We present some new common fixed point theorems for a pair of nonlinear mappings defined on an ordered Banach space. Our results extend several earlier works. An application is given to show the usefulness and the applicability of the obtained results.
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 [
In this paper we prove some new fixed point theorems in r-normed and locally r-convex spaces. Our conclusions generalize many well-known results and provide a partial affirmative answer to Schauder's conjecture. Based on the obtained results, we prove the analogue of a Von Neumann's theorem in locally r-convex spaces. In addition, an application to game theory is presented.
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