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

Full-text PDF

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.

**1.**R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina, and B. N. Sadovskiĭ,*Measures of noncompactness and condensing operators*, Operator Theory: Advances and Applications, vol. 55, Birkhäuser Verlag, Basel, 1992. Translated from the 1986 Russian original by A. Iacob. MR**1153247****2.**Józef Banaś and Kazimierz Goebel,*Measures of noncompactness in Banach spaces*, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekker, Inc., New York, 1980. MR**591679****3.**Edoardo Beretta, Fortunata Solimano, and Yasuhiro Takeuchi,*A mathematical model for drug administration by using the phagocytosis of red blood cells*, J. Math. Biol.**35**(1996), no. 1, 1–19. MR**1478581**, 10.1007/s002850050039**4.**T. A. Burton,*A fixed-point theorem of Krasnoselskii*, Appl. Math. Lett.**11**(1998), no. 1, 85–88. MR**1490385**, 10.1016/S0893-9659(97)00138-9**5.**Cleon S. Barroso and Eduardo V. Teixeira,*A topological and geometric approach to fixed points results for sum of operators and applications*, Nonlinear Anal.**60**(2005), no. 4, 625–650. MR**2109150**, 10.1016/j.na.2004.09.040**6.**Shui Nee Chow,*Existence of periodic solutions of autonomous functional differential equations*, J. Differential Equations**15**(1974), 350–378. MR**0336003****7.**B. C. Dhage,*Remarks on two fixed-point theorems involving the sum and the product of two operators*, Comput. Math. Appl.**46**(2003), no. 12, 1779–1785. MR**2018766**, 10.1016/S0898-1221(03)90236-7**8.**Daniel Henry,*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR**610244****9.**M. A. Krasnosel′skiĭ,*Two remarks on the method of successive approximations*, Uspehi Mat. Nauk (N.S.)**10**(1955), no. 1(63), 123–127 (Russian). MR**0068119****10.**Yongkun Li and Yang Kuang,*Periodic solutions of periodic delay Lotka-Volterra equations and systems*, J. Math. Anal. Appl.**255**(2001), no. 1, 260–280. MR**1813821**, 10.1006/jmaa.2000.7248**11.**Yicheng Liu and Zhixiang Li,*Krasnoselskii type fixed point theorems and applications*, Proc. Amer. Math. Soc.**136**(2008), no. 4, 1213–1220. MR**2367095**, 10.1090/S0002-9939-07-09190-3**12.**Efe A. Ok,*Fixed set theorems of Krasnoselskiĭ type*, Proc. Amer. Math. Soc.**137**(2009), no. 2, 511–518. MR**2448571**, 10.1090/S0002-9939-08-09332-5**13.**Joseph W.-H. So, Jianhong Wu, and Xingfu Zou,*Structured population on two patches: modeling dispersal and delay*, J. Math. Biol.**43**(2001), no. 1, 37–51. MR**1854000**, 10.1007/s002850100081**14.**Sehie Park,*Generalizations of the Krasnoselskii fixed point theorem*, Nonlinear Anal.**67**(2007), no. 12, 3401–3410. MR**2350896**, 10.1016/j.na.2006.10.024**15.**Jianhong Wu,*Theory and applications of partial functional-differential equations*, Applied Mathematical Sciences, vol. 119, Springer-Verlag, New York, 1996. MR**1415838****16.**Tian Xiang and Rong Yuan,*A class of expansive-type Krasnosel′skii fixed point theorems*, Nonlinear Anal.**71**(2009), no. 7-8, 3229–3239. MR**2532845**, 10.1016/j.na.2009.01.197**17.**Eberhard Zeidler,*Nonlinear functional analysis and its applications. I*, Springer-Verlag, New York, 1986. Fixed-point theorems; Translated from the German by Peter R. Wadsack. MR**816732**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (2010):
47H08,
47H10,
37C25

Retrieve articles in all journals with MSC (2010): 47H08, 47H10, 37C25

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.