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 systemsMSC
34K14 34K30 35R10 47D06 93B051 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
References
- Hino, Y, Murakami, S: A generalization of processes and stabilities in abstract functional differential equations. Funkc. Ekvacioj 41, 235-255 (1998) MATHMathSciNetGoogle Scholar
- Murakami, S: Periodic solutions of some functional differential equations with diffusion. Funkc. Ekvacioj 40, 1-17 (1997) MATHMathSciNetGoogle Scholar
- Murakami, S, Nagabuchi, Y: Invariant manifolds for abstract functional differential equations and related Volterra difference equations in a Banach space. Funkc. Ekvacioj 50, 133-170 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Wu, J: Theory and Applications of Partial Functional Differential Equations. Springer, New York (1996) View ArticleMATHGoogle Scholar
- Higham, DJ, Sardar, TK: Existence and stability of fixed points for a discretised nonlinear reaction-diffusion equation with delay. Appl. Numer. Math. 18, 155-173 (1995) View ArticleMATHMathSciNetGoogle Scholar
- Gu, W, Wang, P: A Crank-Nicolson difference scheme for solving a type of variable coefficient delay partial differential equations. J. Appl. Math. 2014, Article ID 560567 (2014) MathSciNetGoogle Scholar
- Jin, YF, Jiang, J, Hou, CM, Guan, DH: New difference scheme for general delay parabolic equations. J. Inf. Comput. Sci. 9(18), 5579-5586 (2012) Google Scholar
- Sardar, TK, Higham, DJ: Dynamics of constant and variable stepsize methods for a nonlinear population model with delay. Appl. Numer. Math. 24, 425-438 (1997) View ArticleMATHMathSciNetGoogle Scholar
- Tanabe, H: Functional Analytic Methods for Partial Differential Equations. Dekker, New York (1997) MATHGoogle Scholar
- Ashyralyev, A, Sobolevskii, PE: On the stability of the delay differential and difference equations. Abstr. Appl. Anal. 6(5), 267-297 (2001) View ArticleMATHMathSciNetGoogle Scholar
- Ashyralyev, A, Sobolevskii, PE: New Difference Schemes for Partial Differential Equations. Birkhäuser, Basel (2014) MATHGoogle Scholar
- Di Blasio, G: Delay differential equations with unbounded operators acting on delay terms. Nonlinear Anal. TMA 52(1), 1-18 (2003) View ArticleMATHMathSciNetGoogle Scholar
- Shakhmurov, VB, Sahmurova, A: Abstract parabolic problems with parameter and application. Appl. Math. Comput. 219(17), 9561-9571 (2013) View ArticleMATHMathSciNetGoogle Scholar
- Ashyralyev, A, Agirseven, D: Finite difference method for delay parabolic equations. In: Numerical Analysis and Applied Mathematics ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP Conference Proceedings, vol. 1389, pp. 573-576 (2011) Google Scholar
- Agirseven, D: Approximate solutions of delay parabolic equations with the Dirichlet condition. Abstr. Appl. Anal. 2012, Article ID 682752 (2012) View ArticleMathSciNetMATHGoogle Scholar
- Ashyralyev, A, Agirseven, D: On convergence of difference schemes for delay parabolic equations. Comput. Math. Appl. 66(7), 1232-1244 (2013) View ArticleMathSciNetGoogle Scholar
- Ashyralyev, A, Agirseven, D: Well-posedness of delay parabolic difference equations. Adv. Differ. Equ. 2014, Article ID 18 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Ashyralyev, A, Agirseven, D: Well-posedness of delay parabolic equations with unbounded operators acting on delay terms. Bound. Value Probl. 2014, Article ID 126 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Ashyralyev, A, Agirseven, D: Note on the stability of delay parabolic equations with unbounded operators acting on delay terms. Electron. J. Differ. Equ. 2014, 160 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Ashyralyev, A, Agirseven, D: Stability of delay parabolic difference equations. Filomat 28(5), 995-1006 (2014) View ArticleMathSciNetMATHGoogle Scholar
- Hino, Y, Naito, T, Van Minh, N, Shin, JS: Almost Periodic Solutions of Differential Equations in Banach Spaces. Taylor & Francis, London (2002) MATHGoogle Scholar
- Boukli-Hacene, N, Ezzinbi, K: Weighted pseudo almost periodic solutions for some partial functional differential equations. Nonlinear Anal. 71, 3612-3621 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Cuevas, C, Hernández, EM: Pseudo-almost periodic solutions for abstract partial functional differential equations. Appl. Math. Lett. 22, 534-538 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Ezzinbi, K, Fatajou, S, N’Guérékata, GM: Pseudo almost automorphic solutions for some partial functional differential equations with infinite delay. Appl. Anal. 87, 591-605 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Ezzinbi, K, N’Guérékata, GM: Almost automorphic solutions for some partial functional differential equations. J. Math. Anal. Appl. 328, 344-358 (2007) View ArticleMATHMathSciNetGoogle Scholar
- Henríquez, HR: Asymptotically almost periodic solutions of abstract retarded functional differential equations of first order. Nonlinear Anal., Real World Appl. 10, 2441-2454 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Liu, Q, Van Minh, N, N’Guérékata, G, Yuan, R: Massera type theorems for abstract functional differential equations. Funkc. Ekvacioj 51, 329-350 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Liu, Q, Yuan, R: Asymptotic behavior of solutions to abstract functional differential equations. J. Math. Anal. Appl. 356, 405-417 (2009) View ArticleMATHMathSciNetGoogle Scholar
- Ruess, WM: Compactness and asymptotic stability for solutions of functional differential equations with infinite delay. In: Ferreyra, G, Ruiz Goldstein, G, Neubrander, F (eds.) Evolution Equations. Lect. Notes Pure and Appl. Math., vol. 168, pp. 361-374. Dekker, New York (1995) Google Scholar
- Zhang, L, Xu, Y: Weighted pseudo almost periodic solutions for functional differential equations. Electron. J. Differ. Equ. 2007, 146 (2007) MathSciNetMATHGoogle Scholar
- Engel, KJ, Nagel, R: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000) MATHGoogle Scholar
- Pazy, A: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) View ArticleMATHGoogle Scholar
- Goldberg, S: Unbounded Linear Operator. Dover, New York (1985) MATHGoogle Scholar
- Travis, CC, Webb, GF: Existence and stability for partial functional differential equations. Trans. Am. Math. Soc. 200, 395-418 (1974) View ArticleMATHMathSciNetGoogle Scholar
- Henríquez, HR: On non-exact controllable systems. Int. J. Control 42, 71-83 (1985) View ArticleMATHGoogle Scholar
- Hino, Y, Murakami, S, Naito, T, Van Minh, N: A variation of constants formula for abstract functional differential equations in the phase space. J. Differ. Equ. 179, 336-355 (2002) View ArticleMATHMathSciNetGoogle Scholar
- Hale, J, Verduyn Lunel, SM: Introduction to Functional Differential Equations. Springer, New York (1993) View ArticleMATHGoogle Scholar
- Corduneanu, C: Almost Periodic Functions. Chelsea, New York (1989) MATHGoogle Scholar
- Zaidman, SD: Almost-Periodic Functions in Abstract Spaces. Pitman, London (1985) MATHGoogle Scholar
- Curtain, RF, Zwart, HJ: An Introduction to Infinite-Dimensional Linear Systems Theory. Springer, New York (1995) View ArticleMATHGoogle Scholar
- Pritchard, AJ, Zabczyk, J: Stability and stabilizability of infinite dimensional systems. SIAM Rev. 23, 25-52 (1981) View ArticleMATHMathSciNetGoogle Scholar
- Triggiani, R: On the stabilizability problem in Banach space. J. Math. Anal. Appl. 52, 383-403 (1975) View ArticleMATHMathSciNetGoogle Scholar
- Henríquez, HR: Approximate controllability of linear distributed control systems. Appl. Math. Lett. 21, 1041-1045 (2008) View ArticleMATHMathSciNetGoogle Scholar
- Curtain, RF, Pritchard, AJ: Infinite Dimensional Linear Systems Theory. Lect. Notes in Control and Information Sciences, vol. 8. Springer, Berlin (1978) View ArticleMATHGoogle Scholar
- Benchimol, CD: Feedback stabilizability in Hilbert spaces. Appl. Math. Optim. 4, 225-248 (1978) View ArticleMATHMathSciNetGoogle Scholar
- Salsa, S: Partial Differential Equations in Action. From Modelling to Theory. Springer, Milan (2008) MATHGoogle Scholar
- Murray, JD: Mathematical Biology. I. An Introduction, 3rd edn. Springer, Berlin (2002) MATHGoogle Scholar
- Balakrishnan, AV: Applied Functional Analysis, 2nd edn. Springer, New York (1981) MATHGoogle Scholar
- Edwards, RE: Functional Analysis. Theory and Applications. Dover, New York (1995) Google Scholar
- Martin, RH: Nonlinear Operators and Differential Equations in Banach Spaces. Krieger, Melbourne (1987) Google Scholar