Research Article | Open Access
Generalized Systems of Variational Inequalities and Projection Methods for Inverse-Strongly Monotone Mappings
We introduce an iterative sequence for finding a common element of the set of fixed points of a nonexpansive mapping and the solutions of the variational inequality problem for three inverse-strongly monotone mappings. Under suitable conditions, some strong convergence theorems for approximating a common element of the above two sets are obtained. Moreover, using the above theorem, we also apply to find solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. As applications, at the end of the paper we utilize our results to study some convergence problem for strictly pseudocontractive mappings. Our results include the previous results as special cases extend and improve the results of Ceng et al., (2008) and many others.
Variational inequalities are known to play a crucial role in mathematics as a unified framework for studying a large variety of problems arising, for instance, in structural analysis, engineering sciences and others. Roughly speaking, they can be recast as fixed-point problems, and most of the numerical methods related to this topic are based on projection methods. Let be a real Hilbert space with inner product and , and let be a nonempty, closed, convex subset of . A mapping is called -inverse-strongly monotone if there exists a positive real number such that (see [1, 2]). It is obvious that every -inverse-strongly monotone mapping is monotone and Lipschitz continuous. A mapping is called nonexpansive if We denote by the set of fixed points of and by the metric projection of onto . Recall that the classical variational inequality, denoted by , is to find an such that The set of solutions of is denoted by . The variational inequality has been widely studied in the literature; see, for example, [3–6] and the references therein.
For finding an element of , Takahashi and Toyoda  introduced the following iterative scheme: for every , where is a sequence in , and is a sequence in . On the other hand, for solving the variational inequality problem in the finite-dimensional Euclidean space , Korpelevich (1976)  introduced the following so-called extragradient method: for every , where . Many authors using extragradient method for approximating common fixed points and variational inequality problems (see also [9, 10]). Recently, Nadezhkina and Takahashi  and Zeng et al.  proposed some iterative schemes for finding elements in by combining (1.4) and (1.5). Further, these iterative schemes are extended in Y. Yao and J. C. Yao  to develop a new iterative scheme for finding elements in .
Consider the following problem of finding such that (see cf. Ceng et al. ): which is called general system of variational inequalities (GSVI), where and are two constants. In particular, if , then problem (1.6) reduces to finding such that which is defined by [15, 16], and is called the new system of variational inequalities. Further, if , then problem (1.7) reduces to the classical variational inequality , that is, find such that , .
We can characteristic problem, if , then it follows that , where is a constant.
In 2008, Ceng et al.  introduced a relaxed extragradient method for finding solutions of problem (1.6). Let the mappings be -inverse-strongly monotone and -inverse-strongly monotone, respectively. Let be a nonexpansive mapping. Suppose and is generated by where , and are three sequences in such that . First, problem (1.6) is proven to be equivalent to a fixed point problem of a nonexpansive mapping.
In this paper, motivated by what is mentioned above, we consider generalized system of variational inequalities as follows.
Let be a nonempty, closed, convex subset of a real Hilbert space . Let be three mappings. We consider the following problem of finding such that which is called a general system of variational inequalities where and are three constants.
In particular, if , then problem (1.9) reduces to finding such that Next, we consider some special classes of the GSVI problem (1.9) reduce to the following GSVI.(i)If , then the GSVI problems (1.9) reduce to GSVI problem (1.6).(ii)If , then the GSVI problems (1.9) reduce to classical variational inequality VI(A,E) problem.
The above system enters a class of more general problems which originated mainly from the Nash equilibrium points and was treated from a theoretical viewpoint in [17, 18]. Observe at the same time that, to construct a mathematical model which is as close as possible to a real complex problem, we often have to use constraints which can be expressed as one several subproblems of a general problem. These constrains can be given, for instance, by variational inequalities, by fixed point problems, or by problems of different types.
This paper deals with a relaxed extragradient approximation method for solving a system of variational inequalities over the fixed-point sets of nonexpansive mapping. Under classical conditions, we prove a strong convergence theorem for this method. Moreover, the proposed algorithm can be applied for instance to solving the classical variational inequality problems.
Let be a nonempty, closed, convex subset of a real Hilbert space . For every point , there exists a unique nearest point in , denoted by , such that is call the metric projection of onto .
Recall that, is characterized by following properties: and for all and .
Lemma 2.1 (see cf. Zhang et al. ). The metric projection has the following properties: (i) is nonexpansive;(ii) is firmly nonexpansive, that is, (iii)for each ,
Lemma 2.2 (see Osilike and Igbokwe ). Let be an inner product space. Then for all and with , one has
Lemma 2.3 (see Suzuki ). Let and be bounded sequences in a Banach space and let be a sequence in with . Suppose for all integers and . Then, .
Lemma 2.4 (see Xu ). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (i), (ii) or . Then, .
Lemma 2.5 (Goebel and Kirk ). Demiclosedness Principle. Assume that is a nonexpansive self-mapping of a nonempty, closed, convex subset of a real Hilbert space . If has a fixed point, then is demiclosed; that is, whenever is a sequence in converging weakly to some (for short, ), and the sequence converges strongly to some (for short, ), it follows that . Here, is the identity operator of .
The following lemma is an immediate consequence of an inner product.
Lemma 2.6. In a real Hilbert space , there holds the inequality
Remark 2.7. We also have that, for all and , So, if , then is a nonexpansive mapping from to .
3. Main Results
In this section, we introduce an iterative precess by the relaxed extragradient approximation method for finding a common element of the set of fixed points of a nonexpansive mapping and the solution set of the variational inequality problem for three inverse-strongly monotone mappings in a real Hilbert space. We prove that the iterative sequence converges strongly to a common element of the above two sets.
In order to prove our main result, the following lemmas are needed.
Lemma 3.1. For given is a solution of problem (1.9) if and only if is a fixed point of the mapping defined by where and .
Lemma 3.2. The mapping defined by Lemma 3.1 is nonexpansive mappings.
Proof. For all , This shows that is a nonexpansive mapping.
Throughout this paper, the set of fixed points of the mapping is denoted by .
Now, we are ready to proof our main results in this paper.
Theorem 3.3. Let be a nonempty, closed, convex subset of a real Hilbert space . Let the mapping be -inverse-strongly monotone, -inverse-strongly monotone, and -inverse-strongly monotone, respectively. Let be a nonexpansive mapping of into itself such that . Let be a contraction of into itself and given arbitrarily and is generated by where , and are three sequences in such that (i),(ii) and ,(iii). Then, converges strongly to , where and is a solution of problem (1.9), where
Proof. Let . Then, and , that is,
Put and . Then, implies that , where . Since , and are nonexpansive mappings. We obtain that
Substituting (3.13) into (3.12), we have
and by (3.11) we also have
Since and by Lemma 2.2, we compute
By induction, we get
where . Therefore, is bounded. Consequently, by (3.11), (3.12) and (3.13), the sequences ,, and are also bounded. Also, we observe that
Let . Thus, we get
It follows that
Combining (3.19) and (3.21), we obtain
This together with (i), (ii), and (iii) implies that
Hence, by Lemma 2.3, we have
From (3.18) and (3.19), we also have and as . Since
it follows by (ii) and (3.25) that
Since , from (3.15) and Lemma 2.2, we get
Therefore, we have
From (ii), (iii), and , as , we get as .
Since , from (3.11) and Lemma 2.2, we get
Thus, we also have By again (ii), (iii), and (3.25), we also get as .
Let ; again from (3.12), (3.13) and Lemma 2.2, we get
Again, we have Similarly again by (ii), (iii), and as , and from (3.33), we also that .
On the other hand, we compute that
So, we obtain Hence, by (3.12), it follows that which implies that By (ii), (iii), , and as , from (3.37) and we get as . Now, observe that