Solutions to impulsive integrodifferential evolution equations under a noncompact evolution system
 Shaochun Ji^{1}Email author and
 Min Wang^{2}
https://doi.org/10.1186/s1366201506389
© Ji and Wang 2015
Received: 22 April 2015
Accepted: 15 September 2015
Published: 26 September 2015
Abstract
In this paper, we study the existence of mild solutions to impulsive integrodifferential evolution equations in Banach spaces. Based on a measure of noncompactness and important properties of semicompact sets, new existence results are obtained. Here the evolution system is only supposed to be strongly continuous, without any compact or equicontinuous assumptions. Some applications are given to illustrate the effectiveness of our results.
Keywords
MSC
1 Introduction
On the other hand, the abstract nonlocal initial problem was initiated by Byszewski and Lakshmikantham [10, 11], where the existence and uniqueness of solutions to semilinear nonlocal differential equations were discussed. The importance of the problem consists in the fact that it is more general and has a better effect than the classical initial conditions \(u(0)=u_{0}\). Therefore the nonlocal Cauchy problem has been studied extensively under various conditions on \(A(t)\) and f, g, by several authors [12–16]. Ntouyas and Tsamatos [14] studied the nonlocal semilinear differential equations with compact conditions. Xue [15] discussed the semilinear nonlocal differential equations when the semigroup \(T(t)\) generated by the coefficient operator is compact and the nonlocal function g is not compact. Some classes of integrodifferential equations with nonlocal conditions have been investigated by Balachandran et al. [17, 18]. Wang and Wei [19] and Machado et al. [20] discussed the existence of a class of impulsive integrodifferential evolution equations, where the evolution system is supposed to be equicontinuous.
In the above work, we find that the compactness of the evolution system plays a key role in this type of impulsive nonlocal Cauchy problem. However, sometimes it is difficult to satisfy. For example, let \(X=L^{2}(\infty,+\infty)\). The ordinary differential operator \(A=\mathrm{d}/\mathrm{d}x\) with \(D(A)=H^{1}(\infty,+\infty)\), generates a semigroup \(T(t)\) defined by \(T(t)u(s)=u(t+s)\), for every \(u\in X \). The \(C_{0}\)semigroup \(T(t)\) is not compact on X.
Recently, by using a new twocomponent measure of noncompactness, Benchohra and Ziane [21] proved the existence of mild solutions for a class of impulsive semilinear evolution differential inclusions with statedependent delay when \(A(t)\) generates a strongly continuous evolution operator. It is an interesting result. In this paper we explain that the existence results of differential systems under a noncompact evolution system can also be obtained via the classical Hausdorff measure of noncompactness. By applying the property of semicompact sets (see Lemma 2.5), we discuss the existence of mild solutions to (1.1) without the compactness of evolution system \(U(t,s)\), even its equicontinuity. This is one motivation of the present work. Note that the assumption on the evolution system here is weaker than that in [9, 19, 20], and no more conditions are added. The Banach space here is nonseparable. Another motivation of the present work is the exact controllability problem of the differential system. Our method can also be applied to an impulsive control system and can deal with the technical error on the exact controllability of differential system caused by the compactness of the evolution system (see Remark 4.2).
The paper is organized as follows. In Section 2 we recall some preliminary facts that we need in the sequel. In Section 3 we prove our results when the evolution system \(U(t,s)\) is strongly continuous. Some applications of our results are given in Section 4.
2 Preliminaries
Let \((X,\\cdot\)\) be a real Banach space. We denote by \(C([0,b];X)\) the space of Xvalued continuous functions on \([0,b]\) with the norm \(\x\=\sup\{\x(t)\, t\in[0,b]\}\) and by \(L^{1}([0,b];X)\) the space of Xvalued Bochner integrable functions on \([0,b]\) with the norm \(\f\_{L^{1}}=\int_{0}^{b}\f(t)\ \, \mathrm{d}t\).
For the sake of simplicity, we put \(J=[0,b]\); \(J_{0}=[0,t_{1}]\); \(J_{i}=(t_{i},t_{i+1}]\), \(i=1,\ldots,p\). In order to define the mild solution of problem (1.1), we introduce the set \(\mathit{PC}([0,b];X)\) = {\(u:[0,b]\rightarrow X\) such that \(u(\cdot)\) is continuous except for a finite number of points \(t_{i}\), at which \(u(t_{i}^{+})\), \(u(t_{i}^{})\) exist and \(u(t_{i})=u(t_{i}^{})\), \(i=1,\ldots,p\)}. It is easy to verify that \(\mathit{PC}([0,b];X)\) is a Banach space with the norm \(\u\_{\mathit{PC}}=\sup\{\u(t)\, t\in[0,b]\}\).
Lemma 2.1
([22])
 (1)
B is relatively compact if and only if \(\beta(B)=0\);
 (2)
\(\beta(B)=\beta(\overline{B})=\beta(\operatorname{conv} B)\), where B̅ and convB mean the closure and convex hull of B, respectively;
 (3)
\(\beta(B)\leq\beta(C)\) when \(B\subseteq C\);
 (4)
\(\beta(B+C)\leq\beta(B) + \beta(C)\), where \(B + C=\{x+y: x\in B, y\in C\}\);
 (5)
\(\beta(B\cup C)\leq\max\{\beta(B), \beta(C)\}\);
 (6)
\(\beta(\lambda B )\leq\lambda\beta(B)\) for any \(\lambda\in \mathbb{R}\);
 (7)
if the map \(Q:D(Q)\subseteq X \rightarrow Z\) is Lipschitz continuous with constant k, then \(\beta_{Z}(QB)\leq k\beta(B)\) for any bounded subset \(B \subseteq D(Q)\), where Z is a Banach space.
 (i)
\(U(s,s)=I\), \(U(t,r)U(r,s)=U(t,s)\) for \(0\leq s\leq r\leq t\leq b\);
 (ii)
\((t,s)\rightarrow U(t,s)\) is strongly continuous for \(0\leq s\leq t\leq b\).
In a natural way, we can consider the respective evolution operator \(U:J\times J\rightarrow L(X)\), where \(L(X)\) is the space of all bounded linear operators in X. Since the evolution system \(U(t,s)\) is strongly continuous on the compact set \(J\times J\), there exists \(M>0\) such that \(\U(t,s)\\leq M\) for any \((t,s)\in J\times J\). More details as regards this evolution system can be found in Pazy [23].
Definition 2.2
Definition 2.3

the sequence \(\{f_{n}(t)\}_{n=1}^{+\infty}\) is relatively compact in X for a.a. \(t\in[0,b]\);

there is a function \(\mu\in L^{1}([0,b];\mathbb{R}^{+})\) satisfying \(\sup_{n\geq1}\ f_{n}(t) \ \leq\mu(t) \) for a.e. \(t\in[0,b]\).
Lemma 2.4
([24], Theorem 4.2.2)
The following lemma can be found in Theorem 2 of [25] and Theorem 5.1.1 of [24].
Lemma 2.5
Let \((Gf)(t)=\int_{0}^{t} U(t,s)f(s)\, \mathrm{d}s\). If \(\{f_{n}\}_{n=1}^{+\infty}\subset L^{1}([0,b];X)\) is semicompact, then the set \(\{Gf_{n}\}_{n=1}^{+\infty}\) is relatively compact in \(C([0,b];X)\) and, moreover, if \(f_{n}\rightharpoonup f_{0}\), then for all \(t\in[0,b]\), \((Gf_{n})(t)\rightarrow(Gf)(t)\) as \(n\rightarrow +\infty\).
Lemma 2.6
([26])
Lemma 2.7
([9])
 (1)
W is equicontinuous on \(J_{0}=[0,t_{1}]\) and each \(J_{i}=(t_{i},t_{i+1}]\), \(i=1,\ldots,p\);
 (2)
W is equicontinuous at \(t=t_{i}^{+}\), \(i=1,\ldots,p\).
Throughout this paper, we denote \(M=\sup\{\U(t,s)\: (t,s)\in J\times J \}\), \(W_{r}=\{u\in \mathit{PC}([0,b]; X):\u(t)\\leq r, \forall t\in[0,b] \}\). Without loss of generality, we let \(u_{0}=0\).
3 Main results
 (H_{1}):

\(A(t)\) is a family of linear (not necessarily bounded) operators and \(A(t):D(A)\rightarrow X\) generates a strongly continuous evolution system \(\{U(t,s):0\leq s\leq t\leq b\}\), \(D(A)\) not depending on t and a dense subset of X (see [23]).
 (H_{2}):

\(g:\mathit{PC}([0,b];X)\rightarrow X\) is continuous and compact.
 (H_{3}):

\(I_{i}:X \rightarrow X\) is continuous and compact for each \(i=1,2,\ldots,p\).
 (H_{4}):

The function \(f:[0,b]\times X\times X \rightarrow X\) satisfies the following:
 (1)
For a.e. \(t\in[0,b]\), the function \(f(t,\cdot,\cdot):X\times X\rightarrow X\) is continuous and for all \(x,y\in X\times X\), the function \(f(\cdot,x,y):[0,b]\rightarrow X\) is strongly measurable.
 (2)There exists a function \(\theta\in L^{1}(J;\mathbb{R}^{+})\) such thatfor a.e. \(t\in[0,b]\) and all \(x,y\in X\).$$\bigl\Vert f(t,x,y)\bigr\Vert \leq\theta(t) \bigl(\Vert x\Vert +\y\ \bigr) $$
 (3)There exists a function \(l\in L^{1}(J;R^{+})\) such that for every bounded set \(A,B\subset X\),for a.e. \(t\in[0,b]\).$$\beta\bigl(f(t,A,B)\bigr)\leq l(t)\bigl[\beta(A)+\beta(B)\bigr] $$
 (1)
 (H_{5}):

The function \(h:T\times X\rightarrow X\) satisfies the following:
 (1)
For a.e. \((t,s)\in T\), the function \(h(t,s,\cdot):X\rightarrow X\) is continuous and for all \(x\in X\), the function \(h(\cdot,\cdot,x):T\rightarrow X\) is strongly measurable.
 (2)There exists a function \(m\in L^{1}(T;\mathbb{R}^{+})\) such thatLet us take \(m^{*}=\max_{(t,s)\in T} \int_{0}^{t} m(t,s) \, \mathrm{d}s\).$$\bigl\Vert h(t,s,x)\bigr\Vert \leq m(t,s)\x\. $$
 (3)There exist functions \(\zeta_{1},\zeta_{2}\in L^{1}(J;\mathbb{R}^{+})\) such thatfor a.e. \((t,s)\in T\), \(D\subset X\) a bounded set.$$\beta\bigl(h(t,s,D)\bigr)\leq\zeta_{1}(t)\zeta_{2}(s) \beta(D) $$
 (1)
Now, we give the existence result under the above hypotheses.
Theorem 3.1
Proof
It is easy to see that the fixed point of K is the mild solution of nonlocal impulsive problem (1.1). Subsequently, we will prove that K has a fixed point by using the Schauder fixed point theorem.
We denote by \(W_{0}=\{u\in \mathit{PC}([0,b];X):\u(t)\\leq r\text{ for all }t\in[0,b]\}\). Then \(W_{0}\subset \mathit{PC}([0,b]; X)\) is bounded and convex.
Now we shall prove that W is relatively compact in \(\mathit{PC}([0,b];X)\).
Therefore, W is convex compact and nonempty in \(\mathit{PC}([0,b];X)\), and \(K(W)\subset W\). By the Schauder fixed point theorem, there exists at least one mild solution u of the problem (1.1), where \(u\in W\) is a fixed point of the continuous map K. This completes the proof of Theorem 3.1. □
Remark 3.2
If the function f is compact or Lipschitz continuous (see, e.g., [10, 16]), then \(l(t)\) in hypothesis (H_{4}) becomes zero or a Lipschitz constant. In our proof, the measure of noncompactness (MNC) is used to get rid of the compactness of the evolution system. Note that in [6, 9, 19, 20], MNC is adopted to discuss the differential and integrodifferential system in Banach spaces when the operation semigroup (evolution system) is compact or equicontinuous. Here the condition on the evolution system \(U(t,s)\) is only supposed to be strongly continuous and they are weak in essence compared with the previous results. In our recent paper [27], we get some existence results of fractional differential equations with nonlocal conditions when the semigroup is strongly continuous. There the work is based on a new regular measure of noncompactness defined by us (see Lemma 3.1 of [27]). We conjecture that the two different approaches in [27] and in the present paper may be considered from a unified point of view in some way. It is an interesting problem and is worth discussing later.
4 Applications
Example 4.1
Take \(X=L^{2}[0,\pi]\). Define \(A(t)\equiv A:D(A)\subset X\rightarrow X\) by \(Az=z'\) with the domain \(D(A)=\{z \in X: z'\in X, z(0)=z(\pi)=0\}\). It is well known that A is an infinitesimal generator of a semigroup \(T(t)\) defined by \(T(t)z(s)=z(t+s)\) for each \(z \in X\). \(T(t)\) is not a compact semigroup on X and \(\beta(T(t)D)\leq\beta(D)\) for a bounded subset D, where β is the Hausdorff MNC.
 (1)\(f:[0,b]\times X\times X \rightarrow X\) is a continuous function defined byWe take$$\begin{aligned}& f(t,x,h) (\theta)=F\bigl(t,x(t,\theta),h(t,\theta) \bigr), \quad t\in [0,b], 0 \leq\theta\leq\pi, \\& h(t,\theta)= \int_{0}^{t} h_{1} \bigl(t,s,x(s,\theta)\bigr)\,\mathrm{d}s. \end{aligned}$$c is a constant. Then f, h satisfy hypotheses (H_{4}) and (H_{5}) of Section 3.$$F\biggl(t,x(t,\theta), \int_{0}^{t} h_{1}\bigl(t,s,x(s,\theta)\bigr)\,\mathrm{d}s \biggr)=c\sin \bigl(t,x(t,\theta)\bigr)+c\int_{0}^{t} \sqrt{x^{2}(\xi,\theta)+1}\,\mathrm{d}\xi, $$
 (2)\(I_{i}:X\rightarrow X\) is a continuous function for each \(i=1,2,\ldots,p\), defined byWe take$$I_{i}(x) (\theta)=I_{i}\bigl(x(\theta)\bigr). $$\(x\in X\), \(\rho_{i} \in C([0,\pi]\times[0,\pi], \mathbb{R})\), for each \(i=1,2,\ldots,p\). Then \(I_{i}\) is compact and satisfies hypothesis (H_{3}).$$I_{i}\bigl(x(\theta)\bigr)=\int_{0}^{\pi} \rho_{i}(\theta,y)\cos^{2}\bigl(x(y)\bigr) \,\mathrm{d}y, $$
 (3)\(g:\mathit{PC}([0,b];X)\rightarrow X\) is a continuous function defined bywith \(u(s)(\theta)=\omega(s,\theta)\). Then g is a compact operator.$$g(u) (\theta)=\int_{0}^{b} g_{1}(s)\log \bigl(1+\bigl\vert u(s) (\theta)\bigr\vert \bigr)\, \mathrm {d}s, \quad u\in \mathit{PC}\bigl([0,b];X\bigr), $$
Under these assumptions, the above partial differential system can be reformulated as the abstract problem (1.1). Then due to Theorem 3.1, the partial differential system has at least one mild solution on \([0,b]\).
Remark 4.2
Hernández and O’Regan [28] point out that some papers on exact controllability of abstract control system contain a similar technical error when the compactness of semigroup and other hypotheses are satisfied, that is, in this case the applications of controllability results are restricted to a finitedimensional space. In order to fix this problem, Ji et al. [8] and Machado et al. [20] used some measures of noncompactness to weaken the assumptions of compactness on the evolution system \(U(t,s)\), where the evolution system is supposed to be equicontinuous. In this paper, the evolution system is only supposed to be strongly continuous, without any restrictions of compactness or equicontinuity. Since the method used in this paper is also available for controllability of the evolution equations with impulsive conditions, we can improve many controllability results under a noncompact semigroup, like in [8, 18, 20].
Declarations
Acknowledgements
The research is supported by Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No. 14KJB110001), the Natural Science Foundation of Jiangsu Province (BK20150415), the Foundation of Huaiyin Institute of Technology (HGC1229).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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