@i_am_will your requirement is not very clear to me. Is it that you want your list items within collapsible lists and search the collapsible lists? Or is it case 2 where you have the list items within collapsible lists and a search within each collapsible list?
In case your requirement is case 1, I found a couple of fiddles and here are the links. However, these are fiddles based on JQM 1.1.1 and Jquery 1.7.1. You will need to update these to the latest or the versions that you are using.