Asymptotic behavior theorems for non-autonomous functional differential equations via Lyapunov-Razumikhin method (Q1892502)

The functional-differential system (1) \(x'(t)= f(t, x_t)\), \(t_0\geq 0\), \(x_{t_0}= \phi\in C\), where \(f: [0, \infty)\times C([- r,0],\mathbb{R}^n)\to \mathbb{R}^n\) is considered. Sufficient conditions for every bounded solution to approach a closed set \(\Omega\subset \mathbb{R}^n\), \(0\in \Omega\), are demonstrated. These results generalize earlier results of Cooke and Huang and also results proved by Ladas, Sficas and Stavroulakis. When \(\Omega= \{0\}\) and the zero solution of (1) is stable, then the obtained theorem yields an asymptotic stability theorem. Applications of the demonstrated theorems are considered.