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1

Finiteness of entropy for granular media equations

Tugaut Julian

Probability and Mathematical Statistics

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2019

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Vol. 39, Fasc. 1

75--84

EN

The current work deals with the granular media equation whose probabilistic interpretation is the McKean-Vlasov diffusion. It is well known that the Laplacian provides a regularization of the solution. Indeed, for any t > 0, the solution is absolutely continuous with respect to the Lebesgue measure. It has also been proved that all the moments are bounded for positive t. However, the finiteness of the entropy of the solution is a new result which will be presented here.

2

Continuity of real valued subquadratic functions

Kominek Z., Troczka-Pawelec K.

Commentationes Mathematicae

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2011

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Vol. 51, [Z] 1

71-75

EN

The aim of this paper is to prove a regularity theorem for real valued subquadratic mappings that are solutions of the inequality [formula], where X = (X, +) is a topological group.

3

Some inequalities connected with a quadratic functional equation

Troczka-Pawelec K.

Scientific Issues of Jan Długosz University in Częstochowa. Mathematics

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2008

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Vol. 13

73-79

EN

Let (X, +) be an Abelian group. One can show that a mapping f: X R satisfying the inequality f(x + y) + f(x-y)≤2f(x)+2f(y) (1) for all x, y ∈ X also satisfies the inequalities f(2x + y)≤4f(:c) + f (y) + f (x + y) - f (x - y) and f(2x+y) + f(2x -y)≤ 8f(x) + 2f(y) for all x, y ∈ X. A question of finding sufficient conditions under which the inequalities (1), (2) and (3) are equivalent will be considered. In this note, some properties of the solution of (1) will be proved. We also consider another definition of a subquadratic function given in [1].

4

Iteration groups, commuting functions and simultaneous systems of linear functional equations

Matkowski J.

Opuscula Mathematica

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2008

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Vol. 28, no. 4

529-539

EN

Let (ƒt)t∈R be a measurable iteration group on an open interval I. Under some conditions, we prove that the inequalies g o ƒa ≤ ƒa o g and g o ƒb ≤ ƒb o g for some a, b ∈ R imply that g must belong to the iteration group. Some weak conditions under which two iteration groups have to consist of the same elements are given. An extension theorem of a local solution of a simultaneous system of iterative linear functional equations is presented and applied to prove that, under some conditions, if a function g commutes in a neighbourhood of ƒ with two suitably chosen elements ƒa and ƒb of an iteration group of ƒ then, in this neighbourhood, g coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.

5

Asymptotic properties of solutions of some interative functional inequalities

Brydak D., Choczewski B., Czerni M.

Opuscula Mathematica

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2008

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Vol. 28, no. 4

441-452

EN

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.

6

Baire measurability of (M,N)-Wright convex functions

Lewicki M.

Commentationes Mathematicae

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2008

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Vol. 48, [Z] 1

75-83

EN

Let I ⊆R be an open interval and M,N : I^2 →1be means on I. Let [formula] We give sufficient conditions on M,N and the function &fi;such that for every Baire measurable solution R of the functional inequality [formula].

7

On the generalization of Wright-convexity

Sobek B.

Journal of Mathematics and Applications

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2006

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Vol. 28

121-126

8

General solutions of two functional inequalities and converses to contraction theorems

Jachymski J.

Bulletin of the Polish Academy of Sciences. Mathematics

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2003

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Vol. 51, no 2

147-156

EN

We give necessary and sufficient conditions for the existence of some general solutions of the following two functional inequalities with an unknown real function phi: phi(Fx) is less than or equal to eta(phi(x)) and gamma(phi(Fx)) is less than or equal to phi(x). As an application we establish reciprocals to fixed point theorems of Matkowski and Wong. This extends an earlier result of Bessaga on a converse to the Banach contraction principle.