Time averaging for functional differential equations
The paper presents a result on averaging for functional-differential equations of the form \[ x'(t)= f(t/\varepsilon, x_t) \] on a finite interval. To formulate the assumptions, denote by \(C= C([-r,0],\mathbb{R}^d)\) the space of all continuous functions from \([-r,0]\) into \(\mathbb{R}^d\) with the standard maximum norm. For \(t_0\), \(T\in\mathbb{R}\), \(t_0< T\), for a continuous function from \(C([t_0- r,T],\mathbb{R}^d)\) and for a fixed \(t\in [t_0, T]\), denote by \(x_t\) the function defined by \(x_t(s)= x(t+ s)\) for \(s\in [-r, 0]\). Assume that: (i) \(f:\mathbb{R}\times C\to \mathbb{R}^d\) is continuous, (ii) \(f\) is Lipschitzian with respect to the second variable, (iii) for all \(u\in C\) there exists \[ \lim_{T\to\infty} (1/T) \int^T_0 f(\tau, u)\,d\tau. \] The above mentioned result has been proved under the assumptions (i)--(iii) by means of methods of nonstandard analysis.
scientific article; zbMATH DE number 1884078