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
© Onjai-uea et al.; licensee Springer 2012
Received: 26 April 2012
Accepted: 27 June 2012
Published: 20 July 2012
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.
Keywordsinverse-strongly accretive operator fixed point general system of finite variational inequalities sunny nonexpansive retraction weak contraction nonexpansive semigroups
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]).
If , then f is called to be contractive with the contractive coefficient k. If , then f is said to be nonexpansive.
for all ;
for all ;
for all and ;
for all , is continuous.
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.  proved the strong convergence theorem in the framework of Banach spaces which is generalized by Iiduka et al.  from Hilbert spaces.
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).
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. , Katchang and Kumam  and Yao et al. , 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. , Katchang and Kumam , Wangkeeree and Preechasilp , Yao et al.  and many other authors.
We always assume that E is a real Banach space and C is a nonempty closed convex subset of E.
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.
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 ()
Q is sunny and nonexpansive;
, , .
Proposition 2.2 ()
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.
By , 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 ()
Lemma 2.4 ()
Letandbe bounded sequences in a Banach space X and letbe a sequence inwith. Supposefor all integersand. Then, .
Lemma 2.5 (Lemma 2.2 in )
whereis a continuous and strict increasing function for allwith. Then.
Lemma 2.6 ()
Lemma 2.7 ()
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.
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.
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
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.
where the sequences, andare inand satisfy, , , and, are positive real numbers. The following conditions are satisfied:
(C4) , bounded subset of C.
Thenconverges strongly toandis a solution of the problem (1.7) whereis the sunny nonexpansive retraction of C onto.
This implies that is bounded, so are , , and .
and hence it follows that .
- (a)First, we show that . Put , , , and for , let be such that
- (b)Next, we show that . From Lemma 3.1, we know that is nonexpansive; it follows that
By Lemma 2.7 and (3.7), we have . Therefore, .
Now, from (C1) and applying Lemma 2.5 to (3.11), we get as . This completes the proof. □
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. □
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
Moreover, we know that A is -inverse strongly monotone and (see also ).
, bounded subset of C.
converges strongly to, whereis the sunny nonexpansive retraction of E onto.
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:
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.
, bounded subset of C.
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.
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.
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