State-dependent Delay Differential Equations on $H^1$

TL;DR

This paper develops a new solution theory for state-dependent delay differential equations in the Sobolev space $H^1$, allowing for Lipschitz continuous pre-histories and right-hand sides, broadening applicability beyond previous methods.

Contribution

It introduces a Lipschitz-based solution framework in $H^1$ using contraction mapping, independent of prior approaches, and accommodates a wider class of functional differential equations.

Findings

Solution theory valid for arbitrary Lipschitz pre-histories.

Applicable to a broad class of functional differential equations.

Does not require initial pre-histories to be in solution manifolds.

Abstract

Classically, solution theories for state-dependent delay equations are developed in spaces of continuous or continuously differentiable functions. The former can be technically challenging to apply in as much as suitably Lipschitz continuous extensions of mappings onto the space of continuous functions are required; whereas the latter approach leads to restrictions on the class of initial pre-histories. Here, we establish a solution theory for state-dependent delay equations for arbitrary Lipschitz continuous pre-histories and suitably Lipschitz continuous right-hand sides on the Sobolev space $H^1$. The provided solution theory is independent of previous ones and is based on the contraction mapping principle on exponentially weighted spaces. In particular, initial pre-histories are not required to belong to solution manifolds and the generality of the approach permits the consideration of a large class of functional differential equations even for which the continuity of the right-hand side has constraints on the derivative.