semigroup of contractions
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2021 ◽  
Vol 40 (6) ◽  
pp. 1507-1519
Author(s):  
Abdelkhalek El amrani ◽  
Jawad Ettayb ◽  
Aziz Blali

Let A ∈ B(X) be a spectral operator on a non-archimedean Banach space over Cp. In this paper, we give a necessary and sufficient condition on the resolvent of A so that the discrete semigroup consisting of powers of A is contractions.



Author(s):  
M. Marin ◽  
A. Öchsner ◽  
R. Ellahi ◽  
M. M. Bhatti


2018 ◽  
Vol 36 (4) ◽  
pp. 1073-1087
Author(s):  
Rachid El Ayadi ◽  
Mohamed Ouzahra

Abstract In this paper, we deal with the distributed bilinear system $ \frac{d z(t)}{d t}= A z(t) + v(t)Bz(t), $ where A is the infinitesimal generator of a semigroup of contractions on a real Hilbert space H. The linear operator B is supposed bounded with respect to the graph norm of A. Then we give sufficient conditions for weak and strong stabilizations. Illustrating examples are provided.



2015 ◽  
Vol 17 (02) ◽  
pp. 1350035 ◽  
Author(s):  
Ioan I. Vrabie

We consider the delay differential equation u′(t) ∈ Au(t) + f(t, ut), t ∈ ℝ+, where A is the infinitesimal generator of a nonlinear semigroup of contractions in a Banach space X and f is continuous, subjected to a general mixed nonlocal + local initial condition of the form u(t) = g(u)(t) + ψ(t), t ∈ [-τ, 0]. We prove that under natural conditions on A, f, g and ψ the problem above has at least one C0-solution. Applications to T-periodic and to T-anti-periodic problems, as well as an illustrative example referring to the nonlinear diffusion equation are also included.





1999 ◽  
Vol 60 (3) ◽  
pp. 459-468 ◽  
Author(s):  
Arnaud Simard

We give explicit constructions of semigroups and operators with particular properties. First we build a bounded C0-semigroup which is invertible and which is not similar to a semigroup of contractions. Afterwards we exhibit operators which admit bounded imaginary powers of angle ω > 0 on a Hilbert space but which do not admit a bounded functional calculus on the sector of angle ω. (This gives the limit of McIntosh's fundamental result.) Finally we build, in the 2-dimensional Hilbert space, an operator which is not the negative generator of a semigroup of contractions, although its imaginary powers are bounded by eπ|s|/2.



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