Some fixed point theorems of the Schauder and the Krasnosel'skii type and application to nonlinear transport equations

In [J. Math. Phys. 37 (1996) 1336-1348] the existence of solutions to the boundary value problem (1.1)-(1.2) was analyzed for isotropic scattering kernels oil L-p spaces for p E (1, 00). Due to the lack of compactness in L-1 spaces, the problem remains open for p = 1. The purpose of this work is to extend this analysis to the case p = 1 for anisotropic scattering kernels. Our strategy consists in establishing new variants of the Schauder and the Krasnosel'skii fixed point theorems in general Banach spaces involving weakly compact operators. In L-1 context these theorems provide an adequate tool to attack the problem. Our analysis uses the specific properties of weakly compacts sets on L-1 spaces and the weak compactness results for one-dimensional transport equations established in [J. Math. Anal. Appl. 252 (2000) 767-789]. (c) 2005 Elsevier Inc. All rights reserved.