Browsing by Subject "functional equation"

Now showing 1 - 6 of 6

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 ,
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: Restricted ,
Pewne twierdzenie o średniej prowadzące do pewnego równania funkcyjnego. Równanie Sincova
(Data obrony: 2010-07-08) Fiema, Natalia
Wydział Matematyki Stosowanej
Access status: Restricted ,
Równanie funkcyjne Abela
(Data obrony: 2010-07-08) Jaśko, Mateusz
Wydział Matematyki Stosowanej
Access status: Open Access ,
Stability of the equation of homomorphism and completeness of the underlying space
(2008) Moszner, Zenon
We prove that all assumptions of a Theorem of Forti and Schwaiger (cf. [4]) on the coherence of stability of the equation of homomorphism with the completeness of the space of values of all these homomorphisms, are essential. We give some generalizations of this theorem and certain examples of applications.