PSP also didn't have to deal with mobile/smartphone gaming. The Vita did.
The Dedicated Handheld Gaming market went from around 230-240 million units sold between the DS & PSP to just under 91 million units sold between the 3DS & PSP; Japan went from 53 million units sold between the DS & PSP to roughly 30-31 million units between the 3DS & Vita.
Sony shooting themselves in the foot with the Vita with things like the memory card issue and losing the support it had with the PSP (i.e. Monster Hunter) certainly didn't help matters. But with the eradication of the handheld market, the Vita wasn't going to do much better than it did anyways.
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