# Twin semigroups and delay equations

## TL;DR

This paper develops a new framework for delay equations by enlarging the state space, allowing the fundamental solution to reside within it, and introduces the Stieltjes-Pettis integral to handle unbounded perturbations.

## Contribution

It introduces a novel approach to delay equations by enlarging the state space and employing the Stieltjes-Pettis integral, addressing longstanding issues in the theory.

## Key findings

- Enlarged state space allows the fundamental solution to be within the space.
- The Stieltjes-Pettis integral effectively handles unbounded perturbations.
- The framework applies to retarded and neutral delay equations.

## 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.