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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Critical type of Krasnosel'skii fixed point theorem


Authors: Tian Xiang and Rong Yuan
Journal: Proc. Amer. Math. Soc. 139 (2011), 1033-1044
MSC (2010): Primary 47H08, 47H10, 37C25
Published electronically: August 2, 2010
MathSciNet review: 2745654
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Abstract: In this paper, by means of the technique of measures of noncompactness, we establish a generalized form of the fixed point theorem for the sum of $ T+S$, where $ S$ is noncompact, $ I-T$ may not be injective, and $ T$ is not necessarily continuous. The obtained results unify and significantly extend a number of previously known generalizations of the Krasnosel'skii fixed point theorem. The analysis presented here reveals the essential characteristics of the Krasnosel'skii type fixed point theorem in strong topology setups. Further, the results are used to prove the existence of periodic solutions of a nonlinear neutral differential equation with delay in the critical case.


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Additional Information

Tian Xiang
Affiliation: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing 100875, People’s Republic of China
Email: tianx@mail.bnu.edu.cn

Rong Yuan
Affiliation: Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University, Ministry of Education, Beijing 100875, People’s Republic of China
Email: ryuan@bnu.edu.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10517-8
PII: S 0002-9939(2010)10517-8
Keywords: Fixed point, noncompact mapping, multi-valued mapping
Received by editor(s): March 14, 2009
Received by editor(s) in revised form: March 29, 2010
Published electronically: August 2, 2010
Additional Notes: This work was supported by National Natural Science Foundation of China
Communicated by: Yingfei Yi
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.