On the stability and boundedness of a class of higher order delay differential equations

We establish some sufficient conditions which guarantee asymptotic stability of the null solution and boundedness of all the solutions of the following nonlinear differential equation of third order with the variable delay, r(t) t ) + g ( x ′ ( t − r ( t ) ) ) x ″ ( t ) + ψ ( x ′ ( t ) ) + f ( x ′ ( t − r ( t ) ) ) + h ( x ( t − r ( t ) ) ) = p ( t , x ( t ) , x ′ ( t ) , x ( t − r ( t ) ) , x ′ ( t − r ( t ) ) , x ″ ( t ) ) , (t, x(t), x′(t), x(t−r(t)), x′(t−r(t)), x′′(t))=0 and ≠0, respectively. By defining an appropriate Lyapunov functional, we prove two new theorems on the stability and boundedness of the solutions of the above equation. We also give an example to illustrate the theoretical analysis in this work. Our results improve a stability result in the literature, which was obtained for nonlinear differential equations of third order without delay, to the above differential equation with delay for stability and boundedness of the solutions.