Well, Nintendo has/will have an indirect competitor which is the mobile market which if Sony was smart, thats what their next handheld should be... A phone with a controller accessory! A lot of people, specially on this site, love the and Sony phones so if it could play games, bam! It would help their market share in the mobile market (which is super low as it is)

http://www.ibtimes.com/worldwide-smartphone-sales-q3-2013-samsung-android-maintain-lead-apples-market-share-slips-1471754

I'm lazy, but I'll give you my thoughts anyway.

I don't think Sony will bother. With all the cuts elsewhere I can't imagine they will take such a risk as a new handheld in this market.

The handheld market without a competitor to Nintendo will pretty much look like it does now frankly.

Not sure what Nintendo will do. I think they will definitely make another handheld but I think they will have to diversify and/or think of something really innovative, otherwise their hardware sales will decline significantly again next gen as well.

- Nintendo will continue to compete with mobile.
- It will be Nintendo vs mobile until Nintendo gives up. No sane company would now try to release a dedicated gaming handheld with propriatary software.
- Vita will die with this generation. No new handheld from Sony. They may want to push their Xperia devices though with exclusive games and controller peripherals.