Twin semigroups and delay equations

2019/06/08 by Diekmann, Odo, Lunel, Sjoerd Verduyn · 1 citation
#34F #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.1906.03409

Abstract

In the standard theory of delay equations, the fundamental solution does not 'live' in the state space. To eliminate this age-old anomaly, we enlarge the state space. As a consequence, we lose the strong continuity of the solution operators and this, in turn, has as a consequence that the Riemann integral no longer suffices for giving meaning to the variation-of-constants formula. To compensate, we develop the Stieltjes-Pettis integral in the setting of a norming dual pair of spaces. Part I provides general theory, Part II deals with "retarded" equations, and in Part III we show how the Stieltjes integral enables incorporation of unbounded perturbations corresponding to neutral delay equations.

Cited by

• Age-Structured Population Dynamics

Related

• A class of abstract delay differential equations in the light of suns and stars
• Perturbation theory and linear partial differential equations with delay
• Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument
• Stationarity of Stochastic Linear Equations with Additive Noise and\n Delays in the Unbounded Drift Terms
• Galerkin-Koornwinder approximations of delay differential equations for physicists