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 on Lp spaces for p∈(1,∞). Due to the lack of compactness in L1 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 L1 context these theorems provide an adequate tool to attack the problem. Our analysis uses the specific properties of weakly compacts sets on L1 spaces and the weak compactness results for one-dimensional transport equations established in [J. Math. Anal. Appl. 252 (2000) 767–789].