How to sync my iphone with operating system IOS 7 iTune version 11.1 with Macbook OSX version 10.5.8

Just to make sure, which OS X is installed on your Mac ?

Click the Apple () menu top left in your screen. From the drop down menu click About This Mac. The version is noted there.

And which iPhone??

v10.6.8 is minimum requriement > Apple - iPhone 5s - Technical Specifications

That also applies to an iPhone 4 > iPhone 4S - Technical Specifications

If your Mac is running v10.5.8, you will need to upgrade to Snow Leopard v10.6.

Snow Leopard is available from the Apple Store > Mac OS X 10.6 Snow Leopard - Apple Store (U.S.)

Before purchasing Snow Leopard v10.6, make sure your Mac meets the requirements > Mac OS X v10.6 Snow Leopard - Technical Specifications

I'm pretty sure that you have to have at least OS X Snow Leopard which is 10.6 to be able to update iTunes to a version high enough to sync with your iOS 7 devices. You are going to need to purchase an upgrade and you are going to need to make sure that your computer can handle running that version of OS X to be able to get everything able to sync properly