Maintenance of oscillation of neutral differential equations under the effect of a forcing term.

*(English)*Zbl 0838.34082The authors consider the forced neutral differential equation
\[
\Biggl[ x(t)+ \sum^s_{i= 1} p_i(t) x(r_i(t))\Biggr]^{(n)}+ \sum^m_{j= 1} q_j(t) h_j(x(\sigma_j(t)))= f(t)\tag{1}
\]
and the corresponding homogeneous equation. They restrict their attention to the sublinear case, i.e. when each function \(h_j\) satisfies the condition
\[
\int^c_0 {du\over h_j(u)}< \infty\quad\text{for every}\quad c> 0.
\]
In the first part of the paper sufficient conditions are obtained such that all solutions of the homogeneous equation oscillate or tend to zero. In the second part of the paper, the authors show that the behaviour of the solutions of the homogeneous equation is maintained under the influence of a small or periodic forcing term. More precisely they consider the cases when \(\lim_{t\to \infty} F(t)= 0\) or \(F(t)\) is \(\tau\)-periodic with \(r(t)= t- \tau\), \(\tau> 0\), where \(F\) is an \(n\)-times continuously differentiable function such that \(F^{(n)}(t)= f(t)\). The results obtained for equation (1) are of special interest since there are very few papers considering neutral equations with a forcing term.

Reviewer: V.Petrov (Plovdiv)