Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings
© Z. Zuo and Y. Cui. 2009
Received: 2 July 2009
Accepted: 3 December 2009
Published: 13 January 2010
We show some sufficient conditions on a Banach space concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbagănu constant, the coefficient , the weakly convergent sequence coefficient WCS( ), and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These fixed point theorems improve some previous results in the recent papers.
In 1969, Nadler  established the multivalued version of Banach contraction principle. Since then the metric fixed point theory of multivalued mappings has been rapidly developed. Some classical fixed point theorems for singlevalued nonexpansive mappings have been extended to multivalued nonexpansive 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) for multivalued nonexpansive mappings?"
Since weak normal structure is implied by different geometric properties of Banach spaces, it is natural to study whether those properties imply the FPP for multivalued mappings. Dhompongsa et al. [3, 4] introduced the DL condition and property (D) which imply the FPP for multivalued nonexpansive mappings. A possible approach to the above problem is to look for geometric conditions in a Banach space which imply either the DL condition or property (D). In this setting the following results have been obtained.
implies the DL condition .
(ii)Saejung  showed that a Banach space has property (D) whenever .
In this paper, we show some sufficient conditions on a Banach space concerning the generalized James constant, the generalized Jordan-von Neumann constant, the generalized Zbăganu constant, the coefficient , the weakly convergent sequence coefficient , and the coefficient of weak orthogonality, which imply the existence of fixed points for multivalued nonexpansive mappings. These theorems improve the above results.
Before going to the result, let us recall some concepts and results which will be used in the following sections. Let be a Banach space with the unit ball and the unit sphere . The two constants of a Banach space
are called the von Neumann-Jordan  and James constants , respectively, and are widely studied by many authors [10–20]. Recently, both constants are generalized in the following ways for (see [12, 13]):
Recently, Gao and Saejung in  define a new constant for :
The modulus of convexity of (see ) is a function defined by
In  the author introduces a modulus that scales the 3-dimensional convexity of the unit ball: he considers the number
It is evident that for all and in consequence . Moreover this last inequality can be strict, since it was shown in  the existence of Banach spaces with which are not uniformly nonsquare.
The WORTH property was introduced by Sims in  as follows. A Banach space has the WORTH property if
Let be a nonempty subset of a Banach space . We shall denote by the family of all nonempty closed bounded subsets of and by the family of all nonempty compact convex subsets of . A multivalued mapping is said to be nonexpansive if
Dhompongsa et al.  introduced the property (D) if there exists such that for any nonempty weakly compact convex subset of , any sequence which is regular asymptotically uniform relative to , and any sequence which is regular asymptotically uniform relative to we have
The Domínguez-Lorenzo condition, DL condition in short form, introduced in  is defined as follows: if there exists such that for every weakly compact convex subset of and for every bounded sequence in which is regular with respect to we have,
It is clear from the definition that property (D) is weaker than the DL condition. The next results show that property (D) is stronger than weak normal structure and also implies the existence of fixed points for multivalued nonexpansive mappings .
3. Main Results
satisfies the DL condition.
satisfies the DL condition.
Repeating the arguments in the proof of Theorem 3.4, we can easily get the following conclusion.
satisfies the DL condition.
Theorem 3.9 strengthens the result of Saejung  and has property (D) whenever .
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