On the Qualitative Behaviors of Solutions To a Kind of Nonlinear Third Order Differential Equations With Retarded Argument

Abstract

In this paper, with use of a Lyapunov functional, we discuss stability and boundedness of solutions to a kind of nonlinear third order differential equation with retarded argument: x'''(t)+ h(x(t),x''(t),x(t - r(t - r(t)),x''(t - r(t)))x''(t) +g(x(t - r(t)),x'(t - r(t))) + f(x(t - r(t))) = p(t,x(t),x'(t),x(t - r(t)),x'(t - r(t)),x''(t)), when p(t,x(t),x'(t),x(t - r(t)),x'(t - r(t)),x''(t)) = 0 and not equal 0, respectively. Our results include and improve some well-known results in the literature. An example is also given to illustrate the importance of results obtained and the topic. Keywords: stability, boundedness, Lyapunov functional, nonlinear third order differential equations, retarded argument.