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Growth of semigroups in discrete and continuous time

Alexander Gomilko, Hans Zwart, Niels Besseling (2011)

Studia Mathematica

We show that the growth rates of solutions of the abstract differential equations ẋ(t) = Ax(t), ẋ ( t ) = A - 1 x ( t ) , and the difference equation x d ( n + 1 ) = ( A + I ) ( A - I ) - 1 x d ( n ) are closely related. Assuming that A generates an exponentially stable semigroup, we show that on a general Banach space the lowest growth rate of the semigroup ( e A - 1 t ) t ≥ 0 is O(∜t), and for ( ( A + I ) ( A - I ) - 1 ) ⁿ it is O(∜n). The similarity in growth holds for all Banach spaces. In particular, for Hilbert spaces the best estimates are O(log(t)) and O(log(n)), respectively. Furthermore, we give conditions...

Hille-Yosida theory in convenient analysis.

Josef Teichmann (2002)

Revista Matemática Complutense

A Hille-Yosida Theorem is proved on convenient vector spaces, a class, which contains all sequentially complete locally convex spaces. The approach is governed by convenient analysis and the credo that many reasonable questions concerning strongly continuous semigroups can be proved on the subspace of smooth vectors. Examples from literature are reconsidered by these simpler methods and some applications to the theory of infinite dimensional heat equations are given.

Identification problems for degenerate parabolic equations

Fadi Awawdeh, Hamed M. Obiedat (2013)

Applications of Mathematics

This paper deals with multivalued identification problems for parabolic equations. The problem consists of recovering a source term from the knowledge of an additional observation of the solution by exploiting some accessible measurements. Semigroup approach and perturbation theory for linear operators are used to treat the solvability in the strong sense of the problem. As an important application we derive the corresponding existence, uniqueness, and continuous dependence results for different...

Isolated points of some sets of bounded cosine families, bounded semigroups, and bounded groups on a Banach space

Adam Bobrowski, Wojciech Chojnacki (2013)

Studia Mathematica

We show that if the set of all bounded strongly continuous cosine families on a Banach space X is treated as a metric space under the metric of the uniform convergence associated with the operator norm on the space 𝓛(X) of all bounded linear operators on X, then the isolated points of this set are precisely the scalar cosine families. By definition, a scalar cosine family is a cosine family whose members are all scalar multiples of the identity operator. We also show that if the sets of all bounded...

Linear and nonlinear abstract differential equations of high order

Veli B. Shakhmurov (2015)

Open Mathematics

The nonlocal boundary value problems for linear and nonlinear degenerate abstract differential equations of arbitrary order are studied. The equations have the variable coefficients and small parameters in principal part. The separability properties for linear problem, sharp coercive estimates for resolvent, discreetness of spectrum and completeness of root elements of the corresponding differential operator are obtained. Moreover, optimal regularity properties for nonlinear problem is established....

Currently displaying 121 – 140 of 335