On oscillation and asymptotic behaviour of a neutral differential equation of first order with positive and negative coefficients

• Main Authors: Prayag Prasad Mishra, Radhanath Rath, Laxmi Narayan Padhy
• Format: Article
• Language: English
• Published: Texas State University 2007-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: Oscillatory solution; nonoscillatory solution; asymptotic behaviour.
• Online Access: http://ejde.math.txstate.edu/Volumes/2007/01/abstr.html

In this paper sufficient conditions are obtained so that every solution of $$ (y(t)- p(t)y(t-au))'+ Q(t)G(y(t-sigma))-U(t)G(y(t-alpha)) = f(t) $$ tends to zero or to $pm infty$ as $t$ tends to $infty$, where $au ,sigma ,alpha$ are positive real numbers, $p,fin C([0,infty),R),Q,Uin C([0,infty),[0,infty))$, and $Gin C(R,R)$, $G$ is non decreasing with $xG(x)>0$ for $ x eq 0$. The two primary assumptions in this paper are $int_{t_0}^{infty}Q(t)=infty$ and $int_{t_0}^{infty}U(t)<infty$. The results hold when $G$ is linear, super linear,or sublinear and also hold when $f(t) equiv 0$. This paper generalizes and improves some of the recent results in [5,7,8,10].