Necessary and sufficient conditions for oscillatory behaviour of solutions of a forced nonlinear neutral equation of first order with positive and negative coefficients

Rok, strany: 2004, 525 - 541

In this paper necessary and sufficient conditions are obtained so that every nonoscillatory solution of

$$ (y(t)-p(t)y(t-τ))' +Q(t)G (y(t-σ)) -R(t)G (y(t-α))=f(t) $$

tends to zero or to $∞$ as $t \to ∞$, where $p, f \in C ([0,∞),\Bbb R)$, $Q, R \in \mathbreak C([0,∞),[0,∞))$, $G \in C(\Bbb R,\Bbb R)$, $τ, σ, α ≥ 0$. $p(t)$ is considered in various ranges and the nonlinear function $G$ could be linear, sublinear, or super linear. This work indicates that the non-linearity of $G$ depends on $α $ and $σ $. The results also hold when $f(t) \equiv 0$. This paper improves and generalizes some recent results. (See [DAS, P.—Misra, N.: A necessary and sufficient condition for the solution of a functional differential equation to be oscillatory or tend to zero, J. Math. Anal. Appl. 204 (1997), 78–87], [Parhi, N.—Chand, S.: On forced first order neutral differential equations with positive and negative coefficients, Math. Slovaca 50 (2000), 81–94], [Parhi, N.—Rath, R. N.: On oscillation criteria for a forced neutral differential equation, Bull. Inst. Math. Acad. Sinica 28 (2000), 59–70], [Parhi, N.—Rath, R. N.: Oscillation criteria for forced first order neutral differential equations with variable coefficients, J. Math. Anal. Appl. 256 (2001), 525–541]).

ISO 690:

Rath, R., Misra, N. 2004. Necessary and sufficient conditions for oscillatory behaviour of solutions of a forced nonlinear neutral equation of first order with positive and negative coefficients. In Mathematica Slovaca, vol. 54, no.3, pp. 525-541. 0139-9918.

APA:

Rath, R., Misra, N. (2004). Necessary and sufficient conditions for oscillatory behaviour of solutions of a forced nonlinear neutral equation of first order with positive and negative coefficients. Mathematica Slovaca, 54(3), 525-541. 0139-9918.