Boundedness of solutions for a class of retarded Liénard equation
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.
scientific article; zbMATH DE number 2006336
scientific article