Measures of noncompactness in spaces of regulated functions with application to semilinear measure driven equations
 Yueju Cao^{1, 2} and
 Jitao Sun^{1}Email author
Received: 5 August 2015
Accepted: 24 January 2016
Published: 8 February 2016
Abstract
We investigate the existence of mild solutions for abstract semilinear measure driven equations with nonlocal conditions. We first establish some results on Kuratowski measure of noncompactness in the space of regulated functions. Then we obtain some existence results for the abstract measure system by using the measure of noncompactness and a corresponding fixed point theorem. The usual Lipschitztype assumptions are avoided, and the semigroup related to the linear part of the system is not claimed to be compact, which improves and generalizes some known results in the literature.
Keywords
MSC
1 Introduction
Measure driven differential equations are also called differential equations with measures or measure differential equations; they arise in many areas of applied mathematics such as nonsmooth mechanics, game theory, etc. (see [3–7]). This type of systems covers some wellknown cases up to the difference of g. When g is an absolutely continuous function, a step function, or the sum of an absolutely continuous function with a step function, the system corresponds to ordinary differential equations, difference equations, or impulsive differential equations, respectively. On the other hand, since measure differential equations admit discontinuous paths that may exhibit infinitely many discontinuities in a finite interval, they can model some complex behaviors in dynamic systems, for example, Zeno trajectories [8].
The nonlocal problem was considered by Byszewski [9]. This type of systems is more appropriate than the classical initial value problem to describe real phenomena because it allows us to consider additional information. In the past few years theorems about the existence and controllability of differential and functional differential abstract evolution systems with nonlocal conditions have been fully studied (see [10–17] and the references therein).
Measure differential equations were investigated early by [18–21]. We can refer to the review paper [22] for a complete introduction of measure differential systems. Recently, the theory of measure differential equations in \(\mathbb{R}^{n}\) space has been developed to some extent (see [2, 23–25]). However, to the best of our knowledge, little literatures has been devoted to measure differential equations in infinitedimensional spaces except [1] and [26, 27]. In separable Banach spaces, by applying Hausdorff measure of noncompactness, the paper [1] discussed the existence of solutions for nonlinear measure driven system in the Kurzweil integral setting (a kind of nonabsolutely convergent integral generalizing the Lebesgue integral), in which the system can be viewed as a particular case of system (1) in this paper with \(A=0\) and \(p(x)=x_{0}\). Although some properties of Hausdorff measure of noncompactness in the space of \(G(J;X)\) were provided in [1], those properties are not intrinsic for \(G(J;X)\) since the function g in the system was involved. Under Lipschitztype conditions, [23, 24] studied the retarded version of nonlinear measure driven system by means of generalized ordinary differential equations when \(X=\mathbb{R}^{n}\) and in the Kurzweil integral setting. The authors in [26] investigated the existence of mild solutions for abstract semilinear measure driven system without consideration of nonlocal conditions, where the compactness of the \(C_{0}\)semigroup related to the linear part of the system is claimed. In this paper, for a general Banach space X, we first establish some useful properties of the Kuratowski measure of noncompactness in the space of regulated functions \(G(J;X)\). Then we obtain some existence results for semilinear measure driven system with nonlocal conditions (1) by applying the Kuratowski measure of noncompactness and a corresponding fixed point theorem. The compactness of the \(C_{0}\)semigroup is not demanded in this paper. In addition, without any assumptions of Lipschitztype as those in [23, 24], a similar analysis to system (1) can lead to the existence result for nonlinear measure retarded equations in the Lebesgue integral setting.
This paper is organized as follows. In Section 2, we review some concepts and results about the LebesgueStieltjes integral and regulated functions and the Kuratowski measure of noncompactness, which will be used throughout this paper. In Section 3, some results of the Kuratowski measure of noncompactness and regulated functions are established and applied to investigate the existence for the semilinear measure system (1). An example that illustrates our results is presented in Section 4. Finally, some conclusions are drawn in Section 5.
2 Preliminaries
In this section, we recall some concepts and basic results about the LebesgueStieltjes integral and regulated functions and the Kuratowski measure of noncompactness. For the properties of operator semigroups, we refer the reader to [28, 29].
The finite sets \(d=\{t_{0},t_{1},\ldots,t_{n}\}\) of points in the closed interval \([a,b]\) such that \(a=t_{0}< t_{1}<\cdots<t_{n}=b\) are called partitions of \([a,b]\). For \(\delta>0\), we say that a partition of \([a,b]\) is δfine if \(t_{i}t_{i1}<\delta\) for all \(i=1,2,\ldots,n\).
The following result holds by Proposition 3 in [1] since the Kurzweil integral is more general than the Lebesgue integral.
Proposition 2.1
The readers can refer to [31, 32] for the theory of LebesgueStieltjes integral and other types of integrals together with the relations among them.
Definition 2.2
([33])
 (i)
If \(x\in\mathcal{A}\), \(t\in[a,b]\), and \(t_{0}\delta< t< t_{0}\), then \(\x({t_{0}}^{})x(t)\<\varepsilon\).
 (ii)
If \(x\in\mathcal{A}\), \(t\in[a,b]\), and \(t_{0}< t< t_{0}+\delta \), then \(\x(t)x({t_{0}}^{+})\<\varepsilon\).
Lemma 2.3
([33])
Remark 2.4
Lemma 2.5
([33])
Let \(\{x_{n}\}_{n=1}^{\infty}\) be a sequence of functions from \([a,b]\) to X. If \(x_{n}\) converges pointwise to \(x_{0}\) as \(n\to\infty\) and the sequence \(\{x_{n}\}_{n=1}^{\infty}\) is equiregulated, then \(x_{n}\) converges uniformly to \(x_{0}\).
Lemma 2.6
Let \(W\subset G(J;X)\). If W is bounded and equiregulated, then the set \(\overline{\operatorname{co}}(W)\) is also bounded and equiregulated.
Proof
According to the boundedness of W, it is evident that \(\overline {\operatorname{co}}(W)\) is bounded.

\(\alpha(S)\) = inf{\(\delta>0: S\) can be expressed as the union of a finite number of sets such that the diameter of each set does not exceed δ, i.e., \(S=\bigcup_{i=1}^{m} S_{i}\) with \(\operatorname{diam}(S_{i})\leq\delta\), \(i=1,2,\ldots,m\)},
Lemma 2.7
([34])
 (i)
\(\alpha(S)=0\) if and only if S is relatively compact;
 (ii)
\(S\subseteq T\) implies \(\alpha(S)\leq\alpha(T)\);
 (iii)
\(\alpha(\overline{S})=\alpha(S)\);
 (iv)
\(\alpha(S\cup T)= \max\{\alpha(S),\alpha(T)\}\);
 (v)
\(\alpha(\lambda S)=\lambda\alpha(S)\), where \(\lambda S=\{ x=\lambda z: z\in S\}\);
 (vi)
\(\alpha(S+T)\leq\alpha(S)+\alpha(T)\), where \(S+T=\{x=y+z: y\in S, z\in T\}\);
 (vii)
\(\alpha(\operatorname{co}S)=\alpha(S)\);
 (viii)\(\alpha(S)\alpha(T)\leq2d_{h}(S,T)\), where \(d_{h}(S,T)\) denotes the Hausdorff metric of S and T, that is,and \(d(\cdot,\cdot)\) is the distance from an element of X to a subset of X.$$d_{h}(S,T)=\max \Bigl\{ \sup_{x\in S} d(x,T), \sup _{x\in T} d(x,S) \Bigr\} , $$
Lemma 2.8
([35])
Let X be a Banach space, and \(D\subseteq X\) a bounded set. Then there exists a countable subset \(D_{0}\) of D such that \(\alpha(D)\leq2\alpha(D_{0})\).
Let μ be a regular Borel measure on J and \(L_{\mu}^{1}(J;X)\) denote the set of μintegrable functions from J to X.
Lemma 2.9
([36])
Corollary 2.10
Proof
Remark 2.11
Since the LebesgueStieltjes measure is a regular Borel measure, the result of Corollary 2.10 holds for the LebesgueStieltjes measure.
Lemma 2.12
([37])
3 Main results
In this section, we show the main results of this paper that are divided into two parts. Some properties of the Kuratowski measure of noncompactness in the space of regulated functions \(G(J;X)\) are established in the first part. These properties are then applied to discuss the existence for the semilinear measure system (1) in the second part.
3.1 Measure of noncompactness in \(G(J;X)\)
Let W be a subset of \(G(J;X)\). For each fixed \(t\in J\), we denote \(W(t)=\{x(t): x\in W\}\). Further, let \(W(J)=\bigcup_{t\in J} W(t)=\{x(t): x\in W, t\in J\}\). Next, we will provide some results on the Kuratowski measure of noncompactness in the space of regulated functions \(G(J;X)\), which generalize those in the space of continuous functions \(C(J;X)\) in [34].
Theorem 3.1
Let \(W\subset G(J;X)\) be bounded and equiregulated on J. Then \(\alpha (W(t))\) is regulated on J.
Proof
Theorem 3.2
Proof
Step 1. We first prove that \(\alpha(W)=\alpha(W(J))\).
(i) Let us first show that \(\alpha(W(J))\leq\alpha(W)\). For every \(\varepsilon>0\), let \(W=\bigcup_{j=1}^{m} W_{j}\) be such that \(\operatorname{diam}(W_{j})<\alpha(W)+\varepsilon\), \(j=1,2,\ldots,m\).
Let P be the finite set of all maps \(i\to\mu(i)\) of \(\{1,2,\ldots,n\}\) into \(\{1,2,\ldots,m\}\). Let Q be the finite set of all maps \(i\to\nu (i)\) of \(\{0,1,2,\ldots,n\}\) into \(\{1,2,\ldots,m\}\). Fixing arbitrarily \(\tau _{i} \in J_{i}\), \(i=1,2,\ldots,n\), for \(\mu\in P\), \(\nu\in Q\), let \(L_{\mu}=\{ x\in W: x(\tau_{i})\in T_{\mu(i)}, i=1,2,\ldots,n\}\), \(L_{\nu}=\{x\in W: x(t_{i})\in T_{\nu(i)}, i=0,1,\ldots,n\}\), and let \(L_{\mu\nu}=L_{\mu}\cap L_{\nu}\). It is clear that \(W=\bigcup_{\mu\in P, \nu\in Q} L_{\mu\nu}\).
(i) and (ii) show that \(\alpha(W)=\alpha(W(J))\).
Step 2. Now we prove that \(\alpha(W(J))=\sup\{\alpha(W(t)): t\in J\}\).
3.2 Existence for semilinear measure driven equations
In this part, we provide existence results for the abstract measure system (1). We first give the definition of mild solutions for system (1).
Definition 3.3
 (H)
The \(C_{0}\)semigroup \(T(t)\) generated by A is equicontinuous, that is, \(\{T(t)x: x\in B\}\) is equicontinuous at any \(t>0\) for any bounded subset \(B\subset X\) (cf. [38]). Let \(M=\sup_{t\in J}\T(t)\\).
 (cf1)
For every \(x\in G(J;X)\), the function \(f(\cdot,x(\cdot))\in \mathcal{LS}_{g}(J;X)\).
 (cf2)
The map \(x\mapsto f(\cdot,x(\cdot))\) from \(G(J;X)\) to \(\mathcal {LS}_{g}(J;X)\) is continuous.
 (cf3)There exist a function \(m \in\mathcal{LS}_{g}(J;\mathbb{R}^{+})\) and a nondecreasing continuous function \(\Phi: \mathbb{R}^{+} \to\mathbb {R}^{+}\) such thatfor all \(x\in X\) and almost all \(t\in J\). In addition,$$\bigl\Vert f(t,x) \bigr\Vert \leq m(t)\Phi \bigl(\Vert x\Vert \bigr) $$$$\liminf_{l\to+\infty} \frac{\Phi(l)}{l}=\gamma< +\infty. $$
 (cf4)There exists a function \(L \in\mathcal{LS}_{g}(J;\mathbb{R}^{+})\) such thatfor almost all \(t\in J\) and every bounded set \(B\subseteq X\).$$\alpha \bigl(f(t,B) \bigr)\leq L(t)\alpha(B) $$
 (cp)\(p: G(J;X)\to X\) is continuous and compact, and there exist positive constants c and d such that$$\bigl\Vert p(x) \bigr\Vert \leq c\Vert x\Vert _{\infty}+d \quad \mbox{for all } x\in G(J;X). $$
Theorem 3.4
Proof
Let \(l>0\) be a constant, and \(B_{l}=\{x\in G(J;X): \x\_{\infty} \leq l\} \). For every positive number l, \(B_{l}\) is clearly a bounded closed convex set in \(G(J;X)\). Write \(N(B_{l})=\{N(x): x(\cdot)\in B_{l}\}\).
Step 1. There exists a positive number l such that \(\{ N(B_{l})\}\subseteq B_{l}\).
Step 2. \(N(B_{l})\) is an equiregulated set of functions.
We can use a similar procedure to show that \(\N(x)({t_{0}}^{})N(x)(t)\ \to0\) as \(t\to{t_{0}}^{}\) for each \(t_{0}\in(0,a]\). Therefore, \(N(B_{l})\) is equiregulated on J in terms of Definition 2.2.
Step 3. N is a continuous operator on \(B_{l}\).
Step 4. There exist a constant \(0\leq\gamma<1\) and a positive integer \(n_{0}\) such that for any \(B\subseteq F\), we have \(\alpha (\widetilde{N}^{n_{0}}(B))\leq\gamma\alpha(B)\), where \(F=\overline {\operatorname{co}}(N(B_{l}))\) and \(\widetilde{N}^{n}\), \(n\geq1\), are defined as in Lemma 2.12.
First, since \(p(B_{l})\) is relatively compact in X and \(T(\cdot)\) is strongly continuous, applying the ArzelàAscoli theorem, we infer that the set \(\{T(\cdot)p(x): x\in B_{l}\}\) is relatively compact in \(C(J;X)\). Therefore, we have \(\alpha(\{T(\cdot)p(x): x\in B_{l}\})=0\).
Since \(N(B_{l})\subseteq B_{l}\), we have \(F\subseteq\overline {\operatorname{co}}(B_{l})=B_{l}\). Hence, \(N(F)\subseteq N(B_{l})\subseteq F\). This implies that \(N: F\to F\) and \(N(F)\) is a bounded set in \(G(J;X)\).
For the nonlinear measure driven system (3), we have the following existence result. The proof of it is similar to that of Theorem 3.4 (in view of \(T(t)=I\) and \(p(x)=x_{0}\) in this case). So we omit it.
Theorem 3.5
Remark 3.6
In the Kurzweil integral setting, paper [1] investigated the nonlinear measure equation (3) in separable Banach spaces. For system (3), Theorem 3.5 can be applied to general Banach spaces in the Lebesgue integral setting. On the other hand, Theorem 3.5 provides existence criteria different from those of Theorem 10 in [1] due to the different results on the measure of noncompactness in the space of regulated functions \(G(J;X)\).
For the special case \(X=\mathbb{R}^{n}\), the papers [23, 24] studied nonlinear measure retarded differential equations in the Kurzweil integral setting, where the Lipschitztype conditions are demanded (see Theorem 5.3 in [23] and Theorem 6.1 in [24], respectively). Theorem 3.5 can be generalized to the retarded version with suitable modifications. Thus, the Lipschitztype conditions in the literature cited are unnecessary here. On the other hand, it is well known that the compactness condition (cf4) on f in Theorem 3.5 is much weaker than the Lipschitz condition on f in Theorem 5.3 in [23]. From this point of view, Theorem 3.5 is less restrictive than that in [23].
Remark 3.7
Let \(PC(J;X)\) denote the space of piecewise continuous functions on J (see [39]). Since \(C(J;X)\subset PC(J;X)\subset G(J;X)\), in view of the relation of differential equations with impulses and measure driven equations [24], our results generalize those in [12, 17], where existence criteria were provided for abstract semilinear differential equations with or without impulsive effects, and correspondingly, the solutions belong to \(C(J;X)\) or \(PC(J;X)\).
4 An example

(cf1) clearly holds;

(cf2) is satisfied because$$\begin{aligned}& \bigl\Vert f(t,x_{n})f(t,x) \bigr\Vert \\& \quad = \int_{0}^{1} \biggl( \int_{0}^{\pi} c_{0}^{2} \bigl\vert \sin \bigl(x_{n}(\omega) \bigr)\sin \bigl(x(\omega ) \bigr) \bigr\vert ^{2}\,d\omega \biggr)^{1/2}\,dg(t) \\& \quad \leq c_{0} \bigl(g(1)g(0) \bigr) \biggl( \int_{0}^{\pi} \bigl\vert x_{n}( \omega)x(\omega ) \bigr\vert ^{2}\,d\omega \biggr)^{1/2} \\& \quad = \frac{1}{2}c_{0}\Vert x_{n}x\Vert ; \end{aligned}$$

(cf3) is checked with \(m(t)=c_{0}\), \(\Phi(l)=l\), and hence, \(\gamma =1\) since$$\begin{aligned} \bigl\Vert f(t,x) \bigr\Vert =& \biggl( \int_{0}^{\pi} c_{0}^{2} \sin^{2} \bigl(x(\omega) \bigr)\,d\omega \biggr)^{1/2} \\ \leq& \biggl( \int_{0}^{\pi} c_{0}^{2}x^{2}( \omega)\,d\omega \biggr)^{1/2}=c_{0}\x\; \end{aligned}$$

(cf4) is checked with \(L(t)=c_{0}\) since \(\alpha(f(t,B))\leq c_{0}\alpha(B)\) for any bounded subset B of X.
Hence, under these assumptions, the partial differential system (4) can be reformulated as the abstract measure system (1) and there exists at least one mild solution for system (4) by Theorem 3.4.
5 Conclusions
In this paper, the issue on abstract semilinear measure driven equations in Banach spaces with nonlocal conditions has been addressed for the first time, which can model a large class of hybrid systems with Zeno behavior. We first establish some useful results on the Kuratowski measure of noncompactness in the space of regulated functions. Then the existence criteria of mild solutions for the discussed measure system are obtained by using the tools of measure of noncompactness and a corresponding fixed point theorem. The results obtained in this paper are also applicable to abstract semilinear dynamic equations on time scales. As shown in [23, 24], this type of equations can be transformed to abstract measure driven equations. Moreover, the issue on the existence for abstract semilinear measure driven equations is relatively new, and we can further develop its investigation inspired, for example, by [43].
Declarations
Acknowledgements
This work is supported by the National Natural Science Foundation of China under Grant 61174039. The authors thank the reviewer very much for important comments and suggestions on this paper.
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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