Tenant issue

We had given our premise to a company. Although their payment was on time and regular there were issues regarding renewal as they were forcing us to get into agreement with other companies of their group. We do not have a proof in writing that they were forcing us to renew as per their terms. They stated this in meetings only. We refused their terms but they did not vacate at the end of the agreement period. After multiple legal notices sent to them asking them to vacate the premises and clear our dues, they sent us a letter that they will be vacating a part of our premises. On the date they were supposed to hand over the partly vacated premises to us, there was no communication from their end. In following weeks, we called them and also went to their other office to meet management (as the management was not coming to our premises) but their management didn't meet and respond. Meanwhile they were keeping our premises occupied by keeping their furniture there (which we are sure of as we could see from outside but do not know how to prove in court) and probably also using for office work (which we are not sure of). Now after all these months they have vacated but they refuse to accept dues, for the part they said they would vacate, for the period over and above the date mentioned in their notice of vacating the premise. They are still using another part of the premise. And ever since the previous agreement ended, they have stopped paying rent, taxes or maintenance fee. I want to know 1) Isn't it required for both parties to sign some paper/document for handing back of the premises at the time of vacation just as they sign at the time of renting? 2) Does a tenant has the legal right to with-hold a part of the premises and vacate it part by part? 3) If the matter goes to court, will the court accept their argument that a notice of vacation was sent to us, so it was our responsibility to take possession? Our argument is that we couldn't just 'break into' that part of the premises that they said they were vacating. 4) How can we prove in court, that we tried contacting them but they didn't respond?