Some stability and boundedness conditions for non-autonomous differential equations with deviating arguments

C. Tunc, Yüzüncü Yil University, Van, Turkey

E. J. Qualitative Theory of Diff. Equ., No. 1. (2010), pp. 1-12.

Abstract: In this article, the author studies the stability and boundedness of solutions for the non-autonomous third order differential equation with a deviating argument, $r$:
\begin{equation*}
\begin{array}{c}
x^{\prime \prime \prime }(t)+a(t)x^{\prime \prime }(t)+b(t)g_{1}(x^{\prime}(t-r))+g_{2}(x^{\prime}(t))+h(x(t-r)) \\
=p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime }(t-r),x^{\prime \prime }(t)),
\end{array}
\end{equation*}
where $r>0$ is a constant. Sufficient conditions are obtained; a stability result in the literature is improved and extended to the preceding equation for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))=0,$ and a new boundedness result is also established for the case $p(t,x(t),x(t-r),x^{\prime }(t),x^{\prime}(t-r),x^{\prime \prime }(t))\neq 0.$

You can download the full text of this paper in DVI, PostScript or PDF format.