Existence results of secondorder impulsive neutral functional differential inclusions in Banach spaces
 Taotao Li^{1} and
 Jianli Li^{1}Email author
https://doi.org/10.1186/s1366201506478
© Li and Li 2015
Received: 14 April 2015
Accepted: 27 September 2015
Published: 7 October 2015
Abstract
In this paper, we investigate the existence of solutions for a class of secondorder impulsive neutral functional differential inclusions in Banach spaces. Sufficient conditions for the existence are derived with the help of the fixed point theorem for multivalued maps due to Dhage.
Keywords
1 Introduction
For any continuous function y defined on \([r,T]\backslash\{ t_{1},t_{2},\ldots, t_{m}\}\) and \(t\in J\), we denote by \(y_{t}\) the element of \(\mathcal{D}\) defined by \(y_{t}(\theta)=y(t+\theta) \), \(\theta\in[r,0]\). Here \(y_{t}(\cdot)\) represents the history of the state from time \(tr\), up to the present time t.
In recent years, the theory of impulsive differential equations or inclusions has become an active area of investigation due to their applications in the fields of mechanics, electrical engineering, medicine biology, ecology, and so on. It has attracted great interest of researchers. For example, with the aid of Schaefer’s theorem, an existence result for first and secondorder impulsive neutral functional differential equations in Banach spaces has been given by the authors in [1]. By means of a fixed point theorem for condensing multivalued map, solvability of impulsive neutral evolution differential inclusions with statedependent delay has been given by the authors in [2]. There are many other methods such as in [3, 4] that have been for various initial and boundary value problems for impulsive differential inclusions. However, recently much attention has been paid to using a fixed point theorem for multivalued maps due to Dhage to solve the problem for impulsive differential inclusions. One can refer to [5, 6] and the references therein. Motivated by the previous mentioned paper, we will study the existence of solutions for a class of secondorder impulsive neutral functional differential inclusions in Banach spaces. Sufficient conditions for the existence are given by means of the fixed point theorem for multivalued maps due to Dhage [7]. For the IVP (1.1) we refer to [8].
2 Preliminaries
In this section, we shall introduce some basic definitions and lemmas which are used throughout this paper.
For \(\psi\in\mathcal{D}\), the norm of ψ is defined by \(\\psi\_{\mathcal{D}}=\sup\{\\psi(\theta )\: r\leq\theta\leq0\}\).
\(AC^{i}(J,E)\) is the space of itimes differentiable functions \(y: J\rightarrow E\), whose ith derivative, \(y^{i}\), is absolutely continuous.
Let \(P(X)\) denote the class of all nonempty subsets of X. Let \(P_{\mathrm{bd},\mathrm{cl}}(X)\), \(P_{\mathrm{cp},\mathrm{cv}}(X)\), \(P_{\mathrm{bd},\mathrm{cl},\mathrm{cv}}(X)\), and \(P_{\mathrm{cd}}(X)\) denote, respectively, the family of all nonempty boundedclosed, compactconvex, boundedclosedconvex and compactacyclic (see [9]) subset of X. For \(x\in X\) and \(Y,Z\in P_{\mathrm{bd},\mathrm{cl}}(X)\), we define \(D(x,Y)=\inf\{\ xy\:y\in Y\}\), \(\rho(Y,Z)=\sup_{a\in Y}D(a,Z)\), and the Hausdorff metric \(H:P_{\mathrm{bd},\mathrm{cl}}(X)\times P_{\mathrm{bd},\mathrm{cl}}(X)\rightarrow R^{+}\) by \(H(A,B)=\max\{\rho(A,B),\rho(B,A)\}\).
F is called upper semicontinuous (for brevity: u.s.c.) on X, if for each \(x_{\ast}\in X\), the \(F(x_{\ast})\) is nonempty, closed subset of X, and if, for each open of V of X containing \(F(x_{\ast})\), there exists an open neighborhood N of \(x_{\ast}\) such that \(F(N)\subseteq V\). F is said to be complete if \(F(N)\) is relatively compact, for every bounded subset \(V\subseteq X\).
If the multivalued map F is completely continuous with nonempty compact values, then F is u.s.c. if and only if F has a closed graph (i.e. \(x_{n}\rightarrow x_{\ast}\), \(y_{n}\rightarrow y_{\ast}\), \(y_{n}\in F(x_{n})\) imply \(y_{\ast}\in F(x_{\ast})\).
A point \(x_{0}\in X\) is called a fixed point of the multivalued map G if \(x_{0}\in F(x_{0})\). For more details of the multivalued maps, see the books of Deimling [10].
Definition 2.1
Lemma 2.1
Let E be a Banach space. Let \(F: J\times E\rightarrow P_{\mathrm{cp},\mathrm{cv}}(E)\) be an \(L^{1}\)Carathéodory multivalued map with \(S_{F,y}:=\{f\in L^{1}(J,E):f(t)\in F(t,y(t)) \textit{ for a.e. } t\in J\}\neq\varnothing\) and let Γ be a linear continuous mapping from \(L^{1}(J,E)\) to \(C(J,E)\), then the operator \(\Gamma\circ S_{F}:C(J,E)\rightarrow P_{\mathrm{cp},\mathrm{cv}}(C(J,E))\), \(y\mapsto (\Gamma\circ S_{F})(y):=\Gamma(S_{F,y})\) is a closed graph operator in \(C(J,E)\times C(J,E)\).
Theorem 2.1
 (i)
\(\Phi_{1}\) is a contraction with a contraction constant k, and
 (ii)
\(\Phi_{2}\) is completely continuous.
 (1)
the operator inclusion \(x\in\Phi_{1}x+\Phi_{2}x\) has a solution, or
 (2)
the set \(G=\{x\in E:x\in\lambda\Phi_{1}x+ \lambda\Phi _{2}x\}\) is unbounded for \(\lambda\in(0,1)\).
Definition 2.2
 (i)
\(t\mapsto F(t,u)\) is measurable for each \(u\in E\);
 (ii)
\(u\mapsto F(t,u)\) is upper semicontinuous on E for all \(t\in J\);
 (iii)for each \(\rho> 0\), there exists \(\varphi_{\rho}\in L^{1}(J,R^{+})\) such that$$\bigl\Vert F(t,u)\bigr\Vert _{P(E)}=\sup\bigl\{ \vert v\vert :v \in F(t,u)\bigr\} \leq \varphi_{\rho}(t), \quad \forall\ u\\leq\rho \text{ and for a.e. } t\in J. $$
3 Main result
Definition 3.1
A function \(y\in\Omega\cap AC^{1}((t_{k},t_{k+1}),E)\), \(k=0,\ldots,m\), is said to be a solution of (1.1) if y satisfies the differential inclusion \(\frac {d}{dt}[y'(t)g(t,y_{t})]\in F(t,y_{t})\) a.e. on \(J\backslash\{ t_{1},t_{2},\ldots, t_{m}\}\), the conditions \(\triangle y _{t=t_{k}}=I_{k}(y(t_{k}^{}))\), \(\triangle y' _{t=t_{k}}=\overline{I}_{k}(y(t_{k}^{}))\), \(k=1,\ldots,m\), \(y(t)=\phi(t)\), \(t\in[r,0]\), and \(y'(0)=\eta\).
Theorem 3.1
 (H_{1}):

\(\ g(t,u)g(t,\overline{u})\\leq p\ u\overline{u}\\), for each \(u, \overline{u}\in \mathcal{D}\), where p is a nonnegative constant.
 (H_{2}):

\(\ I_{k}(y)I_{k}(\overline{y})\ \leq c_{k}\ y\overline{y}\\), for each \(y, \overline {y}\in E\), \(k=1,\ldots,m\), where \(c_{k}\) are nonnegative constants, and there exist constants \(c_{k}'\) such that \( I_{k}(y)\leq c_{k}'\), \(k=1,\ldots,m\), for each \(y\in E\).
 (H_{3}):

\(\\overline{I}_{k}(y)\overline {I}_{k}(\overline{y})\\leq d_{k}\ y\overline {y}\\), for each \(y, \overline{y}\in E\), \(k=1,\ldots,m\), where \(d_{k}\) are nonnegative constants, and there exist constants \(d_{k}'\) such that \(\overline{I}_{k}(y)\leq d_{k}'\), \(k=1,\ldots,m\), for each \(y\in E\).
 (H_{4}):

The function g is completely continuous and there exist constants \(0\leq c_{1}^{\ast}\leq1\) and \(c_{2}^{\ast}\geq0\) such that \( g(t,u)\leq c_{1}^{\ast}\ u\ +c_{2}^{\ast}\), \(t\in J\), \(u\in\mathcal{D}\) are satisfied.
 (H_{5}):

\(F:J\times\mathcal{D}\rightarrow P_{\mathrm{b},\mathrm{cp},\mathrm{cv}}(E)\) is an L ^{1}Carathéodory function.
 (H_{6}):

\(\ F(t,u)\\leq p_{1}(t)\psi (\ u\_{\mathcal{D}})\) for almost all \(t\in J\) and all \(u\in\mathcal{D}\), where \(p_{1}\in L^{1}(J,R_{+})\) and \(\psi: R_{+}\rightarrow(0,\infty)\) is continuous and increasing withwhere \(\overline{c}=\\phi\_{\mathcal{D}}+ [\\eta\+c_{1}^{\ast}\\phi\ _{\mathcal{D}}+2c_{2}^{\ast} ]T +\sum_{k=1}^{m}[c_{k}'+(Tt_{k})d_{k}']\) and \(M(t)= \max\{ c_{1}^{\ast}, p_{1}(t) \}\).$$\int_{0}^{T}M(s)\, ds< \int_{\overline{c}}^{\infty} \frac{ds}{s+\psi(s)}, $$
Proof
Step 1. \(N_{1} \) is a contraction.
Step 2. \(N_{2}(y) \) is convex for each \(y \in\Omega\).
Step 3. \(N_{2} \) maps bounded sets into bounded sets in Ω.
Step 4. \(N_{2} \) maps bounded sets into equicontinuous sets of Ω.
Step 5. \(N_{2}y \) is a compact multivalued map.
From the above claims, we see that \(N_{2}B_{q} \) is a uniformly bounded and equicontinuous collection. Therefore, it suffices to show by the ArzeláAscoli theorem that \(N_{2} \) maps \(B_{q} \) into a precompact set into Ω. That is, for each fixed \(t\in J\), the set \(V(t)=\{h(t):h\in B_{q}\}\) is precompact in E.
Step 6. \(N_{2} \) has a closed graph.
We have \(\[h_{n}(t)\phi(0)][h_{\ast}(t)\phi (0)]\_{\Omega}\rightarrow0\), as \(n\rightarrow\infty\). Consider the linear continuous operator \(\Gamma: L^{1}(J,E)\rightarrow C(J,E)\), \(v\mapsto\Gamma(v)(t)=\int_{0}^{t}\int_{0}^{s}v(u)\,du\,ds\). From Lemma 2.1, it follows that \(\Gamma\circ S_{F}\) is a closed graph operator. Moreover, we have \((h_{n}(t)\phi(0))\in\Gamma(S_{F,y_{n}})\).
Step 7. A priori estimate.
We consider the function μ defined by \(\mu(t):=\sup\{ y(s): r\leq s\leq t\}\), \(0\leq t\leq T\). Let \(t^{\ast}\in[r,t]\) be such that \(\mu(t)= y(t^{\ast})\).
If \(t^{\ast}\in[r,0]\), then \(\mu=\\phi\ _{\mathcal{D}}\) and the above inequality holds.
As a consequence of Theorem 2.1, we deduce that \(N_{1}+N_{2}\) has a fixed point which is the mild solution of the problem (1.1). □
4 Example
Declarations
Acknowledgements
This work is supported by the NNSF of China (No. 11471109, No. 11571088) and a project supported by Scientific Research Fund of Hunan Provincial Education Department (No. 14A098, No. 14A028) and Hunan Provincial Natural Science Foundation of China (14JJ7083).
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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