On the Globally Asymptotically Stability of Periodic Solutions of a Certain Class of Non-Linear Delay Differential Equations

The objective of this paper is to investigate and give sufficient conditions that will guarantee globally asymptotically stable periodic solutions of the non-linear differential equations with delay of the form (1.1). the Razumikhin’s technique was improve upon, to enhance better results equation (1.2) was studied along side with equation (1.1). Equation (1.2) is an integro-differential equations with delay kernel. The coefficients of (1.2) are periodic, and the equation can be rewritten as in form of (3.1), where a,b and c ≥ 0 and -periodic continuous function on R.G ≥ 0, is a normalized kernel from equation (1.2). Equation (1.2) enable ？us to defined equation (3.1) as a fixed point. Since the defined operator “B” for equation (3.1) are not empty, claim ( 1-iv) enable us to use the fixed point theorem to investigate and established our defined properties. (Theorem 3.1 Lemma 3.1 and Theorem 3.2) was used to prove for periodic and asymptotically stability and the Liapunovis direct (second) method was used to prove our main result. See, (Theorem 3.3, 3.4 and 3.5) ？which ？established the objective of this study