Math. J. Okayama Univ. 55 (2013), 157–166 UNIFORM STABILITY AND BOUNDEDNESS OF SOLUTIONS OF NONLINEAR DELAY DIFFERENTIAL EQUATIONS OF THE THIRD ORDER ADEMOLA, A.T. and ARAWOMO, P.O. Abstract. In this paper, a complete Lyapunov functional was con- structed and used to obtain criteria (when p = 0) for uniform asymptotic stability of the zero solution of the nonlinear delay differential equation (1.1). When p =0, sufficient conditions are also established for uni- form boundedness and uniform ultimate boundedness of solutions of this equation. Our results improve and extend some well known results in the literature. 1. Introduction Many works have been done by several authors on the properties of so- lutions of ordinary differential equations of the second, third, fourth, fifth and higher order in which the unknown functions and its derivatives are all evaluated at the same instant, t, see for instance Reissig et. al., [10], a survey book and Ademola et. al., [1], Ezeilo [6], Omeike [8], Tejumola [12], Tun¸ c [17] and the references cited therein to mention few. A more general type of differential equation often called a functional differential equation is one in which the unknown function occurs with various different arguments. This means an equation expressing some derivatives of x at time t in term of x (and its lower order derivative if any) at time t and at earlier instants. With respect to our observation in the relevant literature, interesting re- sults have been obtained on the properties of solutions of these classes of equations see for examples Burton [4, 5], Hale [7] and Yoshizawa [18] which contain general results on the subject matter. Notable authors that have contributed immensely to the qualitative properties of solutions (in partic- ular stability and boundedness of solutions) of nonlinear third order delay differential equations include Afuwape and Omeike [2], Omeike [9], Sadek [11], Tun¸ c [13, 14, 15, 16, 17] and Zhu [19]. These authors dealt with the problems by constructing Lyapunov functionals and obtain criteria for sta- bility and boundedness of solutions. The purpose of this paper is to obtain criteria for uniform stability, uni- form boundedness and uniform ultimate boundedness of solutions for a more Mathematics Subject Classification. Primary 34K12; Secondary 34K20. Key words and phrases. Uniform stability, Uniform boundedness, Uniform ultimate boundedness, Lyapunov functional, Delay differential equation. 157 brought to you by CORE View metadata, citation and similar papers at core.ac.uk provided by Okayama University Scientific Achievement Repository