Stability in Totally Nonlinear Neutral Dynamic Equations on Time Scales

Abstract

Let T be a time scale which is unbounded above and below and such that 0∈T. Let id-τ:[0,∞)∩T→T be such that (id-τ)([0,∞)∩T) is a time scale. We use the Krasnoselskii-Burton's fixed point theorem to obtain stability results about the zero solution for the following totally nonlinear neutral dynamic equation with variable delay x^{△}(t)=-a(t)h(x^{σ}(t))+c(t)x^{△}(t-τ(t))+b(t)G(x(t),x(t-τ(t))), t∈[0,∞)∩T, where f^{△} is the △-derivative on T and f^{△} is the △-derivative on (id-τ)(T). The results obtained here extend the work of Ardjouni, Derrardjia and Djoudi [2].