Existence of periodic solutions for a state dependent delay differential equation
The authors consider a scalar differential equation with delay of the form \[ \dot x(t)=- f\biggl(x\bigl(t-\tau(t) \bigr)\biggr), \quad\dot \tau (t)= h\bigl(x(t), \tau (t)\bigr),\tag{1} \] where \(f:\mathbb{R} \to\mathbb{R}\) and \(h:\mathbb{R} \times [\tau_1,\tau_2] \to\mathbb{R}\) (with \(0<\tau_1 <\tau_2)\) are \(C^1\) maps. By applying a semiejective fixed-point theorem (see theorem 1.1, A semiejective fixed-point theorem (1998), working paper by P. Magal and O. Arino) the authors prove the existence of a nontrivial slowly oscillating periodic solution to (1). This result stands an extension of the result by \textit{D. Arino}, \textit{K. P. Hadeler} and \textit{M. L. Hbid} [J. Differ. Equations 144, No. 2, 263-301 (1998; Zbl 0913.34057)] due to relaxation of the assumptions made in the above paper.
scientific article; zbMATH DE number 1538230
scientific article