Boundedness criteria for a class of second order nonlinear differential equations with delay

Abstract

We consider certain class of second order nonlinear nonautonomous delay differential equations of the form a(t)x^{\prime\prime} + b(t)g(x,x^\prime) + c(t)h(x(t-r))m(x^\prime) = p(t,x,x^\prime) and (a(t)x^\prime)^\prime+ b(t)g(x,x^\prime) + c(t)h(x(t-r))m(x^\prime) = p(t,x,x^\prime), where $a$, $b$, $c$, $g$, $h$, $m$ and $p$ are real valued functions which depend at most on the arguments displayed explicitly and $r$ is a positive constant. Different forms of the integral inequality method were used to investigate the boundedness of all solutions and their derivatives. Here, we do not require construction of the Lyapunov-Krasovskiǐ functional to establish our results. This work extends and improve on some results in the literature.

Keywords

• boundedness
• nonlinear
• differential equation of third order
• integral inequality