19.3.13 Linear operators

condition: Prove that the operator \( A x(t)=\lambda \int_{0}^{t} x(s) d s+1 \) for \( |\lambda|<1 \) is contractive in \( C[0,1] \). Find the fixed point of this operator at \( \lambda=0.5 \). Do three iterations using the method of successive approximations \( \left(x_{0}(t)=0\right) \). Find the relative and absolute errors of the found approximate solutions.