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

Abstract

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 L p spaces for p ∈ ( 1 , ∞ ) . 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].