Under what conditions this function has a fixed point?

Question

I wanted to know that what is the condition under which the function of the type $$k:L^2([a, b]:\mathbb{R}^n)\times L^2([a, b]:\mathbb{R}^m)\rightarrow L^2([a, b]:\mathbb{R}^n)\times L^2([a, b]:\mathbb{R}^m)$$ has a fixed point?