Hy guys!
I'm studying some fixed point results for multi-valued function on a b-metric space $(X,d,s)$. I'm looking for the proof of a theorem wich is just a generalization of Nadler's results for multi-valued function on a metric space.
Please note that:
Def. Let X be a set, and let $s≥1$ be a given real number. A function $d:X×X→R^{+}$ is said to be a $b$-metric if and only if for all $x,y,z\in X$, the following conditions are satisfied:
$d(x,y)=0$ if and only if $x=y$.
$d(x,y)=d(y,x)$.
$d(x,z)≤s[d(x,y)+d(y,z)]$.
A pair $(X,d)$ is called a $b$-metric space.
Here is the theorem I'm looking for.
Theorem
Let $(X,d)$ be a $b$-complete metric space, and let $T:X→CB(X)$ be a multi-valued mapping such that $T$ satisfies the inequality
$$H(Tx,Ty)≤rd(x,y)$$ for all $x,y\in X$, where $0<r<\frac{1}{s}$. Then $T$ has a fixed point.
Anyone can help me finding the proof of this theorem? I know it is probably by Czerwik [9], as you can see here.
Thanks in advance!