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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 $$\mathbb{R}=(-\infty, +\infty)$$, and $$e(t)$$ is a continuous function on $$\mathbb{R}^+=[0, +\infty)$$. 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) $$f_1(x)\geq hL$$ for all $$x\in \mathbb{R}$$ or when $$| x|$$ is large;
(H2)  there exists a constant $$N>1$$ such that $g(x)\left(\int^x_0 f_1(u)\,du- Nhg(x)\right)\geq 0\quad \text{for $$x\in \mathbb{R}$$ or when $$x$$ is large};$ (H3) $$xg(x)\geq 0$$ for all $$x\in \mathbb{R}$$ or when $$| x|$$ is large. Two illustrative examples are also included.

MSC:
 34K12 Growth, boundedness, comparison of solutions to functional-differential equations
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References:
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