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), x(t - r(t)), x'(t - r(t)), x''(t - r(t)))x''(t)
                                +g(x(t - r(t)), x' 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), x0(t), x(t ¡ r(t)), x0(t ¡ r(t)), x00(t)) = 0 and ≠ 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.