This paper is concerned with the behaviour of Runge-Kutta methods for some dissipative nonlinear initial value problems for delay differential systems. The delay differential equations have the form \( y'(t) = f(y(t), y(t-\tau))\), \(t \geq 0\) where the delay \(\tau\) is a positive constant, \(y(t)\) belongs to the finite-dimensional space \(\mathbb{C}^N\) and \(f\) satisfies \( \text{Re}\langle u, f(u,v)\rangle \leq \gamma + \alpha \|u \|^2+ \beta \|v \|^2\) for some real constants \(\gamma, \alpha, \beta \).
First of all the author shows that for \( \alpha + \beta<0\) these systems have for all \( \varepsilon >0\) an open ball around the origin whose radius depend on \(\varepsilon\) which is an absorbing set for the differential system. Then he considers the discretization of these systems by means of Runge-Kutta methods that are \((k,l)\) algebraically stable in the sense of K. Burrage and J. C. Butcher [BIT 20, 189-203 (1980; Zbl 0431.65051)] supplemented with a linear interpolation procedure for the retarded argument. Then some dissipativity results are proved, in particular it is shown that under the DJ-irreducibility assumption on the coefficients of the Runge-Kutta method they retain the dissipativity behaviour.
The paper also considers the extension of the above results for delay differential systems in infinite-dimensional Hilbert spaces as well as systems with several delays.