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

DOI:
https://doi.org/10.1090/S0002-9939-2010-10517-8

Published electronically:
August 2, 2010

MathSciNet review:
2745654

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Abstract | References | Similar Articles | Additional Information

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 , where is noncompact, may not be injective, and 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:
https://doi.org/10.1090/S0002-9939-2010-10517-8

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.