scholarly journals Approximation theorems for the propagators of higher order abstract Cauchy problems

2007 ◽  
Vol 360 (04) ◽  
pp. 1723-1740 ◽  
Author(s):  
Jin Liang ◽  
Rainer Nagel ◽  
Ti-Jun Xiao
Keyword(s):  
Higher Order ◽  
Cauchy Problems ◽  
Approximation Theorems ◽  
Abstract Cauchy Problems

scholarly journals Regularization of some classes of ultradistribution semigroups and sines

2010 ◽  
Vol 87 (101) ◽  
pp. 9-37 ◽  
Author(s):  
Marko Kostic
Keyword(s):  
Linear Operator ◽  
Higher Order ◽  
Cosine Function ◽  
Closed Linear Operator ◽  
Cauchy Problems ◽  
Ultradifferentiable Functions ◽  
Injective Operator ◽  
Regularized Semigroups ◽  
Abstract Cauchy Problems

We systematically analyze regularization of different kinds of ultradistribution semigroups and sines, in general, with nondensely defined generators and contemplate several known results concerning the regularization of Gevrey type ultradistribution semigroups. We prove that, for every closed linear operator A which generates an ultradistribution semigroup (sine), there exists a bounded injective operator C such that A generates a global differentiable C-semigroup (C-cosine function) whose derivatives possess some expected properties of operator valued ultradifferentiable functions. With the help of regularized semigroups, we establish the new important characterizations of abstract Beurling spaces associated to nondensely defined generators of ultradistribution semigroups (sines). The study of regularization of ultradistribution sines also enables us to perceive significant ultradifferentiable properties of higher-order abstract Cauchy problems.

scholarly journals Propagation relations for solutions of some higher order cauchy problems

1996 ◽  
Vol 38 (2) ◽  
pp. 229-239 ◽  
Author(s):  
L. R. Bragg
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Higher Order ◽  
Solution Properties ◽  
Cauchy Problems ◽  
Wave Problem ◽  
Basic Algorithm ◽  
Solution Behaviour ◽  
Abstract Cauchy Problems ◽  
Addition Formulas

AbstractThe Huygens' property is exploited to study propagation relations for solutions of certain types of linear higher order Cauchy problems. Motivated by the solution properties of the abstract wave problem, addition formulas are developed for the solution operators of these problems. The application of these alternative forms of the solution operators to data leads to connecting operator relations between distinct solutions of the problems at different times. We examine this solution behaviour for both analytic and abstract Cauchy problems. A basic algorithm for constructing addition formulas for solutions of ordinary differential equations is included.

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