Abstract
In the literature, various iterative methods have been proposed for finding a common solution of the classical variational inequality problem and a fixed point problem. Research along these lines is performed either by relaxing the assumptions on the mappings in the settings (for instance, commonly seen assumptions for the mapping involved in the fixed point problem are nonexpansive or strictly pseudocontractive) or by adding a general system of variational inequalities into the settings. In this paper, we consider both possible ways in our settings. Specifically, we propose an iterative method for finding a common solution of the classical variational inequality problem, a general system of variational inequalities and a fixed point problem of a uniformly continuous asymptotically strictly pseudocontractive mapping in the intermediate sense. Our iterative method is hybridized by utilizing the well-known extragradient method, the CQ method, the Mann-type iterative method and the viscosity approximation method. The iterates yielded by our method converge strongly to a common solution of these three problems. In addition, we propose a hybridized extragradient-like method to yield iterates converging weakly to a common solution of these three problems.
MSC:49J30, 47H09, 47J20.
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1 Introduction
Let H be a real Hilbert space with the inner product and the norm , let C be a nonempty closed convex subset of H, and let be the metric projection of H onto C. Let be a self-mapping on C. We denote by the set of fixed points of S and by R the set of all real numbers. A mapping is called L-Lipschitz continuous if there exists a constant such that
In particular, if , then A is called a nonexpansive mapping [1]; if , then A is called a contraction. Also, a mapping is called monotone if for all . A is called η-strongly monotone if there exists a constant such that
A is called α-inverse-strongly monotone if there exists a constant such that
It is obvious that if A is α-inverse-strongly monotone, then A is monotone and -Lipschitz continuous.
For a given nonlinear operator , we consider the variational inequality problem (VIP) of finding such that
The solution set of VIP (1.1) is denoted by . VIP (1.1) was first discussed by Lions [2] and now has many applications in computational mathematics, mathematical physics, operations research, mathematical economics, optimization theory, and other fields; see, e.g., [3–6]. It is well known that if A is a strongly monotone and Lipschitz-continuous mapping on C, then VIP (1.1) has a unique solution.
In the literature, there is a growing interest in studying how to find a common solution of . Under various assumptions imposed on A and S, iterative algorithms were derived to yield iterates which converge strongly or weakly to a common solution of these two problems.
1.1 Finding a common element and weak convergence
Consider that a set is nonempty, closed and convex, a mapping is nonexpansive and a mapping is α-inverse-strongly monotone. Takahashi and Toyoda [7] introduced the Mann-type iterative scheme:
where is a sequence in and is a sequence in . They proved that if , then the sequence generated by (1.2) converges weakly to some .
Motivated by Korpelevich’s extragradient method [8], Nadezhkina and Takahashi [9] proposed an extragradient iterative method and showed the iterates converge weakly to a common element of :
where is a monotone, L-Lipschitz continuous mapping and is a nonexpansive mapping and for some and for some . See also Zeng and Yao [10], in which a hybridized iterative method was proposed to yield a new weak convergence result.
1.2 Finding a common element and strong convergence
Let be a nonempty closed convex subset, let be a nonexpansive mapping, and let be an α-inverse strongly monotone mapping. Iiduka and Takahashi [11] introduced the following hybrid method:
where and . They showed that if , then the sequence , generated by this iterative process, converges strongly to . Recently, the method proposed by Nadezhkina and Takahashi [12] also demonstrated the strong convergence result. However, note that they assumed that A is monotone and L-Lipschitz-continuous while S is nonexpansive. For another strong convergence result, see Ceng and Yao [13] whose method is based on the extragradient method and the viscosity approximation method.
As we have seen, most of the papers were based on the different assumptions imposed on A while the mapping S is nonexpansive. In the following, we shall relax the nonexpansive requirement on S (for instance, κ-strictly pseudocontractive, asymptotically κ-strictly pseudocontractive mapping in the intermediate sense, etc.). Furthermore, we also consider adding a general system of variational inequalities to our settings.
1.3 Relaxation on nonexpansive S
Definition 1.1 Let C be a nonempty subset of a normed space X, and let be a self-mapping on C.
-
(i)
S is asymptotically nonexpansive (cf. [14]) if there exists a sequence of positive numbers satisfying the property and
-
(ii)
S is asymptotically nonexpansive in the intermediate sense [15] provided S is uniformly continuous and
-
(iii)
S is uniformly Lipschitzian if there exists a constant such that
It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [14] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [14] as follows.
Theorem GK (see [[14], Theorem 1])
If C is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping has a fixed point in C.
Let C be a nonempty closed convex bounded subset of a Hilbert space H. An iterative method for the approximation of fixed points of an asymptotically nonexpansive mapping with sequence was developed by Schu [16] via the following Mann-type iterative scheme:
where () for some . He proved the weak convergence of to a fixed point of S if . Moreover, iterative methods for approximation of fixed points of asymptotically nonexpansive mappings have been further studied by other authors (see, e.g., [16–18] and references therein).
The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [15] and iterative methods for the approximation of fixed points of such types of non-Lipschitzian mappings were studied by Bruck et al. [15], Agarwal et al. [19], Chidume et al. [20], Kim and Kim [21] and many others.
Recently, Kim and Xu [22] introduced the concept of asymptotically κ-strictly pseudocontractive mappings in a Hilbert space as follows.
Definition 1.2 Let C be a nonempty subset of a Hilbert space H. A mapping is said to be an asymptotically κ-strictly pseudocontractive mapping with sequence if there exists a constant and a sequence in with such that
They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptotically κ-strictly pseudocontractive mapping with sequence is a uniformly ℒ-Lipschitzian mapping with .
Very recently, Sahu et al. [23] considered the concept of asymptotically κ-strictly pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.
Definition 1.3 Let C be a nonempty subset of a Hilbert space H. A mapping is said to be an asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence if there exists a constant and a sequence in with such that
Put . Then (), () and (1.5) reduces to the relation
Whenever for all in (1.6), then S is an asymptotically κ-strictly pseudocontractive mapping with sequence .
For S to be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded, Sahu et al. [23] proposed an iterative Mann-type CQ method in which the iterates converge strongly to a fixed point of S.
Theorem SXY (see [[23], Theorem 4.1])
Let C be a nonempty closed convex subset of a real Hilbert space H, and let be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let be a sequence in such that for all . Let be a sequence in C generated by the following (CQ) algorithm:
where and . Then converges strongly to .
1.4 Common solution of three problems
Let be two mappings. Recently, Ceng et al. [24] introduced and considered the problem of finding such that
which is called a general system of variational inequalities (GSVI), where and are two constants. The set of solutions of GSVI (1.8) is denoted by . In particular, if , then GSVI (1.8) reduces to the new system of variational inequalities (NSVI), introduced and studied by Verma [25]. Further, if additionally, then the NSVI reduces to VIP (1.1). Moreover, they transformed GSVI (1.8) into a fixed point problem in the following way.
Lemma CWY (see [24])
For given , is a solution of GSVI (1.8) if and only if is a fixed point of the mapping defined by
where .
In particular, if the mapping is -inverse strongly monotone for , then the mapping G is nonexpansive provided for .
Utilizing Lemma CWY, they introduced and studied a relaxed extragradient method for solving GSVI (1.8). Throughout this paper, the set of fixed points of the mapping G is denoted by Ξ. Based on the relaxed extragradient method and the viscosity approximation method, Yao et al. [26] proposed and analyzed an iterative algorithm for finding a common solution of GSVI (1.8), and the fixed point problem of a κ-strictly pseudocontractive mapping (namely, there exists a constant such that for all ).
The main theme of this paper is to study the problem of finding a common element of the solution set of VIP (1.1), the solution set of GSVI (1.8) and the fixed point set of a self-mapping . Ceng et al. [27] analyzed this problem by assuming the mapping S to be strictly pseudocontractive as follows.
Theorem CGY (see [[27], Theorem 3.1])
Let C be a nonempty closed convex subset of a real Hilbert space H. Let be α-inverse strongly monotone and be -inverse strongly monotone for . Let be a κ-strictly pseudocontractive mapping such that . Let be a ρ-contraction with . For given arbitrarily, let the sequences , and be generated iteratively by
where for , and such that
-
(i)
and for all ;
-
(ii)
and ;
-
(iii)
and ;
-
(iv)
;
-
(v)
and .
Then the sequence generated by (1.9) converges strongly to and is a solution of GSVI (1.8), where .
In this paper, we study the problem of finding a common element of the solution set of VIP (1.1), the solution set of GSVI (1.8) and the fixed point set of a self-mapping , where the mapping S is assumed to be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Not surprisingly, our main points of proof come from the ideas in [[23], Theorem 4.1] and [[27], Theorem 3.1]. Our major contribution ensures a strong convergence result to the extent of involving uniformly continuous asymptotically κ-strictly pseudocontractive mappings in the intermediate sense. Moreover, in Section 4 we extend Ceng, Hadjisavvas and Wong’s hybrid extragradient-like approximation method given in [[28], Theorem 5] to establish a new weak convergence theorem for finding a common solution of VIP (1.1), GSVI (1.8) and the fixed point problem of S.
2 Preliminaries
Let H be a real Hilbert space whose inner product and norm are denoted by and , respectively. Let C be a nonempty closed convex subset of H. We write to indicate that the sequence converges weakly to x and to indicate that the sequence converges strongly to x. Moreover, we use to denote the weak ω-limit set of the sequence , i.e.,
The metric (or nearest point) projection from H onto C is the mapping which assigns to each point the unique point satisfying the property
Some important properties of projections are gathered in the following proposition.
Proposition 2.1 For given and :
-
(i)
, ;
-
(ii)
, ;
-
(iii)
, .
Consequently, is nonexpansive and monotone.
We need some facts and tools which are listed as lemmas below.
Lemma 2.1 (see [[29], demiclosedness principle])
Let C be a nonempty closed and convex subset of a Hilbert space H, and let be a nonexpansive mapping. Then the mapping is demiclosed on C. That is, whenever is a sequence in C such that and , it follows that . Here I is the identity operator of H.
Lemma 2.2 ([[19], Proposition 2.4])
Let be a bounded sequence on a reflexive Banach space X. If , then .
Lemma 2.3 Let be a monotone mapping. In the context of the variational inequality problem, the characterization of the projection (see Proposition 2.1(i)) implies
Lemma 2.4 Let H be a real Hilbert space. Then the following hold:
-
(a)
for all ;
-
(b)
for all and for all ;
-
(c)
If is a sequence in H such that , it follows that
Lemma 2.5 ([[23], Lemma 2.5])
Let H be a real Hilbert space. Given a nonempty closed convex subset of H and points , and given also a real number , the set
is convex (and closed).
Lemma 2.6 ([[23], Lemma 2.6])
Let C be a nonempty subset of a Hilbert space H, and let be an asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence . Then
for all and .
Lemma 2.7 ([[23], Lemma 2.7])
Let C be a nonempty subset of a Hilbert space H, and let be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence . Let be a sequence in C such that and as . Then as .
Lemma 2.8 (Demiclosedness principle [[23], Proposition 3.1])
Let C be a nonempty closed convex subset of a Hilbert space H, and let be a continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence . Then is demiclosed at zero in the sense that if is a sequence in C such that and , then .
Lemma 2.9 ([[23], Proposition 3.2])
Let C be a nonempty closed convex subset of a Hilbert space H, and let be a continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that . Then is closed and convex.
Remark 2.1 Lemmas 2.8 and 2.9 give some basic properties of an asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence . Moreover, Lemma 2.8 extends the demiclosedness principles studied for certain classes of nonlinear mappings in Kim and Xu [22], Gornicki [30], Marino and Xu [31] and Xu [32].
To prove a weak convergence theorem by the hybrid extragradient-like method for finding a common solution of VIP (1.1), GSVI (1.8) and the fixed point problem of an asymptotically κ-strictly pseudocontractive mapping in the intermediate sense, we need the following lemma by Osilike et al. [33].
Lemma 2.10 ([[33], p.80])
Let , and be sequences of nonnegative real numbers satisfying the inequality
If and , then exists. If, in addition, has a subsequence which converges to zero, then .
Corollary 2.1 ([[34], p.303])
Let and be two sequences of nonnegative real numbers satisfying the inequality
If converges, then exists.
We need a technique lemma in the sequel, whose proof is an immediate consequence of Opial’s property [35] of a Hilbert space and is hence omitted.
Lemma 2.11 Let K be a nonempty closed and convex subset of a real Hilbert space H. Let be a sequence in H satisfying the properties:
-
(i)
exists for each ;
-
(ii)
.
Then is weakly convergent to a point in K.
A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if its graph is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for , for all implies . Let be a monotone and Lipschitzian mapping, and let be the normal cone to C at , i.e., . Define
It is known that in this case T is maximal monotone, and if and only if ; see [36].
3 Strong convergence theorem
In this section, we prove a strong convergence theorem for a hybrid viscosity CQ iterative algorithm for finding a common solution of VIP (1.1), GSVI (1.8) and the fixed point problem of a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense. This iterative algorithm is based on the extragradient method, the CQ method, the Mann-type iterative method and the viscosity approximation method.
Theorem 3.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be α-inverse strongly monotone, and let be -inverse strongly monotone for . Let be a ρ-contraction with , and let be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let and be defined as in (1.6). Let , and be the sequences generated by
where for , ,
is a sequence in and , , are three sequences in such that for all . Assume that the following conditions hold:
-
(i)
;
-
(ii)
for some ;
-
(iii)
for all ;
-
(iv)
for all .
Then the sequences , and converge strongly to .
Proof It is obvious that is closed and is closed and convex for every . As the defining inequality in is equivalent to the inequality
by Lemma 2.5 we also have that is convex for every . As , we have for all and, by Proposition 2.1(i), we get .
Next, we divide the rest of the proof into several steps.
Step 1. for all .
Indeed, take arbitrarily. Then , and
Since is α-inverse strongly monotone and , we have, for all ,
For simplicity, we write , and
for all . Since is -inverse strongly monotone and for , we know that for all ,
Hence we get
Therefore, from (3.5), , and , we have
for every , and hence . So, for every . Now, let us show by mathematical induction that is well defined and for every . For , we have . Hence we obtain . Suppose that is given and for some integer . Since is nonempty, is a nonempty closed convex subset of C. So, there exists a unique element such that . It is also obvious that there holds for every . Since , we have for every , and hence . Therefore, we obtain .
Step 2. is bounded and .
Indeed, let . From and , we have
for every . Therefore, is bounded. From (3.3)-(3.6) we also obtain that , , , and are bounded. Since and , we have
for every . Therefore, there exists . Since and , using Proposition 2.1(ii), we have
for every . This implies that
Since , we have
which implies that
Hence we get
for every . From and , we have .
Step 3. .
Indeed, from (3.1), (3.4) and (3.6) it follows that
which hence implies that
Since , , and , from the boundedness of and we obtain that
Step 4. .
Indeed, utilizing Proposition 2.1(iii), we deduce from (3.1) that
Thus,
Similarly to the above argument, utilizing Proposition 2.1(iii), we conclude from that
that is,
Substituting (3.10) in (3.11), we have
Similarly to the above argument, utilizing Proposition 2.1(iii), we conclude from that
that is,
Substituting (3.12) in (3.13), we have
This together with (3.4) and (3.8) implies that
So, we have
Since , , , , , and , from the boundedness of , and we obtain that
and hence
Step 5. .
Indeed, it follows from (3.1) that
Since and , from the boundedness of and we know that as . Also, from we also have . Since , we have . Then
and hence . Furthermore, observe that
Utilizing Lemma 2.6, we have
for every . Hence it follows from that . Thus from (3.16) and we get . Since , as and S is uniformly continuous, we obtain from Lemma 2.7 that as .
Step 6. .
Indeed, by the boundedness of , we know that . Take arbitrarily. Then there exists a subsequence of such that converges weakly to . We can assert that . First, note that S is uniformly continuous and . Hence it is easy to see that for all . By Lemma 2.8, we obtain . Now let us show that . We note that
Since and , it follows that . Thus, according to Lemma 2.1 we get . Furthermore, we show . Since and , we have and . Let
where is the normal cone to C at . We have already mentioned that in this case the mapping T is maximal monotone, and if and only if ; see [36] for more details. Let be the graph of T, and let . Then we have , and hence . So, we have for all . On the other hand, from and we have
and hence
Therefore, from for all and , we have
Thus, we obtain as . Since T is maximal monotone, we have and hence . Consequently, . This implies that .
Step 7. .
Indeed, from , and (3.7), we have
So, we obtain
From we have due to the Kadec-Klee property of a real Hilbert space [29]. So, it is clear that . Since and , we have
As , we obtain by and . Hence we have . This implies that . It is easy to see that and . This completes the proof. □
4 Weak convergence theorem
In this section, we prove a new weak convergence theorem by the hybrid extragradient-like method for finding a common element of the solution set of VIP (1.1), the solution set of GSVI (1.8) and the fixed point set of a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense.
Theorem 4.1 Let C be a nonempty closed convex subset of a real Hilbert space H. Let be α-inverse strongly monotone, let be -inverse strongly monotone for , let be a ρ-contraction with , and let be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that is nonempty and bounded. Let and be defined as in (1.6). Let and be the sequences generated by
where for , is a sequence in and , , are three sequences in such that for all . Assume that the following conditions hold:
-
(i)
;
-
(ii)
for some ;
-
(iii)
and ;
-
(iv)
and for all , , and for some .
Then the sequences and converge weakly to an element of .
Proof First of all, take arbitrarily. Then, repeating the same arguments as in (3.3) and (3.5), we deduce from (4.1) that
and
Repeating the same arguments as in (3.6), we can obtain that
Since , and it follows that
So, by Lemma 2.10 we know that
This implies that is bounded and hence , are bounded due to (4.2) and (4.3).
Repeating the same arguments as in (3.8), we can conclude that
which hence implies that
Since , , and exists, from the boundedness of we conclude that
Repeating the same arguments as in (3.14), we can conclude that
which hence implies that
Since , , , , , and exists, from the boundedness of and we obtain that
and hence
On the other hand, it follows from (4.1) that
Since and , from the boundedness of and we know that as . Also, from we also have . Repeating the same arguments as in (3.6), we have
which hence implies that
Since exists, , , and the sequence is bounded, we obtain that
Also, since , we have . Then
and hence . Furthermore, observe that
Utilizing Lemma 2.6, we have
for every . Hence it follows from that . Thus from (4.8) and we get . Since , as and S is uniformly continuous, we obtain from Lemma 2.7 that as .
Further, repeating the same arguments as in the proof of Theorem 3.1, we can derive that . Utilizing Lemma 2.11, from the existence of for each , we infer that converges weakly to an element . Since as , it is clear that converges weakly to . □
In the following, we present a numerical example to illustrate how Theorem 4.1 works.
Example 4.1 Let with the inner product and the norm which are defined by
for all with and . Let . Clearly, C is a nonempty closed convex subset of a real Hilbert space . Let be α-inverse strongly monotone, let be -inverse strongly monotone for , let be a ρ-contraction with , and let be a uniformly continuous asymptotically κ-strictly pseudocontractive mapping in the intermediate sense with sequence such that is nonempty bounded; for instance, putting , , , and . It is easy to see that , , and that A is α-inverse strongly monotone with , that and are -inverse strongly monotone, f is a -contraction, S is a nonexpansive mapping, i.e., a uniformly continuous asymptotically 0-strictly pseudocontractive mapping in the intermediate sense with sequences () and (). Moreover, it is clear that , and . Hence, . In this case, from iterative scheme (4.1) in Theorem 4.1, we obtain that for any given ,
Whenever , , and , we have
This shows that converges to the unique element 0 of . Note that as ,
Hence, also converges to the unique element 0 of .
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Acknowledgements
Lu-Chuan Ceng was partially supported by the National Science Foundation of China (11071169), Innovation Program of Shanghai Municipal Education Commission (09ZZ133) and Ph.D. Program Foundation of Ministry of Education of China (20123127110002), Sy-Ming Guu was partially supported by NSC 100-2221-E-182-072-MY2, and Jen-Chih Yao was partially supported by the grant NSC 99-2115-M-037-002-MY3.
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The authors declare that they have no competing interests.
Authors’ contributions
LC conceived of the study and drafted the manuscript initially. SM participated in its design, coordination and finalized the manuscript. JC outlined the scope and design of the study. All authors read and approved the final manuscript.
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Ceng, LC., Guu, SM. & Yao, JC. Hybrid viscosity CQ method for finding a common solution of a variational inequality, a general system of variational inequalities, and a fixed point problem. Fixed Point Theory Appl 2013, 313 (2013). https://doi.org/10.1186/1687-1812-2013-313
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DOI: https://doi.org/10.1186/1687-1812-2013-313