I'm trying to do a exercise of Kreyszig book of functional analysis but I'm stuck, I'm trying to solve the integral equation \begin{equation} x(s)-\mu \int_{0}^{2\pi}sin(s)cos(t)x(t)dt =\hat{y}(s) \end{equation} Where $\hat{y}$ is a given function in $C[0,2\pi]$.
My attempt consist of define $\displaystyle Tx(s):=\int_{0}^{2\pi}sin(s)cos(t)x(t)dt$, and note that if we take $\lambda>2\pi$ where $\lambda=\frac{1}{\mu}$ then $\|\frac{T}{\lambda}\|<1$ so $(I-\frac{T}{\lambda})^{-1}=\sum_{i=0}^{\infty}{(\frac{T}{\lambda})^{i}}$ and then easyly we get a solution but what occurs if $\lambda\leq 2\pi$?