In this passage, Jesus told his disciples a story to address the exact same issue we have been discussing - not giving up. Notice in the story how persistent the woman was. She kept coming to the judge constantly until he finally gave in and granted her request for justice. She was determined and even though at first the judge was unwilling, this did not deter her at all. As I read this story over and over, I pondered on why this woman was so determined. The thought that stuck in my mind is that generally people ask for help from people who have the ability to help them. It's no use asking someone for something you know they don't have. I believe this woman knew that the judge was the only person who could help her out of her situation. Her faith enabled her to overcome.
We see that Jesus wants us to take note of what the unjust judge said. The woman came to him so often and boldly that he thought she would wear him out! What about us? Do we believe that God is ultimately the only one who can really help us?  Will we constantly seek Him and ask Him for what we need? Jesus didn't use the example of the unjust judge to portray God as hard and unwilling. Jesus in fact was saying that if an unjust judge could eventually grant the request of this woman who needed him, how much more will God our loving Father provide and protect and guide His own people who come to Him? He is infinitely more willing, but we must have faith, the kind of faith that will not give up.
God is so kind and willing to do good to those who seek Him. He openly invites us to come to Him. His word - the Bible is filled with great promises to those who would put their trust in Him.