- Research
- Open access
- Published:
Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings
Fixed Point Theory and Applications volume 2012, Article number: 114 (2012)
Abstract
In this paper, we prove a strong convergence theorem for finding a common solution of a general system of finite variational inequalities for finite different inverse-strongly accretive operators and solutions of fixed point problems for a nonexpansive semigroup in a Banach space based on a viscosity approximation method by using weak contraction mappings. Moreover, we can apply the above results to find the solutions of the class of k-strictly pseudocontractive mappings and apply a general system of finite variational inequalities into a Hilbert space. The results presented in this paper extend and improve the corresponding results of Ceng et al. (2008), Katchang and Kumam (2011), Wangkeeree and Preechasilp (2012), Yao et al. (2010) and many other authors.
MSC:47H05, 47H10, 47J25.
1 Introduction
Let E be a real Banach space with norm and C be a nonempty closed convex subset of E. Let be the dual space of E and denote the pairing between E and . For , the generalized duality mapping is defined by for all . In particular, if , the mapping is called the normalized duality mapping and, usually, write . Further, we have the following properties of the generalized duality mapping : (i) for all with ; (ii) for all and ; and (iii) for all . It is known that if E is smooth, then J is single-valued, which is denoted by j. Recall that the duality mapping j is said to be weakly sequentially continuous if for each weakly, we have weakly-*. We know that if E admits a weakly sequentially continuous duality mapping, then E is smooth (for the details, see [24, 25, 29]).
Let be a k-contraction mapping if there exists such that , . Let a nonlinear mapping. We use to denote the set of fixed points of S, that is, . A mapping S is called nonexpansive if , . A mapping f is called weakly contractive on a closed convex set C in the Banach space E if there exists is a continuous and strictly increasing function such that φ is positive on , , and
If , then f is called to be contractive with the contractive coefficient k. If , then f is said to be nonexpansive.
A family of mappings of C into itself is called a nonexpansive semigroup (see also [14]) on C if it satisfies the following conditions:
-
(i)
for all ;
-
(ii)
for all ;
-
(iii)
for all and ;
-
(iv)
for all , is continuous.
We denote by the set of all common fixed points of, that is,
It is known that is closed and convex. Moreover, for the study of nonexpansive semigroup mapping, see [5, 14–16, 26] for more details.
In 2002, Suzuki [21] was the first one to introduce the following implicit iteration process in Hilbert spaces:
for the nonexpansive semigroup. In 2007, Xu [28] established a Banach space version of the sequence (1.2) of Suzuki [21]. In [4], Chen and He considered the viscosity approximation process for a nonexpansive semigroup and proved another strong convergence theorems for a nonexpansive semigroup in Banach spaces, which is defined by
where is a fixed contractive mapping. Recall that an operator is said to be accretive if there exists such that
for all . A mapping is said to be β-strongly accretive if there exists a constant such that
An operator is said to be β-inverse strongly accretive if, for any
for all . Evidently, the definition of the inverse strongly accretive operator is based on that of the inverse strongly monotone operator. To convey an idea of the variational inequality, let C be a closed and convex set in a real Hilbert space H. For a given operator A, we consider the problem of finding such that
for all , which is known as the variational inequality, introduced and studied by Stampacchia [22] in 1964 in the field of potential theory. In 2006, Aoyama et al. [1] first considered the following generalized variational inequality problem in a smooth Banach space. Let A be an accretive operator of C into E. Find a point such that
for all . This problem is connected with the fixed point problem for nonlinear mappings, the problem of finding a zero point of an accretive operator and so on. For the problem of finding a zero point of an accretive operator by the proximal point algorithm, see Kamimura and Takahashi [10, 11]. In order to find a solution of the variational inequality (1.4), Aoyama et al. [1] proved the strong convergence theorem in the framework of Banach spaces which is generalized by Iiduka et al. [8] from Hilbert spaces.
Motivated by Aoyama et al. [1] and also Ceng et al. [3], Qin et al. [18] and Yao et al. [29] first considered the following new general system of variational inequalities in Banach spaces:
Let be a β-inverse strongly accretive mapping. Find such that
Let C be nonempty closed convex subset of a real Banach space E. For two given operators , consider the problem of finding such that
where λ and μ are two positive real numbers. This system is called the general system of variational inequalities in a real Banach spaces. If we add up the requirement that , then the problem (1.6) is reduced to the system (1.5).
By the following general system of variational inequalities, we extend into the general system of finite variational inequalities which is to find and is defined by
where is a family of mappings, , . The set of solutions of (1.7) is denoted by . In particular, if , , , , , and , then the problem (1.7) is reduced to the problem (1.6).
In this paper, motivated and inspired by the idea of Ceng et al. [3], Katchang and Kumam [12] and Yao et al. [29], we introduce a new iterative scheme with weak contraction for finding solutions of a new general system of finite variational inequalities (1.7) for finite different inverse-strongly accretive operators and solutions of fixed point problems for nonexpansive semigroups in a Banach space. Consequently, we obtain new strong convergence theorems for fixed point problems which solve the general system of variational inequalities (1.6). Moreover, we can apply the above theorem to finding solutions of zeros of accretive operators and the class of k-strictly pseudocontractive mappings. The results presented in this paper extend and improve the corresponding results of Ceng et al. [3], Katchang and Kumam [12], Wangkeeree and Preechasilp [26], Yao et al. [29] and many other authors.
2 Preliminaries
We always assume that E is a real Banach space and C is a nonempty closed convex subset of E.
Let . A Banach space E is said to be uniformly convex if, for any , there exists such that, for any implies . It is 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 . It is also said to be uniformly smooth if the limit is attained uniformly for . The modulus of smoothness of E is defined by
where is a function. It is known that E is uniformly smooth if and only if . Let q be a fixed real number with . A Banach space E is said to be q-uniformly smooth if there exists a constant such that for all : see, for instance, [1, 24].
We note that E is a uniformly smooth Banach space if and only if is single-valued and uniformly continuous on any bounded subset of E. Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is min-uniformly smooth for every . Note also that no Banach space is q-uniformly smooth for ; see [24, 27] for more details.
Let D be a subset of C and . Then Q is said to be sunny if
whenever for and . A subset D of C is said to be a sunny nonexpansive retract of C if there exists a sunny nonexpansive retraction Q of C onto D. A mapping is called a retraction if . If a mapping is a retraction, then for all z in the range of Q. For example, see [1, 23] for more details. The following result describes a characterization of sunny nonexpansive retractions on a smooth Banach space.
Proposition 2.1 ([19])
Let E be a smooth Banach space and let C be a nonempty subset of E. Letbe a retraction and let J be the normalized duality mapping on E. Then the following are equivalent:
-
(i)
Q is sunny and nonexpansive;
-
(ii)
, ;
-
(iii)
, , .
Proposition 2.2 ([13])
Let C be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, and let T be a nonexpansive mapping of C into itself with. Then the setis a sunny nonexpansive retract of C.
A Banach space E is said to satisfy Opial’s condition if for any sequence in E () implies
By [7], Theorem 1], it is well known that, if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial’s condition and E is smooth.
We need the following lemmas for proving our main results.
Lemma 2.3 ([27])
Let E be a real 2-uniformly smooth Banach space with the best smooth constant K. Then the following inequality holds:
Lemma 2.4 ([20])
Letandbe bounded sequences in a Banach space X and letbe a sequence inwith. Supposefor all integersand. Then, .
Lemma 2.5 (Lemma 2.2 in [17])
Letandbe two nonnegative real number sequences anda positive real number sequence satisfying the conditions: and. Let the recursive inequality
whereis a continuous and strict increasing function for allwith. Then.
Lemma 2.6 ([6])
Let E be a uniformly convex Banach space andbe a closed ball of E. Then there exists a continuous strictly increasing convex functionwithsuch that
for allandwith.
Lemma 2.7 ([2])
Let C be a nonempty bounded closed convex subset of a uniformly convex Banach space E and let T be nonexpansive mapping of C into itself. Ifis a sequence of C such thatweakly andstrongly, then x is a fixed point of T.
Lemma 2.8 (Yao et al. [29], Lemma 3.1]; see also [1], Lemma 2.8])
Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Let the mappingbe β-inverse-strongly accretive. Then, we have
If, thenis nonexpansive.
3 Main results
In this section, we prove a strong convergence theorem. In order to prove our main results, we need the following two lemmas.
Lemma 3.1 Let C be a nonempty closed convex subset of a real 2-uniformly smooth Banach space E. Letbe the sunny nonexpansive retraction from E onto C. Let the mappingbe a-inverse-strongly accretive such thatwhere. Ifis a mapping defined by
thenis nonexpansive.
Proof Taking , and , where I is the identity mapping on E, we have . For any , we have
Therefore, is nonexpansive. □
Lemma 3.2 Let C be a nonempty closed convex subset of a real smooth Banach space E. Letbe the sunny nonexpansive retraction from E onto C. Letbe nonlinear mapping, where. For, , is a solution of problem (1.7) if and only if
that is
Proof From (1.7), we rewrite as
Using Proposition 2.1(iii), the system (3.2) is equivalent to (3.1). □
Throughout this paper, the set of fixed points of the mapping is denoted by .
The next result states the main result of this work.
Theorem 3.3 Let E be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Letbe a nonexpansive semigroup on C andbe a sunny nonexpansive retraction from E onto C. Letbe a-inverse-strongly accretive such that, where, and K be the best smooth constant. Let f be a weakly contractive mapping on C into itself with function φ. Suppose, whereis defined by Lemma 3.1. For arbitrary given, the sequenceis generated by
where the sequences, andare inand satisfy, , , and, are positive real numbers. The following conditions are satisfied:
(C1) and;
(C2) ;
(C3) ;
(C4) , bounded subset of C.
Thenconverges strongly toandis a solution of the problem (1.7) whereis the sunny nonexpansive retraction of C onto.
Proof First, we prove that is bounded. Let , taking
, where I is the identity mapping on E. From the definition of is nonexpansive then , also. We note that
From (3.3) and (3.4), we also have
This implies that is bounded, so are , , and .
Next, we show that . Notice that
Setting for all , we see that . Then we have
Therefore,
It follows from the conditions (C1), (C2) and (C4), which implies that
Applying Lemma 2.4, we obtain and also
as . Therefore, we have
Next, we show that . Since , from Lemma 2.6, we obtain
Therefore, we have
From the condition (C1) and (3.6), this implies that as . Now, we note that
Therefore, we get
From the conditions (C1), (C2) and (3.6), this implies that as . Since
and hence it follows that .
Next, we prove that . By the reflexivity of E and boundedness of the sequence , we may assume that for some .
-
(a)
First, we show that . Put , , , and for , let be such that
Fix . Notice that
For all , we have
Since the Banach space E with a weakly sequentially continuous duality mapping satisfies Opial’s condition, this implies . Therefore, .
-
(b)
Next, we show that . From Lemma 3.1, we know that is nonexpansive; it follows that
Thus . Since is nonexpansive, we get
and so
By Lemma 2.7 and (3.7), we have . Therefore, .
Next, we show that , where . Since is bounded, we can choose a sequence of where such that
Now, from (3.8), Proposition 2.1(iii) and the weakly sequential continuity of the duality mapping J, we have
From (3.6), it follows that
Finally, we show that converges strongly to . We compute that
By (3.5) and since is bounded, i.e., there exists such that , which implies that
Now, from (C1) and applying Lemma 2.5 to (3.11), we get as . This completes the proof. □
Corollary 3.4 Let E be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Letbe a nonexpansive semigroup on C andbe a sunny nonexpansive retraction from E onto C. Letbe a β-inverse-strongly accretive such thatwhere K is the best smooth constant. Let f be a weakly contractive mapping of C into itself with function φ. Let the sequences, andbe inwith, , and satisfy the conditions (C 1)-(C 4) in Theorem 3.3. Suppose, whereis defined by
and λ be a positive real number. For arbitrary given, the sequencesare generated by
Thenconverges strongly to, whereis the sunny nonexpansive retraction of C onto.
Proof Putting , and in Theorem 3.3, we can conclude the desired conclusion easily. This completes the proof. □
Corollary 3.5 Let E be a uniformly convex and 2-uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping and C be a nonempty closed convex subset of E. Letbe a nonexpansive semigroup on C andbe a sunny nonexpansive retraction from E onto C. Letbe a-inverse-strongly accretive such that, whereand K be the best smooth constant. Let f be a weakly contractive mapping of C into itself with function φ. Let the sequences, andbe inwith, , and satisfy the conditions (C 1)-(C 4) in Theorem 3.3. Suppose, whereis defined by
and, are positive real numbers. For arbitrary given, the sequencesare generated by
Thenconverges strongly toandis a solution of the problem (1.6), whereis the sunny nonexpansive retraction of C onto.
Proof Taking in Theorem 3.3, we can conclude the desired conclusion easily. This completes the proof. □
4 Some applications
4.1 (I) Application to strictly pseudocontractive mappings
Let E be a Banach space and let C be a subset of E. Recall that a mapping is said to be k-strictly pseudocontractive if there exist and such that
for all . Then (4.1) can be written in the following form:
Moreover, we know that A is -inverse strongly monotone and (see also [9]).
Theorem 4.1 Let E be a uniformly convex and 2-uniformly smooth Banach space and C be a nonempty closed convex subset of E. Letbe a nonexpansive semigroup on C andbe a-strictly pseudocontractive mapping with, . Let f be a weakly contractive mapping of C into itself with function φ and suppose the sequences, andinsatisfy, and. Supposeand let, be positive real numbers. If the following conditions are satisfied:
-
(i)
and;
-
(ii)
;
-
(iii)
;
-
(iv)
, bounded subset of C.
Then the sequence is generated by and
converges strongly to, whereis the sunny nonexpansive retraction of E onto.
Proof Putting . From (4.2), we get is -inverse strongly accretive operator. It follows that and is the solution of the problem (1.7) (see also Ceng et al. [3], Theorem 4.1, pp.388-389] and Aoyama et al. [1], Theorem 4.1, p.10]).
Therefore, by Theorem 3.3, converges strongly to some element of . □
4.2 (II) Application to Hilbert spaces
In real Hilbert spaces H, by Lemma 3.2, we obtain the following lemma:
Lemma 4.2 For given, a solution of the problem is as follows:
if and only if
is a fixed point of the mapping defined by
whereis a metric projection H onto C.
It is well known that the smooth constant in Hilbert spaces. From Theorem 3.3, we can obtain the following result immediately.
Theorem 4.3 Let C be a nonempty closed convex subset of a real Hilbert space H. Letbe a-inverse-strongly monotone mapping with, . Letbe a nonexpansive semigroup on C and f be a weakly contractive mapping of C into itself with function φ. Assume that, whereis defined by Lemma 4.2 and let, be positive real numbers. Let the sequences, andinwith, and the following conditions be satisfied:
-
(i)
and;
-
(ii)
;
-
(iii)
;
-
(iv)
, bounded subset of C.
For arbitrary given, the sequencesare generated by
Thenconverges strongly toandis a solution of the problem (4.4).
Remark 4.4 We can replace a contraction mapping f to a weak contractive mapping by setting . Hence, our results can be obtained immediately.
References
Aoyama K, Iiduka H, Takahashi W: Weak convergence of an iterative sequence for accretive operators in Banach spaces. Fixed Point Theory Appl. 2006., 2006:
Browder FE: Nonlinear operators and nonlinear equations of evolution in Banach spaces. Proc. Symp. Pure Math. 1976, 18: 78–81.
Ceng L-C, Wang C-Y, Yao J-C: Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities. Math. Methods Oper. Res. 2008, 67: 375–390. 10.1007/s00186-007-0207-4
Chen R, He H: Viscosity approximation of common fixed points of nonexpansive semigroups in Banach spaces. Appl. Math. Lett. 2007, 20: 751–757. 10.1016/j.aml.2006.09.003
Chen RD, He HM, Noor MA: Modified iterations for nonexpansive semigroups in Banach space. Acta Math. Sin. Engl. Ser. 2010, 26(1):193–202. 10.1007/s10114-010-7446-7
Cho YJ, Zhou HY, Guo G: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2
Gossez JP, Dozo EL: Some geometric properties related to the fixed point theory for nonexpansive mappings. Pac. J. Math. 1972, 40: 565–573.
Iiduka H, Takahashi W, Toyoda M: Approximation of solutions of variational inequalities for monotone mappings. Panam. Math. J. 2004, 14(2):49–61.
Iiduka H, Takahashi W: Strong convergence theorems for nonexpansive mapping and inverse-strong monotone mappings. Nonlinear Anal. 2005, 61: 341–350. 10.1016/j.na.2003.07.023
Kamimura S, Takahashi W: Approximating solutions of maximal monotone operators in Hilbert space. J. Approx. Theory 2000, 106: 226–240. 10.1006/jath.2000.3493
Kamimura S, Takahashi W: Weak and strong convergence of solutions to accretive operator inclusions and applications. Set-Valued Anal. 2000, 8(4):361–374. 10.1023/A:1026592623460
Katchang P, Kumam P: An iterative algorithm for finding a common solution of fixed points and a general system of variational inequalities for two inverse strongly accretive operators. Positivity 2011, 15: 281–295. 10.1007/s11117-010-0074-8
Kitahara S, Takahashi W: Image recovery by convex combinations of sunny nonexpansive retractions. Topol. Methods Nonlinear Anal. 1993, 2: 333–342.
Lau AT-M: Amenability and fixed point property for semigroup of nonexpansive mapping. In Fixed Point Theory and Applications. Edited by: Thera MA, Baillon JB. Longman, Harlow; 1991:303–313.
Lau AT-M: Invariant means and fixed point properties of semigroup of nonexpansive mappings. Taiwan. J. Math. 2008, 12: 1525–1542.
Lau AT-M, Miyake H, Takahashi W: Approximation of fixed points for amenable semigroups of nonexpansive mappings in Banach spaces. Nonlinear Anal. 2007, 67: 1211–1225. 10.1016/j.na.2006.07.008
Li S, Su Y, Zhang L, Zhao H, Li L: Viscosity approximation methods with weak contraction for L -Lipschitzian pseudocontractive self-mapping. Nonlinear Anal. 2011, 74: 1031–1039. 10.1016/j.na.2010.07.024
Qin X, Cho SY, Kang SM: Convergence of an iterative algorithm for systems of variational inequalities and nonexpansive mappings with applications. J. Comput. Appl. Math. 2009, 233: 231–240. 10.1016/j.cam.2009.07.018
Reich S: Asymptotic behavior of contractions in Banach spaces. J. Math. Anal. Appl. 1973, 44(1):57–70. 10.1016/0022-247X(73)90024-3
Suzuki T: Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter nonexpansive semigroups without Bochner integrals. J. Math. Anal. Appl. 2005, 305(1):227–239. 10.1016/j.jmaa.2004.11.017
Suzuki T: On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces. Proc. Am. Math. Soc. 2002, 131: 2133–2136.
Stampacchi G: Formes bilineaires coercivites sur les ensembles convexes. C. R. Math. Acad. Sci. Paris 1964, 258: 4413–4416.
Takahashi W: Convex Analysis and Approximation Fixed Points. Yokohama Publishers, Yokohama; 2000. Japanese
Takahashi W: Nonlinear Functional Analysis. Fixed Point Theory and Its Applications. Yokohama Publishers, Yokohama; 2000.
Takahashi W: Viscosity approximation methods for resolvents of accretive operators in Banach spaces. J. Fixed Point Theory Appl. 2007, 1: 135–147. 10.1007/s11784-006-0004-3
Wangkeeree R, Preechasilp P: Modified Noor iterations for nonexpansive semigroups with generalized contraction in Banach spaces. J. Inequal. Appl. 2012., 2012:
Xu HK: Inequalities in Banach spaces with applications. Nonlinear Anal. 1991, 16: 1127–1138. 10.1016/0362-546X(91)90200-K
Xu HK: A strong convergence theorem for contraction semigroups in Banach spaces. Bull. Aust. Math. Soc. 2005, 72: 371–379. 10.1017/S000497270003519X
Yao Y, Noor MA, Noor KI, Liou Y-C, Yaqoob H: Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl. Math. 2010, 110(3):1211–1224. 10.1007/s10440-009-9502-9
Acknowledgements
The authors thank the “Hands-on Research and Development Project”, Rajamangala University of Technology Lanna (RMUTL), Tak, Thailand (under grant No. UR1-005). Moreover, the authors would like to thank the Higher Education Research Promotion and National Research University Project of Thailand, Office of the Higher Education Commission (NRU-CSEC No. 55000613) for financial support. Finally, the authors are grateful to the reviewers for careful reading of the paper and for the suggestions which improved the quality of this work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Onjai-uea, N., Katchang, P. & Kumam, P. Strong convergence theorems for solving a general system of finite variational inequalities for finite accretive operators and fixed points of nonexpansive semigroups with weak contraction mappings. Fixed Point Theory Appl 2012, 114 (2012). https://doi.org/10.1186/1687-1812-2012-114
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/1687-1812-2012-114