Bulletin of Mathematical Analysis and Applications ISSN: 1821-1291, URL: http://www.bmathaa.org Volume 9 Issue 4(2017), Pages 31-41. ON THE GLOBAL STABILITY OF A NEUTRAL DIFFERENTIAL EQUATION WITH VARIABLE TIME-LAGS YENER ALTUN, CEMIL TUNC ¸ Abstract. In this work, we get assumptions that guaranteeing the global ex- ponential stability (GES) of the zero solution of a neutral differential equation (NDE) with time-lags. By help of the Liapunov-Krasovskii functional (LKF) approach, we obtain a new result related (GES) of the zero solution of the studied (NDE). An example is given to illustrate the applicability and cor- rectness of the obtained result by MATLAB-Simulink. The obtained result includes and improves the results found in the literature. 1. Introduction In 2014, Keadnarmol and Rojsiraphisal [10] considered the first order neutral differential equation (NDE) with two variable time-lags, d dt [x + px(t − τ (t))] = −ax + b tan hx(t − σ(t)). (1.1) Using Lyapunov functionals, the authors established some sufficient conditions for the (GES) of solutions of (NDE) (1.1). By this work, the authors [10] established an improved criterion for the (GES) of solutions of (NDE) (1.1). In this paper, instead of (NDE) (1.1), we consider the first order non-linear (NDE) with two variable time-lags: d dt [x + p(t)x(t − τ (t)] = − a(t)h(x) − b(t)g(x(t − τ (t))) + c(t) tan hx(t − σ(t)),t ≥ 0, (1.2) where ” ′ ” represents d dt ,a,b,c,p :[t 0 , ∞) → [0, ∞),t 0 ≥ 0, and g,h : ℜ→ℜ are continuous functions with g(0) = 0,h(0) = 0; the function c is continuous and differentiable, and |p(t)|≤ p 0 < 1( p 0 -constant). The variable time-lags τ and σ are continuous and differentiable, defined by τ (t) : [0, ∞) → [0,τ 0 ] and σ(t) : [0, ∞) → [0,σ 0 ] satisfying 0 ≤ τ (t) ≤ τ 0 , 0 ≤ σ(t) ≤ σ 0 , τ ′ (t) ≤ δ 1 ,σ ′ (t) ≤ δ 2 < 1, (1.3) where τ 0 > 0,σ 0 > 0,δ 1 > 0,δ 2 (> 0) ∈ℜ. 2000 Mathematics Subject Classification. 34K20, 34K40, 47N20. Key words and phrases. (NDE); (GES); (LKF); matrix inequality; multiple time-lags. c 2017 Universiteti i Prishtin¨ es, Prishtin¨ e, Kosov¨ e. Submitted September 30, 2017. Published October 30, 2017. Communicated by Tongxing Li. 31