19.8.4 Integral equations

Condition: Determine for which \( \alpha \) the mapping \( A \) of the form \( \quad(A x)(t)=\alpha \int_{a}^{b} K(t, s) x(s) d s+g(t) \) is contractive in the space \( C[a, b] \) c uniform norm, find its fixed point exactly and also approximately (by doing four iterations). \[ \begin{array}{l} K(t, s)=2 t^{2} s+3 s, \quad g(t)=2 t^{2}-3 \\ a=0, \quad b=1 . \end{array}\]