Almost automorphic solutions of some semilinear dynamic equations on time scales (Q2804101)

The paper is devoted to the semilinear nonautonomous dynamic equation NEWLINE\[NEWLINEu^\Delta(t)=A(t)u(t)+f\Big(t,u(t),\int_0^t \varphi(s,u(s))\mathrm{d}s\Big),\quad t\in\mathbb{T}, \tag{1}NEWLINE\]NEWLINE where \(\mathbb{T}\) is a time scale which is symmetric and invariant under translation, \(A:\mathbb{T}\to\mathbb{R}^{n\times n}\) is a regressive matrix-valued function, \(\varphi:\mathbb{T}\times\mathbb{R}^n\to\mathbb{R}^n\) and \(f:\mathbb{T}\times\mathbb{R}^n\times\mathbb{R}^n\to\mathbb{R}^n\) are rd-continuous functions. By using the Banach fixed-point theorem, sufficient conditions (exponential dichotomy, Lipschitz continuity, etc.) for the existence and uniqueness of an almost automorphic solution of equation (1) are given in the main result, see Theorem 3.3.