##
**On the stability of continuous quadrature rules for differential equations with several constant delays.**
*(English)*
Zbl 0771.65053

This paper is concerned with the stability of (non-confluent) Runge-Kutta methods for solving the pure delay differential equation
\[
y'(t)=f(t,y(t- \tau),y(t-2\tau),\dots,y(t-R\tau)),\;t>0,
\]
where \(\tau>0\) is a constant delay and \(y(t)=\varphi(t)\) on \([-R\tau,0]\). The method is described by a Butcher array and an appropriate natural continuous extension; it is shown that this gives rise to the same continuous quadrature rule regardless of the choice of the matrix \(A\). The stability analysis is based on the test equation
\[
y'(t)=-\sum^ R_{r=1}b_ r(t)y(t-r\tau), \tag{1}
\]
where the \(b_ r\) are nonnegative (smooth) functions. If \(\beta(t):=\sum^ R_{r=1}b_ r(t)\) satisfies
\[
\int^{x+R\tau}_ x\beta(s)ds\leq 1\quad\text{for all } x\geq 0
\]
and \(\beta(t)>0\) for every \(t\geq R\tau\) then \(| y(t)|\leq 2\max\{|\varphi(x)|:- R\tau\leq x\leq 0\}\), \(t\geq 0\), holds for every solution \(y(t)\) of (1). Thus, if the given method, when applied to (1), generates a numerical solution \(\{y_ n\}\) (on a uniform, constrained mesh with \(h=\tau/N\) \((N\in\mathbb{N}))\) which satisfies the analogous inequality \(| y_ n|\leq 2\max\{|\varphi(t)|:-R\tau\leq t\leq 0\}\) whenever \(\beta(t)\leq 1/(R\tau)\) \((t\geq 0)\), then it is said to be \(QN_ 0\)- stable. The authors characterize \(QN_ 0\)-stable continuous quadrature rules and present several concrete examples of such methods.

Reviewer: H.Brunner (St.John’s)

### MSC:

65L20 | Stability and convergence of numerical methods for ordinary differential equations |

65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |

34K05 | General theory of functional-differential equations |