 Solution

For $\alpha \ne 0$, the Fredholm alternative gives a unique solution exactly when $\alpha \ne -{\lambda }_n$ for all $n\ge 0$, unless a resonant problem could exceptionally be compatible. Here none is: with ${\kappa }_n=n+1/2$,At every resonance the compatibility condition therefore fails.

The case $\alpha =0$ must be checked separately because $L$ is a compact first-kind integral operator. For every admissible $y$, $Ly$ is continuous and $(Ly)(\pi )=0$, whereas $\cos \pi =-1$. Hence $Ly=\cos x$ has no solution. Combining these cases,In the nonresonant nonzero case the shifted eigenvalues stay away from zero, since ${\lambda }_n\to 0$ and $\alpha \ne 0$, so inversion is bounded on the Hilbert space.