Open Access

Some Sufficient Conditions for Fixed Points of Multivalued Nonexpansive Mappings

Fixed Point Theory and Applications20102009:319804

DOI: 10.1155/2009/319804

Received: 2 July 2009

Accepted: 3 December 2009

Published: 13 January 2010

Abstract

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.

1. Introduction

In 1969, Nadler [1] 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 [2], 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.

(i)Kaewkhao [5] proved that a Banach space with
(1.1)
satisfies the DL condition. He also showed that the condition
(1.2)

implies the DL condition [6].

(ii)Saejung [7] 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.

2. Preliminaries

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

(2.1)

are called the von Neumann-Jordan [8] and James constants [9], respectively, and are widely studied by many authors [1020]. Recently, both constants are generalized in the following ways for (see [12, 13]):

(2.2)

It is clear that and .

Recently, Gao and Saejung in [6] define a new constant for :

(2.3)
which is inspired by Zbăganu paper [21]. It is clear that
(2.4)

The modulus of convexity of (see [22]) is a function defined by

(2.5)

The function is strictly increasing on . Here is the characteristic of convexity of , and the space is called uniformly nonsquare if .

In [23] the author introduces a modulus that scales the 3-dimensional convexity of the unit ball: he considers the number

(2.6)

and defines the function by

(2.7)
He also considers the coefficient corresponding to this modulus:
(2.8)

It is evident that for all and in consequence . Moreover this last inequality can be strict, since it was shown in [23] the existence of Banach spaces with which are not uniformly nonsquare.

The weakly convergent sequence coefficient of is defined as follows: where the infimum is taken over all weakly null sequences in such that and exist.

The WORTH property was introduced by Sims in [24] as follows. A Banach space has the WORTH property if

(2.9)
for all and all weakly null sequences . In [25], Jiménez-Melado and Llorens-Fuster defined the coefficient of weak orthogonality, which measures the degree of WORTH wholeness, by
(2.10)

where the infimum is taken over all and all weakly null sequence . It is known that has the WORTH property if and only 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

(2.11)
where denotes the Hausdorff metric on defined by
(2.12)

Let be a bounded sequence in . The asymptotic radius and the asymptotic center of in are defined by

(2.13)
(2.14)

respectively. It is known that is a nonempty weakly compact convex set whenever is.

The sequence is called regular with respect to if for all subsequences of , and is called asymptotically uniform with respect to if for all subsequences of .

Lemma 2.1.
  1. (i)

    (See Goebel [26] and Lim [27]) There always exists a subsequence of which is regular with respect to .

     
  2. (ii)

    (See Kirk [28]) If is separable, then contains a subsequence which is asymptotically uniform with respect to .

     

If is a bounded subset of , then the Chebyshev radius of relative to is defined by

(2.15)

Dhompongsa et al. [4] 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

(2.16)

The Domínguez-Lorenzo condition, DL condition in short form, introduced in [3] 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,

(2.17)

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 [4].

Theorem 2.2.

Let be a Banach space satisfying property (D). Then has weak normal structure.

Theorem 2.3.

Let be a nonempty weakly compact convex subset of a Banach space which satisfies the property (D). Let be a nonexpansive mapping, then has a fixed point.

3. Main Results

Theorem 3.1.

Let be a weakly compact convex subset of a Banach space and let be a bounded sequence in regular with respect to . Then for every ,
(3.1)

Proof.

Denote and . We can assume that . By passing to a subsequence if necessary, we can also assume that is weakly convergent to a point . Since is regular with respect to , then passing through a subsequence does not have any effect to the asymptotic radius of the whole sequence . Let , then we have that
(3.2)
Denote . By the definition of we have that
(3.3)
Convexity of implies that and thus, we obtain
(3.4)
On the other hand, by the weak lower semicontinuity of the norm, we have that
(3.5)

For every there exists such that

(1) ,

(2)

(3)

(4)

Now, put , and . Using the above estimates, we obtain , and
(3.6)
Thus,
(3.7)
By the weak lower semicontinuity of the norm again, we conclude that and hence,
(3.8)
Therefore Since the above inequality is true for every and every , we obtain
(3.9)
and therefore,
(3.10)

Corollary 3.2.

Let be a nonempty bounded closed convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

Proof.

When , then satisfies the DL condition by Theorem 3.1. So has a fixed point by Theorem 2.3.

Remark 3.3.

In particular, when , we get the result of Kaewkhao; a Banach space with
(3.11)

satisfies the DL condition.

Theorem 3.4.

Let be a weakly compact convex subset of a Banach space and let be a bounded sequence in regular with respect to . Then for every ,
(3.12)

Proof.

Let , and be as in the proof of the previous theorem. Thus,
(3.13)
Since and by the definition of , we obtain
(3.14)
On the other hand, by the weak lower semicontinuity of the norm, we have that
(3.15)

For every , there exists such that

(1)

(2)

(3)

(4)

Now, put , and and use the above estimates to obtain , , , and , so that
(3.16)
By the definition of , we get
(3.17)
Let ; we obtain that . Then we have
(3.18)
This holds for arbitrary ; hence, we have that
(3.19)

Corollary 3.5.

Let be a nonempty bounded closed convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

Proof.

When , then satisfies the DL condition by Theorem 3.4. So has a fixed point by Theorem 2.3.

Remark 3.6.

In particular, when , we get the result of Kaewkhao; a Banach space with
(3.20)

satisfies the DL condition.

Repeating the arguments in the proof of Theorem 3.4, we can easily get the following conclusion.

Theorem 3.7.

Let be a nonempty bounded closed convex subset of a Banach space such that and let be a nonexpansive mapping. Then has a fixed point.

Remark 3.8.

In particular, when , we get that
(3.21)
satisfies the DL condition which improves the result of Kaewkhao; a Banach space with
(3.22)

satisfies the DL condition.

Theorem 3.9.

A Banach space has property (D) whenever .

Proof.

Let be a nonempty weakly compact convex subset of . Suppose that and are regular asymptotically uniforms relative to . Passing to a subsequence, we may assume that is weakly convergent to a point and exists. Let . Again, passing to a subsequence of , still denoted by , we assume in addition that
(3.23)
for all . Now, put
(3.24)
It is easy to see that , , ,  and . This implies that   or Now we estimate as follows:
(3.25)
Hence,
(3.26)
Remark 3.10.
  1. (1)

    Theorem 3.9 strengthens the result of Saejung [7] and has property (D) whenever .

     
  2. (2)

    Theorem 3.9 also improves the result implying that the Banach space has normal structure from Theorem 2.2.

     

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Chongqing Three Gorges University
(2)
Department of Mathematics, Harbin University of Science and Technology

References

  1. Nadler SB Jr.: Multi-valued contraction mappings. Pacific Journal of Mathematics 1969, 30: 475–488.MathSciNetView ArticleMATHGoogle Scholar
  2. Kirk WA: A fixed point theorem for mappings which do not increase distances. The American Mathematical Monthly 1965, 72: 1004–1006. 10.2307/2313345MathSciNetView ArticleMATHGoogle Scholar
  3. Dhompongsa S, Kaewcharoen A, Kaewkhao A: The Domínguez-Lorenzo condition and multivalued nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):958–970. 10.1016/j.na.2005.05.051MathSciNetView ArticleMATHGoogle Scholar
  4. Dhompongsa S, Domínguez Benavides T, Kaewcharoen A, Kaewkhao A, Panyanak B: The Jordan-von Neumann constants and fixed points for multivalued nonexpansive mappings. Journal of Mathematical Analysis and Applications 2006,320(2):916–927. 10.1016/j.jmaa.2005.07.063MathSciNetView ArticleMATHGoogle Scholar
  5. Kaewkhao A: The James constant, the Jordan-von Neumann constant, weak orthogonality, and fixed points for multivalued mappings. Journal of Mathematical Analysis and Applications 2007,333(2):950–958. 10.1016/j.jmaa.2006.12.001MathSciNetView ArticleMATHGoogle Scholar
  6. Gao J, Saejung S: Normal structure and the generalized James and Zbăganu constants. Nonlinear Analysis: Theory, Methods & Applications 2009,71(7–8):3047–3052. 10.1016/j.na.2009.01.216MathSciNetView ArticleMATHGoogle Scholar
  7. Saejung S: Remarks on sufficient conditions for fixed points of multivalued nonexpansive mappings. Nonlinear Analysis: Theory, Methods & Applications 2007,67(5):1649–1653. 10.1016/j.na.2006.07.037MathSciNetView ArticleMATHGoogle Scholar
  8. Clarkson JA: The von Neumann-Jordan constant for the Lebesgue spaces. Annals of Mathematics 1937,38(1):114–115. 10.2307/1968512MathSciNetView ArticleMATHGoogle Scholar
  9. Gao J, Lau K-S: On two classes of Banach spaces with uniform normal structure. Studia Mathematica 1991,99(1):41–56.MathSciNetMATHGoogle Scholar
  10. Alonso J, Martín P: A counterexample to a conjecture of G. Zbăganu about the Neumann-Jordan constant. Revue Roumaine de Mathématiques Pures et Appliquées 2006,51(2):135–141.MATHGoogle Scholar
  11. Yunan C, Zhanfei Z: Geometric properties concerning some parameters. Journal of Natural Science of Heilongjiang University 2008, 6: 711–718.MATHGoogle Scholar
  12. Dhompongsa S, Piraisangjun P, Saejung S: Generalised Jordan-von Neumann constants and uniform normal structure. Bulletin of the Australian Mathematical Society 2003,67(2):225–240. 10.1017/S0004972700033694MathSciNetView ArticleMATHGoogle Scholar
  13. Dhompongsa S, Kaewkhao A, Tasena S: On a generalized James constant. Journal of Mathematical Analysis and Applications 2003,285(2):419–435. 10.1016/S0022-247X(03)00408-6MathSciNetView ArticleMATHGoogle Scholar
  14. Jiménez-Melado A, Llorens-Fuster E, Saejung S: The von Neumann-Jordan constant, weak orthogonality and normal structure in Banach spaces. Proceedings of the American Mathematical Society 2006,134(2):355–364.MathSciNetView ArticleMATHGoogle Scholar
  15. Saejung S: On James and von Neumann-Jordan constants and sufficient conditions for the fixed point property. Journal of Mathematical Analysis and Applications 2006,323(2):1018–1024. 10.1016/j.jmaa.2005.11.005MathSciNetView ArticleMATHGoogle Scholar
  16. Zuo Z, Cui Y: On some parameters and the fixed point property for multivalued nonexpansive mappings. Journal of Mathematical Sciences: Advances and Applications 2008,1(1):183–199.MathSciNetMATHGoogle Scholar
  17. Zuo Z, Cui Y: A note on the modulus of -convexity and modulus of -convexity. Journal of Inequalities in Pure and Applied Mathematics 2008,9(4):1–7.MathSciNetMATHGoogle Scholar
  18. Zuo Z, Cui Y: Some modulus and normal structure in Banach space. Journal of Inequalities and Applications 2009, Article ID 676373, 2009:-15.Google Scholar
  19. Zuo Z, Cui Y: A coefficient related to some geometric properties of a Banach space. Journal of Inequalities and Applications 2009, Article ID 934321, 2009:-14.Google Scholar
  20. Zuo Z, Cui Y: The application of generalization modulus of convexity in fixed point theory. Journal of Natural Science of Heilongjiang University 2009, 2: 206–210.MATHGoogle Scholar
  21. Zbăganu G: An inequality of M. Rădulescu and S. Rădulescu which characterizes the inner product spaces. Revue Roumaine de Mathématiques Pures et Appliquées 2002,47(2):253–257.MATHGoogle Scholar
  22. Day MM: Normed Linear Spaces, Ergebnisse der Mathematik und Ihrer Grenzgebiete. Volume 2. 3rd edition. Springer, New York, NY, USA; 1973:viii+211.Google Scholar
  23. Jiménez-Melado A: The fixed point property for some uniformly nonoctahedral Banach spaces. Bulletin of the Australian Mathematical Society 1999,59(3):361–367. 10.1017/S0004972700033037MathSciNetView ArticleMATHGoogle Scholar
  24. Sims B: Orthogonality and fixed points of nonexpansive maps. In Workshop/Miniconference on Functional Analysis and Optimization (Canberra, 1988), Proceedings of the Centre for Mathematical Analysis, Australian National University. Volume 20. Australian National University, Canberra; 1988:178–186.Google Scholar
  25. Jiménez-Melado A, Llorens-Fuster E: The fixed point property for some uniformly nonsquare Banach spaces. Bollettino della Unione Matemàtica Italiana 1996,10(3):587–595.MathSciNetMATHGoogle Scholar
  26. Goebel K: On a fixed point theorem for multivalued nonexpansive mappings. Annales Universitatis Mariae Curie-Skłodowska 1975, 29: 69–72.MathSciNetMATHGoogle Scholar
  27. Lim TC: A fixed point theorem for multivalued nonexpansive mappings in a uniformly convex Banach space. Bulletin of the American Mathematical Society 1974, 80: 1123–1126. 10.1090/S0002-9904-1974-13640-2MathSciNetView ArticleMATHGoogle Scholar
  28. Kirk WA: Nonexpansive mappings in product spaces, set-valued mappings and -uniform rotundity. In Nonlinear Functional Analysis and Its Applications, Part 2 (Berkeley, 1983), Proceedings of Symposia in Pure Mathematics. Volume 45. Edited by: Browder FE. American Mathematical Society, Providence, RI, USA; 1986:51–64.View ArticleGoogle Scholar

Copyright

© Z. Zuo and Y. Cui. 2009

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