Say a person borrows 50 cents from a fellow Jew while out of town. The borrower cut his trip short and left without being able to pay back. Can one rely that the lender would give up hope of getting back the 50 cents once realizing that the person he lent the money to has left? Is there any halachic source for what one should do in such a scenario--assuming contact information was never given?
There is a disagreement amongst the authorities what happens if the lender has given up hope ("ye'ush) of retrieving the debt, is the debt cancelled out, or does it remains. I have written an article on this in Hebrew. You can read it in the following link: https://docs.google.com/document/d/1lUYqb6ScHifOUAYC1CbYZtHczMnFYKrVQ_7A3yFGxJU/edit?usp=sharing However, in the case you described where the amount was so small and it was given to a person from out of town, it is possible that the lender never intended to receive it back and gave it as a present. But one would need to be familiar with the specifics of the case to determine that.