ABSTRACT
A continuous semigroup of linear contractions (S(t)) t>0 on a Banach space X can be generated through the product formula () ∀ x ∈ X , ∀ t ≥ 0 , S ( t ) x = lim n → ∞ U ( t n ) n x , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq2882.tif"/>
where (U(h)) h>0 is any family of contractions on X whose right-derivative at h = 0 coincides with the infinitesimal generator −A of (S(t)), that is () ∀ x ∈ D ( A ) lim h ↓ 0 x − U ( h ) x h = A x ( = lim t ↓ 0 x − S ( t ) x t ) . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq2883.tif"/>
Actually formula (0.1) extends to the case when (S(t)) is a semigroup of nonlinear contractions “generated” by an m-accretive operator A and condition (0.2) may also be weakened to () I − U ( h ) h → h ↓ 0 A in the sense of graphs . https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq2884.tif"/>
338We look here at the same formula (0.1) when the regular step-size t/n is replaced by a variable step-size, namely () lim n → ∞ U ( h N n n ) U ( h N n − 1 n ) … U ( h 1 n ) x = S ( t ) x ∀ x ∈ D ( A ) ¯ , where ∑ 1 ≤ i ≤ N n h i n = t and lim n → ∞ ( max 1 ≤ i ≤ N n h i n ) = 0. https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187599/b1a4101e-e019-448a-8402-e9c5cf3b70e4/content/eq2885.tif"/>
Surprisingly, it turns out that (0.4) fails under assumption (0.3): there may be lack of stability and even lack of consistency. We exhibit examples in this direction. However, we prove that (0.4) holds true under the stronger assumption (0.2).
