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Integral equivalence of two systems of differential equations. (English) Zbl 0643.34042
The aim of the author is to give some new necessary and sufficient conditions for the nonlinear equation of the form $$x''+f(x)h(x')x'+g(x)k(x')=0,$$ which is known as the generalized Liénard equation, to have only bounded solutions. Here f and g are continuous functions on $${\mathbb{R}}$$ while h and k are continuous positive functions on $${\mathbb{R}}$$. It is noted that the same problem has previously studied by several authors and some results requiring also the signum condition $$xg(x)>0$$ among the others have been obtained. The objective of the present paper is now to eliminate this condition. The result is given in the form of a theorem whose statement is rather simple while whose proof depends on a long sequence of lemmas and propositions.
Reviewer: M.Ideman
##### MSC:
 34C11 Growth and boundedness of solutions to ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34A34 Nonlinear ordinary differential equations and systems
##### Keywords:
generalized Liénard equation; bounded solutions