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Journal: Annales Mathematicae Silesianae

Remarks connected with the weak limit of iterates of some random-valued functions and iterative functional equations

Karol Baron

Language: EN | Published: 08-05-2020 | Abstract | pp. 36-44

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 B of all its Borel subsets we consider the set R_c of all B ⊗A measurable and contractive in mean functions f:X×Ω→X with finite integral ∫_Ωρ(f(x,ω),x)P(dω) for x∈X, the weak limit π^f of the sequence of iterates of f∈R_c, and investigate continuity-like property of the function f↦π^f, f∈?R_c, and Lipschitz solutions ϕ that take values in a separable Banach space of the equation
ϕ(x) = ∫_Ωϕ(f(x,ω))P(dω) + F(x).
Next, assuming that X is a real separable Hilbert space, Λ:X→X is linear and continuous with ‖Λ‖<1, and μ is a probability Borel measure on X with finite first moment we examine continuous at zero solutions ϕ:X→ℂ of the equation
ϕ(x) = μ^{^}(x)ϕ(Λx)
which characterizes the limit distribution π^f for some special f∈R_c.

2020-05-08



Journal: Annales Mathematicae Silesianae

Generalization of Titchmarsh's theorem for the Bessel transform in the space L_{p,α}(ℝ_+)

Mohamed El Hamma , Radouan Daher

Language: EN | Published: 15-01-2013 | Abstract | pp. 55-59

In this paper, we prove a generalization of Titchmarsh’s theorem for the Bessel transform in the space L_p,α(ℝ⁺) for functions satisfying the (ψ,p)-Bessel Lipschitz condition.

2013-01-15



Journal: Annales Mathematicae Silesianae

A Kneser theorem for ordinary differential equations in Banach spaces

Marc Mitschele

Language: EN | Published: 30-09-2010 | Abstract | pp. 71-85

We show that the set of solutions of the initial-value problem
u(τ)=a,   u'(t) = g(t,u(t)) + k(t,u(t)),   τ≤t≤T,
in a Banach space is compact and connected, whenever g and k are bounded and continuous functions such that g is one-sided Lipschitz and k is Lipschitz with respect to the Kuratowski measure of noncompactness. The existence of solutions is already known from Sabina Schmidt [10].

2010-09-30



Journal: Annales Mathematicae Silesianae

Extending the applicability of the super-Halley-like method using ω-continuous derivatives and restricted convergence domains

Joannis K. Argyros , Santhosh George

Language: EN | Published: 27-10-2018 | Abstract | pp. 21-40

We present a local convergence analysis of the super-Halley-like method in order to approximate a locally unique solution of an equation in a Banach space setting. The convergence analysis in earlier studies was based on hypotheses reaching up to the third derivative of the operator. In the present study we expand the applicability of the super-Halley-like method by using hypotheses up to the second derivative. We also provide: a computable error on the distances involved and a uniqueness result based on Lipschitz constants. Numerical examples are also presented in this study.

2018-10-27



Journal: Annales Mathematicae Silesianae

On estimates for the Bessel transform

Ahmed Abouelaz , Radouan Daher , Mohamed El Hamma

Language: EN | Published: 30-09-2013 | Abstract | pp. 59-66

Using a Bessel translation operator, we obtain a generalization of Theorem 2.2 in [3] for the Bessel transform for functions satisfying the (ψ,δ,β)-Bessel Lipschitz condition in the space L_2,α(ℝ⁺).

2013-09-30