The first picture shows what appears to be Sweet violet foliage (Viola odorata), though I'm not sure this is something which grows readily where you are - link below contains a couple of pics of the foliage
http://www.wildheartpermaculture.co.uk/?p=2177
If it is violets, you'd be seeing small lilac coloured flowers a bit later on, or in spring anyway. If you've already got a violet or two somewhere, its likely that, because they do seed themselves readily.
There is something else in the same picture with different leaves, just four leaves are visible in the centre of the pic, not sure what that is, but its not the same as the violet leaves.
The last pic, showing the much rounder leaves, could be Asara, as already suggested by Michelle, but could also be Violet, hard to be certain at this stage. Either way, you probably don't want quite so many in the border...