Abstract
The purpose of this paper is to study the viscosity iterative schemes for approximating a fixed point of an asymptotically nonexpansive semigroup on a compact convex subset of a smooth Banach space with respect to a sequence of strongly asymptotic invariant means defined on an appropriate space of bounded real valued functions of the semigroup. Our results extend and improve the result announced by Lau et al. (Nonlinear Anal. 67(4):1211-1225, 2007) and many others.
Similar content being viewed by others
1 Introduction
Let be the topological dual of a real Banach space E and C be a nonempty closed and convex subset of E. The value of at will be denoted by or . With each , we associate the set
Using the Hahn-Banach theorem, it is immediately clear that for each . The multi-valued mapping J from E into is said to be the (normalized) duality mapping. Let . A Banach space E is said to be uniformly convex, if for any , there exists a such that, for any , implies . It is well known that a uniformly convex Banach space is reflexive and strictly convex. A Banach space E is said to be smooth if the limit exists for all . As is well known, the duality mapping is norm to weak-star continuous when E is smooth; see [1]. Recall that a mapping T of C into itself is said to be:
-
(1)
Lipschitzian with Lipschiz constant if
-
(2)
nonexpansive if
-
(3)
asymptotically nonexpansive if there exists a sequence of positive numbers such that and
Iteration processes are often used to approximate a fixed point of a nonexpansive mapping T. The first one is introduced by Halpern [2] and is defined as follows: Take an initial guess arbitrarily and define recursively by
where is a sequence in .
In 2007, Lau et al. [3] introduced Halpern’s iterative schemes for approximating fixed point of semigroup of nonexpansive mappings on a nonempty compact convex subset C of Smooth (and strictly convex) Banach space and introduced the following iteration process. Let and
where is a sequence of left strong regular invariant means defined on an appropriate invariant subspace of .
A semigroup S is called left reversible if any two right ideals of S have nonvoid intersection, i.e., for . In this case, is a directed set when the binary relation ⪯ on S is defined by if and only if for . is called a Lipschitzian semigroup on C if be a Lipschitzian mapping of C into C with Lipschitz constant for each , for each and . A Lipschitzian semigroup is called nonexpansive (or a contractive) semigroup if , for each , and asymptotically nonexpansive semigroup if . Left revisable semigroup of nonexpansive mappings were first studied by Lau [4] and Holmes and Lau [5].
In this paper, motivated and inspired by Lau et al. [3], Katchang and Kumam [6], Kumam et al. [7], Razani and Yazdi [8], Piri [9], Piri and Badali [10], Piri and Kumam [11], Piri et al. [12], Saewan and Kumam [13], we introduce the composite explicit viscosity iterative schemes as follows:
where f is a weakly contractive mapping and A is a strongly positive bounded linear operator on E with coefficient and , for an asymptotically nonexpansive semigroup on compact convex subset C of a smooth Banach space E with respect to finite family of left strongly asymptotically invariant sequences of means defined on an appropriate invariant subspace of . We prove, under certain appropriate assumptions on the sequences , , and , that and defined by (3) converges strongly to , which is the unique solution of the variational inequality:
Our results improve and extend many previous results of Lau et al. [3], Saeidi [14], Saeidi and Naseri [15], Katchang and Kumam [6] and Piri and Kumam [11] and many others.
2 Preliminaries
Let S be a semigroup and let be the space of all bounded real valued functions defined on S with supremum norm. For and , we define elements and in by
Let X be a closed subspace of containing 1 and let be its topological dual. An element μ of is said to be a mean on X if . We often write instead of for and . Let X be left invariant (resp. right invariant), i.e., (resp. ) for each . A mean μ on X is said to be left invariant (resp. right invariant) if (resp. ) for each and . X is said to be left (resp. right) amenable if X has a left (resp. right) invariant mean. X is amenable if X is both left and right amenable. As is well known, is amenable when S is a commutative semigroup; see [3]. A net of means on X is said to be left strongly asymptotically if
for each , where is the adjoint operator of .
Let C be a nonempty closed and convex subset of E. Throughout this paper, S will always denote a semigroup with an identity e. S is called left reversible if any two right ideals in S have nonvoid intersection, that is, , for . In this case, we can define a partial ordering ≺ on S by if and only if . It is easy too see , . Further, if then for all . If a semigroup S is left amenable, then S is left reversible (see [16]). But the converse is false. is called a Lipschitzian semigroup on C if be a Lipschitzian mapping of C into C with Lipschitz constant for each , for each and . A Lipschitzian semigroup is called nonexpansive (or a contractive) semigroup if , for each , and asymptotically nonexpansive semigroup if . We denote by the set of common fixed points of φ, and by the set of almost periodic elements in C, that is, all such that is relatively compact in the norm topology of E. We will call a subspace X of , φ-stable if the functions and on S 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 each . We denote such a point by . Note that , for each (see [17]).
Lemma 2.1 [18]
Let S be a left reversible semigroup and be an asymptotically nonexpansive semigroup on weakly compact convex subset C of a Banach space E. Let X be a left invariant and φ-stable subspace of and μ be an asymptotically left strongly asymptotically invariant means on X. Then .
Lemma 2.2 [14]
Let S be a left reversible semigroup and be an asymptotically nonexpansive semigroup on weakly compact convex subset C of a Banach space E into C. Let X be a left invariant and φ-stable subspace of and be an asymptotically left invariant sequence of means on X. If and , then z is a common fixed point of φ.
Let D be a subset of B, where B is a subset of a Banach space E and let P be a retraction of B onto D, that is, for each . Then P is said to be sunny [19] if for each and with , . A subset D of B is said to be a sunny nonexpansive retract of B, if there exists a sunny nonexpansive retraction P of B into D.
Lemma 2.3 [14]
Let S be a left reversible semigroup and be an asymptotically nonexpansive semigroup on a nonempty compact convex subset C of a Banach space E into C. Let X be a left invariant and φ-stable subspace of and μ be a left invariant sequence of means on X. Then is nonexpansive and . Moreover, if E is smooth, then is a sunny nonexpansive retract of C and the sunny nonexpansive retraction of C onto C onto is unique.
Lemma 2.4 [1]
Let C be a nonempty convex subset of smooth Banach space E, D a nonempty subset of C, and a retraction. Then the following are equivalent:
-
(a)
P is the sunny nonexpansive.
-
(b)
for all and .
-
(c)
for all .
In a smooth Banach space, an operator A is strongly positive if there exists a constant with the property that
where I is the identity mapping and J is the normalized duality mapping.
Lemma 2.5 [20]
If A is a strongly positive bounded linear operator on a smooth Banach space E with coefficient and , then .
Definition 2.6 [21]
A self-mapping is called weakly contractive of the class if there exists a continuous and nondecreasing function such that , , , , and for any ,
Remark 2.7 Clearly a contractive mapping with constant k must be a weakly contractive mapping, where , but the converse is not true. For example the mapping from to is weakly contractive with . But f is not a contractive mapping (see [22]).
Lemma 2.8 [23]
Let and be bounded sequences in a Banach space X and let be a sequence in such that . If for all integers and
then .
Lemma 2.9 [24]
Let E be a real smooth Banach space and J be the duality mapping. Then
Lemma 2.10 [25]
Let be a sequence of nonnegative real numbers such that
where and are sequences of real numbers satisfying the following conditions:
-
(i)
, ,
-
(ii)
either or .
Then .
Lemma 2.11 [1]
Let be a metric space. A subset C of X is compact if and only if every sequence in C contains a convergent subsequence with limit in C.
3 The main result
In this section, we establish a strong convergence theorem for finding a common fixed point of an asymptotically nonexpansive semigroup in a smooth Banach space.
Theorem 3.1 Let S be a left reversible semigroup, and let be an asymptotically nonexpansive semigroup on a nonempty compact convex subset C of a smooth Banach space E such that . Let f be a weakly contractive mapping of the class , and let A be a strongly positive linear operator on E with coefficient . Let γ be a real number such that , and let X be a left amenable and φ-stable subspace of containing 1 and the function is an element of X for each and . Let be a finite family of left strongly asymptotically invariant sequence of mean on X such that for , , and let be a sequence in , be a sequence in and be sequences in satisfying in the following conditions:
(B1) , ,
(B2) ,
(B3) , .
If and are sequences generated by and
then and converge strongly to , which is the unique solution of the variational inequality
Equivalently, , where P denotes the unique sunny nonexpansive retraction of C onto .
Proof Since C is a compact convex subset of a Banach space E from Lemma 2.1, we have
From Lemma 2.3 and definition of , for every , we have
Therefore
Since C is compact, it is bounded. So we assume that
First, we show that for any sequence ,
We have
Since for , . So, we get (7). Next, we claim that . For , from the definition of and Lemma 2.3, we have
which implies that
Setting , we see that . Then we compute
It follows that
Therefore, we observe that
So from (7), (B1), and (B2), we obtain
Applying Lemma 2.8, we obtain . We also have , therefore, we get
We note that
Thus, we have the following:
By (8), (B1), and (B2), we obtain the following:
We consider
By (9) and (B3), we have the following:
Next, we prove that , where
From Lemma 2.11, we get . Let . Then there exists a subsequence of such that
It follows from Lemma 2.3 that
Thus, due to (10), (11), and Lemma 2.2, we get . Since E is smooth, from Lemma 2.3 there exists a unique sunny nonexpansive retraction P of C onto . Since A is bounded, without loss of generality, we may assume that . So from Lemma 2.5, we get . Since A is linear and f is a weak contraction, we have
Since , is a contraction of C into itself, therefore is contraction. Then the Banach contraction theorem guarantees that has a unique fixed point z. By Lemma 2.4, z is the unique solution of the variational inequality
Next, we prove that
Indeed, we can choose a subsequence of such that
Since C is compact, we may assume, with no loss of generality, that converges strongly to some . Since and duality mapping J is norm to weak-star continuous from (12) and (13), we have
Finally, we show that converges strongly to z. Using Lemma 2.3, Lemma 2.9, and (6), we have
and consequently,
Then we have
where and
It follows from condition (B1) and (14) that
Therefore, applying Lemma 2.10 to (15), we see that converges strongly to z and since for , , converges strongly to z. This completes the proof. □
4 Applications
Theorem 4.1 [14]
Let S be a left reversible semigroup and be a representation of S as Lipschitzian mapping from nonempty compact convex subset C of a smooth Banach space E into C, with the uniform Lipschitzian condition and g be an α-contraction on C for some . Let X be a left invariant φ-stable subspace of containing 1, be a sequence of left strongly asymptotically invariant means defined on X such that and be the sequence defined by
Let , , and be sequences in such that
(C1) , ,
(C2) ,
(C3) ,
(C4) ,
(C5) .
Let be the sequence generated by and
Then the sequence converges strongly to some , the set of common fixed points of φ, which is the unique solution of the variational inequality
Equivalently, one has , where P is the unique sunny nonexpansive retraction of C onto .
Proof It is sufficient to take , , , for and in Theorem 3.1. □
Theorem 4.2 [11]
Let be a representation of S as a Lipschitzian mappings from a nonempty compact convex subset C of a smooth Banach space E into C, with the uniform Lipschitzian constant on the Lipschiz constant of mappings, such that , and g be a contraction of C into itself with constant . Let X be a left invariant and φ-stable subspace of containing 1 and the function is an element of X for each and and be a finite family of left strongly asymptotically invariant means on X such that for , . Let , and be sequences in satisfy in conditions (C 1)-(C 4) and be a sequence in satisfies in condition
() , .
If and are sequences generated by and
then and converge strongly to which is the unique solution of the variational inequality
Equivalently, , where P denotes the unique sunny nonexpansive retraction of C onto .
Proof It is sufficient to take , , and in Theorem 3.1. □
Theorem 4.3 [6]
Let S be a left reversible semigroup and be a representation of S as Lipschitzian mapping from nonempty compact convex subset C of a smooth Banach space E into C, with the uniform Lipschitzian condition and g be an α-contraction on C for some . Let X be a left invariant φ-stable subspace of containing 1, is a sequence of left strong regular invariant means defined on X such that and be the sequence defined by
Let , , , and be sequences in such that
(C1) , ,
(C2) ,
(C3) ,
(C4) ,
(C5) ,
(C6) .
Let be the sequence generated by and
Then the sequence converges strongly to some , which is the unique solution of the variational inequality.
Equivalently, one has , where P is the unique sunny nonexpansive retraction of C onto .
Proof It is sufficient to take , , for all and for in Theorem 3.1. □
Theorem 4.4 [3]
Let S be a left reversible semigroup and be a representation of S as nonexpansive mappings from a compact convex subset C of a strictly convex and smooth Banach space E into C such that , let X be an amenable and S-stable subspace of and let be a strongly left regular sequence of means on X. Let be a sequence in such that and . Let and let be the sequence defined by
Then converges strongly to Px, where P denotes the unique sunny nonexpansive retraction of C onto .
Proof It is sufficient to take , , for all and for in Theorem 3.1. □
Theorem 4.5 [15]
Let be a nonexpansive semigroup on a Hilbert space H such that . Let X be a left invariant subspace of such that , and the function is an element of X for each . Let be a left regular sequence of means on X and let be a sequence in such that and . Let A be a strongly positive linear bounded operator on H with coefficient and f be an α-contraction on H for some . Let and let be generated by and
Then the sequence converges strongly to some , the set of common fixed points of φ, which is the unique solution of the variational inequality
Equivalently, one has .
Proof It is sufficient to take for all and for , in Theorem 3.1. □
Remark 4.6 Theorem 3.1 improves and extends Theorem 3.1 of [14], Theorem 3.1 of [6], Theorem 4.1 of [3] and Theorem 3.1 of [15] in the following aspects.
-
(1)
Theorem 3.1 extends the theorem and Theorem 3.1 of [14] forms one sequence of means to a finite family of sequences of means and gives all consequences of this theorem without assumption (C5) used in its proof.
-
(2)
Theorem 3.1 extends the theorem and Theorem 3.1 of [6] forms one sequence of means to a finite family of sequences of means and gives all consequences of this theorem without assumption (C5) used in its proof.
-
(3)
Theorem 3.1 extends the theorem and Theorem 4.1 of [3] forms one sequence of means to a finite family of sequence of means and gives all consequences of this theorem without the assumption of strict convexity of Banach spaces used in its proof.
-
(4)
Theorem 3.1 extends the theorem and Theorem 3.1 of [15] forms one sequence of means to a finite family of sequence of means and gives all consequences of this theorem from Hilbert spaces to Banach spaces.
References
Agarwal RP, O’Regan D, Sahu DR Topological Fixed Point Theory and Its Applications 6. In Fixed Point Theory for Lipschitzian-Type Mappings with Applications. Springer, New York; 2009.
Halpern B: Fixed points of nonexpanding maps. Bull. Am. Math. Soc. 1967, 73(6):957–961. 10.1090/S0002-9904-1967-11864-0
Lau AT, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. 2007, 67(4):1211–1225. 10.1016/j.na.2006.07.008
Lau AT: Invariant means on almost periodic functions and fixed point properties. Rocky Mt. J. Math. 1973, 3: 69–76. 10.1216/RMJ-1973-3-1-69
Holmes RD, Lau AT: Asymptotically non-expansive actions of topological semigroups and fixed points. Bull. Lond. Math. Soc. 1971, 3: 343–347. 10.1112/blms/3.3.343
Katchang P, Kumam P: A composite explicit iterative process with a viscosity method for Lipschitzian semigroup in smooth Banach space. Bull. Iran. Math. Soc. 2011, 37: 143–159.
Kumam P, Plubtieng S, Katchang P: Viscosity approximation to a common solution of variational inequality problems and fixed point problems for Lipschitzian semigroup in Banach spaces. Math. Sci. 2013., 7: Article ID 28 10.1186/2251-7456-7-28
Razani A, Yazdi M: An iterative method for family of nonexpansive mappings. Math. Rep. 2014, 16(66)(1):7–23.
Piri H: Hybrid pseudo-viscosity approximation schemes for systems of equilibrium problems and fixed point problems of infinite family and semigroup of non-expansive mappings. Nonlinear Anal. 2011, 74: 6788–6804. 10.1016/j.na.2011.06.056
Piri H, Badali AH: Strong convergence theorem for amenable semigroups of nonexpansive mappings and variational inequalities. Fixed Point Theory Appl. 2011., 2011: Article ID 55
Piri H, Kumam P: Approximating fixed points for a reversible semigroup of Lipschitzian mappings in a smooth Banach space. J. Inequal. Appl. 2013., 2013: Article ID 555
Piri H, Kumam P, Sitthithakerngkiet K: Approximating fixed points for Lipschitzian semigroup and infinite family of nonexpansive mappings with the Meir-Keeler type contraction in Banach spaces. Dyn. Contin. Discrete Impuls. Syst. 2014, 21: 201–229.
Saewan S, Kumam P: Explicit iterations for Lipschitzian semigroups with the Meir-Keeler type contraction in Banach spaces. J. Inequal. Appl. 2012., 2012: Article ID 279 10.1186/1029-242X-2012-279
Saeidi S: Approximating common fixed points of Lipschitzian semigroup in smooth Banach spaces. Fixed Point Theory Appl. 2008., 2008: Article ID 363257
Saeidi S, Naseri S: Iterative methods for semigroups of nonexpansive mappings and variational inequalities. Math. Rep. 2010, 12(62)(1):59–70.
Lau AT, Zhang Y: Fixed point properties of semigroups of non-expansive mappings. J. Funct. Anal. 2008, 254(10):2534–2554. 10.1016/j.jfa.2008.02.006
Hirano N, Kido K, Takahashi W: Nonexpansive retractions and nonlinear ergodic theorems in Banach spaces. Nonlinear Anal. 1988, 28: 1269–1281.
Saeidi S: Strong convergence of Browder’s type iterations for left amenable semigroups of Lipschitzian mappings in Banach spaces. J. Fixed Point Theory 2009, 5: 93–103. 10.1007/s11784-008-0092-3
Reich S: Asymptotic behavior of contraction in Banach spaces. J. Math. Anal. Appl. 1973, 44: 57–70. 10.1016/0022-247X(73)90024-3
Cai G, Hu CS: Strong convergence theorems of a general iterative process for a finite family of -strict pseudo-contractions in q -uniformly smooth Banach spaces. Comput. Math. Appl. 2010, 59: 149–160. 10.1016/j.camwa.2009.07.068
Combettes PL: The foundations of set theoretic estimation. Proc. IEEE 1993, 81: 182–208.
Alber YI, Guerre-Delabriere S: Principles of weakly contractive maps in Hilbert spaces. Oper. Theory, Adv. Appl. 1997, 98: 7–22.
Zhang SS, Yang L, Liu JA: Strong convergence theorems for nonexpansive mappings in Banach spaces. Appl. Math. Mech. 2007, 28: 1287–1297. 10.1007/s10483-007-1002-x
Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.
Yaoa Y, Lioub Y-C, Chena R: A general iterative method for an infinite family of nonexpansive mappings. Nonlinear Anal. 2008, 69: 1644–1654. 10.1016/j.na.2007.07.013
Acknowledgements
The second author was supported by the Thailand Research Fund and the King Mongkut’s University of Technology Thonburi (Grant No. RSA5780059).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Piri, H., Kumam, P. Strong convergence theorems for fixed points of asymptotically nonexpansive semigroups in Banach spaces. Fixed Point Theory Appl 2014, 225 (2014). https://doi.org/10.1186/1687-1812-2014-225
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2014-225