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Leray-Schauder alternatives for weakly sequentially continuous mappings and application to transport equation. (English) Zbl 1193.47056

The paper deals with the weakly sequentially continuous operators in a Banach space. The study of condensing contractive operators has been one of the main objects of research in nonlinear functional analysis. One of the most important results here is due to B. N. Sadovskii. Let \(Q\) be a nonempty closed convex bounded subset of a Banach space \(E\). The celebrated Sadovskii fixed point principle states that any condensing self-mapping of \(Q\) has a fixed point in \(Q\). That result was extended by Browder to a 1-set contraction mapping \(F\) by assuming the additional condition that \(((I- F)(Q))\) is closed, where \(I\) denotes the identity map. The Leray-Schauder principle, one of the most important theorems in nonlinear analysis, was first proved in the context of degree theory. Other variations of this principle are due to Browder, Schaefer, Petryshyn and Potter.
The mentioned results are very useful for the investigations of of nonlinear differential and integral equations in Banach spaces. Many of these equations can be transformed into fixed point problems involving nonlinear weakly compact operators. In the present paper, some new variants of Leray-Schauder type fixed point theorems for the considered operators are established. With the help of the obtained fixed point theorems, the authors investigate nonlinear stationary radiative transport equations and prove new existence and locality principles for a source problem in \(L^1\) setting with general boundary conditions.

MSC:

47H10 Fixed-point theorems
47N20 Applications of operator theory to differential and integral equations
35P05 General topics in linear spectral theory for PDEs
35R09 Integro-partial differential equations
85A25 Radiative transfer in astronomy and astrophysics
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