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2014-03-11 Nonlinear Seminar

Dr Jacek Gulgowski
(Institute of Mathematics, GdaƄsk University)
On the continuity of superposition operator in the spaces of functions of bounded variation

It is a well-know fact that the superposition of a function $$ f\colon \mathbb{R}\to \mathbb{R} $$ satisfying a local Lipschitz condition with a function $$ x(t) $$ of bounded variation leads to the function $$ (f\circ x)(t) $$ which is also of bounded variation. During the seminar we are going to present a few results concerning the continuity of such a superposition operator $$ F \colon BV[0,1]\to BV[0,1] $$. Furthermore, we are going to discuss several generalizations of the concept of bounded variation in the sense of Jordan and investigate the issue of continuity of the superposition operator $$ F \colon BV[0,1]\to E $$, where $$ E $$ denotes the space of functions of bounded variation in the generalized sense.