Levitan Almost Periodic Solutions of Linear Differential Equations

2019/07/03 by David Cheban, Cheban, David
Mathematics · Engineering · #Nonlinear Differential Equations Analysis #Advanced Differential Equations and Dynamical Systems #Stability and Controllability of Differential Equations

paper · pdf · doi:10.48550/arxiv.1907.02512

Abstract

The known Levitan's Theorem states that the linear differential equation x'=A(t)x+f(t) (*) with Bohr almost periodic coefficients A(t) and f(t) admits at least one Levitan almost periodic solution if it has a bounded solution. The main assumption in this theorem is the separation among bounded solutions of homogeneous equations x'=A(t)x . (**) In this paper we prove that linear differential equation (*) with Levitan almost periodic coefficients has a Levitan almost periodic solution, if it has at least one bounded solution. In this case, the separation from zero of bounded solutions of equation (**) is not assumed. The analogue of this result for difference equations also is given. We study the problem of existence of Bohr/Levitan almost periodic solutions for equation (*) in the framework of general nonautonomous dynamical systems (cocycles).

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