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 Local K-convoluted C-cosine functions and abstract cauchy problems

Filomat ◽  
2016 ◽  
Vol 30 (9) ◽  
pp. 2583-2598 ◽  
Author(s):  
Chung-Cheng Kuo
Keyword(s):  
Linear Operator ◽  
Abstract Cauchy Problem ◽  
Cosine Function ◽  
Equivalence Relations ◽  
Cauchy Problems ◽  
Bounded Linear ◽  
Unique Existence ◽  
Locally Integrable Function ◽  
Cosine Functions ◽  
Abstract Cauchy Problems

Let K : [0,T0)? F be a locally integrable function, and C : X ? X a bounded linear operator on a Banach space X over the field F(=R or C). In this paper, we will deduce some basic properties of a nondegenerate local K-convoluted C-cosine function on X and some generation theorems of local Kconvoluted C-cosine functions on X with or without the nondegeneracy, which can be applied to obtain some equivalence relations between the generation of a nondegenerate local K-convoluted C-cosine function on X with subgenerator A and the unique existence of solutions of the abstract Cauchy problem: U''(t)=Au(t)+f(t) for a.e. t ? (0, T0), u(0) = x, u'(0) = y when K is a kernel on [0, T0), C : X ? X an injection, and A : D(A) ? X ? X a closed linear operator in X such that CA ? AC. Here 0 < T0 ? ?, x,y ? X, and f ? L1,loc([0,T0),X).

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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