Let
- $(E,\mathcal E,\lambda)$ be a measure space
- $I$ be a finite nonempty set
- $p,q_j$ be probabiliy density functions on $(E,\mathcal E,\lambda)$ for $j\in I$
- $\mu:=p\lambda$
- $\sigma_{ij}:E^2\to[0,\infty)$ be $\mathcal E^{\otimes2}$-measurable for $i,j\in I$ with $\sigma_{ij}(x,y)=\sigma_{ji}(y,x)$ for all $x,y\in E$ and $i,j\in I$ and $$\sum_{j\in I}\int\lambda({\rm d}y)q_j(y)\sigma_{ij}(x,y)=1\tag1$$ for all $x\in E$ and $i\in I$
- $g\in\mathcal L^2(\mu)$
- $\Lambda,w_i\in\mathcal L^2(\mu)$ for $i\in I$
- $\alpha>0$
I want to solve the following system of equations for $(w_i)_{i\in I}$: $$\sum_{j\in I}\int\lambda({\rm d}y)q_j(y)\sigma_{ij}(x,y)\frac2{e^{\alpha\left(w_i(x)p(x)q_j(y)-w_j(y)p(y)q_i(x)\right)}+1}|g(x)-g(y)|^2+\Lambda(x)=0\tag2$$ for all $x\in E$ and $i\in I$ and $$\sum_{i\in I}w_i=1\tag3.$$
How can we do that? Is there an easy trick to separate the $w_i(x)$ term in $(1)$?