Oscillatory behaviour of solutions of forced neutral differential equations. (English) Zbl 0874.34063

Parhi, N.; Mohanty, P. K.

Ann. Pol. Math. 65, No. 1, 1-10 (1996).

The authors study some qualitative properties of solutions of forced neutral delay differential equations of the form \[ \left[x(t)+\sum_{i=1}^l p_{i}(t)x(t-\tau_{i})\right]^{(n)} + \delta\sum_{j=1}^m q_{j}(t)x(t-\sigma_{j}) = f(t) \tag{1} \] and \[ \left[x(t) +\sum_{i=1}^{l} p_{i}(t)g_{i}(x(t-\tau_{i}))\right]^{(n)} + \delta\sum_{j=1}^m q_{j}(t)h_{j}(x(t-\sigma_{j})) = f(t)\tag{2} \] where \(p_{i}, q_{j}, f \in C([t_{0},\infty),\mathbb{R})\), \(g_{i}, h_{j} \in C(\mathbb{R},[0,\infty))\) are such that \(xg_{i}(x) > 0,\;xh_{j}(x) > 0\), for \(x \neq 0\) and \(\tau_{i}, \sigma_{j} \geq 0.\) They establish some sufficient conditions for oscillation of all solutions of (1) and (2). Also, in the case of a nonoscillatory situation, they provide some results on their asymptotic behaviour.

Reviewer: A.Cañada (Granada)

MSC:

34K40	Neutral functional-differential equations
34C10	Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations
34K11	Oscillation theory of functional-differential equations

Keywords:

neutral delay differential equations; oscillatory behaviour; linear; nonlinear; asymptotic behaviour; forced equations