The question is as follows:
Find in $C[0,\frac{\pi}{2}]$ and $L_p[0,\frac{\pi}{2}]$, for $1 \leq p \leq +\infty$, the solution (depending on $\lambda$) of the integral equation of the First kind $$x(t) = \lambda \int_{0}^{\frac{\pi}{2}} \cos(t-s)x(s) ds $$.
$\textbf{Some idea:}$
As we see $\int \cos(t-s)x(s) ds$ looks like the convolution product of $k(s,t)=\cos(t-s)$ and $x(s)$. So maybe we can somehow transform this integral such that be able to use the Laplace Transform. (Because we know that $L[k \star x] = L[k]L[x]$, and it will make the job easier.) But I really do not know how to solve such equations!
Can someone give some idea on how to solve it?
Thanks!