×

zbMATH — the first resource for mathematics

Exponential stability and uniform boundedness of solutions for nonautonomous periodic abstract Cauchy problems. An evolution semigroup approach. (English) Zbl 1279.47063
Let \(\mathcal{U}=\{U(t,s)\}_{t\geq s}\) be a strongly continuous and \(q\)-periodic evolution family on a Banach space \(X\). The authors show that, if \[ \sup_{\mu\in \mathbb R}\sup_{t\geq \tau\geq 0}\left\|\int^t_\tau e^{i\mu s}U(t,s)x\, ds\right\|<\infty \text{ for all } x\in X, \] then \[ \sup_{\mu\in \mathbb R}\sup_{t\geq 0}\left\|\int^t_\tau e^{-i\mu s}\mathcal{T}(s)f\, ds\right\|<\infty \text{ for all } f\in \mathrm{span}\{\cup_{t\geq 0}\mathcal{A}_t\}, \] where \[ \mathcal{T}(t)f(s):=\left\{ \begin{matrix} U(s,s-t)f(s-t), & s\geq t \\ 0, & s<t, \\ \end{matrix} \right. \;\;t\geq 0, \;s\in \mathbb R, \;f\in \overline{\mathrm{span}}\{\cup_{t\geq 0}\mathcal{A}_t\}, \] and \(\mathcal{A}_t\) is the set of all \(X\)-valued functions \(f\) on \(\mathbb R\) for which there exists a function \(F\in P_q(R,X)\cap AP_1(R,X)\) such that \(F(t)=0\), \(f=F|_{[t,\infty)}\) and \(f(s)=0\) for \(s<t\). Here, \(P_q(\mathbb R,X)\) denotes the \(q\)-periodic, \(X\)-valued, uniformly continuous functions on \(\mathbb R\) and \(AP_1(\mathbb R,X)\) is a certain space of almost periodic functions (cf. [C. Corduneanu, Almost periodic oscillations and waves. New York, NY: Springer (2009; Zbl 1163.34002)]).

MSC:
47D06 One-parameter semigroups and linear evolution equations
47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
47A30 Norms (inequalities, more than one norm, etc.) of linear operators
47A10 Spectrum, resolvent
35B15 Almost and pseudo-almost periodic solutions to PDEs
35B10 Periodic solutions to PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Arendt W., Batty C.J.K., Hieber M., Neubrander F.: Vector Valued Laplace transform. Birkhäuser, Basel (2001) · Zbl 0978.34001
[2] Arshad S., Buşe C., Saierli O.: Connections between exponential stability and boundedness of solutions of a couple of differential time depending and periodic systems. Electron. J. Qualitative Theory Differ. Equ. 90, 1–16 (2011) · Zbl 1340.34178
[3] Balint, S.: On the Perron–Bellman theorem for systems with constant coefficients, Ann. Univ. Timisoara, vol 21, fasc. 1–2, 3–8 (1983) · Zbl 0536.34038
[4] Baroun M., Maniar L., Schnaubelt R.: Almost periodicity of parabolic evolution equations with inhomogeneous boundary values. Integr. Equ. Oper. Theory 65(2), 169–193 (2009) · Zbl 1197.47055 · doi:10.1007/s00020-009-1704-z
[5] Buşe C.: On the Perron–Bellman theorem for evolutionary processes with exponential growth in Banach spaces. NZ J. Math. 27, 183–190 (1998) · Zbl 0972.47027
[6] Buşe C., Cerone P., Dragomir S.S., Sofo A.: Uniform stability of periodic discrete system in Banach spaces. J. Differ. Equ. Appl. 11(12), 1081–1088 (2005) · Zbl 1094.47040 · doi:10.1080/10236190500331271
[7] Buşe C., Pogan A.: Individual exponential stability for evolution families of bounded and linear operators. NZ J. Math. 30, 15–24 (2001) · Zbl 0990.35020
[8] Chicone, C., Latushkin, Y.: Evolution semigroups in dynamical systems and differential equations. Mathematical Surveys and Monographs, vol. 70. American Mathematical Society, Providence (1999) · Zbl 0970.47027
[9] Clark S., Latushkin Y., Montgomery-Smith S., Randolph T.: Stability radius and internal versus external stability in banach spaces: an evolution semigroup approach. SIAM J. Control Optim. 38(6), 1757–1793 (2000) · Zbl 0978.47030 · doi:10.1137/S036301299834212X
[10] Corduneanu C.: Almost Periodic Oscilations and Waves. Springer Sciences+Business Media LLC, Berlin (2009) · Zbl 1163.34002
[11] Daners D., Medina K.P.: Abstract evolution equations, periodic problems and applications. Pitman Research Notes in Mathematics Series, vol. 279. Longman Scientific &amp; Technical, (1992) · Zbl 0789.35001
[12] Engel K., Nagel R.: One-Parameter Semigroups for Linear Evolution Equations. Springer, New York (2000) · Zbl 0952.47036
[13] Greiner G., Voight J., Wolff P.M.: On the spectral bound of the generator of semigroups of positive operators. J. Oper. Theory 5, 245–256 (1981) · Zbl 0469.47032
[14] Howland S.J.: On a theorem of Gearhart. Integral Equ. Oper. Theory 7, 138–142 (1984) · Zbl 0535.47025 · doi:10.1007/BF01204917
[15] Huang F.: Exponential stability of linear systems in Banach spaces. Chin. Ann. Math. 10, 332–340 (1989) · Zbl 0694.47027
[16] Mather J.: Characterization of Anosov diffeomorphisms. Indag. Math. 30, 479–483 (1968) · Zbl 0165.57001
[17] Pazy A.: Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983) · Zbl 0516.47023
[18] Phong, V.Q.: On stability of C 0-semigroups. Proc. Am. Math. Soc. 129, No. 10, 2871–2879 (2001) · Zbl 0998.47022
[19] Nguyenm T.L.: On nonautonomous functional differential equations. J. Math. Anal. Appl. 239(1), 158–174 (1999) · Zbl 0949.34063 · doi:10.1006/jmaa.1999.6568
[20] Nguyen T.L.: On the wellposedness of nonautonomous second order Cauchy problems. East-West J. Math. 1(2), 131–146 (1999) · Zbl 0949.34045
[21] Van Minh N., Räbiger F., Schnaubelt R.: Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half-line. Integr. Equ. Oper. Theory 32, 332–353 (1998) · Zbl 0977.34056 · doi:10.1007/BF01203774
[22] Neubrander F.: Laplace transform and asymptotic behavior of strongly continuous semigroups. Houston Math. J. 12(4), 549–561 (1986) · Zbl 0624.47031
[23] Reghiş M., Buşe C.: On the Perron-Bellman theorem for strongly continuous semigroups and periodic evolutionary processes in Banach spaces. Italian J. Pure Appl. Math. 4, 155–166 (1998) · Zbl 0981.47030
[24] Schnaubelt, R.: Well-posedness and asymptotic behavior of non-autonomous linear evolution equations, Evolution equations, semigroups and functional analysis. Progr. Nonlinear Differential Equations Appl., vol. 50. Birkhäuser, Basel, pp. 311–338 (2002) · Zbl 1044.34016
[25] Stein E.M., Shakarchi R.: Fourier analysis: An introduction. Princeton University Press, Princeton (2003) · Zbl 1026.42001
[26] van Neerven J.M.A.M.: Individual stability of C 0 semigroups with uniformly bounded local resolvent. Semigroup Forum 53(1), 155–161 (1996) · Zbl 0892.47040 · doi:10.1007/BF02574130
[27] Weis L., Wrobel V.: Asymptotic behavior of C 0-semigroups in Banach spaces. Proc. Am. Math. Soc. 124(12), 3663–3671 (1996) · Zbl 0863.47027 · doi:10.1090/S0002-9939-96-03373-4
[28] Wrobel V.: Asymptotic behavior of C 0-semigroups in B-convex spaces. Indiana Univ. Math. J. 38, 101–114 (1989) · Zbl 0653.47018 · doi:10.1512/iumj.1989.38.38004
[29] Zabczyk J.: Mathematical Control Theory: An Introduction Systems and Control. Birkhäuser, Basel (1992) · Zbl 1071.93500
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.