I have a huge amount of lines indicating boundaries of certains areas but they are most of time "opened" sometimes they do not have common vertices and sometimes are overlapping. I want to create a polygon by these lines that split the "BIG AREA POLYGON" into smaller polygon following the lines. How to manage this? I've tried the poligonize tool and the lines to polygons but results are not as expected.
What you're looking at here are significant topology errors. Here's a good article on ways to fix them in QGIS. Once you've fixed the topology errors, the polygonizing tool will work as expected, assuming the topology fixes have created the correct boundaries for the polygons.
I know this isn't a direct answer to your question but since this has been unanswered for a while now, I thought pointing you into a direction might help.