Yes, I'd say give him food and shelter and do
your best to try find the owner, including asking around and walking him to see if he heads for somewhere. It might be a family pet and beeing really missed
If no one claims him, and he likes being with you, congrats! You've got yourself a dog
(Don't know how laws are at your place, but usually, if you have a dog for some time and no one claimed him, you can legally own it)
Also, and this is more fuzzy, you should be able to ask the owner reinbursment for any vet and care charges you got.
So first thing I'd do is go to the vet, check for Id chip, then med check and basic deworming, yadayadayada... Tell him you are searching for the owner, keep the recipes and... you have proof of when you found him