I am trying to solve the equation f(x+t)=f(x)+f(t)+2√f(x)√f(t) - as in find a function that satisfies this equation. I notice that the RHS is (√f(x)+√f(t))2 but I am stuck after this.
Answer
The presence of √f(x) in your functional equation implies that the range of f is nonnegative.
If you are looking for a continuous function, Simon's comment shows that g must be linear. (With real functions, continuity and additivity imply full linearity. This is mentioned here although I'd prefer to show a link to an actual proof.)
Therefore √f(x)=ax
for some a.
- If a=0, then f is the zero function.
- If a>0, then f is only defined for x≥0, and f(x)=a2x2.
- If a<0, then f is only defined for x≤0, and f(x)=a2x2.
That is, there are three families of solutions. They all have the same form f(x)=a2x2, but either you must restrict the domain to non-negatives, non-positives, or a is necessarily 0.
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