In this paper we present a two-norms version of Krasnoselskii’s fixed point theorem in cones. The abstract result is then applied to prove the existence of positive \(L_p\) solutions of Hammerstein integral equations with better integrability properties on the kernels.
Department of Mathematics, National University of Ireland, Galway, Ireland
Department of Mathematics Babes-Bolyai University, Cluj-Napoca, Romania
D. O’Regan, R. Precup, Compression-expansion fixed point theorem in two norms and applications, J. Math. Anal. Appl. 309 (2005), 383-391, https://doi.org/10.1016/j.jmaa.2005.01.043
Journal of Mathematical Analysis and Applications
MR2154122, Zbl 1078.47017
google scholar link
 R. Agarwal, M. Meehan, D. O’Regan, R. Precup, Location of nonnegative solutions for differential equations on finite and semi-infinite intervals, Dynam. Systems Appl. 12 (2003) 323–341.
 L.H. Erbe, H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120 (1994) 743–748.
 L.H. Erbe, S. Hu, H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994) 640–648.
 A. Granas, J. Dugundji, Fixed Point Theory, Springer-Verlag, New York, 2003.
 M. Meehan, D. O’Regan, Positive Lp solutions of Hammerstein integral equations, Arch. Math. 76 (2001) 366–376.
 D. O’Regan, R. Precup, Theorems of Leray–Schauder Type and Applications, Taylor & Francis, London, 2002.
 R. Precup, Positive solutions of evolution operator equations, Austral. J. Math. Anal. Appl. 2 (2005) 1–10.