New Criterion that Guarantees Sufficient Conditions for Globally Asymptotically Stable Periodic Solutions of Non-Linear Differential Equations with Delay

The objective of this paper is to investigate and give sufficient conditions that we guarantees globally asymptotically stable periodic solutions, of non-linear differential Equations with Delay of the form (1.1). The Razumikhin’s technique was improve upon to enhance better result’s hence equation (1.2), was studied along side with equation (1.1). Equation (1.2) is an integro-differential equations with delay kernel. Since the coefficients of (1.2) are periodic, it is re-written as equation (3.1), where a ,b, and c ≥ 0, and ω- periodic continuious function on R. G ≥ 0, is a normalized kernel from equation (1.2), which enable us to defined equation (3.1) as a fixed point. Since the defined operator B, for equation (3.1) are not empty, claim1 -1V enable us to used the fixed point theorem to investigate and established our defined properties. See, (Theorem 3.1, Lemma 3.1 and Theorem 3.2) and the Liapunov’s direct (second) method to prove our main results. See, (Theorem3.3, 3.4, and 3.5) which established the objective of this study.