Stability and bounded of solutions to non-autonomous delay differential equations of third order

Abstract

In this paper, we obtain some sufficient conditions to guarantee the uniform asymptotic stability of zero solution and bounded of all solutions to non-autonomous delay differential equation of third order

$$\begin{array}{@{}ll}\vspace*{-1pt}&\dddot{x}(t)+a(t)\varphi\bigl(\dot{x}(t)\bigr)\ddot{x}(t)+b(t)\psi\bigl(\dot{x}(t)\bigr)\\[5pt]&\qquad{}+c(t)h\bigl(x(t-r)\bigr)\\[5pt]&\quad =p\bigl(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t)\bigr),\vspace*{-1pt}\end{array}$$

when $p(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t))=0$ and $p(t,x(t),x(t-r),\dot{x}(t),\dot{x}(t-r),\ddot{x}(t))\ne 0$, respectively. By using the Liapunov functional approach, we prove two new results on the subject.