Differential equations with locally bounded delay
The initial value problem with state-dependent delays \[ x'(t)=g(x_t), \quad x_0=\phi, \] is studied for a continuously differentiable map \(g: U\to \mathbb R,\) where \(U\) is a non-empty open subset of a Banach space. It is proved that the set \(M:=\{\phi\in U: \phi'(0)=g(\phi)\}\) is a continuously differentiable submanifold with codimension \(n\) and that the above initial value problem defines a continuous semiflow of continuously differentiable solution operators on \(M.\) It is also verified that the derivative of the solution operators are given by solutions of linear variational equations with segments tangent to \(M.\) Local stable and unstable manifolds at a stationary point of the semiflow on \(M\) are provided.
scientific article; zbMATH DE number 6012777