I upgraded the software on my iPhone 4 recently which appeared to work as expected. When I tried using any of the Applications installed on the phone after the update, I found that the apps were not opening. I tried opening a number of the apps, but the same issue was happening with all of them, where they would start opening, but then close straight away, returning to the home screen.
I did a bit of a search, and found the following article which explained a similar issue and a solution that fixed the problem with my Apps not opening on iPhone 4.
iPhone 4 Apps Not Opening (Closing Straight Away)
The solution that fixed this problem allowing apps to open again was to download a new App from the store. I selected a free app to dowload, which worked fine. Once it had installed, all other apps were then opening without closing straight away.