# Almost periodic solutions of periodic second order linear evolution equations

## Main Article Content

## Abstract

The paper is concerned with periodic linear evolution equations of the form $x''(t)=A(t)x(t)+f(t)$, where $A(t)$ is a family of (unbounded) linear operators in a Banach space $X$, strongly and periodically depending on $t$, $f$ is an almost (or asymptotic) almost periodic function. We study conditions for this equation to have almost periodic solutions on ${\mathbb R}$ as well as to have asymptotic almost periodic solutions on ${\mathbb R}^+$. We convert the second order equation under consideration into a first order equation to use the spectral theory of functions as well as recent methods of study. We obtain new conditions that are stated in terms of the spectrum of the monodromy operator associated with the first order equation and the frequencies of the forcing term $f$.

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## References

[1] C. J. K. Batty, W. Hutter, and F. R ̈abiger, Almost periodicity of mild solutions of inhomogeneous periodic Cauchy problems, J. Differential Equations 156 (1999), 309–327. Google Scholar

[2] W. Arendt, C.J.K. Batty, M. Hieber, and F. Neubrander, Vector-valued Laplace transforms and Cauchy problems. Second edition. Monographs in Mathematics, 96. Birkhauser/Springer Basel AG, Basel, 2011. Google Scholar

[3] A.M. Fink, Almost Periodic Differential Equations, Lecture Notes in Math., 377, Springer, Berlin-New York, 1974. Google Scholar

[4] H.R. Henriquez, Existence of solutions of the nonautonomous abstract Cauchy problem of second order, Semigroup Forum 87 (2) (2013), 277–297. Google Scholar

[5] H.R. Henriquez, V. Poblete, and J.C. Pozo, Mild solutions of non-autonomous second order problems with nonlocal initial conditions, J. Math. Anal. Appl. 412 (2) (2014), 1064–1083. Google Scholar

[6] Y. Hino, T. Naito, N.V. Minh, and J.S. Shin, Almost Periodic Solutions of Differential Equations in Banach Spaces, Taylor & Francis. LOndon & New York 2001. Google Scholar

[7] M. Kozak, A fundamental solution of a second-order differential equation in a Banach space, Univ. Iagel. Acta Math., 32 (1995), 275–289. Google Scholar

[8] B. M. Levitan, and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Moscow Univ. Publ. House 1978. English translation by Cambridge University Press 1982. Google Scholar

[9] Vu Trong Luong, Nguyen Huu Tri, and Nguyen Van Minh, Asymptotic behavior of solutions of periodic linear partial functional differential equations on the half line Submitted. Preprint available at https://arxiv.org/abs/1807.03828 Google Scholar

[10] J.L. Massera, The existence of periodic solutions of systems of differential equations, Duke Math. J. 17 (1950). 457–475. Google Scholar

[11] Nguyen Van Minh, Asymptotic behavior of individual orbits of discrete systems, Proceedings of the A.M.S. 137 (9) (2009), 3025–3035. Google Scholar

[12] Nguyen Van Minh, G. N’Guerekata, and S. Siegmund, Circular spectrum and bounded solutions of periodic evolution equations, J. Differential Equations 246 (8) (2009), 3089–3108. Google Scholar

[13] R. Miyazaki, D. Kim, T. Naito and J.S. Shin, Fredholm operators, evolution semigroups, and periodic solutions of nonlinear periodic systems, J. Differential Equations, 257 (2014), 4214–4247. Google Scholar

[14] S. Murakami, T. Naito, and Nguyen Van Minh, Massera’s theorem for almost periodic solutions of functional differential equations, Journal of the Math Soc. of Japan, 47 (2004) (1), 247–268. Google Scholar

[15] T. Naito, and N. V. Minh, Evolution semigroups and spectral criteria for almost periodic solutions of periodic evolution equations, J. Differential Equations 152 (1999), 358–376. Google Scholar

[16] T. Naito, N. V. Minh, and J. S. Shin, New spectral criteria for almost periodic solutions of evolution equations, Studia Mathematica, 145 (2001), 97–111. Google Scholar

[17] W. M. Ruess, and Q. P. Vu, Asymptotically almost periodic solutions of evolution equations in Banach spaces, J. Differential Equations 122 (2) (1995), 282–301. Google Scholar

[18] J.S. Shin and T. Naito, Semi-Fredholm operators and periodic solutions for linear functional differential equations, J. Differential Equations 153 (1999), 407–441. Google Scholar

[19] Q. P. Vu, Stability and almost periodicity of trajectories of periodic processes, J. Differential Equations, 115 (1995), 402–415. Google Scholar