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Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces
Fixed Point Theory and Applications volume 2012, Article number: 116 (2012)
Abstract
In this paper, we provide the existence and convergence theorems of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in general Banach spaces, which extend and improve many recent results in this area.
MSC:47H09, 47H10, 47H20.
1 Introduction
Let E be a Banach space and C a nonempty bounded closed convex subset of E. A mapping T on C is said to be nonexpansive if for all . A well-known result of Browder [1] asserts that if E is uniformly convex, then every nonexpansive mapping on C has a fixed point. Kirk [2], Belluce and Kirk [3] extended this result to the case that X has a normal structure or Opial’s property. Goebel and Kirk [4] proved that if E is a uniformly convex Banach space, then every asymptotically nonexpansive mapping on C has a fixed point.
As is well known, not every semigroup of nonexpansive mappings on a subset of a Banach space has a fixed point [5]. The existence and convergence of fixed points for semigroups of various mappings have been studied extensively [6–10]. Recently, Suzuki and Takahashi [8], Takahashi and Zembayashi [9], Zhu and Li [10] proved the existence theorems of fixed points for semigroups of nonexpansive, asymptotically nonexpansive and asymptotically nonexpansive type mappings, respectively. For instance, in [9], Takahashi and Zembayashi proved the following theorem:
Theorem 1.1 [9]
Let C be a nonempty compact convex subset of a Banach space E and be a semigroup of asymptotically nonexpansive mappings on C, then the set of common fixed points of â„‘ is nonempty.
Many results are known in the case that the semigroup G is commutative, amenable or reversible [11–24]. In the case of an amenable semigroup, the first result was established by Takahashi [21] where he proved:
Theorem 1.2 [21]
Let C be a nonempty compact convex subset of a Banach space E. Let be an amenable semigroup of nonexpansive mappings on C. Then C contains a common fixed point for â„‘.
Theorem 1.2 was proved for a commutative semigroup by DeMarr [11]. Later in [13], Lau showed that the fixed point property is equivalent to the existence of a left invariant mean on , the space of almost periodic functions on the semigroup â„‘. It should be pointed out that if â„‘ is left reversible, then always has a left invariant mean [13], but the converse is false [14]. And in [16], Lau, Miyake and Takahashi gave the following existence theorem:
Theorem 1.3 [16]
Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup (with identity) and be a semigroup of nonexpansive mappings on C. Let X be a left invariant ℑ-stable subspace of containing 1, and μ be a left invariant mean on X. Then , where denotes the set of almost periodic elements in C, i.e., all such that is relatively compact in the norm topology of E. Further, if C is compact, then the set is nonempty.
In [20], Saeidi extended Theorem 1.3 to the case for left reversible semigroups of asymptotically nonexpansive mappings. Inspired and motivated by [8–10, 16, 20, 21, 23], we investigate the existence and convergence of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. We first provide the existence theorem of fixed points for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces. Utilizing this result, we obtain a strong convergence theorem of iterative sequences for left amenable semigroups of asymptotically nonexpansive type mappings. The results obtained in this paper extend and improve many recent results in [8–10, 16, 20, 23].
2 Preliminaries
Let C be a nonempty bounded subset of a Banach space E. Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for the mappings and from G to G are continuous. Let be a continuous representation of G on C, i.e., , , and the mapping from into C is continuous when has the product topology. Recall that â„‘ is said to be
-
(1)
nonexpansive if for all and ,
-
(2)
asymptotically nonexpansive [25–27] if there exists a function with such that for all and ,
-
(3)
asymptotically nonexpansive type [25–27] if for each , there exists a function with such that for all and ,
It is easily seen that and that both inclusions are proper [25–27].
Let be the Banach space of all bounded real valued functions on G with the supremum norm. Then, for each and , we can define in by for all . Let X be a subspace of containing 1 and be its dual space. An element is called a mean on X if . We always denote the value of μ at by . Let X be left invariant, i.e., for all . A mean μ on X is said to be left invariant if for all and . Further, X is called left amenable if X has a left invariant mean. In this case, we also say that G is a left amenable semigroup. Recall that a semigroup G is called left reversible if any two closed right ideals of G have nonvoid intersection. In this case, is a directed system when the binary relation ≤ on G is defined by if and only if , . As is well known, the class of left reversible semigroups includes all commutative semigroups and if a semigroup G is left amenable, then G is left reversible. But the converse is false [28].
Let be an asymptotically nonexpansive type semigroup on C. Let denote the set of all fixed points of ℑ, i.e., . A subspace X of is called ℑ-stable if functions and on G are in X for all and . We know that if μ is a mean on X and if for each the function is contained in X and C is weakly compact, then there exists a unique point of E such that for all . Such a point is always denoted by . Obviously, for each .
3 Main results
Lemma 3.1 Let C be a nonempty weakly compact convex subset of a Banach space E. Let G be a left reversible semigroup and be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition for all . Let X be a left invariant ℑ-stable subspace of containing 1, and μ be a left invariant mean on X. Then .
Proof If is empty, then so is as . Let and define , then for all , we have
i.e., for all ,
Next, we shall show . In fact, if , then for each ,
i.e.,
By , then for any , there exists such that
It follows from (3.2) that we can choose a cluster point of the net in the set C with and there exists satisfying and . Combining it with (3.1) and (3.3), we get
Hence and so . It follows from (3.1) and (3.3) that
for all . Noting
we obtain
This implies that
Thus there exists such that
Since is a relatively compact set, , as a subset of , has a strong convergent subsequence. Without loss of generality, we can assume that . Setting , then , and by (3.6),
On the other hand,
and
Thus we can conclude and . So by (3.7),
Similar to the proof of (3.5), we can prove and
This means
Thus there exists such that
Therefore,
and
Since has a strong convergent subsequence, without loss of generality, we can assume that . Setting , then , ,
and
Thus we have found such that
Now, by mathematical induction, we can find a sequence satisfying
Since , we can seek out with . Thus
which is a contradiction with the relative compactness of . Therefore, we can conclude .
In the following, we shall show . Indeed, for any , is continuous at z, then for all , there exists a () such that for all with ,
By (3.1) and the definition of , we can get
and so we can find a such that , i.e., for all ,
Hence
Since is arbitrary, we get . This completes the proof. □
Now we can give the existence theorem of fixed points for left amenable semigroups of non-Lipschitzian mappings in Banach spaces.
Theorem 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition for all . Let X be a left invariant ℑ-stable subspace of containing 1, and μ be a left invariant mean on X. Then the set is nonempty.
Proof For all and , we have
and so by , we get , i.e., is a nonexpansive mapping from C into itself. Since a nonexpansive mapping of a compact convex subset of a Banach space into itself has a fixed point [29], has a fixed point z. By Lemma 3.1, . This completes the proof. □
Remark 3.1 Theorem 3.1 is an extension of the main results in [8–10, 16, 20, 23] to the case for left amenable semigroups of asymptotically nonexpansive type mappings in Banach spaces.
Recall that for each , we define a point evaluation on X by for every . A convex combination of point evaluation is called a finite mean on G. If λ is a finite mean on G, say , where , , , and , then for all . For convenience, we denote it by . A net of finite means on G is said to be strongly left regular if
for all , where A is a directed system and is the conjugate operator of .
Corollary 3.1 Let C be a nonempty compact convex subset of a Banach space E. Let G be a left reversible semigroup and be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition for all . Let X be a left invariant â„‘-stable subspace of containing 1 and be a net of strongly left regular finite means on G. If satisfies
then .
Proof Since and , we can find a subnet of such that and , where A is a directed system. Hence μ is a left invariant mean on X (see [30]) and , which implies . This completes the proof. □
Remark 3.2 Corollary 3.1 is an extension of the main results in [8–10, 23].
Next we shall prove the strong convergence theorem for the iterative sequences of left reversible semigroups of asymptotically nonexpansive type mappings. We need a lemma which plays a crucial role in the proof of Theorem 3.2.
Lemma 3.2 [30]
Let and be bounded sequences in a Banach space X and let be a sequence in with . Suppose that for all and
for all . Then .
Theorem 3.2 Let C be a nonempty compact convex subset of a Banach space X and G be a left reversible semigroup. Let be a continuous representation of G as asymptotically nonexpansive type mappings on C, with the condition for all . Let X be a left invariant ℑ-stable subspace of containing 1, and μ be a left invariant mean on X. Let and define a sequence in C by
for all , where satisfies . Then converges strongly to a fixed point .
Proof It follows from
that exists. By Lemma 3.1, we get and so . Since C is compact, there exists a subsequence such that . Hence, z is a fixed point of . By Lemma 3.1, we have and
Hence exists. Thus , which implies that converges strongly to . This completes the proof. □
In the following, we shall give an example of a semigroup which is asymptotically nonexpansive type but not asymptotically nonexpansive on a compact set.
Example 3.1 [27]
Let Δ be the Cantor ternary set. Define the Cantor ternary function
then is a continuous and increasing but not absolutely continuous function with , (see [31]). Since a Lipschitzian function is absolutely continuous, Ï„ is non-Lipschitzian. For all , we define by
Then is continuous but not Lipschitzian continuous (since Ï„ is non-Lipschitzian) and for all , ,
Therefore, we can conclude that the semigroup is asymptotically nonexpansive type but not an asymptotically nonexpansive on . Also, 0 is a fixed point of â„‘.
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Acknowledgement
This manuscript has benefited greatly from the constructive comments and helpful suggestions of the anonymous referees, the authors would like to express their deep gratitude to them. This research is supported by the Natural Science Foundation of China (10971182, 11201410), the Natural Science Foundation of Jiangsu Province (BK2009179, BK2010309 and BK2012260), the Jiangsu Government Scholarship for Overseas Studies, the Natural Science Foundation of Jiangsu Education Committee (10KJB110012) and the Natural Science Foundation of Yangzhou University.
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Huang, Q., Zhu, L. & Li, G. Fixed point theorems for left amenable semigroups of non-Lipschitzian mappings in Banach spaces. Fixed Point Theory Appl 2012, 116 (2012). https://doi.org/10.1186/1687-1812-2012-116
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DOI: https://doi.org/10.1186/1687-1812-2012-116