An abstract averaging theorem

For each t ? 0, let A(t) generate a contraction semigroup on a Banach space L. Suppose the solution of u_t = ?A(t)u is given by an evolution operator V_?(t, s). Conditions are given under which $\text{V}{}_{\mathrm{?}}\text{(}\text{(t+s)}\text{?}\text{,}\text{s}\text{?}\text{)}$ converges strongly as ? → 0 to a semigroup T(t) generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝}{}_0{}^{\mathrm{T}}\text{A(t)f dt}$.This result is applied to the following situation: Let B generate a contraction group S(t) and the closure of ?A + B generate a contraction semigroup S_?(t). Conditions are given under which $\text{S(?}\text{t}\text{?}\text{) S}{}_{\mathrm{?}}\text{(}\text{t}\text{?}\text{)}$ converges strongly to a semigroup generated by the closure of $\text{A}\text{?}\text{f ≡}\text{lim}{}_{\mathrm{T\to \infty }}\text{(}\text{1}\text{T}\text{) ∝ S(?t) AS(t)f dt}$. This work was motivated by and generalizes a result of Pinsky and Ellis for the linearized Boltzmann Equation.