Problem 1
Let A be a mapping of a metric space \(\mathrm{R}\) into itself. Prove that the condition $$ p(A x, A y)
Problem 7
Consider the nonlinear integral equation $$ f(x)=\lambda \int_{a}^{b} K(x, y ; f(y)) d y+\varphi(x) $$ with continuous \(K\) and \(\varphi\), where \(K\) satisfies a Lipschitz condition of the form $$ \left|K\left(x, y ; z_{1}\right)-K\left(x, y ; z_{2}\right)\right| \leqslant M\left|z_{1}-z_{2}\right| $$ in its "functional" argument. Prove that (21) has a unique solution for all $$ |\lambda|<\frac{1}{M(b-a)} $$ Write the successive approximations to this solution.
