Browsing by Author "Matkowski, Janusz"

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Access status: Open Access ,
Conjugate functions, Lp-norm like functionals, the generalized Hölder inequality, Minkowski inequality and subhomogeneity
(2014) Matkowski, Janusz
For $h:(0,\infty )\rightarrow \mathbb{R}$, the function $h^{\ast }\left( t\right) :=th(\frac{1}{t})$ is called $(∗)$-conjugate to $h$. This conjugacy is related to the Hölder and Minkowski inequalities. Several properties of $(∗)$-conjugacy are proved. If $\varphi$ and $\varphi ^{\ast }$ are bijections of $\left(0,\infty \right)$ then $(\varphi ^{-1}) ^{\ast }=\left( \left[ \left( \varphi ^{\ast }\right) ^{-1}\right] ^{\ast }\right) ^{-1}$. Under some natural rate of growth conditions at $0$ and $infty$, if $\varphi$ is increasing, convex, geometrically convex, then $\left[ \left( \varphi^{-1}\right) ^{\ast }\right] ^{-1}$ has the same properties. We show that the Young conjugate functions do not have this property. For a measure space $(\Omega ,\Sigma ,\mu )$ denote by $S=S(\Omega ,\Sigma ,\mu )$ the space of all $\mu$-integrable simple functions $x:\Omega \rightarrow \mathbb{R}$. Given a bijection $\varphi :(0,\infty )\rightarrow (0,\infty )$, define $\mathbf{P}_{\varphi }:S\rightarrow \lbrack 0,\infty )$ by $\mathbf{P}_{\varphi }(x):=\varphi ^{-1}\bigg( \int\limits_{\Omega (x)}\varphi \circ \left\vert x\right\vert d\mu \bigg),$ where $\Omega(x)$ is the support of $x$. Applying some properties of the $(∗)$ operation, we prove that if $\int\limits_{\Omega }xy\leq \mathbf{P}_{\varphi }(x)\mathbf{P}_{\psi }(y)$ where $\varphi ^{-1}$ and $\psi ^{-1}$ are conjugate, then $\varphi$ and $\psi$ are conjugate power functions. The existence of nonpower bijections $\varphi$ and $\psi$ with conjugate inverse functions $\psi =\left[ ( \varphi ^{-1}) ^{\ast}\right] ^{-1}$ such that $\mathbf{P}_{\varphi }$ and $\mathbf{P}_{\psi }$ are subadditive and subhomogeneous is considered.
Access status: Open Access ,
Extensions of solutions of a functional equation in two variables
(2009) Matkowski, Janusz
An extension theorem for the functional equation of several variables $f(M(x,y))=N(f(x),f(y)),$ where the given functions $M$ and $N$ are left-side autodistributive, is presented.
Access status: Open Access ,
Iteration groups, commuting functions and simultaneous systems of linear functional equations
(2008) Matkowski, Janusz
Let $( f^t )_{t \in \mathbb{R}}$ be a measurable iteration group on an open interval $I$. Under some conditions, we prove that the inequalies $g\circ f^a \leq f^a \circ g$ and $g\circ f^b \leq f^b\circ g$ for some $a,b \in \mathbb{R}$ 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 $f$ with two suitably chosen elements $f^{a}$ and $f^{b}$ of an iteration group of $f$ then, in this neighbourhood, $g$ coincides with an element of the iteration group. Some weak conditions ensuring equality of iteration groups are considered.
Access status: Open Access ,
O operatorach złożenia generowanych przez multifunkcje
(Data obrony: 2009) Ludew, Jakub Jan
Wydział Matematyki Stosowanej
For a given function $h : I \times \mathbb{R} \rightarrow \mathbb{R}, I=[0,1]$ the mapping $H : \mathbb{R} \rightarrow \mathbb{R}$, defined by $(H\varphi(x) = h(x,\varphi (x)), \varphi \in \mathbb{R}^{I}$ is called a composition operator of a generator $h$. In 1982 J. Matkowski proved that a composition operator mapping the Banach space Lip(I) of Lipschitzian functions $\varphi : I \rightarrow \mathbb{R}$ into itself is globally Lipschitzian if and only if there exist functions $a, b \in Lip(I)$ such that postać $h(x,y) = a(x)y + b(x), x \in I, y \in \mathbb{R}$. Then this result has been extended to some other function Banach spaces in papers by J. Matkowski and his students. Composition operators generated by multifunctions in the spaces of Lipschitzian and bounded variation functions and multifunctions have been studied by A. Smajdor, W. Smajdor, G. Zawadzka and others.The main goal of the doctoral thesis is to examine composition operators generated by set-valued functions. In my paper I prove that Lipschitzian composition operators acting in the function spaces of: functions satisfying the Hölder condition, functions of $C^{1}$, absolutely continuous and continuous functions of bounded variation with values in corresponding multifunction spaces have to be generated by a set-valued function of the form $H(x,y)=A(x,y) + B(x)$, where $A(x, \cdot)$ is continuous linear multifunction, and $B, A(\cdot ,y)$ belong to the corresponding multifunction spaces mentioned above. The doctoral thesis also contains opposite theorems. Two of them strengthen respective results in the papers of A. Smajdor, W. Smajdor and G. Zawadzka. The basic tool is Rådström's embedding theorem.
Access status: Open Access ,
On extension of solutions of a simultaneous system of iterative functional equations
(2009) Matkowski, Janusz
Some sufficient conditions which allow to extend every local solution of a simultaneous system of equations in a single variable of the form $\varphi(x) = h (x, \varphi[f_1(x)],\ldots,\varphi[f_m(x)]),$ $\varphi(x) = H (x, \varphi[F_1(x)],\ldots,\varphi[F_m(x)]),$ to a global one are presented. Extensions of solutions of functional equations, both in single and in several variables, play important role (cf. for instance [M. Kuczma, Functional equations in a single variable, Monografie Mat. 46, Polish Scientific Publishers, Warsaw, 1968, M. Kuczma, B. Choczewski, R. Ger, Iterative functional equations, Encyclopedia of Mathematics and Its Applications v. 32, Cambridge, 1990, J. Matkowski, <a href="https://repo.agh.edu.pl/handle/AGH/50094">Iteration groups, commuting functions and simultaneous systems of linear functional equations</a>, Opuscula Math. 28 (2008) 4, 531-541]).
Access status: Open Access ,
Subadditive periodic functions
(2011) Matkowski, Janusz
Some conditions under which any subadditive function is periodic are presented. It is shown that the boundedness from below in a neighborhood of a point of a subadditive periodic (s.p.) function implies its nonnegativity, and the boundedness from above in a neighborhood of a point implies it nonnegativity and global boundedness from above. A necessary and sufficient condition for existence of a subadditive periodic extension of a function $f_{0}:[0,1)\rightarrow \mathbb{R}$ is given. The continuity, differentiability of a s.p. function is discussed, and an example of a continuous nowhere differentiable s.p. function is presented. The functions which are the sums of linear functions and s.p. functions are characterized. The refinements of some known results on the continuity of subadditive functions are presented.
Access status: Open Access ,
Uniformly continuous set-valued composition operators in the spaces of functions of bounded variation in the sense of Wiener
(2010) Azócar Bates, Luis Antonio; Guerrero, José Atilio; Matkowski, Janusz; Merentes Díaz, Nelson José
We show that the one-sided regularizations of the generator of any uniformly continuous and convex compact valued composition operator, acting in the spaces of functions of bounded variation in the sense of Wiener, is an affine function.