Theorem of Frèchet and Asymptotically Almost Periodic Solutions of Some Nonlinear Equations of Hyperbolic Type

Vladimír Lovicar
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Vladimír Lovicar: Matematický ústav ČSAV

A chapter in Nonlinear Evolution Equations and Potential Theory, 1975, pp 107-115 from Springer

Abstract: Abstract The aim of this lecture is to prove the existence of asymptotically almost periodic, solutions of a second order equation with nonlinear dissipative term by an easy method based on a theorem of Fréchet. For the sake of simplicity, very restrictive assumptions on the coefficients of the equation are introduced. A more general situation is considered in paper [5], where the reader can find also the proofs of Lemmas 1–4, used in this lecture. The existence of asymptotically almost periodic solutions gives us the possibility to prove the existence of almost periodic solutions. For an other method for obtaining almost periodic solutions of equations of the same type as in this lecture see [2]. As in the method mentioned above, the proof of existence of bounded solutions plays here the crucial role. It is very well known that their existence may be derived from local existence of solutions and from appropriate apriori estimates. The local existence of solutions will be supposed here, since this problem is treated in many other papers.

Keywords: Banach Space; Periodic Solution; Periodic Function; Order Equation; Nonlinear Evolution Equation (search for similar items in EconPapers)
Date: 1975
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4613-4425-4_8

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DOI: 10.1007/978-1-4613-4425-4_8

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