Stability in mixed linear delay Levin-Nohel integro-dynamic equations on time scales

Abstract

In this paper we use the contraction mapping theorem to obtain asymptotic stability results about the zero solution for the following mixed linear delay Levin-Nohel integro-dynamic equation

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

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