Stability of neutral integro-dynamic equations on time scales

Abstract

   

Let T be a time scale not bounded below and above such that t₀∈T. In this paper, we provide new sufficient conditions and use the Banach fixed point theorem to establish the stability results about the zero solution for the following Levin-Nohel intego-dynamic equation with variable delay

x^{Δ}(t)+∫_{t-r(t)}^{t}a(t,s)x(s)Δs+(g(t,x(t-r(t))))^{Δ}=0, t∈[t₀,∞)∩T,

where x^{Δ} is Δ-derivative on T. An asymptotic stability theorem with a necessary and sufficient condition is proved. In addition, the case of the equation with several delays is studied. The results obtained here extend the work of Bessioud, Ardjouni, Djoudi <cite>b0</cite>.