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Existence results for impulsive semilinear differential inclusions with nonlinear boundary conditions
 Yan Luo^{1}Email authorView ORCID ID profile
 Received: 22 May 2018
 Accepted: 22 October 2018
 Published: 29 October 2018
Abstract
In this paper, we discuss the nonlinear boundary problem for firstorder impulsive semilinear differential inclusions. We establish existence results by using Martelli’s fixed point theorem with upper and lower solutions method. We find that by giving different definitions of lower and upper solutions we can get all existence results. We also present an example.
Keywords
 Impulsive semilinear differential inclusions
 Nonlinear boundary value
 Fixed point theorem
 Lower and upper solutions
MSC
 34A60
 34A37
 47H10
1 Introduction
The evolving process of dynamics is often subjected to abrupt changes such as shocks, harvesting, and natural disasters. These shortterm perturbations are often treated as having acted instantaneously or in the form of impulses. For example, (1.1) subjects to impulse effects (1.2). Impulsive differential equations have been developed in modeling impulsive problems in physics, population dynamics, biotechnology, pharmacokinetics, industrial robotics, and so forth. In the case where the righthand side of (1.1) has discontinuities and differential inclusions, \(F(t,x(t))\) has played an important role in modeling phenomena, especially in scenarios involving automatic control systems.
Variational methods and critical point theory plays a major role in discussing the existence of solutions for boundary problem for impulsive differential inclusions; see [1–9]. There are many other methods such as in [10–12]. In [13], the authors considered periodic boundary conditions \(g(x,y)=xy\), that is, \(x(0)=x(b)\). Those results are applicable in some important cases. However, they are not valid for antiperiodic boundary conditions, for example, \(x(0)=x(b)\), which corresponds to \(g(x,y)=x+y\). Note that, in this case, g is nondecreasing in the second variable, and hence the results are not applicable. To the author’s best knowledge, there is no paper discussing such a boundary problem for impulsive differential inclusions.
Motivated by the works mentioned, the aim of this paper is to study the existence of solutions for nonlinear boundary problem (1.1)–(1.3) by Martelli’s fixed point theorem with upper and lower solutions method. The rest of the paper is organized as follows. In Sect. 2, we briefly introduce some notations and necessary preliminaries. In Sect. 3, we prove existence results of solutions for system (1.1)–(1.3), and we give some corollaries in Sect. 4. Finally, in Sect. 5, we present an example to illustrate the main result.
2 Preliminaries
We introduce some notations, definitions, and preliminary facts.
Let X be a Banach space, and let Z be a subset of X. We denote \(P(X)=\{Z\subset X\mid Z\neq\emptyset\}\), \(P_{cv}(X)=\{Z\subset P(X)\mid Z\mbox{ is convex}\}\), \(P_{cp}(X)=\{Z\subset P(X)\mid Z \mbox{ is compact}\}\), \(P_{cv,cp}(X)=P_{cv}(X)\cap P_{cp}(X)\), and so forth.
Let \(L(R)=\{N:R\rightarrow R\mid N\mbox{ is linear bounded}\}\), and for \(N\in L(R)\), we define \(\N\_{L(R)}=\inf\{r>0\mid\forall x\in R, N(x)< rx\}\). Then \((L(R),\\cdot\_{L(R)})\) is a Banach space.
By \(AC(J,R)\) we denote the space of all absolutely continuous functions \(x:J\rightarrow R\).
Definition 2.1
 (i)
\(t\rightarrow F(t,x)\) is measurable for each \(x\in R\),
 (ii)
\(x\rightarrow F(t,x)\) is upper semicontinuous on R for almost all \(t\in J\),
 (iii)for each \(\rho>0\), there exists \(\varphi_{\rho}\in L^{1}(J,[0,+\infty))\) such that$$\bigl\Vert F(t,x) \bigr\Vert _{P(R)}=\sup\bigl\{ \vert v \vert : v\in F(t,x)\bigr\} \leq\varphi_{\rho}(t),\quad \forall \vert x \vert \leq \rho \mbox{ and a.e. } t\in J. $$
Definition 2.2
Definition 2.3
A function \(x\in PC(J,R)\cap AC(J',R)\) is said to be a solution of (1.1)–(1.3) if \(g(x(0),x(b))=0\), \(\Delta x(t_{k})=I_{k}(x(t_{k}))\), \(k=1,\ldots,m\), and there exists a function \(v\in L^{1}(J,R)\) such that \(v(t)\in F(t,x(t))\) a.e. on J, \(x'(t)=Ax(t)+v(t)\).
Lemma 2.4
(see [14])
Lemma 2.5
(Martelli’s fixed point theorem [15])
Let X be a Banach space, and let \(G:X\rightarrow P_{cv,cp}(X)\) be an upper semicontinuous and condensing map. If the set \(\Re=\{x\in X:\lambda x\in G(x)\textit{ for some }\lambda>1\}\) is bounded, then G has a fixed point.
Remark 2.6
 (i)
If a multivalued map F is completely continuous with nonempty compact values, then F is upper semicontinuous 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 \(y^{\ast}\in F(x^{\ast})\)).
 (ii)
If a multivalued map F is completely continuous, then F is condensing. For general information, see [16].
Let \(J_{0}=[0,t_{1}]\), \(J_{k}=(t_{k},t_{k+1}]\), \(k=1,\ldots,m\), \(t_{m+1}=b\).
Definition 2.7
(See [17])
Lemma 2.8
(Compactness criterion; see [17])
 (i)
S is bounded, that is, \(\x\< c\) for each \(x\in S\) and some \(c>0\),
 (ii)
S is quasiequicontinuous on J.
Definition 2.9
Let X be a Banach space. A multivalued map F is said to be completely continuous if \(F(U)\) is relatively compact for every bounded subset \(U\subseteq X\).
Lemma 2.10
Proof
3 Main result
Theorem 3.1
 (H1):

\(F:J\times R\rightarrow P_{cv,cp}(R)\) is an \(L^{1}\)Carathéodory multivalued map.
 (H2):

Functions \(\alpha,\beta\in PC(J,R)\cap AC(J',R)\) are related lower and upper solutions of problem (1.1)–(1.3), which are given in Definition 2.2 and satisfy \(\alpha(t)\leq\beta(t)\), \(t\in J\).
 (H3):

\(I_{k}\in C(R,R)\), \(k=1,\ldots,m\).
 (H4):

g is a continuous singlevalued map in \((x,y)\in [\alpha(0),\beta(0)]\times[\alpha(b),\beta(b)]\) and nondecreasing in \(y\in[\alpha(b),\beta(b)]\).
 (H5):

A is the infinitesimal generator of a linear bounded semigroup \(T(t)\), \(t\geq0\), and there exists \(M>0\) such that \(\ T(t)\_{L(R)}\leq M\).
Proof
Evidently, if x is a solution of (3.1), \(\alpha(t)\leq x(t)\leq\beta (t)\), and \(\alpha(0)\leq x(0)g(\tau(0,x), \tau(T,x))\leq\beta(0)\), then x is a solution of (1.1)–(1.3).
Next, we will show that N has a fixed point by applying Lemma 2.5. The proof will be given in several steps. We first show that N is a completely continuous multivalued map, upper semicontinuous with convex closed values.
Step 1. \(N(x)\) is convex for each \(x\in PC(J,R)\).
Step 2. N is completely continuous.
Step 3. N has a closed graph.
As a consequence of Steps 1 to 3, N is a completely continuous multivalued upper semicontinuous map with convex closed values.
Step 4. The set \(\Re=\{x\in PC(J,R):\lambda x\in N(x) \mbox{ for some } \lambda>1\}\) is bounded.
Similarly, we can prove that \(\alpha(t)\leq x(t)\) on J. This shows that (3.9) holds.
According to Steps 1 to 5, the solution x of (3.1) is also a solution of (1.1)–(1.3). The proof is complete. □
4 Corollary
Definition 4.1
Corollary 4.2
 (H7):

Functions \(\alpha,\beta\in PC(J,R)\cap AC(J',R)\) are lower and upper solutions of problem (1.1)–(1.3) given in Definition 4.1 and satisfying \(\alpha(t)\leq\beta(t)\), \(t\in J\).
 (H8):

g is a continuous singlevalued map in \((x,y)\in [\alpha(0),\beta(0)]\times[\alpha(b),\beta(b)]\) and nonincreasing in \(y\in[\alpha(b),\beta(b)]\).
The proof is similar to that of Theorem 3.1, and we omit it.
Remark 4.3
If \(g(x(0),x(b))=x(0)x(b)\) in (1.1)–(1.3), that is, \(x(0)=x(b)\), which satisfies (H8), then (1.1)–(1.3) become a periodic boundary value problem for impulsive semilinear differential inclusions.
Definition 4.4
Corollary 4.5
Proof
The rest of the proof of Corollary 4.5 is similar to the proof of Theorem 3.1, and we omit it. □
Definition 4.6
Corollary 4.7
The proof is similar to that of Corollary 4.5, and we omit it.
5 An example
 (a)
For \(x\in D(A)\), \(Ax=\sum_{n=1}^{\infty}n^{2}\langle x,z_{n}\rangle z_{n}\).
 (b)
For \(x\in X\), \(C(t)x=\sum_{n=1}^{\infty}\cos(nt)\langle x,z_{n}\rangle z_{n}\).
Consequently, \(\C(t)\\leq1\) for all \(t\in R\), that is, condition (H5) in Theorem 3.1 is satisfied. More about the cosine family can be found in [19, 20]. Hence the partial differential inclusions (5.1) can be rewritten in an abstract form as system (1.1)–(1.3).
If we assume that conditions (H1), (H2), and (H6) in Theorem 3.1 hold, then system (5.1) has at least one mild solution.
Declarations
Acknowledgements
The author is very grateful to the editor and referees for their useful suggestions, which have improved the paper.
Availability of data and materials
Not applicable.
Funding
Not applicable.
Authors’ contributions
The author read and approved the final manuscript.
Competing interests
The author declares that she has no competing interests.
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