Periodic solutions of nonlinear hyperbolic problems
type: Item
instance of: scholarly article
title: Periodic solutions of nonlinear hyperbolic problems
zbMATH Open document ID: 0549.35085
DOI: 10.1016/0898-1221(83)90061-5
publication date: 1983-01-01T00:00:00Z
review text: This paper sketches a rather brief survey of methods employed in establishing existence of periodic solutions of nonlinear partial differential equations of the form \(u_{tt}+Au+f(t,x,u,u_ t)=p(t,x)\) where A is a linear operator with domain and range in \(L_ 2(0,\pi)\), the unknown function \(u:[0,\infty)\times [0,\pi]\to {\mathbb{R}},\) f is the prescribed nonlinearity and p(t,x) is periodic in t. Homogeneous boundary conditions are prescribed for A. Assuming solutions as finite combinations of eigenfunctions of A with functions of t as coefficients, there results a finite system of ordinary differential equations for these coefficients as in the classical elementary method of separation of variables. One method then is to obtain estimates for these approximate solutions and their derivatives that will help passage to the limit along a strongly convergent subsequence of these solutions to obtain a weak solution of the original problem. Another method is to constrct a Liapunov function for each of the ordinary differential equations with the aid of which it could be shown that the ordinary differential system is dissipative thereby establishing the existence of periodic solutions. A different approach to the treatment of the partial differential equation problem is to rewrite it as an operator equation \(Eu=Nu\) where E is a linear operator and N is a nonlinear operator from one Banach space to another. With appropriate decomposition properties for these spaces, the available theory of nonlinear operator equations in abstract spaces could be invoked to establish existence of a weak solution of the original problem. The problem of establishing existence of periodic solutions of the ordinary differential systems is described in some detail but the paper contains no theorems or proofs specifically indicated as such. Some of the details not provided here are promised to appear elsewhere and it is remarked that the methods indicated could be modified to deal with nonlin…
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B10
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C25
zbMATH DE Number: 3875864
zbMATH Keywords: periodic solutions
zbMATH Keywords: finite system of ordinary differential equations
zbMATH Keywords: estimates
zbMATH Keywords: approximate solutions
zbMATH Keywords: weak solution
zbMATH Keywords: Liapunov function
zbMATH Keywords: operator equation
author: R. Kannan
author: V. Lakshmikantham
MaRDI profile type: Publication
full work available at URL: 0898-1221(83)90061-5
OpenAlex ID: W2023065251
cites work: Soluzioni quasi-periodiche, o limitate, di sistemi differenziali non lineari quasi-periodici, o limitati
cites work: Forced vibrations for a nonlinear wave equation
cites work: Q3872872
cites work: Solutions of nonlinear hyperbolic equations at resonance
cites work: Periodic solutions of nonlinear wave equations with damping
cites work: Solutions of nonlinear wave equations with nonlinear damping
cites work: Q4075818
cites work: Solutions in the large of Lienard systems with forcing terms
cites work: An Abstract Existence Theorem at Resonance
cites work: Periodic solutions of a class of hyperbolic equations containing a small parameter
cites work: Nonlinear boundary value problems and operators \(TT^*\)
cites work: Existence of periodic solutions of semilinear parabolic equations and the method of upper and lower solutions
cites work: Existence of periodic solutions of nonlinear boundary value problems and the method of upper and lower solutions
cites work: Addendum
cites work: Q3233048
cites work: Q5616377
cites work: Q5583770
cites work: Soluzioni periodiche dell'equazione delle onde con termine dissipativo non lineare
cites work: Soluzioni periodiche dell'equazione delle onde non omogenea con termine dissipativo quadratico
cites work: Q5529795
cites work: Q3252847
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 3875864