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We consider nonlinear delay differential and renewal equations with infinitedelay. We extend the work of Gyllenberg et al, Appl. Math. Comput. (2018) byintroducing a unifying abstract framework and derive a finite-dimensionalapproximating system via pseudospectral discretization. For renewal equations,via integration we consider a reformulation in a space of absolutely continuousfunctions that ensures that point evaluation is well defined. We prove theone-to-one correspondence of equilibria between the original equation and itsapproximation, and that linearization and discretization commute. Our mostimportant result is the proof of convergence of the characteristic roots of thepseudospectral approximation of the linear(ized) equations, which ensures thatthe finite-dimensional system correctly reproduces the stability properties ofthe original linear equation if the dimension of the approximation is largeenough. This result is illustrated with several numerical tests, which alsodemonstrate the effectiveness of the approach for the bifurcation analysis ofequilibria of nonlinear equations.

Equations with infinite delay: pseudospectral discretization for numerical stability and bifurcation in an abstract framework