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hcentering]{geometry} +\usepackage{setspace} %Pakiet do lokalnej zmiany interlinii +\usepackage{enumerate} %Pakiet do robienia niestandardowych list. +\usepackage{ulem} %Pakiet do tekstu z kropkami pod nim +\usepackage{xcolor} +%dodatkowe operatory matematyczne +\DeclareMathOperator{\sgn}{sgn} %funkcja signum +\newcommand{\norm}[1]{\left\Vert #1\right\Vert} %norma \norm{x}=||x|| +\newcommand{\ilsk}[2]{\left\langle #1 , #2\right\rangle} %iloczyn skalarny \ilsk{x}{y}= + +%definicje różnych oznaczeń zbiorów +\newcommand{\R}{\mathbb{R}} %zb. liczb rzeczywistych +\newcommand{\N}{\mathbb{N}} %zb. liczb naturalnych +\newcommand{\C}{\mathbb{C}} %zb. liczb zespolonych +\newcommand{\Z}{\mathbb{Z}} %zb. liczb całkowitych +\newcommand{\K}{\mathbb{K}} %oznaczenie na ciało K +\newcommand{\CB}{\mathcal{B}} %oznaczenie na B pisane, np w przestrzeni zbiorów ograniczonych lub sigma ciałach borelowskich + +%style potrzebne dla zmiany domyślnych styli twierdzeń i definicji +\newtheoremstyle{crr1}% styl dla twierdzeń + {\topsep}% Space above, empty = `usual value' + {\topsep}% Space below + {\itshape}% Body font - czcionka pochylona + {0pt}% Indent amount (empty = no indent, \parindent = para indent) + {\bfseries}% Thm head font + {.}% Punctuation after thm head + { }% Space after thm head: \newline = linebreak or " " + {\thmname{#1}\thmnumber{ #2}\thmnote{ (\textit{#3})}}% Thm head spec + +\newtheoremstyle{crr2}% styl dla definicji + {\topsep}% Space above, empty = `usual value' + {\topsep}% Space below + {\normalfont}% Body font - czcionka normalna + {0pt}% Indent amount (empty = no indent, \parindent = para indent) + {\bfseries}% Thm head font + {.}% Punctuation after thm head + { }% Space after thm head: \newline = linebreak or " " + {\thmname{#1}\thmnumber{ #2}\thmnote{ (\textit{#3})}}% Thm head spec +%definicja różnych matematycznych środowisk +\theoremstyle{crr1} +\newtheorem{tw}{Twierdzenie}[chapter] +\newtheorem{lem}[tw]{Lemat} +\newtheorem{wn}[tw]{Wniosek} +\theoremstyle{crr2} +\newtheorem{df}[tw]{Definicja} %chapter - numeracja wg rozdziału, section - wg "sekcji" +\newtheorem{prz}[tw]{Przykład} %Dodanie [df] sprawia, że przykłady i definicje będą miały wspólną numerację +\newtheorem{uw}[tw]{Uwaga} +%definiowanie środowiska dla dowodu +\newenvironment{dwd}{\noindent{\bf Dowód:}\rm }{\hfill\qed\par} +% potrzebne gdy wyświetla się "Spis rzeczy" zamiast "Spis treści" i "Literatura" zamiast "Bibliografia" +\addto\captionspolish{\renewcommand{\contentsname}{Spis treści}} +\addto\captionspolish{\renewcommand{\bibname}{Bibliografia}} +%----------------------------------------------------------------------------------- +\begin{document} +\setlength{\mathindent}{20pt}% wcięcie wzorów matematycznych +\begin{titlepage} +\begin{center} +{\LARGE Uniwersytet Śląski w Katowicach} +\vskip 1cm +{\LARGE Wydział Matematyki, Fizyki i Chemii} +\vskip 1cm +{\LARGE Instytut Matematyki} +\vskip 3cm +{\Large \bf Imię i nazwisko}% wstaw swoje dane +\vskip 1cm +{\large Numer albumu 012345}% numer albumu +\vskip 2.5cm +\begin{doublespace} +{\huge \bf RÓWNANIA FUNKCYJNE ORAZ ICH STABILNOŚĆ}% tytuł pracy +\end{doublespace} +\vskip 2.5cm +{\large PRACA DYPLOMOWA\\LICENCJACKA (lub MAGISTERSKA/INŻYNIERSKA)} +\end{center} +\vspace{3.0cm} +\begin{flushright} +\parbox{4.5cm}{Promotor: \\ +Dr Radosław Łukasik} +\end{flushright} +\vspace{3cm} +\begin{center} +Katowice 2019% rok +\end{center} +\newpage +\thispagestyle{empty} +\noindent +Słowa kluczowe: \dotuline{równanie Cauchy'ego, Jensena, stabilność\hfill}\\% wstaw swoje słowa kluczowe max 2 linijki +\par \dotuline{\hfill}\\[1cm] +\textbf{Oświadczenie autora pracy}\\[1cm] +Ja niżej podpisany/a:\\ +\par imię (imiona) i nazwisko \dotuline{Jan Kowalski\hfill}\\% wstaw swoje dane +\par autor pracy dyplomowej pt. \dotuline{Klasyczne równania funkcyjne\hfill}\\% wstaw swój tytuł +\par\dotuline{\hfill}\\% +\par\dotuline{\hfill}\\% razem 3 linie na tytuł jeśli jest ich więcej usuń + +\noindent Numer albumu: 012345\\% wstaw numer albumu + +\noindent Student/ka Wydziału Matematyki, Fizyki i Chemii Uniwersytetu Śląskiego w Katowicach\\ + +\noindent kierunku studiów\\ +\dotuline{Matematyka\hfill}\newline + +\noindent specjalności\\ +\dotuline{Matematyka nauczycielska\hfill}\\% wstaw swoją specjalność + +\noindent Oświadczam, że ww. praca dyplomowa: +\begin{itemize} +\item +została przygotowana przeze mnie samodzielnie\footnote{uwzględniając merytoryczny wkład promotora (w ramach prowadzonego seminarium dyplomowego)}, +\item +nie narusza praw autorskich w rozumieniu ustawy z dnia 4 lutego 1994 r. o prawie autorskim i prawach pokrewnych (tekst jednolity Dz. U. z 2006 r. Nr 90, poz. 631, z późn. zm.) oraz dóbr osobistych chronionych prawem cywilnym, +\item +nie zawiera danych i informacji, które uzyskałem/-am w sposób niedozwolony, +\item +nie była podstawą nadania dyplomu uczelni wyższej lub tytułu zawodowego ani mnie, ani innej osobie. +\end{itemize} +\vskip 1cm +Oświadczam również, że treść pracy dyplomowej zamieszczonej przeze mnie w Archiwum Prac Dyplomowych jest identyczna z treścią zawartą w wydrukowanej wersji pracy.\\[1cm] +\textbf{Jestem świadomy/a odpowiedzialności karnej za złożenie fałszywego oświadczenia.}\\[1cm]% lub Jestem świadomy +\parbox{6cm}{\dotfill\center{Data}}\hfill\parbox{6cm}{\dotfill\center{Podpis autora pracy}} +\end{titlepage} +\setcounter{page}{1} +\tableofcontents %wyświetlanie spisu treści + +\chapter*{Wstęp} +\addcontentsline{toc}{chapter}{Wstęp} +We wstępie opiszemy z grubsza co robimy w tej pracy. + +\chapter{Pierwszy rozdział} +\begin{df} +\label{def1.1} +Funkcję $f \colon \R^N \to \mathbb{R} $ nazywamy addytywną +wtedy i tylko wtedy, gdy spełnia ona równanie Cauchy'ego, to znaczy +\begin{equation}%jednolinijkowy wzór z numerem +\label{eq1.1} +f(x+y)=f(x)+f(y),\ x,y\in\R^N. +\end{equation} +\end{df} +Do równania \eqref{eq1.1} się tutaj odwołamy (\ref{eq1.1}). Jest to równanie z definicji \ref{def1.1} + +\begin{lem} +Jeżeli $f \colon \R^N \to \R$ jest addytywna to: +\begin{equation*}%jednolinijkowy wzór bez numeru +f(\sum \limits_{i=1}^{n} x_{i})=\sum \limits_{i=1}^{n} f(x_{i}) +\end{equation*} +dla każdego $n \in \N$ i dla wszystkich $x_1,x_2,...,x_n \in\N^N$. +\end{lem} +\begin{dwd} +Zauważmy, że +\begin{align*}%wielolinjkowe obliczenia/wzory bez numeracji +f(\sum \limits_{i=1}^{n+1}x_i)= f(\sum\limits_{i=1}^{n}x_i + x_{n+1})= f(\sum\limits_{i=1}^{n}x_i) + f(x_{n+1}) += \sum\limits_{i=1}^{n}f(x_i) +f(x_{n+1}) = \sum \limits_{i=1}^{n+1} f(x_i) +\end{align*} +\end{dwd} + +\begin{lem} +Jeżeli %$f_{1} \colon \mathbb{R}^N \to \mathbb{R}$ i +$f_1,f_2 \colon \mathbb{R}^N \to \mathbb{R}$ +są funkcjami addytywnymi, to dla wszystkich $a,b\in\mathbb{R}$ funkcja +$f=af_{1}+bf_{2}$ jest addytywna. +\end{lem} +\begin{dwd} +Ustalmy $x,y \in \R^N$. +Wówczas +\begin{align*} +f(x+y) &=af_1(x+y) + bf_2(x+y)%= \\ +=a\Big( f_1(x) +f_1(y)\Big) + b\Big(f_2(x)+f_2(y)\Big) \\ +&=af_1(x) + bf_2(x) +af_1(y) + bf_2(y) + = f(x)+f(y). +\end{align*} +\end{dwd} + +\chapter{Rozdział drugi} + +Rozważmy równania funkcyjne: +\begin{eqnarray}%zamiast align +E(x+y)&=& E(x) \cdot E(y) \label{eq:20a} \\ %E +L(xy)&=& L(x) + L(y) \label{eq:20b} \\ %L +M(xy)&=& M(x) \cdot M(y) \label{eq:20c} % M +\end{eqnarray} +znane jako warianty równania Cauchy'ego.\\ +Będziemy je odpowiednio nazywać: wykładnicze, logarytmiczne i multiplikatywne. + +\section{Równanie wykładnicze} +Coś piszemy w podrozdziale. + +\section{Równanie logarytmiczne} +\begin{uw} +Rozwiązania tego równania można znaleźć w \cite{Kuczma2009}. Pewne uogólnienia znajdują się za to w pracy R. Łukasika (patrz~\cite[Theorem 2]{Lukasik2014}). +\end{uw} + +Rozpatrzmy następujący wykres funkcji (lepiej robić w zewnętrznym programie w formacie eps):\\ +\setlength{\unitlength}{0.1mm}%definicja jednostki +\begin{picture}(1000,500) +\linethickness{1mm}%grubość linii +\put(100,100){\textcolor{red}{\line(1,1){300}}} +\linethickness{0.5mm} +\linethickness{0.1mm}\put(0,250){\vector(1,0){1000}} +\linethickness{0.1mm}\put(500,0){\vector(0,1){500}} +\thicklines +\put(25,25){\textcolor{blue}{\circle{30}}} +\put(500,250){\textcolor{green}{\circle*{30}}} +\put(950,200){\mbox{$t$}} +\put(520,450){\mbox{$f(t)$}} +\qbezier(500,250)(550,400)(1000,500) +\end{picture} + +$$\ilsk{x}{x}=\norm{x}^2$$ +$$n^2 \underset{n\to\infty}{\longrightarrow} \infty$$ +%bibliografia +\begin{thebibliography}{99} +\addcontentsline{toc}{chapter}{Bibliografia} %dodajemy ją do spisu treści +\bibitem{AD1989} +J. Acz{\'e}l, J. Dhombres, Functional Equations in Several Variables, Cambridge University Press, Cambridge (1989). +\bibitem{Kannappan} +Pl. Kannappan, Functional Equations and Inequalities with Applications, Springer, 2009 \\ +\bibitem{Kuczma2009} +M. Kuczma, Introduction to the Theory of Functional Equations and Inequalities, second edition, Birkhäuser, 2009. +\bibitem{Lukasik2014} +Radosław Łukasik, Some generalization of the quadratic and Wilson’s functional equation, \textit{Aequat. Math.} \textbf{87} (2014), 105--123. + +\end{thebibliography} +\end{document}