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Boundedness of solutions for a class of retarded Liénard equation. (English) Zbl 1044.34023
The authors consider the retarded Liénard equation $$x''+f_1(x) x+f_2(x) x^2+g(x(t-h))=e(t),$$ where $h$ is a nonnegative constant, $f_1, f_2$, and $g$ are continuous functions on $\bbfR=(-\infty, +\infty)$, and $e(t)$ is a continuous function on $\bbfR^+=[0, +\infty)$. They obtain some new sufficient conditions, as well as some new necessary and sufficient conditions for all solutions and their derivatives to be bounded avoiding the following traditional conditions: (H1) $f_1(x)\ge hL$ for all $x\in \bbfR$ or when $\vert x\vert $ is large; (H2)  there exists a constant $N>1$ such that $$ g(x)\left(\int^x_0 f_1(u)\,du- Nhg(x)\right)\ge 0\quad \text{for $ x\in \bbfR$ or when $x$ is large}; $$ (H3) $xg(x)\ge 0$ for all $x\in \bbfR$ or when $\vert x\vert $ is large. Two illustrative examples are also included.

34K12Growth, boundedness, comparison of solutions of functional-differential equations
Full Text: DOI
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