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Existence and asymptotic stability of periodic solutions for impulsive delay evolution equations
- Qiang Li1Email authorView ORCID ID profile and
- Mei Wei1
https://doi.org/10.1186/s13662-019-1994-7
© The Author(s) 2019
- Received: 17 April 2018
- Accepted: 28 January 2019
- Published: 6 February 2019
Abstract
This paper we devote to considering the periodic problem for the impulsive evolution equations with delay in Banach space. By using operator semigroup theory and the fixed point theorem, we establish some new existence theorems of periodic mild solutions for the equations. In addition, with the aid of an integral inequality with impulsive and delay, we present essential conditions on the nonlinear and impulse functions to guarantee that the equations have an asymptotically stable ω-periodic mild solution.
Keywords
- Evolution equations
- Impulsive and delay
- Periodic solutions
- Existence and uniqueness
- Asymptotic stability
- Operator semigroups
MSC
- 34G20
- 34K30
- 47H07
- 47H08
1 Introduction
Let X be a real Banach space with norm \(\|\cdot\|\), \(\mathcal {L}(X)\) stand for the Banach space of all bounded linear operators from X to X equipped with its natural topology. Let \(r>0\) be a constant, we denote by \(PC([-r, 0], X)\) the Banach space of piecewise continuous functions from \([-r, 0]\) to X with finite points of discontinuity where functions are left continuous and have the right limits, with the sup-norm \(\|\phi\|_{\mathrm{Pr}}=\sup_{s\in[-r,0]}\|\phi(s)\|\).
The theory of partial differential equations with delays has an extensive physical background and leads to a realistic mathematical model, and it has undergone a rapid development in the last 50 years. The evolution equations with delay are more realistic than the equations without delay in describing numerous phenomena observed in nature, hence the numerous properties of their solutions have been studied; see [7, 22] and the references therein for more comments.
One of the important research directions related to the asymptotic behavior of the solutions for the evolution equations with delay is to find conditions for the existence and stability of ω-periodic solutions in the case that the nonlinear mapping is ω-periodic function in t. In the last few decades, the existence and asymptotic stability of periodic solutions have been investigated by some authors (see [5, 8–10, 15–17, 23] and the references therein). In [5], under the assumption that the solutions of the associated homotopy equations were uniformly bounded, Burton and Zhang obtained the existence of periodic solutions of an abstract delay evolution equation. In [23], Xiang and Ahmed showed an existence result of periodic solution to the delay evolution equations in Banach spaces under the assumption that the corresponding initial value problem had a prior estimate. In [15–17], Liu studied periodic solutions by using bounded solutions or ultimate bounded solutions for delay evolution equations in Banach spaces. In [8], Huy and Dang studied the existence, uniqueness and stability of periodic solutions to a partial functional differential equation in Banach space in the case that the nonlinear function satisfied Lipschitz-type condition. Specially, in [10], Li discussed the existence and asymptotic stability of periodic solutions to the evolution equation with multiple delays in a Hilbert space. By using the analytic semigroup theory and the integral inequality with delays, the author obtained the essential conditions on the nonlinearity F to guarantee that the equation has ω-periodic solutions or an asymptotically stable ω-periodic solution.
On the other hand, it is well known that an impulsive evolution equation has an extensive physical, chemical, biological, engineering background and realistic mathematical model, and hence has been emerging as an important area of investigation in the last few decades. Since the end of the previous century, the theory of impulsive evolution equation in Banach space has been largely developed (see [1, 2, 6, 12, 13, 24] and the references therein). We would like to mention that Liang et al. [12] studied the periodic solutions to a kind of impulsive evolution equation with delay in Banach spaces. In the case that the nonlinear function satisfied Lipschitz conditions, the authors found that the evolution equation has a periodic solution by using the ultimate boundedness of solutions and Horn’s fixed point theorem. Recently, in [11, 14], Liang et al. studied nonautonomous evolutionary equations with time delay and being impulsive. With the nonlinear term being continuous and Lipschitzian, they proved the existence theorem for periodic mild solutions to the nonautonomous delay evolution equations.
Although there have been many meaningful results on the delay or impulsive evolution equation periodic problem in Banach space, to the best of our knowledge, these results have a relatively large limitation. First of all, the most popular approach is the use of ultimate boundedness of solutions and the compactness of Poincaré map realized through some compact embeddings. However, in some concrete applications, it is difficult to choose appropriate initial conditions to guarantee the boundedness of the solution. Secondly, we observe that the most popular condition imposed on the nonlinear term F is its Lipschitz-type condition. In fact, for equations arising in complicated reaction–diffusion processes, the nonlinear function F represents the source of material or population, which dependents on time in diversified manners in many contexts. Thus, we may not hope to have the Lipschitz-type condition of F. Finally, there are few papers studying the asymptotically stable of periodic solutions for the impulsive evolution equations with delay.
Motivated by the papers mentioned above, we consider the periodic problem for the impulsive delay evolution equation (1.1) in Banach space. By using a periodic extension and the fixed point theorem, we study the existence of ω-periodic mild solutions for Eq. (1.1). It is worth mentioning that the assumption of prior boundedness of solutions is not employed and the nonlinear term F satisfies some growth condition, which is weaker than Lipschitz-type condition. On the other hand, by means of an integral inequality with impulsive and delay, we present the asymptotic stability result for Eq. (1.1), which will make up the research in this area blank.
The rest of this paper is organized as follows. In Sect. 2, we collect some well-known definitions and notions, and then provide preliminary results which will be used throughout this paper. In Sect. 3, we apply the operator semigroup theory to find the ω-periodic mild solutions for Eq. (1.1) and in Sect. 4, by strengthening the condition, we obtain the global asymptotic stability theorems for Eq. (1.1). In the last section, we give an example to illustrate the applicability of abstract results obtained in Sects. 3 and 4.
2 Preliminaries
Throughout this paper, we assume that X is a Banach space with norm \(\|\cdot\|\).
Lemma 2.1
Let −A generate an exponentially stable \(C_{0}\)-semigroup \(T(t)\) (\(t\geq0\)) in X and \(\nu_{0}\) be a growth index of the semigroup \(T(t)\) (\(t\geq0\)). Then the linear impulsive evolution equation (2.5) exists a unique ω-periodic mild solution \(u:=Ph\in PC_{\omega}(\mathbb{R},X)\). Furthermore, the operator \(P:PC_{\omega}(\mathbb{R},X)\to PC_{\omega}(\mathbb{R},X)\) is a bounded linear operator.
Proof
3 Existence and uniqueness
In this section, we discuss the existence of ω-periodic mild solution to Eq. (1.1) for the case that the semigroup \(T(t)\) (\(t\geq0\)) generated by −A is a compact semigroup, which implies that \(T(t)\) is a compact operator for any \(t>0\).
Now, we are in a position to state and prove our main results of this section.
Theorem 3.1
- (H1)there exist nonnegative constants \(c_{0}\), \(c_{1}\), \(c_{2}\) such that$$\bigl\Vert F(t,x,\phi) \bigr\Vert \leq c_{0}+c_{1} \Vert x \Vert +c_{2} \Vert \phi \Vert _{\mathrm{Pr}},\quad t \in \mathbb{R},x\in X,\phi\in PC\bigl([-r,0],X\bigr), $$
- (H2)for every \(I_{k}\), \(I_{k}(\theta)=\theta\), and there exist positive constants \(a_{k}\) such that$$\bigl\Vert I_{k}(x)-I_{k}(y) \bigr\Vert \leq a_{k} \Vert x-y \Vert , \qquad a_{k+p}=a_{k}, \quad x,y\in X,k\in \mathbb{Z}, $$
- (H3)
\((c_{1}+c_{2})+\frac{1}{\omega}\sum_{k=1}^{p}a_{k}<\frac{|\nu_{0}|}{M}\),
Proof
Thus, the Arzela–Ascoli theorem guarantees that \(Q_{1}\) is a compact operator.
Therefore, by the famous Sadovskii fixed point theorem [19], we know that Q has a fixed point \(u\in\overline {\varOmega}_{R}\), that is, Eq. (1.1) has an ω-periodic mild solution. The proof is completed. □
- (H1′):
-
there are positive constants \(c_{1}\) , \(c_{2}\) , such that for every \(t\in \mathbb{R}\) , \(x_{0},x_{1}\in X\) and \(\phi,\psi\in PC([-r,0],X)\)$$\bigl\Vert F(t,x,\phi)-F(t,y,\psi) \bigr\Vert \leq c_{1} \Vert x-y \Vert +c_{2} \Vert \phi-\psi \Vert _{\mathrm{Pr}}, $$
Theorem 3.2
Let X be a Banach space, −A generates an exponentially stable compact semigroup \(T(t)\) (\(t\geq0\)) in X. Assume that \(F:\mathbb{R}\times X\times PC([-r, 0], X)\rightarrow X\) is continuous and \(F(t,\cdot,\cdot)\) is ω-periodic in t, \(I_{k}\in C(X,X)\) (\(k\in\mathbb{Z}\)). If the conditions (H1′), (H2) and (H3) hold, then Eq. (1.1) has unique ω-periodic mild solution u.
Proof
4 Asymptotic stability
In order to obtain the results as regards asymptotic stability, we need the following integral inequality of Gronwall–Bellman type with delay and being impulsive.
Lemma 4.1
Proof
For the initial value problem (4.1), we have the following result.
Theorem 4.1
Let X be a Banach space, −A generates a \(C_{0}\)-semigroup \(T(t)\) (\(t\geq0\)) in X. Assume that \(F:J\times X\times PC([-r, 0], X)\rightarrow X\) is continuous, \(I_{k}\in C(X,X)\) (\(k=1,2,\ldots\)), and \(\varphi\in PC([-r,0],X)\). If the conditions (H1′) and (H2) hold, then the initial value problem (4.1) has a unique mild solution \(u\in PC([-r,\infty),X)\).
Proof
From Lemma 4.1, it follows that \(\|u(t)-v(t)\|=0\) for every \(t\geq0\). Hence, \(u\equiv v\). This completes the proof of Theorem 4.1. □
Theorem 4.2
- (H3′):
-
\((c_{1}+c_{2}e^{-\nu_{0}r})+\frac{1}{\omega}\sum_{k=1}^{p}a_{k}<\frac{|\nu_{0}|}{M}\),
Proof
From the condition (H3′), it follows that the condition (H3) holds. By Theorem 3.2, the periodic problem (1.1) has a unique ω-periodic mild solution \(u^{*}\in PC_{\omega}(\mathbb{R},X)\). For any \(\phi\in PC([-r,0],X)\), the initial value problem (4.1) has a unique global mild solution \(u=u(t,\phi)\in PC([-r,\infty),X)\) by Theorem 4.1.
5 Application
In this section, we present one example, which does not aim at generality, but indicates how our abstract results can be applied to concrete problems.
Let \(\overline{\varOmega}\in \mathbb{R}^{n}\) be a bounded domain with a \(C^{2}\)-boundary ∂Ω for \(n\in N\). Let \(\nabla^{2}\) is a Laplace operator, and \(\lambda_{1}\) is the smallest eigenvalue of operator \(-\nabla^{2}\) under the Dirichlet boundary condition \(u|_{\partial\varOmega}=0\). It is well known [3, Theorem 1.16], that \(\lambda_{1}>0\).
Declarations
Acknowledgements
The authors are most grateful to the editor and anonymous referees for the careful reading of the manuscript and valuable suggestions that helped in significantly improving an earlier version of this paper.
Availability of data and materials
Not applicable.
Funding
Research supported by NNSF of China (11261053, 11501455, 11561040), the Nature Science Foundation for Young Scientists (201701D221007), Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi Province, China (2017149).
Authors’ contributions
QL and MW contributed equally and significantly in writing this article. Both authors read and approved the final manuscript.
Competing interests
QL and MW declare that they have no competing interests.
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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