There are two* logical, discrete ways to solve this Futoshiki puzzle, each with its own unique solution. You can solve it using the letters given, or use the numbers 1-5, but if you map the letters to numbers, there must be a justification for how you map them (ie. you can't arbitrarily assign a letter to a number). The rules of Futoshiki are:
- Each row and column must contain one of each digit from 1-5 (or letter in "DUCKY").
- All inequality constraints between boxes must be satisfied (ie. 1 < 5; you'll have to figure out what the comparison between letters is depending on which route you take).
Here is a penpa-edit (without the letters entered in case you want to use numbers instead), and here is a transcription:
[ ] [ ] [D] [ ] [ ] v v ^ [ ] [ ] [U]<[ ] [ ] [D] [U] [C] [K] [Y] v [ ] [ ] [K] [ ]<[ ] v [ ]>[ ] [Y] [ ]<[ ]
A full answer of both solutions is preferable, but if you want to submit a partial answer of only one solution, I will accept the other solution (since it was left to last, it seems that it was harder), if that makes sense.
*Technically, there are three, but two of these methods are simply "inverses" of each other, so only one of them will count.