Sentence examples for be a continuous semigroup from inspiring English sources

Let { T ( t ) : t ∈ R } be a continuous semigroup of nonexpansive mappings on C. Let { t n } be a sequence in R +.

Theorem 3.2 Let ρ ∈ ℜ and let F = { T t : t ≥ 0 } be a continuous semigroup of mappings on a subset C of L ρ.

Let { Q ( q ) : q ∈ R } be a continuous semigroup of nonexpansive mappings on C. Let { q n } be a sequence in R +.

Theorem 3.4 Let ρ ∈ ℜ be (UUC), and let F = { T t : t ≥ 0 } be a continuous semigroup of ρ-nonexpansive mappings on a ρ-closed, ρ-bounded, convex, nonempty subset of L ρ.

Let ((S_{t})_{tgeq0}) be a continuous semigroup on H, which has the property lVert S_{t}rVert_{H}leqslantexp{ alpha t)}, quadforall t>0 (2.5) for some constant (alpha>0), and with (VertcdotVert_{H}) denoting the operator norm on H, it is called a pseudo-contraction semigroup on H. Let (T>0) be fixed.

0 < α n + 1 < α n for all n ∈ N ; k n ∈ N for all n ∈ N ; α n / α n + 1 ∉ Q for all n ∈ N ; { α n } converges to 0. Theorem 3.3 Let ρ ∈ ℜ and let F = { T t : t ≥ 0 } be a continuous semigroup of mappings on a subset C of L ρ.

Moreover, for all t ≥ 0, if we equip Γ with the topology endowed with the E-norm, then the map S t : Γ → Γ × R + defined as S t ( X ¯ ) = X ( t ). is a continuous semigroup.

It follows from Gronwall's lemma that sup t ∈ [ 0, T Z Z ( t ) < ∞, which implies that S t is a continuous semigroup by the standard O.D.E. theory. □. We transform the global solution of the equivalent system (2.10) into the global conservative solution of the original system (2.1) in this section.

Since T 0 ( z ) = z, it follows that T s n ( z ) = z for every n ∈ N. Because s n → t and ℱ is a continuous semigroup, we conclude, as previously observed, that T t ( z ) = z which concludes the proof of the theorem.

Assume that lim n → ∞ q n = lim n → ∞ α n q n = 0. Then lim n → ∞ x n = x ¯ for some x ¯ ∈ Ω. Proof For each t ≥ 0, let T ( t ) : C → C be defined by T ( t ) x : = x for each x ∈ C. Clearly, { T ( t ) : t ≥ 0 } is a continuous semigroup of nonexpansive mappings on C. Since C is a compact set, Ω : = ⋂ q ≥ 0 F ( Q ( q ) ) ≠ ∅.

Let {μt, t>0} be a continuous convolution semigroup of probability distributions on a Lie group G.