Approximate solutions of delay differential equations (Q1076892)

The author considers the Cauchy problem for delay differential equations of the form \[ (1)\quad x'(t)=F(t,x(t),x(t-\tau)),\quad t\in [0,a],\quad x(t)=\phi (t),\quad t\in [-\tau,0]. \] Let \(P_ k\) be a space of polynomials on [0,a] of degree less than or equal to k, and \(f_ k\in P_ k\) be such that \(\| Sf_ k\| =\inf \{\| Sf\|:f\in P_ k\}\) where S is the usual integral operator of the problem (1). Conditions on the existence of \(f_ k\) are given. It is shown that \(\| f_ k-x\| \to 0\) as \(k\to \infty\). The bounds of the error are found.