An Approximation Theorem of Lax Type for Semigroups of Lipschitz Operators

Abstract

Let X be a Banach space with norm ∥·∥ and D a subset of X. A one-parameter family ${{\{ T(t)\} }_{{te[0,\infty )}}}$ of Lipschitz operators from D into itself is called a semigroup of Lipschitz operators on D if it satisfies the following conditions: (S1) For x є D and t,s ≥ 0,

$$\begin{array}{*{20}{c}} {T(t)T(s)x = T(t + s)x,} & {T(0)x = x.} \\ \end{array}$$

(S2) For x єD and t, ≥ 0,

$$\mathop{{\lim }}\limits_{{s \to t}} \parallel T(t)x - T(s)x\parallel = 0.$$

(S3) For τ > 0, there exists M _τ≥1 such that

$$\parallel T(t)x - T(t)y\parallel \leqslant {{M}_{\tau }}\parallel x - y\parallel$$

for x,y є D and t є[0,T]