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On cubic difference equations with variable coefficients and fading stochastic perturbations
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We consider the stochastically perturbed cubic difference equation with variable coefficients xn+1=xn(1−hnx2n)+ρn+1ξn+1,n∈N,x0∈R. Here (ξn)n∈N is a sequence of independent random variables, and (ρn)n∈N and (hn)n∈N are sequences of nonnegative real numbers. We can stop the sequence (hn)n∈N after some random time N so it becomes a constant sequence, where the common value is an FN -measurable random variable. We derive conditions on the sequences (hn)n∈N , (ρn)n∈N and (ξn)n∈N , which guarantee that limn→∞xn exists almost surely (a.s.), and that the limit is equal to zero a.s. for any initial value x0∈R .
Nonlinear stochastic difference equation , Global almost sure asymptotic stability , Nonuniform timestepping
Baccas, R., Kelly, C. and Rodkina, A. (2019) ‘On cubic difference equations with variable coefficients and fading stochastic perturbations’, in Elaydi, S., Pötzsche, C. and Sasu, A. (eds.) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017, Timișoara, Romania, 24-28 July. Springer Proceedings in Mathematics and Statistics, Vol 287, pp. 171-197. doi: 10.1007/978-3-030-20016-9_6
© 2019, Springer Nature Switzerland AG. This is a post-peer-review, pre-copyedit version of a paper published in Elaydi, S., Pötzsche, C. and Sasu, A. (eds.) Difference Equations, Discrete Dynamical Systems and Applications. ICDEA 2017. Springer Proceedings in Mathematics and Statistics, Vol 287. The final authenticated version is available online at: https://doi.org/10.1007/978-3-030-20016-9_6