Searching for something while being on a https secured page opens a warning dialog. Thats annoying for the visitor of your page. To fix that problem, open a suitable module or create a new one and add the following xml to the modules config.xml.

Suppress warning when searching on https secured pages
You want or need help implementing this solution? Maybe you did not find what you were looking for? Contact me and I will assist you with whatever Magento problem troubles you!