The iBooks behaviour I get is identical to that described by turfstuf. I briefly see the page I had been on then then I go back to a bookcase then the book opens again and I get back to the page I had been on. Except very occasionally, maybe if the book has been idle for hours, the book just closes and I see the active book shelf and the book stays closed.
I will be in Bristol next week and will try and get an appointment to see someone in the Apple store there. I don't hold out much hope of getting anywhere since I suspect the iOS dev team does know about this but is keeping quiet. I expect to be fobbed off with reinstall-everything-from-scratch and I imagine the Geniuses are more versed in configuration issues and app usage than obscure iOS 7 bugs. But it doesn't hurt to ask.
So far in this thread we've seen reports of problems on the iPhone 4, iPad 2 and iPad mini (1st gen, Filmphil?), all of which have 512MB RAM. So at least we are seeing a pattern.
I've been ditching old apps to claw back some space but it has made no difference, as was phaulfromstone's conclusion too.
I won't be the only one to be intensely annoyed if iOS 7 implicitly demands a more recent machine, particularly with no downgrade path to iOS 6 (and to think of the abuse I used to throw at iOS 6 ...)