Periodic solutions of functional differential equations of neutral type
The paper deals with neutral functional differential equations of the form \[ d/dt\left[x(t)-G(t,x_t)\right] =F(t,x_t),\tag{1} \] where \(F\) and \(G\) are continuous and \(T\) periodic in \(t\). Moreover for all \(r>0\) there exists a real function \(W\) defined on \(\mathbb{R}^+\), \(\lim_{b\to 0^+}W(b)=0\), such that \(|G(t,x_t)-G(s,x_s)|\leq W|t-s|\) for all continuous functions \(x\), \(\sup_{t\in [-h,\infty)}|x(t)|\leq r\), \(t,s\in \mathbb{R}\). (1) is said to be convex, if for all solutions of (1), \(x,y\), \(\alpha x+(1-\alpha)y\) is also a solution of (1) for \(\alpha \in [0,1]\). Necessary and sufficient conditions for the existence of periodic solutions of (1) with finite or infinite delay are given.
functional differential equations of neutral type
scientific article; zbMATH DE number 1005607