Is it normal to abandon a friendship if
... if you are the only one putting in effort to keep in touch and meet up?
She clearly enjoys the time we spend and says she cares about me but she never texts me to ask how I am doing or arranges our meet-ups. I am always the one doing that. And she is always 'busy' and makes promise after promise to make time for me that she doesn't keep. I have started telling her that I don't want her to feel obligated, or like I am pestering her. She always says 'No, no, no - I love our meet-ups and all the little things you send me!' and assures me that she appreciates me. But nothing ever changes. If I didn't keep contacting her first, we would completely lose touch.
I keep telling myself that it's the final straw and that this friendship is too emotionally exhausting to maintain. But then I see her and I know she cares. We have a special bond. Would it be normal to abandon this friendship?