On Some Geometric Constants and the Fixed Point Property for Multivalued Nonexpansive Mappings
© J. Zhang and Y. Cui. 2010
Received: 30 July 2010
Accepted: 5 October 2010
Published: 11 October 2010
We show some geometric conditions on a Banach space X concerning the Jordan-von Neumann constant, Zbăganu constant, characteristic of (separation) noncompact convexity, and the coefficient R(1, X), the weakly convergent sequence coefficient, which imply the existence of fixed points for multivalued nonexpansive mappings.
Fixed point theory for multivalued mappings has many useful applications in Applied Sciences, in particular, in game theory and mathematical economics. Thus it is natural to try of extending the known fixed point results for single-valued mappings to the setting of multivalued mappings.
In 1969, Nadler  established the multivalued version of Banach's contraction principle. One of the most celebrated results about multivalued mappings was given by Lim  in 1974. Using Edelstein's method of asymptotic centers, he proved the existence of a fixed point for a multivalued nonexpansive self-mapping where is a nonempty bounded closed convex subset of a uniformly convex Banach space. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. Some other classical fixed point theorems for single-valued mappings have been extended to multivalued mappings. However, many questions remain open, for instance, the possibility of extending the well-known Kirk's theorem, that is, do Banach spaces with weak normal structure have the fixed point property (FPP, in short) for multivalued nonexpansive mappings?
implies property (D).
(3)Satit Saejung  proved that the condition implies property (D).
(4)Gavira  showed that the condition
implies (DL) condition.
In 2007, Domínguez Benavides and Gavira  have established FFP for multivalued nonexpansive mappings in terms of the modulus of squareness, universal infinite-dimensional modulus, and Opia modulus. Attapol Kaewkhao  has established FFP for multivalued nonexpansive mappings in terms of the James constant, the Jordan-von Neumann Constants, weak orthogonality.
Besides, In 2010, Domínguez Benavides and Gavira  have given a survey of this subject and presented the main known results and current research directions.
In this paper, in terms of the Jordan-von Neumann constant, Zb ganu constant, and the coefficient , the weakly convergent sequence coefficient, we show some geometrical properties which imply the property (D) or (DL) condition and so the FPP for multivalued nonexpansive mappings.
Let and be as above; then is called regular relative to if for all subsequence of ; further, is called asymptotically uniform relative to if for all subsequence of . In Banach spaces, we have the following results:
(Kirk ) if is separable, then contains a subsequence which is asymptotically uniform relative to .
In 2006, Dhompongsa et al.  introduced the Domnguez-Lorenzo condition ((DL) condition, in short) in the following way.
Definition 2.1 (see ).
The (DL) condition implies weak normal structure . We recll that a Banach space is said to have a weak normal structure (w-NS) if for every weakly compact convex subset of with there exist such that .
The (DL) condition also implies the existence of fixed points for multivalued nonexpansive mappings.
Theorem 2.2 (see ).
Definition 2.3 (see ).
It was observed that property (D) is weaker than the (DL) condition and stronger than weak normal structure, and Dhompongsa et al.  proved that property (D) implies the w-MFPP.
Theorem 2.4 (see ).
The ultrapower of a Banach space has proved to be useful in many branches of mathematics. Many results can be seen more easily when treated in this setting.
Note that if is nontrivial, then can be embedded into isometrically. For more details see .
3. Main Results
Hence we deduce the desired inequality.
By Theorems 2.2 and 3.1, we have the following result.
We first give the following tool.
Hence we deduce the desired inequality.
Using Theorem 2.2, we obtain the following corollary.
In the following, we present some properties concerning geometrical constants of Banach spaces which also imply the property (D).
Using Theorems 2.4 and 3.7, we obtain the follwing corollary.
It is known that is NUC if and only if . The above-mentioned definitions and properties can be found in .
By Theorem 3.9, we obtain the following Corollary.
Theorem 3.12 (see [18, Theorem ]).
The authors would like to thank the anonymous referee for providing some suggestions to improve the manuscript. This work was supported by China Natural Science Fund under grant 10571037.
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