We consider nonlinear delay differential and renewal equations with infinite delay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) by introducing a unifying abstract framework and derive a finite-dimensional approximating system via pseudospectral discretization. For renewal equations, via integration we consider a reformulation in a space of absolutely continuous functions that ensures that point evaluation is well defined. We prove the one-to-one correspondence of equilibria between the original equation and its approximation, and that linearization and discretization commute. Our most important result is the proof of convergence of the characteristic roots of the pseudospectral approximation of the linear(ized) equations, which ensures that the finite-dimensional system correctly reproduces the stability properties of the original linear equation if the dimension of the approximation is large enough. This result is illustrated with several numerical tests, which also demonstrate the effectiveness of the approach for the bifurcation analysis of equilibria of nonlinear equations.

翻译：我们考虑具有无限时滞的非线性时滞微分方程与更新方程。通过引入统一的抽象框架扩展了Gyllenberg等人（Appl. Math. Comput., 2018）的工作，并利用伪谱离散方法推导出有限维逼近系统。针对更新方程，我们通过积分在绝对连续函数空间中进行重构，确保点估值具有良好定义。我们证明了原方程与其逼近系统之间平衡态的一一对应关系，并验证了线性化与离散化可交换。最重要的结果是对线性化方程伪谱近似特征根收敛性的证明，这保证了当逼近系统维数足够大时，有限维系统能正确再现原线性方程的稳定性特征。该结论通过多项数值实验得到验证，同时展示了该方法对非线性方程平衡态分岔分析的有效性。