Of course they will - it is good technology. And it will be used to make fiat payments more efficient - and it will be the death for crypto currencies.
I think banks will use bitcoin as a possibility. Blockchain technology is growing and hence internet money will grow. People around the world will know and use bitcoin in the most popular and modern way. The traditional bank will also use bitcoin to increase revenue for banks, increasing the number of transactions with customers is likely to happen in the near future.