There are many people who are not on speaking terms. Why? Ahhhhh - always a different story.
Josh offended cousin Chavi who told her mother who called up Josh's mother [her sister] to complain. Josh's mother knew the secret that the best defense is a good offense so she went on the offensive and attacked Chavi - and her mother. Chavi's mother was incensed, told her husband who stopped saying hello to his brother in law in shul and the family is ruptured, they stop inviting each other to family simchas, stop talking and the Satan dances.....
Not talking to another person out of enmity is just like eating trief or driving a car on Rosh Hashana day. Same Torah. The pasuk says לא תשנא את אחיך בלבבך - don't hate your brother in your heart, and not talking to someone out of enmity, hostility or hatred defines one as a שונא [see Mishna Sanhedrin 3/5 about the psul of a שונא לדון when he doesn't speak to him for three days and many derive the general איסור of שנאה from there].
Yet, there are far too many people not on speaking terms.
In a few hours the nine days begins. When is a better time to put an end to this and make up. It is OK. People are human. They often don't like to apologize. I know people who haven't apologized in decades. There is something twisted in their brain chemistry that doesn't allow them to apologize regardless of how much they hurt other people. People also can be selfish. Just about everybody is selfish to a certain extent. That is important information because that means that they weren't out to get YOU but only out to protect and fend for themselves. Don't take it personally. Few people care about you enough to go out of their way to hurt you. It is all a matter of their self interest. They are self interested because they were born that way. Not everybody is a talmid of Rav Yisrael Salanter.
I go through my history and can think of one or two or many more people who offended me in various ways. But I can't think of anybody who did it purely maliciously. Nobody would spend money to hurt me. It is always due emotional inadequaces, a lack of self development, being unconscious [meaning not sufficiently counscious of the feelings, needs and emotions of others] or other factors which are all the hallmark of the imperfect fault ridden human being.
So remember - it has NOTHING to do with you. Let them be. Your job is to develop yourself so that you become more selfless, sensitive, kind, campassionate, caring and giving. Most people don't wake up in the morning with a list of the good qualities that they would like to cultivate that day. That is why they fail miserably at lifes greatest test - personal change and growth.
But if you are reading this then chances are that you want to change. One way to do that is to change your thought processes and allow others to be fault riddled. Not speaking to other people is not the Jewish way....