t1I1xt1-t,t∈t1,t2,⋮W~m+1-x1-Tbx0-gx-∫0bb-sq-1Sb-sfs,xs,Hxsds-∑k=1mTb-tkIkxtk-t,t∈tm,b.On the basis of this control, with a similar proof to Theorem 14, we can conclude that the operator Q defined by (58)Qxt=Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxs+Busds,t∈0,t1,Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxs+Busds+Tt-t1I1xt1-,t∈t1,t2,⋮Ttx0-gx+∫0tt-sq-1St-sfs,xs,Hxs+Busds+∑k=1mTt-tkIkxtk-,t∈tm,b,has a fixed point x(·). This fixed point is a PC-mild solution of system (4), which implies that the system is controllable on I. The proof is complete.Theorem 17.
Assume that hypotheses(H4), (H6)–(H9), (H10′), and (H11) are satisfied. If the inequalities (59)qbqM1Γ1+q∫0bφ2s+N1ds+mM1ψ0+M1κ0<1,ϕ0M1<1hold, where N1=qKM1∫0b‍(b-s)q-1φ2(s)ds/Γ(1+q)+mKM1ψ0 and φ2(s) and ϕ0 are as in Theorem 15, then system (4) is controllable on I.Proof.
The proof is similar to that of Theorem15 and so is omitted.


## 5. Example

Consider the following nonlinear partial integrodifferential equation of the form(60)∂1/3∂t1/3zt,y=∫01y-szs,yds+ft,zt,y,Hzt,y+μt,y,zt,0=zt,1=0,t∈J=0,1,z0,y+ϕtzt,y=z0y,0≤y≤1,Δzt=1/2=I1x12-,where 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 (61)Awt=∫01y-swsds.It is not difficult to get(62)Aw=w∫01y-sds=12-y1-yw≤12wand, clearly, A is the infinitesimal generator of a uniformly continuous semigroup (T(t))t≥0 on X. Put x(t)(y)=z(t,y) and u(t)(y)=μ(t,y), and take (63)ft,x,Hx=k0x+Hx,Hxt=∫0tht,s,xsds,ht,s,x=k1x,I1x=ωtx,gx=ϕtx,where k0 and k1 are positive constants and ω(t) and ϕ(t) are continuous functions. Then f:[0,1]×X×X→X and I1:X→X are continuous functions; f, g, I1, and h satisfy (H6)–(H9), respectively.Fory∈(0,1], we define (64)W1u=∫01/212-s-2/3S12-sBusds,W2u=∫011-s-2/3S1-sBusds,where(65)Ttws=∫0∞ξ1/3θwt1/3θ+sdθ,Stws=13∫0∞θξ1/3θwt1/3θ+sdθ,and, for θ∈(0,∞), (66)ξ1/3θ=3θ-4ϖ1/3θ-3,ϖ1/3θ=1π∑n=1∞-1n-1θ-n+3/3Γn+3/3n!sin⁡nπ3.Moreover, the linear operator Wi from L2Ii,U  (i=1,2) into X induces an invertible operator W~i- defined on L2[Ii,U]/Ker⁡Wi and there exists a positive constant K>0 such that BW~i-≤K; that is, (H11) is satisfied. With the choices of A,f,g,H, and B=I (the identity operator), we see that (60) is an abstract formulation of (4). All conditions of Theorem 17 are able to be fulfilled, so we deduce that (60) is controllable on I. On the other hand, we have (67)ft,x,Hx-ft,y,Hy≤k0x-y+k1x-y,ht,s,x-ht,s,y≤k1x-y,I1x-I1y≤ωtx-y,gx-gy≤ϕtx-y.It is easy to see that all assumptions of Theorem 16 are satisfied when using the suitable choices of k0,k1,ω,ϕ. Hence, Theorem 16 can also yield controllability of (60) on I.


## 6. Conclusions

In this paper, we studied the existence and uniqueness results for a class of impulsive fractional semilinear integrodifferential equations with nonlocal initial conditions in a Banach space. Introducing the concept of PC-mild solutions and using the piecewise continuous control functions and uniformly continuous semigroup, we obtained the controllability results for the corresponding fractional impulsive integrodifferential system. Assuming that the semigroup is compact and utilizing some additional conditions, Hernández and O’Regan [30] showed that some known results on exact controllability (see the references cited therein) are valid if and only if the Banach space is finite dimensional. Recently, Hernández et al. [31] pointed out that some recent results on exact controllability of abstract differential systems with an unbounded linear operator dominated by a sectorial operator were not applicable. Contrary to those results, we do not need in our results conflicting conditions, which, in a certain sense, is a significant improvement compared to the results in the cited papers. An illustrative example is given to demonstrate the effectiveness of the results obtained. Our future work will focus on constrained controllability, nonlocal problems, and their applications in nonlinear dynamical systems (see [32–36]).



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*Source: 6979571-2017-09-20.xml*