Suppose $F(x,y;k)=f(x,y)+kg(x,y)=0$ uniquely defines the solution $y(x;k)$ for $x\in \mathbb{D}$, a compact domain, and $0\leq k \leq 1$ is a parameter. We know that for $k=0,1$, $y(x;0)$ and $y(x,1)$ are strictly increasing and strictly concave functions.

Then, can we say something about the property of $y(x;k)$ for $0<k<1$? For example, is $y(x;k)$ strictly increasing and concave?