Can you have a relationship without trust?

I have been dating my boyfriend for about a year now. And I do not trust him at all. He lies about the smallest and dumbest things so that I feel like I wouldn't be able to trust him with matters of importnace. For example I have found him communicating with other girls numerous times telling them he loves them. When they ask about me he just says we are friends and I forced myself upon him. He sounds like an *** but he does have some good qualities that I adore. he is sweet, gentile, attentive and protective of me and my feelings. But then again who can he do all this to me? Everytime we fight about it he swears he loves me and me alone and that he is just having some moments of weakness. IDK if I should let him go or stay and give him time to change his ways? I love him so much but I'm so tired of all the hurting, and mental torture I go throught because I can't trust him. What do I do?