Quasi-flows and equations with nonlinear differentials
The authors introduce an equation with a nonlinear differential \[ \phi (t+dt)-\phi (t)-D_{t,\phi (t)}(dt)=o(dt),\tag{*} \] where the mapping \(D_{t,x}: \mathbb R^{+}\rightarrow\mathbb R^d\) is defined for suitable \((t,x)\in\mathbb R\times\mathbb R^d\). The initial value problem for (*) is to find a function \(\phi\) satisfying (*) and the initial condition \(\phi(t_0)=x_0\). A local existence and uniqueness theorem on the problem is proved. The motivations come from applied problems that involve nonsmooth evolution processes satisfying the evolution properties, but which are not solutions to a classical differential equation.
scientific article; zbMATH DE number 1836636
scientific article