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On the convergence of sequences of iterates of random-valued vector functions

Rafał Kapica (2007)

Annales Polonici Mathematici

Given a probability space (Ω,,P) and a subset X of a normed space we consider functions f:X × Ω → X and investigate the speed of convergence of the sequence (fⁿ(x,·)) of the iterates f : X × Ω X defined by f¹(x,ω ) = f(x,ω₁), f n + 1 ( x , ω ) = f ( f ( x , ω ) , ω n + 1 ) .

On the difference equation x n + 1 = a 0 x n + a 1 x n - 1 + + a k x n - k b 0 x n + b 1 x n - 1 + + b k x n - k

Elmetwally M. Elabbasy, Hamdy El-Metwally, E. M. Elsayed (2008)

Mathematica Bohemica

In this paper we investigate the global convergence result, boundedness and periodicity of solutions of the recursive sequence x n + 1 = a 0 x n + a 1 x n - 1 + + a k x n - k b 0 x n + b 1 x n - 1 + + b k x n - k , n = 0 , 1 , where the parameters a i and b i for i = 0 , 1 , , k are positive real numbers and the initial conditions x - k , x - k + 1 , , x 0 are arbitrary positive numbers.

On the difference property of families of measurable functions

Rafał Filipów (2003)

Colloquium Mathematicae

We show that, generally, families of measurable functions do not have the difference property under some assumption. We also show that there are natural classes of functions which do not have the difference property in ZFC. This extends the result of Erdős concerning the family of Lebesgue measurable functions.

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