The problem is the following: Find $\lambda$ such that
$ b^{T}A\left[A^{T}A-\lambda L^{T}L\right]^{-1}L^{T}L\left[A^{T}A-\lambda L^{T}L\right]^{-1}A^{T}b-\delta^{2}<0 $
where $\lambda,\delta\in\mathbb{R}$ with $\delta>0,$ and we have that $A\in M_{m\times n}\left(\mathbb{R}\right)$, $L\in M_{n\times n}\left(\mathbb{R}\right)$ and $b\in M_{m\times1}\left(\mathbb{R}\right)$. The idea is find a expression for $\lambda$ in terms of $A,$ $L$ and $b$.
Thank you in advance for your help.