Equivalence of functional-differential equations of neutral type and abstract Cauchy problems
The authors prove equivalence of neutral functional differential equations \((1)\quad (d/dt)Dx_ t=Lx_ t+f(t)\) and abstract Cauchy problems in appropriate state spaces. If \(D\in {\mathcal B}(C;{\mathbb{R}}^ n)\) and \(L\in {\mathcal B}(W^{1,p};{\mathbb{R}}^ n)\) then (1) is equivalent to the abstract Cauchy problem \((2)\quad \dot u(t)={\mathcal A}u(t)+(f(t),0)\) in \({\mathbb{R}}^ n\times L^ p\), where \({\mathcal A}\) is the infinitesimal generator of the \(C_ 0\)-semigroup on \({\mathbb{R}}^ n\times L^ p\) generated by the generalized solutions of (1). D need not to be atomic at zero. If \(D\in {\mathcal B}(L^ p;{\mathbb{R}}^ n)\) then the abstract Cauchy problem (2) is in the state space \(L^ p\) instead of \({\mathbb{R}}^ n\times L^ p\).
neutral functional differential equations
scientific article; zbMATH DE number 3933552