test-fancypagestyle.tex

\documentclass{book}
\usepackage{lipsum}
\usepackage[headings]{fancyhdr}

\fancypagestyle{fancy}{
  \renewcommand{\chaptermark}[1]{\markboth{Chapter \thechapter. ##1}{}}
  \renewcommand{\sectionmark}[1]{\markright{\thesection\ ##1}}
}

\fancypagestyle{myotherfancy}[fancydefault]{
}

\pagestyle{headings}
\markboth{Myheadings Leftmark (Right side of left page)}{Markboth Right?}
\markright{Myheadings Rightmark (Left side of right page)}

\begin{document}

\tableofcontents

\chapter{Introduction}
\thispagestyle{headings}
\lipsum[1]

\section{The Problem}
\label{sec:problem}

\lipsum[2-3]

\section{Evaluation}

\lipsum[3-5]

\newpage
\pagestyle{fancy}
\chapter{Another chapter}

Here we switch to page style \texttt{fancy}. The headers should come out as:

Right header on even page:
\begin{quote}
  \textsl{Chapter 2. Another chapter}
\end{quote}
Left header on odd page:
\begin{quote}
  \textsl{2.1 Another section}
\end{quote}
These come out as \textsl{slanted}, because that is the default for page style \texttt{myheadings} (as well as the default for page style \texttt{fancy}).
The page number comes on the other side of the header.

\lipsum[1]

\section{Another section}

\lipsum

\newpage
\pagestyle{myotherfancy}
\chapter{Third chapter}

Here we switch to page style \texttt{myotherfancy}. Here we use the original page style \texttt{fancy} headers. The headers should come out as:

Right header on even page:
\begin{quote}
  \textsl{CHAPTER 3. THIRD CHAPTER}
\end{quote}
Left header on odd page:
\begin{quote}
  \textsl{3.1. YA SECTION}
\end{quote}
These come out as \textsl{slanted}, because that is the default for page style \texttt{myheadings} (as well as the default for page style \texttt{fancy}).
The page number comes on the other side of the header.

\lipsum[1]

\section{YA section}

\lipsum

\end{document}