© 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).
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.
3. Extension of Nadler and Daffer-Kaneko's Theorems
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.
4. Extension of Rhoades and Zhang-Song's Theorems
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.
5. Conclusion and Future Directions
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
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