Stability in nonlinear neutral differential equations with infinite delay

Abstract

In this paper we use the contraction mapping theorem to obtain asymptotic stability results of the nonlinear neutral differential equation with infinite delay d/dt x(t) = -a(t) x (t - T1(t)) + d/dt Q (t, x (t - T2(t))) + ∫ D(t,s) f (x (s))ds. An asymptotic stability theorem with a necessary and sufficient condition is proved, which improves and generalizes some results due to Burton [6], Zhang [17], Althubiti, Makhzoum, Raffoul [1].

Keywords

• Fixed points
• Stability
• Neutral differential equation
• Integral equation
• Infinite delay