Some fixed point theorems for concentrative mappings between locally convex linear topological spaces. (English) Zbl 0874.47035

The author proves a Leray-Schauder-type alternative for continuous operators on closed convex subsets of locally convex spaces which are condensing with respect to a generating family of seminorms. The abstract result is then applied to a fixed point problem on the locally convex space of continuous vector functions over the semi-axis, equipped with the usual family of “exhaustive” seminorms.


47H10 Fixed-point theorems
47J25 Iterative procedures involving nonlinear operators
46A04 Locally convex Fréchet spaces and (DF)-spaces
Full Text: DOI


[1] Dugundji, J.; Granas, A., Fixed Point Theory, (Monografie Matematyczne (1982), PWN: PWN Groningen) · Zbl 1025.47002
[2] Daneš, J., Some fixed point theorems, Commentat. math. Univ. Carol., 9, 223-235 (1968) · Zbl 0165.49201
[3] Daneš, J., Generalized concentrative mappings and their fixed points, Commentat. math. Univ. Carol., 11, 115-136 (1970) · Zbl 0195.14903
[4] Furi, M.; Pera, P., A continuation method on locally convex spaces and applications to ordinary differential equations on noncompact intervals, Annls Soc. pol. Math., 47, 331-346 (1987) · Zbl 0656.47052
[5] Kothe, G., Topological Vector Spaces I (1983), Springer: Springer Warsaw
[6] Engelking, R., General Topology (1989), Heldermann Verlag: Heldermann Verlag New York · Zbl 0684.54001
[7] Himmelberg, C.; Porter, J.; Van Vleck, F., Fixed point theorems for condensing multifunctions, (Proc. Am. math. Soc., 23 (1969)), 635-641 · Zbl 0195.14902
[8] Kelley, J., General Topology (1955), D. Van Nostrand Company: D. Van Nostrand Company Berlin · Zbl 0066.16604
[9] Treves, F., Topological Vector Spaces, Distributions and Kernels (1967), Academic Press: Academic Press Toronto · Zbl 0171.10402
[10] Daneš, J., Some fixed point theorems in metric and Banach spaces, Commentat. math. Univ. Carol., 12, 37-51 (1971) · Zbl 0224.47032
[11] Banas, J.; Goebel, K., Measures of Noncompactness in Banach Spaces (1980), Marcel Dekker: Marcel Dekker New York · Zbl 0441.47056
[12] Day, M., Normed Linear Spaces (1973), Springer: Springer New York · Zbl 0268.46013
[13] Potter, A., An elementary version of the Leray-Schauder theorem, J. London math. Soc., 5, 414-416 (1972) · Zbl 0242.47037
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.