arXiv subjects

Josef Kreulich

Publications and source records attributed to Josef Kreulich.

6 recordsLinked to original sources

arXiv2021

Short notes on $L^1(Ω,X)$ with infinite measure

This study uses the ideas of \cite{Rieffel} to provide the dual of $L^1(μ,X)$ in the positive and $σ-$ finite cases. This results in elegant necessary and sufficient criteria for weak compactness in $L^1(S,μ,X)$ in the $σ-$finite case, using the ideas of \cite{RuessL1} and \cite{Cooper}. Finally, the result of \cite{NeervenLNM} is extended to compute the sun-dual of $L^1(\re,X)$ with respect to the canonical translation semigroup, dropping the approximation property from $X^*,$, which is applied to obtain almost periodicity for integrals of non-smooth functions. Moreover, for evolution semigroups, it is shown that weak compactness of the orbits implies strong stability.

math.FA↗

arXiv2020

Compactification of bounded semigroup representations

The given study uses the methods to identify compactifications of semigroups $S\subset L(X),$ which reside in the space $L(X).$ This method generalizes in some sense the deLeeuw-Glicksberg-Theory to a greater class of functions. The approach provides an abstract approach to several notions of almost periodicity, which mainly involving right semitopological semigroups \cite{RuppertLNM}, and the adjoint theory. Moreover, the given setting is refined to the case of bounded $C_0-$semigroups.

math.FA↗

arXiv2019

On Compactifications of bounded $C_0-$semigroups

In this study, we refine the compactification presented by Witz \cite{Witz} for general semigroups to the case of bounded $C_0$-semigroups, involving adjoint theory for this class of operators. This approach considerably reduces the operator space in which the compactification is performed. Additionally, this approach leads to a decomposition of $X^{\sun}$ and to an extension of ergodic results to dual semigroups.

math.FA↗

arXiv2017

Existence and Asymptotics of Abstract Functional Differential Equations

It is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear abstract functional differential equations in both, the finite and infinite delay case. A generalization of the integral solution will provide regularity results. Moreover, this method applies to derive uniform convergence on the halfline, and therefore general results on boundedness and various types of asymptotic almost periodicity.

math.DS↗

arXiv2017

Whole Line Solutions to Abstract Functional Differential Equations

In the underlying study it is shown how the linear method of the Yosida-approximation of the derivative applies to solve possibly nonlinear and multivalued functional differential equations like: \begin{eqnarray*} u^\prime(t) &\in& A(t,u_t)u(t) +ωu(t), \ t \in \mathbb{R} \end{eqnarray*} Furthermore, in the case of finite and infinite delay we give an answer about whether the solution is bounded, periodic, almost periodic, or some kind of almost automorphy.

math.DS↗

arXiv2013

Asymptotic Behaviour of Nonlinear Evolution Equations in Banach Spaces

We show how the approach of Yosida approximation of the derivative serves to obtain new results for evolution systems. Using this method we obtain multivalued time dependent perturbation results. Additionally, translation invariant subspaces $Y$ of the bounded and uniformly continuous functions are considered, to obtain criteria for the existence of solutions $u\in Y$ to the equation $$ u^{\prime}(t)\in A(t)u(t)+ \om u(t) + f(t), t\in \re, $$ or of solutions $u$ asymptotically close to $Y$ for the inhomogeneous differential equation \begin{eqnarray*} u^{\prime}(t)&\in& A(t)u(t) + \om u(t) + f(t), \ \ t > 0, u(0)&=&u_0, \end{eqnarray*} in general Banach spaces, where $A(t)$ denotes a possibly nonlinear time dependent dissipative operator. Particular examples for the space $Y$ are spaces of functions with various almost periodicity properties and more general types of asymptotic behavior. Further, an application to functional differential equations is given.

math.DS↗