Doing a search on the YouTube app

No there's no way you can search into a specific channel on YouTube. Basically YouTube does not provide that privilege.

But if you are considering a specific channel and want to search videos related to the channel only than you can perform relative search which includes the basic search attribute with the channel name. By doing so your search results will be narrow downed and you will find easily what you wanted.

For example:- If you like Luis Fonsi vevo channel and want to search within his creation than try something like this:- x+Luis Fonsi vevo(where x=your basic search attribute.)