**General Mathematics, Vol. 5, No. 1 - 4, pp. 109-126, 1995 **

**Abstract:** On a general metric space $(S,\sigma)$ three types of functionals
${\cal E}_h(\sigma ,p)$ $(h=1,2,3)$ [13], which generalize the concept of $p$--energy
$(p\geq 1)$ of a curve, are considered. Conditions in order for them to coincide are
studied. After giving a suitable notion of asymptotically equal generalized distances, one
shows that if $\sigma$ and $\rho$ are so, then ${\cal E}_h(\sigma ,p)(\gamma)={\cal
E}_h(\rho ,p)(\gamma)$ for any $p\geq 1$, when the curve $\gamma$ of $S$ has finite energy
for some $p_{0}>1$ [14]. A remarkable exemple is pointed out.

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