On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations

Abstract

In this paper we consider unbounded solutions of perturbed convolution Volterra summation equations. The equations studied are asymptotically sublinear, in the sense that the state-dependence in the summation is of smaller than linear order for large absolute values of the state. When the perturbation term is unbounded, it is elementary to show that solutions are also. The main results of the paper are mostly of the following form: the solution has an additional unboundedness property U if and only if the perturbation has property U . Examples of property U include monotone growth, monotone growth with fluctuation, fluctuation on R without growth, existence of time averages. We also study the connection between the times at which the perturbation and solution reach their running maximum, and the connection between the size of signed and unsigned running maxima of the solution and forcing term.

Large fluctuations and growth rates of linear Volterra summation equations

“…Before stating and discussing our main results we first provide a brief motivation for our interest in equations such as (1.1) and outline connections to applications (see also [4] where much of the following discourse is elaborated upon).…”

“…In particular, we show that if the unbounded sequence H has an interesting property A which characterises its growth or fluctuation, then x possesses the property A as well; in many situations the converse also holds (cf. Appleby and Patterson [4]).…”