In [J. Math. Phys. 37 (1996) [1336][1337][1338][1339][1340][1341][1342][1343][1344][1345][1346][1347][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 [
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