Leray-Schauder principles for condensing multivalued mappings in topological linear spaces

Schöneberg, Rainald

doi:10.1090/S0002-9939-1978-0507320-9

Leray-Schauder principles for condensing multivalued mappings in topological linear spaces
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by Rainald Schöneberg

Proc. Amer. Math. Soc. 72 (1978), 268-270

DOI: https://doi.org/10.1090/S0002-9939-1978-0507320-9

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Abstract:

By establishing the existence of an equivalent fixed point problem it is shown without any recourse to degree theory or to the theory of homotopy-extension-theorems that all fixed point theorems of Leray-Schauder type for condensing (single- or multi-valued) mappings in topological linear spaces can immediately be deduced from the corresponding fixed point theorems of Schauder type.

References

Felix E. Browder, Problèmes nonlinéaires, Séminaire de Mathématiques Supérieures, No. 15 (Été, vol. 1965, Les Presses de l’Université de Montréal, Montreal, Que., 1966 (French). MR 0250140
D. E. Edmunds and J. R. L. Webb, A Leray-Schauder theorem for a class of nonlinear operators, Math. Ann. 182 (1969), 207–212. MR 254688, DOI 10.1007/BF01350323
P. M. Fitzpatrick and W. V. Petryshyn, Fixed point theorems for multivalued noncompact acyclic mappings, Pacific J. Math. 54 (1974), no. 2, 17–23. MR 405179, DOI 10.2140/pjm.1974.54.17
P. M. Fitzpatrick and W. V. Petryshyn, Fixed point theorems and the fixed point index for multivalued mappings in cones, J. London Math. Soc. (2) 12 (1975/76), no. 1, 75–85. MR 405180, DOI 10.1112/jlms/s2-12.1.75

A remark on a fixed point theorem for condensing set-valued mappings

W. V. Petryshyn, Structure of the fixed points sets of $k$-set-contractions, Arch. Rational Mech. Anal. 40 (1970/71), 312–328. MR 273480, DOI 10.1007/BF00252680
W. V. Petryshyn and P. M. Fitzpatrick, A degree theory, fixed point theorems, and mapping theorems for multivalued noncompact mappings, Trans. Amer. Math. Soc. 194 (1974), 1–25. MR 2478129, DOI 10.1090/S0002-9947-1974-2478129-5
A. J. B. Potter, An elementary version of the Leray-Schauder theorem, J. London Math. Soc. (2) 5 (1972), 414–416. MR 312342, DOI 10.1112/jlms/s2-5.3.414
J. R. L. Webb, A fixed point theorem and applications to functional equations in Banach spaces, Boll. Un. Mat. Ital. (4) 4 (1971), 775–788 (English, with Italian summary). MR 0377631