This monograph serves two primary purposes. First, it provides a comprehensive treatment of bi-continuous semigroup theory and its applications, ranging from network theory to control theory. The exposition proceeds systematically from fundamental theory through approximation methods, extrapolation techniques as well as perturbation theory, concluding with applications to networks, mean ergodicity and control systems. Each chapter includes illustrative examples to reinforce the theoretical concepts. Second, this work aims at supporting emerging researchers and doctoral students by offering a structured foundation for further investigation in operator semigroups. By synthesizing current knowledge, it provides a reference point for new research projects in this domain.
Introduction.- 1 An excursion on C0-semigroups.- 2 Bi-Continuous
Semigroups: Preliminaries and Examples.- 3 The generator of bi-continuous
semigroups.- 4 General Intermediate and Extrapolated Spaces.- 5 Intermediate
and Extrapolation Spaces for Bi-Continuous Semigroups.- 6 Approximation of
bi-continuous semigroups.- 7 Bounded and MiyaderaVoigt perturbations.- 8
DeschSchappacher perturbations of bi-continuous semigroups.- 9 Bi-continuous
Semigroups for Flows on Infinite Networks.- 10 Mean ergodic bi-continuous
semigroups.- 11 Bi-continuous semigroups in control theory.- A The mixed
topology.- B Riesz spaces and Banach lattices.- Bibliography.
Prof. Christian Budde is an Associate Professor in the Department of Mathematics and Applied Mathematics at the University of the Free State, South Africa. He obtained his Ph.D. in Mathematics from the Bergische Universität Wuppertal, Germany, in 2019, following studies at Radboud University Nijmegen. His research lies at the intersection of functional analysis, operator semigroups, and evolution equations, with applications to network dynamics and non-autonomous systems. His research has been supported by numerous national and international grants from the DFG, DAAD, and the South African NRF, among others. He is an active member of the mathematical community. He served as General Secretary of the South African Mathematical Society (SAMS) and is currently the president of SAMS. Moreover, he is Associated Editor for Afrika Matematika, and member of several scientific and review committees, including the Mathematics Reviewing Board of the Heidelberg Laureate Forum. Prof. Budde has organized multiple international conferences and special sessions, most recently his own conference Evolution Equations, Operator Semigroups and Beyond (2025).
