It is well known that imposing a global Lipschitz condition on nonlinear composition operators leads to a strong degeneracy phenomenon in many function spaces. In contrast to this, we show that a local version of Banach's contraction mapping principle is less restrictive and applies to a large variety of nonlinear problems. We illustrate this by means of applications to nonlinear integral equations with bounded or weakly singular kernels.
"Locally Lipschitz composition operators and applica- tions to nonlinear integral equations." J. Integral Equations Applications 25 (3) 321 - 339, FALL 2013. https://doi.org/10.1216/JIE-2013-25-3-321