Abstract
We investigate when a natural iterative method converges to the exact solution of a differential-functional von Foerster-type equation which describes a single population depending on its past time and state densities, and on its total size. On the right-hand side, we assume either Perron comparison conditions or some monotonicity.
Highlights
Von Foerster and Volterra-Lotka equations arise in biology, medicine, and chemistry, [1,2,3,4,5]
We are interested in the first model, which is essentially nonlocal, because it contains the total size of population u(t,x)dx
Existence results for certain von Foerster type problems has been established by means of the Banach contraction principle, the Schauder fixed point theorem, or iterative methods, see [6,7,8,9,10]
Summary
Von Foerster and Volterra-Lotka equations arise in biology, medicine, and chemistry, [1,2,3,4,5]. The independent variables xj and an unknown function u stand for certain features and densities, respectively. It follows from this natural interpretation that xj ≥ 0 and u ≥ 0. Existence results for certain von Foerster type problems has been established by means of the Banach contraction principle, the Schauder fixed point theorem, or iterative methods, see [6,7,8,9,10]. Just because of nonlocal terms, these methods demand very thorough calculations and a proper choice of subspaces of continuous and integrable functions Sometimes, it may cost some simplifications of the real model. Note that an associate result on fast convergent quasilinearization methods has been published in [14]
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