I am looking for a function $f$ that satisfy the following relation $f(y+\alpha*f(x,y),y)=\alpha*f(x,y)$, where $\alpha\in [0,1]$. The trivial one that satisfy this relation is $f(x,y)=x-y$. I tried with several other examples, but failed to find another example. Does there exist any non-trivial function that satisfy the above relation?
For general ways to attack such kind of functional equations, take a look at Evan Chen's Introduction to Functional Equations and Monsters. Introductory, oriented at Math Olympiad competitions, it is quite understandable and thorough (as far as it goes).
First question: Does $f$ have to satisfy your equation for all $\alpha$ (what range of values?) or are you searching for functions that satify the equation for some specific values of $\alpha$?
Next, fool around a bit. What can you deduce for specific $x, y$? Zero, $x = - y$, ...? You have found a solution. Does a multiple of it work? It is a linear combination of $x, y$. Any others?