Time scales calculus was initiated in 1988 by Stefan Hilger. It bridges the gap between continuous and discrete analysis and expands on both theories. Differential equations are defined on an interval of the set of real numbers while difference equations are defined on discrete sets. However, some physical systems are modeled by what is called dynamic equations because they are either differential equations, difference equations or a combination of both. This means that dynamic equations are defined on connected, discrete or combination of both types of sets. Hence, time scales calculus provides a generalization of differential and difference analysis, see [6, 7, 14, 15] and the references therein. Delay dynamic equations arise from a variety of applications including in various fields of science and engineering such as applied sciences, practical problems concerning mechanics, the engineering technique fields, economy, control systems, physics, chemistry, biology, medicine, atomic energy, information theory, harmonic oscillator, nonlinear oscillations, conservative systems, stability and instability of geodesic on Riemannian manifolds, dynamics in Hamiltonian systems, etc. In particular, problems concerning qualitative analysis of delay dynamic equations have received the attention of many authors, see [1]–[15] and the references therein.

Let T be a time scale such that t0∈ T. In this paper, we are interested in the analysis of qualitative theory of the problems of the existence of positive solutions for second-order nonlinear neutral dynamic equations. Inspired and motivated by the references in this paper, we concentrate on the existence of positive solutions for the second-order nonlinear neutral dynamic equation