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Continuous solutions of iterative equations of infinite order

Kapica R., Morawiec J.

Opuscula Mathematica

2009

Vol. 29, no. 2

147-155

Given a probability space (Ω, A, P) and a complete separable metric space X, we consider R continuous and bounded solutions φ: X → R of the equations φ (x) = ∫Ω φ(ƒ(x, ω)) P(dω) and φ(x) = 1 - ∫Ωφ(ƒ(x, ω)) P(dω), assuming that the given function ƒ : X x Ω → X is controlled by a random variable L: Ω → (0, ∞) with -∞ < ∫Ω log L (ω) P (dω) < 0. An application to a refinement type equation is also presented.

Asymptotic properties of solutions of some interative functional inequalities

Brydak D., Choczewski B., Czerni M.

Opuscula Mathematica

2008

Vol. 28, no. 4

441-452

Continuous solutions of iterative linear inequalities of the first and second order are considered, belonging to a class Fτ of functions behaving at the origin as a prescribed function T.

Sequences of iterates of random-valued vector functions and continuous solutions of a linear functional equation of infinite order

Kapica R.

Bulletin of the Polish Academy of Sciences. Mathematics

2002

Vol. 50, no 4

447-455

Given a probability space (Ω, A, P), a separable Banach space X, and measurable functions L: Ω → (0, +∞), M: Ω → X we obtain some theorems on the existence and on the uniqueness of continuous solutions φ: X → R of the equation φ (x) = ∫Ω φ (L(ω) x + M (ω)) P (dω).