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A characterization of evolution operators

Naoki Tanaka (2001)

Studia Mathematica

A class of evolution operators is introduced according to the device of Kato. An evolution operator introduced here provides a classical solution of the linear equation u'(t) = A(t)u(t) for t ∈ [0,T], in a general Banach space. The paper presents a necessary and sufficient condition for the existence and uniqueness of such an evolution operator.

A dynamical system in a Hilbert space with a weakly attractive nonstationary point

Ivo Vrkoč (1993)

Mathematica Bohemica

A differential equation is a Hilbert space with all solutions bounded but with so finite nontrivial invariant measure is constructed. In fact, it is shown that all solutions to this equation converge weakly to the origin, nonetheless, there is no stationary point. Moreover, so solution has a non-empty Ω -set.

A linear condition determining local or global existence for nonlinear problems

John Neuberger, John Neuberger, James Swift (2013)

Open Mathematics

Given a nonlinear autonomous system of ordinary or partial differential equations that has at least local existence and uniqueness, we offer a linear condition which is necessary and sufficient for existence to be global. This paper is largely concerned with numerically testing this condition. For larger systems, principals of computations are clear but actual implementation poses considerable challenges. We give examples for smaller systems and discuss challenges related to larger systems. This...

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