In a problem I am working on, I get the following functional equality: $$a(x)+b(y)=c(x+y)+d\left({x\over y}\right)+w(x,y)$$ Where $x,\ y>0$ and $a,\ b,\ c,\ d,\ w$ are real Borel measurable functions. Also, $w(x,y)$ can be written as $w(x,y)=K\left(\int_0^{x+y}e^{c(z)}{z}\ dz,\ \int_0^{x\over{x+y}}e^{d(z)}\ dz\right)=K(s,t)$ for some measurable function $K$.
Let $r>0$ then: $$a(rx)+b(ry)=c(rx+ry)+a(x)+b(y)-c(x+y)-w(x,y)+w(rx,ry)$$ $$a_r(x)+b_r(y)=c_r(x+y)+w_r(x,y)$$ Where $a_r(x)=a(rx)-a(x)$, $b_r(x)=b(rx)-b(x)$, $c_r(x+y)=c(rx+ry)-c(x+y)$ and $w_r(x,y)=w(rx,ry)-w(x,y)$.
Consequently, $$w_r(0,y)=b_r(y)-c_r(y)$$ $$w_r(x,0)=a_r(x)-c_r(x)$$
My goal is to know a solution technique to get a general form for $a,\ b,\ c,\ d$ given that I have a form for $K(s,t).$
For example if $K(s,t)=0$ then this becomes Olkin-Baker equation which has a nice solution.
Now if $K_{\alpha}(s,t)={(1-\alpha)-\alpha(1-\alpha)(1-s)(1-t)+2\alpha st \over \left(1-\alpha (1-s)(1-t)\right)^{3}}$ for $\alpha \in [-1,1]$
If $x=0$ then $t=0$ and $K_{\alpha}(s,0)={(1-\alpha)-\alpha(1-\alpha)(1-s) \over \left(1-\alpha (1-s)\right)^{3}}.$
And if $y=0$ then $t=1$ and $K_{\alpha}(s,1)={(1-\alpha)+2\alpha s}.$
So for $y=0$ we get $$a_r(x)-c_r(x)=2\alpha \left(\int_0^{rx}e^{c(z)}z\ dz\ -\int_0^x e^{c(z)}z\ dz\right)$$
And similarly for $x=0$ $$b_r(y)-c_r(y)=K_{\alpha}\left(\int_0^{ry}e^{c(z)}{z}\ dz,\ 0\right)-K_{\alpha}\left(\int_0^{y}e^{c(z)}{z}\ dz,\ 0\right)$$
And the original equation $$a_r(x)+b_r(y)=c_r(x+y)+K_{\alpha}\left(\int_0^{rx+ry}e^{c(z)}{z}\ dz,\ \int_0^{x\over{x+y}}e^{d(z)}\ dz\right)-K_{\alpha}\left(\int_0^{x+y}e^{c(z)}{z}\ dz,\ \int_0^{x\over{x+y}}e^{d(z)}\ dz\right)$$
I would get equations involving integrals. What techniques I can use to solve such equations?
Any links to papers that deal with similar problems are appreciated.