MathLabs

Problem 5

Find all functions f:R+→R+f:\mathbb{R}^+\to\mathbb{R}^+ such that (z+1)f(x+y)=f(xf(z)+y)+f(yf(z)+x)(z+1)f(x+y)=f(xf(z)+y)+f(yf(z)+x) for all positive real numbers x,y,zx,y,z.
Step 2 of 8: A linear-system lemma
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cu+v=a, u+cv=b,u=ca−bc2−1, v=cb−ac2−1cu+v=a,\ u+cv=b,\quad u=\frac{ca-b}{c^2-1},\ v=\frac{cb-a}{c^2-1}
Detailed analysis

Lemma: if c>1c>1 exceeds both a/ba/b and b/ab/a, the system cu+v=acu+v=a, u+cv=bu+cv=b has a positive solution u,vu,v, given by the displayed formulas, since both numerators and c2−1c^2-1 are positive under these hypotheses.