Be aware of this possibility, particularly if you tend to avoid frequently. If you notice the other person is avoiding, it might be a clue that she is uncertain and needs time to investigate the situation. Communicate your hopes for resolving the conflict. Try to overcome the person's fear or negativity by reassuring her that a win-win resolution is possible. Appeal to the avoider's sense of honesty.
Most conflicts have several parts.
Break the conflict down and deal with one part at a time. You'll soon find out which parts the other person is avoiding. Dominating is usually power-oriented and delivered with force. The ability to take control can actually be quite helpful when the other person lacks knowledge or expertise, and your opponent may be relieved that you can offer a solution. Don't try to dominate too often, however.