Almost Periodicity of Mild Solutions of Inhomogeneous Periodic Cauchy Problems

Abstract We consider a mild solution u of a well-posed, inhomogeneous, Cauchy problem, u(t)=A(t) u(t)+f(t), on a Banach space X, where A(·) is periodic. For a problem on R+, we show that u is asymptotically almost periodic if f is asymptotically almost periodic, u is bounded, uniformly continuous and totally ergodic, and the spectrum of the monodromy operator V contains only countably many points of the unit circle. For a problem on R, we show that a bounded, uniformly continuous solution u is almost periodic if f is almost periodic and various supplementary conditions are satisfied. We also show that there is a unique bounded solution subject to certain spectral assumptions on V, f and u.

[1]  Josef Kreulich.  Eberlein-weakly almost-periodic solutions of evolution equations in Banach spaces , 1996, Differential and Integral Equations.

[2]  G. Pedersen.  C-Algebras and Their Automorphism Groups , 1979 .

[3]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[4]  C. Batty,et al.  Local spectra and individual stability of uniformly bounded ₀-semigroups , 1998 .

[5]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[6]  A. Fink.  Almost Periodic Differential Equations , 1974 .

[7]  W. Arendt,et al.  Almost Periodic Solutions of First- and Second-Order Cauchy Problems , 1996 .

[8]  W. Ruess,et al.  Asymptotically Almost Periodic Solutions of Evolution Equations in Banach Spaces , 1995 .

[9]  V. V. Zhikov,et al.  Almost Periodic Functions and Differential Equations , 1983 .

[10]  W. Arendt,et al.  Asymptotically Almost Periodic Solutions of Inhomogeneous Cauchy Problems on The Half‐Line , 1999 .

[11]  V. Q. Phóng.  On the Spectrum, Complete Trajectories, and Asymptotic Stability of Linear Semi-dynamical Systems , 1993 .

[12]  René T. Rau,et al.  Hyperbolic evolution semigroups on vector valued function spaces , 1994 .

[13]  Adalbert Kerber,et al.  The Cauchy Problem , 1984 .

[14]  V. Q. Phóng.  Stability and Almost Periodicity of Trajectories of Periodic Processes , 1995 .

[15]  Nguyen Van Minh,et al.  Evolution Semigroups and Spectral Criteria for Almost Periodic Solutions of Periodic Evolution Equations , 1999 .

[16]  Hiroki Tanabe,et al.  Equations of evolution , 1979 .

[17]  B. Basit,et al.  Some problems concerning different types of vector valued almost periodic functions , 1995 .

[18]  Amnon Pazy,et al.  Semigroups of Linear Operators and Applications to Partial Differential Equations , 1992, Applied Mathematical Sciences.

[19]  C. Batty,et al.  Tauberian theorems and stability of solutions of the Cauchy problem , 1998 .

[20]  W. Arendt,et al.  DISCRETE SPECTRUM AND ALMOST PERIODICITY , 1997 .

[21]  J. Moffat,et al.  On groups of automorphisms of operator algebras , 1977, Mathematical Proceedings of the Cambridge Philosophical Society.

[22]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[23]  Daniel Daners,et al.  Abstract evolution equations, periodic problems and applications , 1992 .

[24]  Roland Schnaubelt,et al.  Exponential stability, exponential expansiveness, and exponential dichotomy of evolution equations on the half-line , 1998 .

[25]  W. Ruess,et al.  Ergodic theorems for semigroups of operators , 1992 .

[26]  N. Lloyd,et al.  ALMOST PERIODIC FUNCTIONS AND DIFFERENTIAL EQUATIONS , 1984 .

[27]  E. Davies,et al.  One-parameter semigroups , 1980 .

[28]  J. Prüss.  Evolutionary Integral Equations And Applications , 1993 .

[29]  W. Ruess,et al.  Integration of asymptotically almost periodic functions and weak asymptotic almost periodicity , 1989 .

[30]  Vu Quoc Phong,et al.  The Operator EquationAX−XB=C, Admissibility, and Asymptotic Behavior of Differential Equations , 1998 .

[31]  W. Ruess,et al.  Weak almost periodicity and the strong ergodic limit theorem for periodic evolution systems , 1990 .