Stepanov-almost periodic solutions of non-autonomous neutral functional differential equations with functional delay
The paper deals with the following non-autonomous neutral functional differential equation in a Banach space \(\mathbb{X}\): \[\frac{d}{dt}\left[u(t)-F(t,u(t-\alpha(t)))\right]= A(t)\left[u(t)-F(t,u(t-\alpha(t)))\right] + G(t,u(t),u(t-\alpha(t))),\quad t\in \mathbb{R}\tag{1} \] where \(A(t), \, t\in \mathbb{R}\) is a family of linear operator. The operator \((\gamma I -A(t))^{-1}\) and the real function \(\alpha\) defined on \( \mathbb{R}\) are almost periodic. \(F: \mathbb{R}\times \mathbb{X}\times \mathbb{X} \to \mathbb{X}\) and \(G: \mathbb{R}\times \mathbb{X}\times \mathbb{X} \to \mathbb{X}\) are Stepanov-almost periodic continuous functions satisfying Lipschitz conditions. Under the assumption that family of linear operator \(A(t), \, t\in \mathbb{R}\) satisfies the `Acquistapace-Terreni' condition and the evolution family generated by \(A(t), \, t\in \mathbb{R}\) is exponentially stable, the author proves the existence and uniqueness of Stepanov-almost periodic mild soulution to (1). The result is achieved by means of Banach's fixed point theorem.
neutral functional differential equation
scientific article; zbMATH DE number 6946427
scientific article