Math 6420 • PDE

Consider the nonlinear integral equation \[ f(x)-\lambda \int _a^b k(x,y;f(y))\,dy=\phi (x) \] where \(\lambda \in \mathbb {R}\), \(\phi \) is continuous on \([a,b]\), \(k\) is continuous on \(C=[a,b]\times [a,b]\times \mathbb {R}\), and \(k\) satisfies a Lipschitz condition of the form \[ \lvert k(x,y;z_1)-k(x,y;z_2)\rvert \le L\lvert z_1-z_2\rvert \] in its ‘functional argument’. Show that the nonlinear integral equation has a unique solution, \(f\), for any \(\lambda \) such that \[ \lvert \lambda \rvert <\frac {1}{L(b-a)}. \]