Krasnoselskii-type fixed point theorems with applications to Hammerstein integral equations in \(L^1\) spaces (Q2862558)

The authors prove Krasnoselskij-type fixed point theorems for bounded domains and unbounded domains, respectively. These results improve recent results obtained by \textit{J. Garcia-Falset}, \textit{K. Latrach}, \textit{E. Moreno-Gálvez} and \textit{M.-A. Taoudi} [J. Differ. Equations 252, No. 5, 3436--3452 (2012; Zbl 1252.47047)]. The authors also prove a nonlinear alternative of Krasnoselskij-type for the weak topology and the following application:NEWLINENEWLINE Theorem 4.5. Let \(X\) and \(Y\) be two finite-dimensional Banach spaces and let \(\Omega\) be a bounded domain of \(\mathbb{R}^n\). Then the following generalized Hammerstein integral equation NEWLINE\[NEWLINEy(t)= g(t, p(t))+ \lambda \int_\Omega k(t, s) f(s, y(s))\,ds,\quad t\in\Omega,NEWLINE\]NEWLINE where the functions \(f, g, k\) satisfy some conditions, has at least one solution in \(L^1(\Omega,X)\).