How can I solve this Equation?

Question

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain and $K:\Omega\times\Omega\rightarrow\mathbb{R}$ with $\|K\|_{L^2(\Omega\times\Omega)}<1.$ How can I show that the following equation has only the zero solution? $$u(x)=\int_\Omega K(x,y)u(y)dy$$

Thanks

Answer 1

Take the $L^2$ norm of both sides and estimate the right hand side, using the Cauchy-Schwarz Inequality.

Look for an inequality of the form $z \ge 0, z \le cz$ with $c <1$. Then you can conclude $z = 0$.