On some classes of closed pro-$p$-groups

This paper considers the problem of the finiteness of a pro-$p$-group $G$ with trivial Schur multiplicator (or closed pro-$p$-group) in the following three cases: (i) $G$ has 3 generators and 3 defining relations, $p\geqslant3$ and $G^p\supset(G,G)$; (ii) $G$ has 2 generators and 2 defining relations, $p\geqslant5$ and $G^p\supset(G,G,G)$; (iii) $G$ has 3 generators and 3 defining relations, $p\geqslant5$ and one of the relations lies in the fourth term of the Zassenhaus filtration.
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