Fixed point theory for self maps between Fréchet spaces. (English) Zbl 0997.47044

The compression-expansion Krasnoselskij’s fixed point theorem in a cone of a Banach space is extended to Fréchet spaces. An application to the existence of nontrivial continuous solutions of a nonlinear integral equation on the half-line is presented to illustrate the theory. An analogous result for multivalued upper semicontinuous \(k\)-set contractive mappings is also obtained.


47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H07 Monotone and positive operators on ordered Banach spaces or other ordered topological vector spaces
47H04 Set-valued operators
Full Text: DOI


[1] Agarwal, R.P.; O’Regan, D., A note on the existence of multiple fixed points for multivalued maps with applications, J. differential equations, 160, 389-403, (2000) · Zbl 1008.47055
[2] R. P. Agarwal, and, D. O’Regan, Multivalued nonlinear equations on the half line: a fixed point approach, to appear.
[3] Agarwal, R.P.; O’Regan, D.; Wong, P.J.Y., Positive solutions of differential, difference and integral equations, (1999), Kluwer Academic Dordrecht · Zbl 0923.39002
[4] Corduneanu, C., Integral equations and applications, (1991), Cambridge Univ. Press New York · Zbl 0714.45002
[5] Erbe, L.H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. amer. math. soc., 120, 743-748, (1994) · Zbl 0802.34018
[6] Guo, D.; Lakshmikantham, V., Nonlinear problems in abstract cones, (1988), Academic Press San Diego · Zbl 0661.47045
[7] O’Regan, D.; Meehan, M., Existence theory for nonlinear integral and integrodifferential equations, (1998), Kluwer Academic Dordrecht · Zbl 0932.45010
[8] Tan, K.K.; Yuan, X.Z., Random fixed point theorems and approximation in cones, J. math. anal. appl., 185, 378-390, (1994) · Zbl 0856.47036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.