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On Probability Distribution Solutions of a Functional Equation

Janusz Morawiec, Ludwig Reich (2005)

Bulletin of the Polish Academy of Sciences. Mathematics

Let 0 < β < α < 1 and let p ∈ (0,1). We consider the functional equation φ(x) = pφ (x-β)/(1-β) + (1-p)φ(minx/α, (x(α-β)+β(1-α))/α(1-β)) and its solutions in two classes of functions, namely ℐ = φ: ℝ → ℝ|φ is increasing, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 , = φ: ℝ → ℝ|φ is continuous, φ | ( - ∞ , 0 ] = 0 , φ | [ 1 , ∞ ) = 1 . We prove that the above equation has at most one solution in and that for some parameters α,β and p such a solution exists, and for some it does not. We also determine all solutions of the equation in ℐ and we show the exact connection...

On q -summation and confluence

Lucia Di Vizio, Changgui Zhang (2009)

Annales de l’institut Fourier

This paper is divided in two parts. In the first part we study a convergent q -analog of the divergent Euler series, with q ∈ ( 0 , 1 ) , and we show how the Borel sum of a generic Gevrey formal solution to a differential equation can be uniformly approximated on a convenient sector by a meromorphic solution of a corresponding q -difference equation. In the second part, we work under the assumption q ∈ ( 1 , + ∞ ) . In this case, at least four different q -Borel sums of a divergent power series solution of an irregular singular...

On reverses of some binary operators

Michal Šabo, Peter Strežo (2005)

Kybernetika

The notion of reverse of any binary operation on the unit interval is introduced. The properties of reverses of some binary operations are studied and some applications of reverses are indicated.

On separation theorems for subadditive and superadditive functionals

Zbigniew Gajda, Zygfryd Kominek (1991)

Studia Mathematica

We generalize the well known separation theorems for subadditive and superadditive functionals to some classes of not necessarily Abelian semigroups. We also consider the problem of supporting subadditive functionals by additive ones in the not necessarily commutative case. Our results are motivated by similar extensions of the Hyers stability theorem for the Cauchy functional equation. In this context the so-called weakly commutative and amenable semigroups appear naturally. The relations between...

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