- Research
- Open Access
Almost periodic solutions of partial differential equations with delay
- Hernán R Henríquez^{1},
- Claudio Cuevas^{2}Email author and
- Alejandro Caicedo^{3}
https://doi.org/10.1186/s13662-015-0388-8
© Henríquez et al.; licensee Springer. 2015
- Received: 15 October 2014
- Accepted: 27 January 2015
- Published: 12 February 2015
Abstract
In this paper we study the existence of almost periodic solutions for linear retarded functional differential equations with finite delay and values in a Banach space. We relate the existence of almost periodic solutions with the stabilization of distributed control systems. We apply our results to transport models and to the wave equation.
Keywords
- retarded functional differential equations in abstract spaces
- almost periodic functions
- semigroups of operators
- compact operators
- controllability of systems
MSC
- 34K14
- 34K30
- 35R10
- 47D06
- 93B05
1 Introduction
Motivated by the fact that abstract retarded functional differential equations (abbreviated, ARFDE) arise in many areas of applied mathematics, this type of equations has received much attention in recent years [1–4]. Some applications and numerical methods for equations where the operator acting on the delay term is bounded have been studied in [5–8].
ARFDEs where the operator acting on delay term is unbounded have been studied in some works. For equations on Hilbert spaces, we refer to [9] and the references therein. Moreover, ARFDEs on general Banach spaces where the operator acting on the delay term is unbounded have been studied in a few works. For information of the reader, we refer to [10–13]. These studies are aimed at regular solutions of partial differential equations. Moreover, the study of more general solutions of ARFDEs where the operator acting on delay term is unbounded has been addressed in recent work. Stability aspects and differences schemes for approximate solutions have been treated in [14–20].
In particular, the problem of existence of almost periodic and asymptotically almost periodic solutions has been considered by several authors. We refer the reader to the book [21] and to the papers [22–30] and the references listed therein for recent information on this subject. Our objective is to establish existence of almost periodic mild solutions for a class of linear ARFDE of first order.
For the necessary concepts related with the abstract Cauchy problem and the theory of strongly continuous semigroup of operators we refer to Engel and Nagel [31] and Pazy [32]. We only mention here a few concepts and results directly related with our development. Let \(G(t)\) be a strongly continuous semigroup defined on a Banach space X with infinitesimal generator D. We say that G is strongly stable if \(G(t) x \to0\), \(t\to\infty\), for all \(x \in X\) and we say that G is uniformly stable if \(\|G(t)\| \to0\), \(t \to\infty\). Moreover, we employ the terminology and notations for spectral bound \(s(D)\), growth bound \(\omega _{0}(G)\) and essential growth bound \(\omega _{\mathrm{ess}}(G)\) from [31]. Specifically, \(s(D) = \sup\{ \operatorname{Re}(\lambda) : \lambda\in \sigma (D)\} \); \({ \omega _{0}(G) = \lim_{t \to\infty} \frac{\ln{\|G(t)\|}}{t}} \) and \({ \omega _{\mathrm{ess}}(G) = \lim_{t \to\infty}\frac{\ln{\|G(t)\|_{\mathrm{ess}}}}{t}}\), where the symbol \(\|\cdot\|_{\mathrm{ess}}\) denotes the essential norm of an operator. Consequently, in terms of these notations, G is uniformly stable if, and only if, \(\omega _{0}(G) < 0\).
For completeness we also mention here that a strongly continuous semigroup \(G(t)\) is said to be compact if \(G(t)\) is a compact operator for all \(t > 0\) and that G is said to be quasi-compact if there is \(t_{0} > 0\) and a compact operator R such that \(\|G(t_{0}) - R \| < 1\). We collect in the following lemma a pair of fundamental results [31, Corollary IV.2.11, Proposition V.3.5] for our further development.
Lemma 1.1
- (i)
The semigroup G is quasi-compact if and only if \(\omega _{\mathrm{ess}}(G) < 0\).
- (ii)
\(\omega _{0}(G) = \max\{ \omega _{\mathrm{ess}}(G), s(D) \}\).
Throughout this work we denote by \(\mathcal {L}(X)\) the Banach algebra of bounded linear operators defined on X and by \(X^{*}\) the dual space of X. For a linear operator A with domain \(D(A)\) and range \(\mathcal {R}(A)\) in X, we represent by \(\sigma (A)\) (resp. \(\sigma _{p}(A)\), \(\rho(A)\)) the spectrum (respectively point spectrum, resolvent set) of A. For \(\lambda\in\rho(A)\) we set \(R(\lambda, A) = (\lambda I - A)^{-1}\) for the resolvent operator of A. Finally, if \(D(A)\) is dense in X, then \(A^{\prime}\) denotes the dual operator of A [33].
This paper is organized as follows. In Section 2 we have collected some technical results about spectral properties of ARFDE, most of which are included in [4, 31, 34], in Section 3 we apply these properties to study the existence of almost periodic solutions of ARFDE, and in Section 4 we have included some applications of our results.
2 Preliminaries
Lemma 2.1
We are in a position to establish the first result about asymptotic behavior of the solution semigroup.
Theorem 2.1
Assume that the semigroup \(T(t)\) is uniformly stable and that the operator \(T(t) L : C \to X\) is compact for all \(t > 0\). Then the semigroup \(V(t)\) is quasi-compact.
Proof
There are many interesting situations in which the semigroup \(T(t)\) is not compact but the operator \(T(t) L : C \to X\) is compact for \(t > 0\). Next we mention a pair of general cases:
(i) The operator \(L : C \to X\) is compact. For instance, \(L(\varphi ) = \sum_{i = 1}^{k} A_{i} \varphi (-r_{i})\), where \(A_{i} : X \to X\), \(i = 1, \ldots, k\), are compact linear operators, or \(L(\varphi ) = \int_{-r}^{0} N(\theta ) \varphi (\theta )\,d\theta \), where \(N: [-r, 0] \to \mathcal {L}(C)\) is a map continuous for the norm of operators and \(N(\theta )\) is a compact operator for each \(-r \leq \theta \leq0\). As a matter of fact, this property is verified under more general conditions in N. In Section 4 we present some concrete examples.
(ii) A more general situation is the following. Assume that there exists a topological decomposition of \(X = X_{0} \oplus X_{1}\), where \(X_{i}\) are invariant spaces under \(T(t)\), and \(X_{1}\) has finite dimension. Let \(P_{0}\) be the projection on \(X_{0}\) with kernel \(X_{1}\). If \(T(t) P_{0} L\) is compact, then the product \(T(t) L\) is also compact.
Combining Theorem 2.1 with Theorem V.3.7 in [31] we can establish the following property of asymptotic behavior for the solution semigroup associated to the homogeneous problem (2.5)-(2.6).
Corollary 2.1
Assume that the semigroup \(T(t)\) is uniformly stable and that the operator \(T(t) L : C \to X\) is compact for all \(t > 0\). Then the semigroup \(V(t)\) is uniformly stable if and only if \(\sup \operatorname{Re} \sigma _{p}(A_{V}) < 0\).
Lemma 2.2
Let \(\lambda\in \mathbb {C}\). Then \(\lambda\in \sigma _{p}(A_{V})\) if and only if there is \(u \in D(A)\), \(u \neq0\), such that \(\Delta(\lambda) u = 0\). In this case the function \(\varphi = e^{\lambda \theta } u\) is the eigenvector of \(A_{V}\) corresponding to λ. Moreover, if \(\lambda\in \sigma (A + L_{\lambda})\), then \(\lambda\in \sigma (A_{V})\).
Remark 2.1
To complete this section, we establishes formally the following concept.
3 Existence of almost periodic solutions
In this section we turn our attention to the existence of almost periodic solutions of (1.1). Throughout this section we assume that A and L satisfy the general conditions considered in Section 2 and that \(f : \mathbb {R}\to X\) is a continuous function.
If the semigroup V is quasi-compact, we can apply the properties and notations introduced in Remark 2.1 in relation with the homogeneous equation (2.5). In particular, we set \(\Lambda = \{ \lambda\in \sigma _{p}(A_{V}) : \operatorname{Re}(\lambda) \geq0 \}\). A direct application of [36, Theorem 4.3] gives the following property.
Proposition 3.1
Now we are in a position to establish the main result of this work.
Theorem 3.1
Assume that the semigroup T is uniformly stable and that the operator \(T(t) L\) is compact for \(t > 0\). Let \(f : \mathbb {R}\to X\) be an almost periodic function. If (1.1) has a bounded mild solution on \(\mathbb {R}^{+}\), then it has an almost periodic mild solution.
Proof
Definition 3.1
The semigroup T is said to be stabilizable if there exists a compact linear operator \(K : X \to X\) such that the semigroup generated by \(A + K\) is uniformly stable.
Corollary 3.1
Assume that semigroup \(T(\cdot)\) is stabilizable and the operator \(T(t) L\) is compact for \(t > 0\). Let \(f : \mathbb {R}\to X\) be an almost periodic function. If (1.1) has a bounded mild solution on \(\mathbb {R}^{+}\), then it has an almost periodic mild solution.
Proof
Combining this result with the stabilizability criteria established in [40–42], we can present some results for the existence of almost periodic solutions of equation (1.1) in terms of the controllability of the system (3.3). The system (3.3) is said to be approximately controllable in finite time if for every \(x_{1} \in X\) and \(\varepsilon > 0\) there exist \(t_{1} > 0\) and a control function \(u \in L^{1}([0, t_{1}], \mathbb {C}^{m})\) such that \(\|x(t_{1}) - x_{1}\| \leq \varepsilon \), where \(x(\cdot)\) is the mild solution of (3.3) with initial condition \(x(0) = 0\). In [40, 43] the reader can find criteria for the approximate controllability of special classes of systems of type (3.3).
In the next result we assume that there is a topological decomposition \(X = X_{0} \oplus X_{1}\), where \(X_{i}\) are invariant subspaces under A, and \(X_{1}\) is a finite dimensional space. Let \(T_{i}(t)\) be the restriction of the semigroup \(T(t)\) on \(X_{i}\) for \(i = 0, 1\). Combining Corollary 3.1 with [44, Corollary 3.33] we obtain the following result.
Corollary 3.2
- (a)
The semigroup \(T_{0}(t)\) is uniformly stable.
- (b)
The system (3.3) is approximately controllable in finite time.
- (c)
The operator \(T(t) L\) is compact for \(t > 0\).
- (d)
Equation (1.1) has a bounded mild solution on \(\mathbb {R}^{+}\).
In [40–42] the reader can find many systems that satisfy the conditions considered in the statement of Corollary 3.2. Similarly, combining Corollary 3.1 with the results in [45], we get the following consequence of the controllability.
Corollary 3.3
- (a)
The system (3.3) is approximately controllable in finite time.
- (c)
The operator \(T(t) L\) is compact for \(t > 0\).
- (d)
Equation (1.1) has a bounded mild solution on \(\mathbb {R}^{+}\).
Proof
4 Applications
Lemma 4.1
Proof
This result also holds for some functions g discontinuous (the interested reader can consult [50, Proposition V.4.1]).
Corollary 4.1
Under the preceding conditions, if f is almost periodic and (4.1)-(4.2) has a bounded mild solution on \(\mathbb {R}^{+}\), then it has an almost periodic solution.
As a second application, we apply our results to study the existence of almost periodic solutions of the wave equation with delay. To establish a general result, we consider an abstract version of the wave equation.
Corollary 4.2
Under the above conditions, let \(f : \mathbb {R}\to H\) be an almost periodic function. Assume that \(L_{1}\) is a compact operator, and that (4.8) has a bounded mild solution on \(\mathbb {R}^{+}\). Then (4.8) has an almost periodic mild solution.
Proof
It is clear that L is a compact operator so that \(G(t) L\) is also compact for all \(t > 0\). □
To complete this application, next we will present a pair of concrete examples of compact linear operators \(L_{1} : C([-r, 0], H) \to H\).
(i) Let \(K : H \to H\) be a compact linear operator. We fix \(\theta _{0} \in[-r, 0]\) and define \(L_{1} (\varphi ) = K \varphi (\theta _{0})\) for \(\varphi \in C([-r, 0], H)\). It is immediate that \(L_{1}\) is a compact linear operator.
5 Conclusion
In this work we have studied the existence of almost periodic solutions of ARFDEs where the operator acting on the delay term is bounded. Somewhat superficially, our results are able to reduce the existence of almost periodic solutions of the inhomogeneous equation to the possibility to control or stabilize the system without delay.
Declarations
Acknowledgements
The authors acknowledge the disposition of the reviewer to review the work as well as his numerous comments and suggestions that greatly improved the original text. The first author was supported partially by CONICYT under Grant FONDECYT 1130144 and DICYT-USACH. The second author was supported partially by CNPq/Brazil under Grant 478053/2013-4.
Open Access This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.
Authors’ Affiliations
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