Existence of Cesàro limit of bounded solution of evolution equation in banach space

Abstract

An existence criterion for the Cesàro limit$\left( {\mathop {\lim }\limits_{t \to \infty } \frac{1}{t}\int\limits_0^t {y(\xi )d\xi } } \right)$ of a bounded solution y(t) of the problem dy(t)/dt = Ay(t), y(0)=y₀, t ∈ [O, ∞), where A 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 (O, δ) belonging to the set of the regular points ρ(A) of the operator A.