An iterative method for functional differential equations
The authors consider the problem \[ \dot x(t)=f(t,x(t),x_t),\quad x_0(\theta)=x_{2\pi}(\theta),\quad \theta\in [-r,0], \] where \(f\colon I\times \mathbb{R}\times C\to C\) is a continuous function, \(I=[0,2\pi]\) and \(C=C([-r,0],\mathbb{R})\) is the space of continuous real-valued functions defined on \([-r,0]\). The set \(C\) is a Banach space with supremum norm. Upper and lower functions for this problem are defined and are used for the construction of a monotone iterative method to prove the existence of a solution of the problem considered. As application, the special case \( f(t,x(t),x_t):=g(t,x_t)-Kx(t)\) with \(g\) continuous and positive constant \(K\) is considered for which the existence of at least one solution of the periodic boundary problem is proved.
scientific article; zbMATH DE number 2136142