Krasnoselskii type fixed point theorems and applications

Authors:
Yicheng Liu and Zhixiang Li

Journal:
Proc. Amer. Math. Soc. **136** (2008), 1213-1220

MSC (2000):
Primary 47H10, 34K13

Published electronically:
December 5, 2007

MathSciNet review:
2367095

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Abstract: In this paper, we establish two fixed point theorems of Krasnoselskii type for the sum of , where is a compact operator and may not be injective. Our results extend previous ones. As an application, we apply such results to obtain some existence results of periodic solutions for delay integral equations and then give three instructive examples.

**1.**Cezar Avramescu,*A fixed point theorem for multivalued mappings*, Electron. J. Qual. Theory Differ. Equ. (2004), No. 17, 10 pp. (electronic). MR**2111742****2.**Cezar Avramescu and Cristian Vladimirescu,*Fixed point theorems of Krasnoselskii type in a space of continuous functions*, Fixed Point Theory**5**(2004), no. 2, 181–195. MR**2117331****3.**Cleon S. Barroso,*Krasnoselskii’s fixed point theorem for weakly continuous maps*, Nonlinear Anal.**55**(2003), no. 1-2, 25–31. MR**2001629**, 10.1016/S0362-546X(03)00208-6**4.**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**5.**T. A. Burton,*Integral equations, implicit functions, and fixed points*, Proc. Amer. Math. Soc.**124**(1996), no. 8, 2383–2390. MR**1346965**, 10.1090/S0002-9939-96-03533-2**6.**T. A. Burton,*Krasnoselskii’s inversion principle and fixed points*, Proceedings of the Second World Congress of Nonlinear Analysts, Part 7 (Athens, 1996), 1997, pp. 3975–3986. MR**1603541**, 10.1016/S0362-546X(96)00219-2**7.**T. A. Burton and Colleen Kirk,*A fixed point theorem of Krasnoselskii-Schaefer type*, Math. Nachr.**189**(1998), 23–31. MR**1492921**, 10.1002/mana.19981890103**8.**B. C. Dhage,*A fixed point theorem in Banach algebras involving three operators with applications*, Kyungpook Math. J.**44**(2004), no. 1, 145–155. MR**2040753****9.**Daniel Henry,*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR**610244****10.**Vasile I. Istrăţescu,*Fixed point theory*, Mathematics and its Applications, vol. 7, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. An introduction; With a preface by Michiel Hazewinkel. MR**620639****11.**Yicheng Liu and Zhixiang Li,*Schaefer type theorem and periodic solutions of evolution equations*, J. Math. Anal. Appl.**316**(2006), no. 1, 237–255. MR**2201760**, 10.1016/j.jmaa.2005.04.045**12.**Yi Cheng Liu and Zhi Xiang Li,*New fixed point theorems and applications*, Acta Math. Sinica (Chin. Ser.)**49**(2006), no. 5, 1067–1074 (Chinese, with English and Chinese summaries). MR**2285410****13.**Helmut Schaefer,*Über die Methode der a priori-Schranken*, Math. Ann.**129**(1955), 415–416 (German). MR**0071723****14.**D. R. Smart,*Fixed point theorems*, Cambridge University Press, London-New York, 1974. Cambridge Tracts in Mathematics, No. 66. MR**0467717****15.**A. C. Thompson,*On certain contraction mappings in a partially ordered vector space.*, Proc. Amer. Math. soc.**14**(1963), 438–443. MR**0149237**, 10.1090/S0002-9939-1963-0149237-7**16.**Jianhong Wu,*Theory and applications of partial functional-differential equations*, Applied Mathematical Sciences, vol. 119, Springer-Verlag, New York, 1996. MR**1415838****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**

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Additional Information

**Yicheng Liu**

Affiliation:
Department of Mathematics and System Science, College of Science, National University of Defense Technology, Changsha, 410073, People’s Republic of China.

Email:
liuyc2001@hotmail.com

**Zhixiang Li**

Affiliation:
Department of Mathematics and System Science, College of Science, National University of Defense Technology, Changsha, 410073, People’s Republic of China.

Email:
zhxli02@yahoo.com.cn

DOI:
https://doi.org/10.1090/S0002-9939-07-09190-3

Keywords:
Fixed point theorem,
separate contraction mapping,
periodic solution,
multi-valued mapping,
delay integral equation.

Received by editor(s):
July 28, 2004

Received by editor(s) in revised form:
December 20, 2005

Published electronically:
December 5, 2007

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2007
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.