Continuous-Time Stochastic Averaging on the Infinite Interval for Locally Lipschitz Systems

On time-dependent neutral stochastic evolution equations with a fractional Brownian motion and infinite delays

In this paper, we consider a class of time-dependent neutral stochastic evolution equations with the infinite delay and a fractional Brownian motion in a Hilbert space. We establish the existence and uniqueness of mild solutions for these equations under non-Lipschitz conditions with Lipschitz conditions being considered as a special case. An example is provided to illustrate the theory

Infinite time interval backward stochastic differential equations with continuous coefficients

In this paper, we study the existence theorem for [Formula: see text] [Formula: see text] solutions to a class of 1-dimensional infinite time interval backward stochastic differential equations (BSDEs) under the conditions that the coefficients are continuous and have linear growths. We also obtain the existence of a minimal solution. Furthermore, we study the existence and uniqueness theorem f...

Stochastic reaction - diffusion systems with multiplicative noise and non - Lipschitz reaction term

We study existence and uniqueness of a mild solution in the space of continuous functions and existence of an invariant measure for a class of reaction-diffusion systems on bounded domains of R , perturbed by a multiplicative noise. The reaction term is assumed to have polynomial growth and to be locally Lipschitz-continuous and monotone. The noise is white in space and time if d = 1 and colour...

An effective optimization algorithm for locally nonconvex Lipschitz functions based on mollifier subgradients

Infinite Number of Stable Periodic Solutions for an Equation with Negative Feedback

For all μ > 0, a locally Lipschitz continuous map f with xf (x) > 0, x ∈ R\{0}, is constructed, such that the scalar equation ẋ (t) = −μx (t)−f (x (t− 1)) with delayed negative feedback has an infinite number of periodic orbits. All periodic solutions defining these orbits oscillate slowly around 0 in the sense that they admit at most one sign change in each interval of length of 1. Moreover, i...