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Keywords:
boundedness; nonlinear; differential equation of third order; integral inequality
boundedness; nonlinear; differential equation of third order; integral inequality
Summary:
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-Krasovski\v ı functional to establish our results. This work extends and improve on some results in the literature.
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