- Research Article
- Open Access
© B. Djafari Rouhani and S. Moradi. 2010
- Received: 18 September 2009
- Accepted: 10 January 2010
- Published: 26 January 2010
Fixed point and coincidence results are presented for multivalued generalized -weak contractive mappings on complete metric spaces, where is a lower semicontinuous function with and for all . Our results extend previous results by Zhang and Song (2009), as well as by Rhoades (2001), Nadler (1969), and Daffer and Kaneko (1995).
- Differential Geometry
- Contraction Mapping
- Computational Biology
- Multivalued Mapping
- Common Fixed Point
The concepts of weak and -weak contractive mappings were defined by Daffer and Kaneko  in 1995.
In the following theorem, Nadler  extended the Banach Contraction Principle to multivalued mappings.
Daffer and Kaneko  proved the existence of a fixed point for a multivalued weak contraction mapping of a complete metric space into .
In Section 3 we extend Nadler and Daffer-Kaneko's theorems to multivalued generalized weak contraction mappings (see Definition 2.1).
Rhoades [5, Theorem ] proved the following fixed point theorem for -weak contractive single valued mappings, giving another generalization of the Banach Contraction Principle.
Recently Zhang and Song  proved the following theorem on the existence of a common fixed point for two single valued generalized -weak contraction mappings.
where (i.e., multivalued generalized weak contractions). Then there exists a point such that and (i.e., and have a common fixed point). Moreover, if either or is single valued, then this common fixed point is unique.
First we extend Zhang and Song's theorem (Theorem 1.4) to the case where one of the mappings is multivalued.
Unicity of the common fixed point follows from (4.1).
We break the argument into four steps.
If were unbounded, then by Step 1, and are unbounded. We choose the sequence such that , is even and minimal in the sense that , and , and similarly is odd and minimal in the sense that , and is even and minimal in the sense that and , and is odd and minimal in the sense that and .
In the proof of Theorem in Zhang and Song , the boundedness of the sequence is used, but not proved. Also, for the proof that is a Cauchy sequence, the monotonicity of is used, without being explicitly mentioned.
In our proof of Theorem 4.1, which is different from [8, Theorem ], is not assumed to be nondecreasing.
We have also extended Rhoades' theorem by assuming to be only l.s.c., as well as Zhang and Song's theorem to the case where one of the mappings is multivalued. Future directions to be pursued in the context of this research include the investigation of the case where both mappings in Zhang and Song's theorem are multivalued.
This work is dedicated to Professor W. A. Kirk for his 70th birthday
- Daffer PZ, Kaneko H: Fixed points of generalized contractive multi-valued mappings. Journal of Mathematical Analysis and Applications 1995,192(2):655–666. 10.1006/jmaa.1995.1194MathSciNetView ArticleMATHGoogle Scholar
- Chifu C, Petrusel G: Existence and data dependence of fixed points and strict fixed points for contractive-type multivalued operators. Fixed Point Theory and Applications 2007, 2007:-8.Google Scholar
- Nadler SB: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.MathSciNetView ArticleMATHGoogle Scholar
- Shahzad N, Lone A: Fixed points of multimaps which are not necessarily nonexpansive. Fixed Point Theory and Applications 2005, (2):169–176.MathSciNetMATHGoogle Scholar
- Rhoades BE: Some theorems on weakly contractive maps. Nonlinear Analysis: Theory, Methods & Applications 2001,47(4):2683–2693. 10.1016/S0362-546X(01)00388-1MathSciNetView ArticleMATHGoogle Scholar
- Boyd DW, Wong JSW: On nonlinear contractions. Proceedings of the American Mathematical Society 1969, 20: 458–464. 10.1090/S0002-9939-1969-0239559-9MathSciNetView ArticleMATHGoogle Scholar
- Reich S: Some fixed point problems. Atti della Accademia Nazionale dei Lincei. Rendiconti. Classe di Scienze Fisiche, Matematiche e Naturali 1974,57(3–4):194–198.MathSciNetMATHGoogle Scholar
- Zhang Q, Song Y: Fixed point theory for generalized -weak contractions. Applied Mathematics Letters 2009,22(1):75–78. 10.1016/j.aml.2008.02.007MathSciNetView ArticleMATHGoogle Scholar
- Assad NA, Kirk WA: Fixed point theorems for set-valued mappings of contractive type. Pacific Journal of Mathematics 1972, 43: 553–562.MathSciNetView ArticleMATHGoogle Scholar
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