Karol Baron
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View article: Convergence in Law of Iterates of Weakly Contractive in Mean Random-Valued Functions
Convergence in Law of Iterates of Weakly Contractive in Mean Random-Valued Functions Open
We investigate the asymptotic behaviour of the sequence of forward type iterations of a given random-valued vector function on the state space being a separable and complete metric space. Assuming non-linear contraction in mean we prove th…
View article: Note on an Iterative Functional Equation
Note on an Iterative Functional Equation Open
We study the problem of solvability of the equation ϕ ( x ) = ∫ Ω g ( w ) ϕ ( f ( x , ω ) ) P ( d ω ) + F ( x ) , \varphi \left( x \right) = \int_\Omega {g\left( w \right)} \varphi \left( {f\left( {x,\omega } \r…
View article: Strong Law of Large Numbers for Iterates of Some Random-Valued Functions
Strong Law of Large Numbers for Iterates of Some Random-Valued Functions Open
Assume $$ (\Omega , {\mathscr {A}}, P) $$ is a probability space, X is a compact metric space with the $$ \sigma $$ -algebra $$ {\mathscr {B}} $$ of all its Borel subsets and $$ f: X \times \Omega \rightarrow X $$ is …
View article: From Invariance Under Binomial Thinning to Unification of the Cauchy and the Gołąb–Schinzel-Type Equations
From Invariance Under Binomial Thinning to Unification of the Cauchy and the Gołąb–Schinzel-Type Equations Open
We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Gołąb–Schinzel…
View article: From invariance under binomial thinning to unification of the Cauchy and the Gołąb-Schinzel-type equations
From invariance under binomial thinning to unification of the Cauchy and the Gołąb-Schinzel-type equations Open
We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Gołab-Schinzel…
View article: From invariance under binomial thinning to unification of the Cauchy and the Go{\l}\k{a}b-Schinzel-type equations
From invariance under binomial thinning to unification of the Cauchy and the Go{\l}\k{a}b-Schinzel-type equations Open
We point out to a connection between a problem of invariance of power series families of probability distributions under binomial thinning and functional equations which generalize both the Cauchy and an additive form of the Go{\l}ab-Schin…
View article: Continuous solutions to two iterative functional equations
Continuous solutions to two iterative functional equations Open
Based on iteration of random-valued functions we study the problem of solvability in the class of continuous and Hölder continuous functions $$\varphi $$ of the equations $$\begin{aligned} \varphi (x)=F(x)-\int _{\Omega }\varphi \big (f(…
View article: Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations
Remarks Connected with the Weak Limit of Iterates of Some Random-Valued Functions and Iterative Functional Equations Open
The paper consists of two parts. At first, assuming that (Ω, A, P ) is a probability space and ( X, ϱ ) is a complete and separable metric space with the σ-algebra of all its Borel subsets we consider the set c of all ⊗ 𝒜 - measurabl…
View article: Weak limit of iterates of some random-valued functions and its application
Weak limit of iterates of some random-valued functions and its application Open
Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$, a complete and separable metric space X with the $$ \sigma $$-algebra $$ {\mathcal {B}} $$ of all its Borel subsets, a $$ {\mathcal {B}} \otimes {\mathcal {A}} $$-measurable and…
View article: Weak law of large numbers for iterates of random-valued functions
Weak law of large numbers for iterates of random-valued functions Open
Given a probability space $$ (\Omega , {\mathcal {A}}, P) $$ , a complete and separable metric space X with the $$ \sigma $$ -algebra $$ {\mathcal {B}} $$ of all its Borel subsets and a $$ {\mathcal {B}} \otimes {\mathcal {A}} $$ -measurab…
View article: On the set of orthogonally additive functions with orthogonally additive second iterate
On the set of orthogonally additive functions with orthogonally additive second iterate Open
Let E be a real inner product space of dimension at least 2. We show that both the set of all orthogonally additive functions mapping E into E having orthogonally additive second iterate and its complement are dense in the space of all ort…
View article: Lipschitzian solutions to linear iterative equations revisited
Lipschitzian solutions to linear iterative equations revisited Open
We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions $$\varphi $$ of equations of the form $$\begin{aligned} \varphi (x)=\int _{\Omega }g(\omega )\varphi \big (f(x,\omega )\big )\mu (d\omeg…
View article: On Orthogonally Additive Functions With Big Graphs
On Orthogonally Additive Functions With Big Graphs Open
Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense…
View article: Lipschitzian solutions to inhomogeneous linear iterative equations
Lipschitzian solutions to inhomogeneous linear iterative equations Open
We study the problems of the existence, uniqueness and continuous dependence of Lipschitzian solutions $φ$ of equations of the form $$ φ(x)=\int_Ωg(ω)φ\big(f(x,ω)\big)μ(dω)+F(x), $$ where $μ$ is a measure on a $σ$-algebra of subsets of a s…
View article: On Lipschitzian solutions to an inhomogeneous linear iterative equation
On Lipschitzian solutions to an inhomogeneous linear iterative equation Open
Based on iteration of random-valued functions we study the problems of existence, uniqueness and continuous dependence of Lipschitzian solutions $${\varphi}$$ of the equation $$\varphi(x)=F(x)-\int_{\Omega} \varphi(f(x, \omega))P(d\omega),…
View article: On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions
On The Continuous Dependence Of Solutions To Orthogonal Additivity Problem On Given Functions Open
We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.