New properties of the D-operator and its applications on the problem of periodic solutions to neutral functional differential system
The object of study is a neutral functional differential equation of the form \[ \frac{dDx_t}{dt}=f(t,x_t),\tag{1} \] where \(x_t(\theta)=x(t+\theta),\;\theta\in [-\tau,0],\;\tau>0\), and \(f\in C\left( \mathbb{R}\times (C[-\tau,0],\mathbb{R}^n),\mathbb{R}^n \right)\) is \(\omega\)-periodic in \(t\) and maps bounded sets on bounded sets. Here, the \(D\)-operator is defined as \[ D(x_t)=x(t)-Bx(t-\tau) \] for some given \(n\times n\) real matrix \(B\). It is proved that, if \(|\lambda|\neq 1\) for any eigenvalue \(\lambda\) of \(B\), then any \(\omega\)-periodic solution of (1) is continuously differentiable on \(\mathbb{R}\). Finally, it is given a continuation-type result to assure the existence of some \(\omega\)-periodic solution of (1). The poof is based on coincidence degree theory.
neutral functional differential equation
scientific article; zbMATH DE number 5880257