On Some Limits and Series Arising From Semigroup Theory 1 (original) (raw)
2009
In this paper we introduce a sequence (Mn)n≥n0 of positive linear operators as a modification of the Bernstein-Schnabl operators associated with a positive projection on C(K), where K is a convex compact subset of a locally convex space; moreover we study its main approximation and qualitative properties. Furthermore, we establish an asymptotic formula for those operators, and we prove that to the sequence (Mn)n≥n0 there corresponds a uniquely determined C0-semigroup (in some special case a Feller one) which is representable as a limit of suitable powers of the operators. AMS (MOS) Subject Classification. 47D03, 41A36, 41A65.
On convergence and asymptotic behaviour of semigroups of operators
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 2020
The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed. This article is part of the theme issue ‘Semigroup applications everywhere’.
International Journal of Mathematics and Mathematical Sciences, 2012
Let E be a real reflexive and strictly convex Banach space with a uniformly Gâteaux differentiable norm. Let J {T t : t ≥ 0} be a family of uniformly asymptotically regular generalized asymptotically nonexpansive semigroup of E, with functions u, v : 0, ∞ → 0, ∞. Let F : F J ∩ t≥0 F T t / ∅ and f : K → K be a weakly contractive map. For some positive real numbers λ and δ satisfying δ λ > 1, let G : E → E be a δ-strongly accretive and λ-strictly pseudocontractive map. Let {t n } be an increasing sequence in 0, ∞ with lim n → ∞ t n ∞, and let {α n } and {β n } be sequences in 0, 1 satisfying some conditions. Strong convergence of a viscosity iterative sequence to common fixed points of the family J of uniformly asymptotically regular asymptotically nonexpansive semigroup, which also solves the variational inequality G − γf p, j p − x ≤ 0, for all x ∈ F, is proved in a framework of a real Banach space.
Overiterated Linear Operators and Asymptotic Behaviour of Semigroups
Mediterranean Journal of Mathematics, 2008
The results provided hereby are related to the asymptotic behaviour of certain strongly continuous semigroups, which may be expressed in terms of iterates of positive linear operators, in the sense of Altomare's theory. We present some applications to concrete cases involving continuous and discrete type operators, namely the Beta and the Stancu operators.