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We show that uniqueness results of the kind those obtained for KdV andSchr\"odinger equations ([7], [28]), are not valid for the dispersiongeneralized-Benjamin-Ono equation in the weighted Sobolev spaces  $$H^s(\R)\cap L^2(x^{2r}dx),$$ for appropriated $s$ and $r$.  In particular, we obtain that the uniqueness result proved for the dispersiongeneralized-Benjamin-Ono equation ([13]), is not true for all pairs ofsolutions $u_1\neq 0$ and $u_2\neq 0$. To achieve these results we employ thetechniques present in our recent work [6]. We also improve some Theoremsestablished for the dispersion generalized-Benjamin-Ono equation and for theBenjamin-Ono equation ([13], [12]).

On decay of the solutions for the dispersion generalized-Benjamin-Ono and Benjamin-Ono equations