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As part of a research problem my friend asked me to search for a an example of a complete metric space $(X,d)$ and functions $f,g,h$ from $X$ to $X$ such that:

(a) Range of $h$ contains the range of $f$ and the range of $g$.

(b) There exist non negative real numbers $\alpha,\beta,\gamma$ and $\delta$ with $\alpha+ \beta+ \gamma+ 2\delta <1$ so that:

$d(f(x),g(y))\le \alpha d(h(x),h(y))+\beta d(h(x),f(x))+\gamma d(h(y),g(y))+ \delta\big(d(h(x),g(y))+(h(y),f(x))\big)$

(c) $f,g,h$ should not satisfy $d(f(x),g(y))\le \alpha d(h(x),h(y))$for all $\alpha\in[0,1)$.

My first thought was to turn towards trivial examples like the identity function but that doesn't seem to work. Can someone make a suggestion?


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Have you tried to find such functions on a general 3-point metric space $X$? – user31373 Sep 8 '12 at 0:55

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