For a class of nonlinear higher-order neutral dynamic equations on a time scale, we analyze the existence and asymptotic behavior of nonoscillatory solutions on the basis of hypotheses that allow applications to equations with different integral convergence and divergence of the reciprocal of the coefficients. Two examples are presented to demonstrate the efficiency of new results.

Let $$x_{\mathcal{V},f}$$ be the solution of the discrete Cauchy problem $$\zeta{(n+1)} = \mathcal{B}(n)\zeta{(n)}+ f(n)$$, $$\zeta(0)= 0$$, where $$\mathcal{B}(n)$$ is a sequence of bounded linear operators on a Banach space $$\mathcal{X}$$. We prove that the discrete evolution family $$\mathcal{V}:= \{{\mathbb{V}(n,k): n,k\in \mathcal{Z}_+, n\geq k}\}$$ of bounded linear operators acting on a complex Banach space $$\mathcal{X}$$ is uniformly exponentially stable if and only if for each $$f\in \mathcal{C}_{AAP_0}(\mathcal{Z}_+,\mathcal{X})$$ the solution $$x_{\mathcal{V},f}$$ belongs to $$\mathcal{C}_{AAP_0}(\mathcal{Z}_+,\mathcal{X})$$, where $$\mathcal{C}_{AAP_0}(\mathcal{Z}_+,\mathcal{X})$$ is the space of asymptotically almost periodic sequences. We use the approach of discrete evolution semigroups.