scholarly journals On the quasi-Fredholm and Saphar spectrum of strongly continuous Cosine operator function

2021 ◽  
Vol 7 (1) ◽  
pp. 80-87
Author(s):  
Hamid Boua
Keyword(s):  
Fredholm Operator ◽  
Infinitesimal Generator ◽  
Operator Function ◽  
Cosine Family ◽  
Cosine Operator Function ◽  
Cosine Operator ◽  
Counter Example ◽  
Strongly Continuous Cosine Family

AbstractLet (C(t))t∈𝕉 be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – coshλt is Saphar (resp. quasi-Fredholm) operator and λt /∉iπ𝕑, then A – λ2 is also Saphar (resp. quasi-Fredholm) operator. We show by counter-example that the converse is false in general.

scholarly journals On a cosine operator function framework of approximation processes in Banach space

Filomat ◽  
2019 ◽  
Vol 33 (13) ◽  
pp. 4213-4228
Author(s):  
Andi Kivinukk ◽  
Anna Saksa ◽  
Maria Zeltser
Keyword(s):  
Banach Space ◽  
Operator Function ◽  
Order Of Approximation ◽  
Linear Combinations ◽  
Cosine Operator Function ◽  
Cosine Operator ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
The Given ◽  
General Method

We introduce the cosine-type approximation processes in abstract Banach space setting. The historical roots of these processes go back to W. W. Rogosinski in 1926. The given new definitions use a cosine operator functions concept. We proved that in presented setting the cosine-type operators possess the order of approximation, which coincide with results known in trigonometric approximation. Moreover, a general method for factorization of certain linear combinations of cosine operator functions is presented. The given method allows to find the order of approximation using the higher order modulus of continuity. Also applications for the different type of approximations are given.

The Fourth Order of Accuracy Decomposition Scheme for Abstract Hyperbolic Equation

2008 ◽  
Vol 15 (1) ◽  
pp. 165-175
Author(s):  
Jemal Rogava ◽  
Mikheil Tsiklauri
Keyword(s):  
Approximate Solution ◽  
Hyperbolic Equation ◽  
Fourth Order ◽  
Operator Function ◽  
Order Accuracy ◽  
Order Of Accuracy ◽  
Cosine Operator Function ◽  
Decomposition Scheme ◽  
Cosine Operator ◽  
Abstract Hyperbolic Equation

Abstract Using the rational splitting of a cosine operator-function, the fourth order accuracy decomposition scheme is constructed for hyperbolic equation when the principal operator is self-adjoint positively defined and is represented as a sum of two summands. Stability of the constructed scheme is shown and the error of an approximate solution is estimated.

scholarly journals On approximation processes defined by a cosine operator function

2017 ◽  
Vol 66 (2) ◽  
pp. 214
Author(s):  
A Kivinukk ◽  
A Saksa ◽  
M Zeltser
Keyword(s):  
Operator Function ◽  
Cosine Operator Function ◽  
Cosine Operator

Chaos for Cosine Operator Functions Generated by Shifts

2014 ◽  
Vol 24 (09) ◽  
pp. 1450108 ◽  
Author(s):  
Chung-Chuan Chen
Keyword(s):  
Operator Function ◽  
Weighted Shift ◽  
Necessary Condition ◽  
Weighted Shifts ◽  
Cosine Operator Function ◽  
Cosine Operator ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
Sufficient And Necessary Condition

Let 1 ≤ p < ∞. We give the sufficient and necessary condition for cosine operator functions, generated by bilateral weighted shifts on ℓp(ℤ), to be chaotic. Moreover, such a cosine operator function is chaotic if, and only if, its weighted shift is chaotic.

scholarly journals Spectral Theory For Strongly Continuous Cosine

2021 ◽  
Vol 8 (1) ◽  
pp. 40-47
Author(s):  
Hamid Boua
Keyword(s):  
Spectral Theory ◽  
Infinitesimal Generator ◽  
Invertible Operator ◽  
Cosine Family ◽  
Strongly Continuous Cosine Family ◽  
Drazin Invertible Operator

Abstract Let (C(t)) t∈ℝ be a strongly continuous cosine family and A be its infinitesimal generator. In this work, we prove that, if C(t) – cosh λt is semi-Fredholm (resp. semi-Browder, Drazin inversible, left essentially Drazin and right essentially Drazin invertible) operator and λt ∉ iπℤ, then A – λ 2 is also. We show by counterexample that the converse is false in general.

Recurrence of Cosine Operator Functions on Groups

2016 ◽  
Vol 59 (4) ◽  
pp. 693-704 ◽  
Author(s):  
Chung-Chuan Chen
Keyword(s):  
Operator Function ◽  
Locally Compact Group ◽  
Necessary Condition ◽  
Multiple Recurrence ◽  
Locally Compact ◽  
Cosine Operator Function ◽  
Cosine Operator ◽  
Operator Functions ◽  
Cosine Operator Functions ◽  
Sufficient And Necessary Condition

AbstractIn this note, we study the recurrence and topologically multiple recurrence of a sequence of operators on Banach spaces. In particular, we give a sufficient and necessary condition for a cosine operator function, induced by a sequence of operators on the Lebesgue space of a locally compact group, to be topologically multiply recurrent.

scholarly journals A second-order impulsive Cauchy problem

2002 ◽  
Vol 31 (8) ◽  
pp. 451-461 ◽  
Author(s):  
Eduardo Hernández Morales
Keyword(s):  
Banach Space ◽  
Cauchy Problem ◽  
Infinitesimal Generator ◽  
Continuous Functions ◽  
Second Order ◽  
Linear Operators ◽  
Classical Solutions ◽  
Cosine Family ◽  
Strongly Continuous Cosine Family

We study the existence of mild and classical solutions for an abstract second-order impulsive Cauchy problem modeled in the formu¨(t)=A u(t)+f(t,u(t),u˙(t)),t∈(−T0,T1),t≠ti;u(0)=x0,u˙(0)=y0; △u(ti)=Ii1 (u (ti)). △u˙(ti)=Ii2 (u˙ (ti+))whereAis the infinitesimal generator of a strongly continuous cosine family of linear operators on a Banach spaceXandf,Ii1,Ii2are appropriate continuous functions.

Exact and trajectory controllability of second order nonlinear differential equations with deviated argument

2017 ◽  
Vol 26 (2) ◽  
pp. 181-191
Author(s):  
M. MUSLIM ◽  
AVADHESH KUMAR ◽  
RAVI P. AGARWAL
Keyword(s):  
Differential Equations ◽  
Nonlinear Differential Equations ◽  
Exact Controllability ◽  
Second Order ◽  
Fixed Point Method ◽  
Linear Operators ◽  
Cosine Family ◽  
Bounded Linear ◽  
Deviated Argument ◽  
Strongly Continuous Cosine Family

In this manuscript, we consider a control system governed by a second order nonlinear differential equations with deviated argument in a Hilbert space X. We used the strongly continuous cosine family of bounded linear operators and fixed point method to study the exact and trajectory controllability. Also, we study the exact controllability of the nonlocal control problem. Finally, we give an example to illustrate the application of these results.

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