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Positive periodic solutions of parabolic evolution problems: a translation along trajectories approach

Aleksander Ćwiszewski — 2011

Open Mathematics

A translation along trajectories approach together with averaging procedure and topological degree are used to derive effective criteria for existence of periodic solutions for nonautonomous evolution equations with periodic perturbations. It is shown that a topologically nontrivial zero of the averaged right hand side is a source of periodic solutions for the equations with increased frequencies. Our setting involves equations on closed convex cones, therefore it enables us to study positive solutions...

Approximate smoothings of locally Lipschitz functionals

Aleksander Ćwiszewski;  Wojciech Kryszewski — 2002

Bollettino dell'Unione Matematica Italiana

The paper deals with approximation of locally Lipschitz functionals. A concept of approximation, based on the idea of graph approximation of the generalized gradient, is discussed and the existence of such approximations for locally Lipschitz functionals, defined on open domains in R N , is proved. Subsequently, the procedure of a smooth normal approximation of the class of regular sets (containing e.g. convex and/or epi-Lipschitz sets) is presented.

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