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A generalization of Krasnosel'skii compression fixed point theorem by using star convex sets

Part of: Boundary value problems Nonlinear operators and their properties

Published online by Cambridge University Press:  25 January 2019

Cristina Lois-Prados  and
Rosana Rodríguez-López
Cristina Lois-Prados
Affiliation:
Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain (cristina.lois.prados@usc.es; rosana.rodriguez.lopez@usc.es)
Rosana Rodríguez-López
Affiliation:
Instituto de Matemáticas, Facultade de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela 15782, Spain (cristina.lois.prados@usc.es; rosana.rodriguez.lopez@usc.es)
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Abstract

In the framework of fixed point theory, many generalizations of the classical results due to Krasnosel'skii are known. One of these extensions consists in relaxing the conditions imposed on the mapping, working with k-set contractions instead of continuous and compact maps. The aim of this work if to study in detail some fixed point results of this type, and obtain a certain generalization by using star convex sets.

Keywords

Krasnosel'skii fixed point theoremk-set contractionstar convex set

MSC classification

Primary: 34B05: Linear boundary value problems
Secondary: 34B15: Nonlinear boundary value problems 34B18: Positive solutions of nonlinear boundary value problems 47H08: Measures of noncompactness and condensing mappings, $K$-set contractions, etc. 47H09: Contraction-type mappings, nonexpansive mappings, $A$-proper mappings, etc. 47H10: Fixed-point theorems
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 150 , Issue 1 , February 2020 , pp. 277 - 303
Copyright
Copyright © Royal Society of Edinburgh 2019

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References

1Anderson, D. R., Avery, R. I. and Henderson, J.. Functional expansion-compression fixed point theorem of Leggett-Williams type. Electron. J. Differ. Eq. 2010 (2010), 1–9.Google Scholar
2Avery, R. I., Henderson, J. and O'Regan, D.. Functional compression-expansion fixed point theorem. Electron. J. Differ. Eq. 2008 (2008), 1–12.Google Scholar
3Các, N. P. and Gatica, J. A.. Fixed point theorems for mappings in ordered Banach spaces. J. Math. Anal. Appl. 71 (1979), 547–557.CrossRefGoogle Scholar
4Darbo, G.. Punti uniti transformazioni a condominio non-compactto. Rend. Semin. Mat. U. Pad. 24 (1955), 84–92.Google Scholar
5Güo, D. and Lakshmikhantam, V.. Nonlinear Problems in Abstract Cones (California: Notes and Reports in Mathematics in Science and Engineering, 1998).Google Scholar
6Krasnosel'skii, M. A.. Fixed points of cone-compressing or cone-expanding operators. Soviet Math. Dokl. 1 (1960), 1285–1288.Google Scholar
7Kwong, M. K.. On Krasnoselskii's cone fixed point theorem. Fixed Point Theory A. 2008 (2008), 18, article n 164537.Google Scholar
8Munkres, J. R.. Topology (USA: Prentice Hall, 2000).Google Scholar
9Potter, A. J. B.. A fixed point theorem for positive k-set contractions. P. Edinburgh Math. Soc. 19 (1974), 93–102.CrossRefGoogle Scholar
10Rudin, W.. Functional analysis (New York: McGraw-Hill, 1991).Google Scholar
11Webb, J. R. L.. New fixed point index results and nonlinear boundary value problems. Bull. Lond. Math. Soc. 49 (2017), 534–547.CrossRefGoogle Scholar