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 pert… Show more
“…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).…”
Section: Growth Rates and Economic Applicationsmentioning
confidence: 99%
“…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]).…”
This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples. Finally, we show that our theory can be extended to cover a class of nonlinear equations via a straightforward linearisation argument.
“…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).…”
Section: Growth Rates and Economic Applicationsmentioning
confidence: 99%
“…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]).…”
This paper concerns the asymptotic behaviour of solutions of a linear convolution Volterra summation equation with an unbounded forcing term. In particular, we suppose the kernel is summable and ascribe growth bounds to the exogenous perturbation. If the forcing term grows at a geometric rate asymptotically or is bounded by a geometric sequence, then the solution (appropriately scaled) omits a convenient asymptotic representation. Moreover, this representation is used to show that additional growth properties of the perturbation are preserved in the solution. If the forcing term fluctuates asymptotically, we prove that fluctuations of the same magnitude will be present in the solution and we also connect the finiteness of time averages of the solution with those of the perturbation. Our results, and corollaries thereof, apply to stochastic as well as deterministic equations, and we demonstrate this by studying some representative classes of examples. Finally, we show that our theory can be extended to cover a class of nonlinear equations via a straightforward linearisation argument.
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