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Let T be a time scale which is unbounded above and below and such that 0 ϵ T. Let id - r : T → T be such that (id - r) (T) is a time scale. We use the contraction mapping theorem to obtain stability results about the zero solution for the following neutral nonlinear dynamic equations with unbounded delay χ▵ (t) = -a(t) χσ (t) + b(t)G(χ²(t),χ²(t - r(t))) + c(t)(χ²) ▵ (t - r(t), t ϵ(t))) +c(t) (χ²)▵ (t - r(t)), t ϵ T, and χ▵ (t) = -a(t) χσ (t) + b(t)G(χ²(t),χ²(t - r(t))) + c(t)(χ²) ▵ (t - r(t), t ϵ(t))) +c(t) (χ²)▵ (t - r(t)), t ϵ T, where ƒ▵ is the ▵-derivative on T and ƒ▵ is the ▵-derivative on (id - r) (T) . We provide interesting examples to illustrate our claims. [ABSTRACT FROM AUTHOR]

We study the existence and uniqueness of periodic solutions and the stability of the zero solution of the nonlinear neutral differential equation d/dt x(t) = -a(t)x(t) + d/dt Q(t, x(t - g(t))) + G(t, x(t), x(t - g(t))). In the process we use integrating factors and convert the given neutral differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ Krasnoselskii's fixed point theorem to show the existence of a periodic solution of this neutral differential equation. We also use the contraction mapping principle to show the existence of a unique periodic solution and the asymptotic stability of the zero solution provided that Q(0, 0) = G(t, 0, 0) = 0. [ABSTRACT FROM AUTHOR]

Abstract: We use fixed-point theorems to obtain stability results about the zero solution of thenonlinear neutral differential equation with functional delay, [Copyright &y& Elsevier]