1,568 research outputs found
On the admissibility of unboundedness properties of forced deterministic and stochastic sublinear Volterra summation equations
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 if and only if the
perturbation has property . Examples of property include monotone
growth, monotone growth with fluctuation, fluctuation on 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.Comment: 45 page
Long Memory in a Linear Stochastic Volterra Differential Equation
In this paper we consider a linear stochastic Volterra equation which has a
stationary solution. We show that when the kernel of the fundamental solution
is regularly varying at infinity with a log-convex tail integral, then the
autocovariance function of the stationary solution is also regularly varying at
infinity and its exact pointwise rate of decay can be determined. Moreover, it
can be shown that this stationary process has either long memory in the sense
that the autocovariance function is not integrable over the reals or is
subexponential. Under certain conditions upon the kernel, even arbitrarily slow
decay rates of the autocovariance function can be achieved. Analogous results
are obtained for the corresponding discrete equation
Blow-up and superexponential growth in superlinear Volterra equations
This paper concerns the finite-time blow-up and asymptotic behaviour of
solutions to nonlinear Volterra integrodifferential equations. Our main
contribution is to determine sharp estimates on the growth rates of both
explosive and nonexplosive solutions for a class of equations with nonsingular
kernels under weak hypotheses on the nonlinearity. In this superlinear setting
we must be content with estimates of the form ,
where is the blow-up time if solutions are explosive or
if solutions are global. Our estimates improve on the sharpness of results in
the literature and we also recover well-known blow-up criteria via new methods.Comment: 24 page
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