Mat 540 Week 1 Math Paper

In \cite{Romauera92}, Romaguera pointed out that if $X=\mathbb{R}^+$ and $p : X \times X\to \mathbb{R}^+$ defined by
$p(x, y) = \max\{x, y\}$ for all $x, y \in X$ then ${CB}^p(X)=\emptyset$ and the approach used in Theorem \ref{THM201} and elsewhere has a disadvantage that the fixed point theorems for self-mappings may not be derived from it, when ${CB}^p(X)=\emptyset$. To overcome from this problem he introduced the concept of mixed multi-valued mappings and obtained a different version of Nadler's theorem in a partial metric spaces.
\begin{definition}
Let $(X, p)$ be a partial metric space. A mapping $T : X \to X \cup {CB}^p(X)$ is called a mixed multi-valued mapping on $X$ if $T$ is a multi-valued mapping on $X$ such that for each $x\in X$

\end{theorem}
\begin{proof}
It can be completed using the proofs of Theorem \ref{THM202} above and Theorem 2.2 \cite{Singh91}.
\end{proof}
The following example illustrate our results.

\begin{example}\label{EXM302} \cite{Jleli90}.
Let $X=\{0, 1, 2\}$ and $p : X \times X\to \mathbb{R}^{+}$ defined by
\begin{eqnarray*}
& &p(0,0)=0,\; p(1,1)=p(2,2)=\dfrac{1}{4},\; p(1,0)=\dfrac{1}{3},\; p(2,0)=\dfrac{3}{5},\; p(2,1)=\dfrac{2}{5},\\
& &\text{and $p(x,y)=p(y,x)$ for all } x,y\in X.
\end{eqnarray*}
Clearly, $(X, p)$ is a $0$-complete partial metric space. Now, define the mapping $S:X\rightarrow C^{p}(X)$ such that
\[
Sx= \begin{cases}
\{0\} \text{ if $x \neq 2, $} \\
\{0,1\} \text{ if $x = 2. $}
\end{cases}
\]
It can be easily seen that $S$ asymptotically regular and for all $x, y\in X$,
\[
H_p(Sx, Sy)\leq\varphi(\mu(x,y))
\]
with $\gamma\geq 1$ and $\varphi(t)=\frac{3t}{4}$. Therefore the assumptions of Theorem \ref{THM202} are fulfilled.
\end{example}

The following example shows the generality of our results.
\begin{example}\label{EXM302}
Let $X=\{0, 1, 4\}$ be endowed with the partial metric $p : X \times X\to \mathbb{R}^{+}$ defined by