Uniform Ultimate Boundedness of Solutions of Third-Order Nonlinear Delay Differential Equations

Abstract

In this paper, we study the uniform boundedness and uniform ultimate boundedness of solutions of the third-order delay differential equation (*) ::: x +φ(x, ˙ x)¨ x + g( ˙ x(t r(t))) + f (x(t r(t))) = p(t, x, ˙ x, ¨), where 0 r(t) γ, γ is a positive constant, φ(x, ˙ x), g( ˙ x), f (x) and p(t, x, ˙ x, ¨) are contin- uous functions. We obtain some sufficient conditions which ensure that all the solutions of Eq.(*) are uniformly bounded and uniformly ultimately bounded. Our result revise and improve some earlier results. Mathematics Subject Classication 2000: 34K20.