Is it appropriate to use active and passive voice together in abstract?
I don't think there is any inherent problem with mixing active and passive voice.  However I would offer the following suggestions:


The passive voice is certainly unnecessary here, so you might as well avoid it.  You could simply say "We have also obtained some new results."
I think most people prefer to write abstracts in the present tense.  "In the present paper, we prove theorems... We also obtain some new results."
The last sentence "obtained some new results" is so vague that it doesn't tell the reader anything.  Either add more details ("We also obtain some new results on the reticulation of splines") or just omit the sentence altogether.