Suppose that $y=y(a(\lambda),\lambda,x)$ is a decreasing and twice continuously differentiable functional with respect to $x$ and $a(\lambda)$ is linear mapping from a space $\Lambda$ to $(0,+\infty)$. How can we find the maximum of the following function with respect to $y$, that is,
$$g(y)=-\frac{1}{2\lambda}y^2(a(\lambda),\lambda,x)+\left(\frac{a(\lambda)}{\lambda}-x\right)y(a(\lambda),\lambda,x)$$
Namely, I want to calculate
$$\max_{y(a(\lambda),\lambda,x)}\left\{\underbrace{\frac{1}{2\lambda}y^2(a(\lambda),\lambda,x)+\left(\frac{a(\lambda)}{\lambda}-x\right)y(a(\lambda),\lambda,x)}_{g(y)}\right\}$$
I thing this is not a classic optimization problem since the control is a functional and not just a variable. What does it changes in such a case?