Boundedness, periodicity, and convergence of solutions in a retarded Liénard equation. (English) Zbl 0803.34064

From authors’ introduction: “The paper deals with the retarded Liénard equation \[ x''+ f(x)x'+ g(x(t- h))= e(t),{(*)} \] where \(h\geq 0\) is the delay and \(e\) is a bounded function. With appropriate assumptions on \(f\) and \(g\) the authors obtain necessary and sufficient conditions for solutions of \((*)\) to be uniformly ultimately bounded. Thus by a fixed point theorem, those conditions imply that \((*)\) has a \(T\)-periodic solution whenever \(e\) is \(T\)-periodic. Also are given conditions under which all solutions of \((*)\) converge.
Reviewer: W.M.Oliva (Lisboa)


34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C11 Growth and boundedness of solutions to ordinary differential equations
34C25 Periodic solutions to ordinary differential equations
34D40 Ultimate boundedness (MSC2000)
Full Text: DOI


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