Wyniki wyszukiwania - BazTech

1

Approximate controllability results in α-norm for some partial functional integrodifferential equations with nonlocal initial conditions in Banach spaces

Ndambomve Patrice, Kpoumie Moussa El-Khalil, Ezzinbi Khalil

Journal of Applied Analysis

|

2023

|

Vol. 29, nr 1

127--142

EN

In this work, we discuss the approximate controllability of some nonlinear partial functional integrodifferential equations with nonlocal initial condition in Hilbert spaces.We assume that the corresponding linear part is approximately controllable. The results are obtained by using fractional power theory and α-norm, the measure of noncompactness and theMönch fixed-point theorem, and the theory of analytic resolvent operators for integral equations. As a result, we obtain a generalization of the work of Mahmudov [N. I. Mahmudov, Approximate controllability of evolution systems with nonlocal conditions, Nonlinear Anal. 68 (2008), no. 3, 536-546], without assuming the compactness of the resolvent operator. Our results extend and complement many other important results in the literature. Finally, a concrete example is given to illustrate the application of the main results.

2

Roots and Powers in Regular Languages : Recognizing Nonregular Properties by Finite Automata

Frei Fabian, Hromkovič Juraj, Karhumäki Juhani

Fundamenta Informaticae

|

2020

|

Vol. 175, nr 1-4

173--185

EN

It is well known that the set of powers of any given order, for example squares, in a regular language need not be regular. Nevertheless, finite automata can identify them via their roots. More precisely, we recall that, given a regular language L , the set of square roots of L is regular. The same holds true for the nth roots for any n and for the set of all nontrivial roots; we give a concrete construction for all of them. Using the above result, we obtain decision algorithms for many natural problems on powers. For example, it is decidable, given two regular languages, whether they contain the same number of squares at each length. Finally, we give an exponential lower bound on the size of automata identifying powers in regular languages. Moreover, we highlight interesting behavior differences between taking fractional powers of regular languages and taking prefixes of a fractional length. Indeed, fractional roots in a regular language can typically not be identified by finite automata.

3

Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum

Gil’ M.

Demonstratio Mathematica

|

2017

|

Vol. 50, nr 1

267--277

EN

We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A−A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An(n = 1, 2,...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

4

A note on invariant sets

Schilling R. L.

Probability and Mathematical Statistics

|

2004

|

Vol. 24, Fasc. 1

47--66

EN

A measurable set A is invariant with respect to a not necessarily symmetric sub-Markovian operator T on Lp (X, m) if T1A ≤ 1A, and strongly invariant if T1A = 1A. We show that these definitions accommodate many of the usual definitions of invariance, e.g., those used in Dirichlet form theory, ergodic theory or for stochastic processes. In finite measure spaces or if T∗ is sub-Markovian and recurrent, the notions of invariance and strong invariance coincide. We also show that for certain analytic semigroups of sub-Markovian operators, (strongly) invariant sets are already determined by a single operator, T1.