## 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.

### MSC:

 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
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### References:

 [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
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