The problem, according to sources cited by The Wall Street Journal, is related to "manufacturing difficulties" of an unknown nature. Sharp had planned to begin production by the end of August, but the the Journal's sources say it is now unclear if or when Sharp will be able to start production.
The other two companies Apple uses are Japan Display Inc., and LG Display. These two suppliers have begun production of iPhone 5 displays and are supplying the components to Apple so the device can be assembled.
What we don't know is how many displays Apple is expecting to receive from each supplier. In the past, LG has been its chief display panel supplier. It's possible that LG is responsible for supplying the majority of the iPhone 5's displays, or that LG and Japan Display together cover most of the supply base.
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No matter the mix, Apple will have fewer panels than anticipated during the initial production run. This may or may not impact availability of the iPhone 5 at launch. (Earlier this year, LG faced difficulties manufacturing the new iPad's Retina display. LG's production problems didn't have any noticeable impact on the availability of the new iPad at launch.) Apple typically releases devices in the U.S. first, and then other markets over time. While the production delay may not affect the initial U.S. supply, it could delay the iPhone's launch in other markets.
The iPhone 5 is using a new type of display that includes in-cell technology. The new display tech allows Apple to eliminate a layer of material between the display and the touch panel above it by combining the two. This lets Apple build a thinner display component, which could lead to space savings within the device itself and possibly a thinner device altogether.
Apple is expected to announce the iPhone 5 on September 12, and the device is expected to go on sale as soon as September 21.