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Nonlinear alternatives of Schauder and Krasnosel’skij types with applications to Hammerstein integral equations in \(L^1\) spaces. (English) Zbl 1208.47044

In this interesting paper, by using the De Blasi measure of weak noncompactness, the authors establish some new variants of nonlinear alternatives of Leray-Schauder and Krasnosel’skij type involving the weak topology of Banach spaces. In addition, the authors apply their abstract results to a nonlinear Hammerstein integral equation in \(L^{1}\) spaces.
The main results of this paper complement some recent ones due to K. Latrach, M. A. Taoudi and A. Zeghal [J. Differential Equations 221, 256–2710 (2006; Zbl 1091.47046)] and K. Latrach and M. A. Taoudi [Nonlinear Anal. 66, 2325–2333 (2007; Zbl 1128.45006)].
Reviewer: Long Wei (Jiangxi)

MSC:

47H08 Measures of noncompactness and condensing mappings, \(K\)-set contractions, etc.
47H10 Fixed-point theorems
47H30 Particular nonlinear operators (superposition, Hammerstein, Nemytskiĭ, Uryson, etc.)
47N20 Applications of operator theory to differential and integral equations
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