Let T be a periodic time scale. We use the Krasnoselskii's fixed point theorem to show that the impulsive neutral dynamic equations with infinite delay x ∆ (t) = −A(t)x σ (t) + g ∆ (t, x(t − h(t))) + t −∞ D (t, u) f (x(u)) u, t = t j , t ∈ T, x(t + j) = x(t − j) + I j (x(t j)), j ∈ Z + have a periodic solution. Under a slightly more stringent conditions we show that the periodic solution is unique using the contraction mapping principle.

The nonlinear neutral dynamic equation with periodic coefficients [u(t) − g(u(t − τ (t)))] ∆ =p(t) − a(t)u σ (t) − a(t)g(u σ (t − τ (t))) − h(u(t), u(t − τ (t))) is considered in this work. By using Krasnoselskii's fixed point theorem we obtain the existence of periodic and positive periodic solutions and by contraction mapping principle we obtain the uniqueness. Stability results of this equation are analyzed. The results obtained here extend the work of Mesmouli, Ardjouni and Djoudi [14].

In this paper, we study the existence and uniqueness of positive solutions of the first-order nonlinear Caputo-Hadamard fractional differential equation D α 1 (x (t) − g (t, x (t))) = f (t, x (t)) , 1 < t ≤ e, x (1) = x 0 > g (1, x 0) > 0, where 0 < α ≤ 1. In the process we convert the given fractional differential equation into an equivalent integral equation. Then we construct appropriate mappings and employ the Krasnoselskii and Banach fixed point theorems and the method of upper and lower solutions to show the existence and uniqueness of a positive solution of this equation. Finally, an example is given to illustrate our results.