I have changed a few files name by de-capitalize the first letter, as in Name.jpg to name.jpg. Git does not recognize this changes and I had to delete the files and upload them again. Is there a way that Git can be case-sensitive when checking for changes in file names? I have not made any changes to the file itself.

As long as you're just renaming a file, and not a folder, you can just use git mv:

git mv -f yOuRfIlEnAmE yourfilename

(As of a change in Git 2.0.1, the -f flag in the incantation above is superfluous, but it was needed in older Git versions.)