United have lost their last two Premier League games - 2-1 at Bournemouth and 2-1 at home to Norwich - and travel to Stoke on Boxing Day, followed by a visit from Chelsea two days later.
Manager Louis van Gaal is coming under increased pressure amid speculation recently-sacked Jose Mourinho could take over, with tentative talks already said to have been held at boardroom level over the possibility of replacing the Dutchman with the former Chelsea boss.
However, Mata insists belief is high in the dressing room that United will turn things around.
"We are really disappointed for not giving you what you deserve, because your support is amazing and unconditional, but I'm sure that we will turn this situation around," he wrote in his weekly blog.
"I was confident that we were going to get the three points on Saturday in Old Trafford, but that was not the case. We are in a complicated situation and in order to get out of it we have to stick together and believe in ourselves so we can get a positive result as soon as possible.
"We all feel frustrated, same as last week, and we need to win a game as quickly as possible because it will change our mood and give us confidence, I'm sure of that."
Mata added: "Our next chance to do it will be on Boxing Day at Stoke, and that's not an easy game. No matter how tough the situation can be, I'm not even thinking about giving up.
"I believe that's the only way to get out of this bad streak - keep working hard and sooner or later we will get the results that we want."