If you’ve broached the subject of a rent rise, but your tenants refuse (whether you have a periodic contract or fixed-term contract that has ended), you could issue a Section 21 notice. This notice formally requests that your tenants leave at the end of their contract.
If you issue a Section 21, you must give your tenants at least two months’ notice. Full details can be found at GOV.UK.
However, it’s worth noting that as of May 2023, the Renters’ Reform Bill started making its way through Parliament and seeks to abolish Section 21, so this situation may change in the future.