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Renewal theorems for linear discrete Volterra equations. (English) Zbl 0541.60063

Renewal theorems as they have been proved e.g. by W. Feller [An introduction to probability theory and its applications, Vol. I, 3rd ed. (1968; Zbl 0155.231); Vol. II (1966; Zbl 0138.102)] for scalar Volterra equations are extended to discrete linear Volterra equations in ordered Banach spaces. This permits to show that, in linear discrete models, age- structured populations which are spatially distributed asymptotically exhibit geometric growth and a stationary age-space distribution which is independent of the initial state of the population. Further, as a basis for nonlinear renewal theorems, the non-negative solutions of limiting discrete linear renewal equations are characterized. The methods of this paper combine Feller’s approach to scalar renewal theorems as presented in Vol. II of his quoted book, with techniques developed by M. A. Krasnosel’skij [Positive solutions of operator equations. (1964; Zbl 0121.106)] for the study of special positive linear operators.

MSC:

60H20 Stochastic integral equations
60K05 Renewal theory
60J70 Applications of Brownian motions and diffusion theory (population genetics, absorption problems, etc.)
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