On the retarded Liénard equation. (English) Zbl 0756.34075

Author’s abstract: “We consider the equation \(x''+f(x)x'+g(x(t-h))=0\) in which \(f,g\) are continuous with \(f(x)>0\) for \(x\in R\), \(h\) is a nonnegative constant, and \(xg(x)>0\) if \(| x|\geq k\) for some \(k\geq 0\). Necessary and sufficient conditions are given for boundedness of all solutions and their derivatives. When \(k=0\) we give necessary and sufficient conditions for all solutions and their derivatives to converge to zero”.


34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
34D20 Stability of solutions to ordinary differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C11 Growth and boundedness of solutions to ordinary differential equations
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