Existence of the boundary in the Chesaro sense of a limited solution of the evolution equation in the Banach space

Abstract

An existence criterion for the Cesàro limit $\Bigl(\lim_{t \to \infty} \frac{1}{t}\int_{0}^{t} y(\xi) {\rm d}\xi \Bigr)$  of a bounded solution $y(t)$ of the problem ${\rm d}y (t)/{\rm d}t = Ay (t)$, $y(0) = y_0$, $t\in [0, \infty)$, where $А$  is a closed linear operator with dense domain of definition $D (A)$ in a reflexive Banach space $E$, is obtained under the condition that there exists a sufficiently small interval $(0, \delta)$ belonging to the set of the regular points $\rho(А)$ of the operator $A$.