How can I prove?.
Prove that if $[(b-a)|\lambda|sup_{t,s \in [a,b]}|\kappa (t,s)|]<1$, then a integral linear equation of Fredholm has a unique solution in $C[a,b]$.
We know by definition that; in a integral equation of fredholm $K(t,s,u)=\lambda \kappa(t,s)u, \lambda \in \mathbb{R}$, we obtain the integral linear equation of fredholm $\psi (t)= \lambda \int _{a} ^b \kappa (t,s) \psi(s) ds + \varphi(t), t\in [a,b]$.