Abstract

We introduce an iterative scheme by the viscosity approximation to find the set of solutions of the generalized system of relaxed cocoercive quasivariational inclusions and the set of common fixed points of an infinite family of strictly pseudocontractive mappings problems in Hilbert spaces. We suggest and analyze an iterative scheme under some appropriate conditions imposed on the parameters; we prove that another strong convergence theorem for the above two sets is obtained. The results presented in this paper improve and extend the main results of Li and Wu (2010) and many others.

1. Introduction and Preliminaries

Let be a real Hilbert space with inner product and norm being denoted by and , respectively, and let be a nonempty closed convex subset of . Recall that is the metric projection of onto ; that is, for each there exists the unique point in such that A mapping is called nonexpansive if and the mapping is called a contraction if there exists a constant such that A point is a fixed point of provided . We denote by the set of fixed points of ; that is, . If is bounded, closed and convex and is a nonexpansive mappings of into itself, then is nonempty (see [1]). Recall that a mapping is said to be (i)monotone if (ii)-Lipschitz continuous if there exists a constant such that if , then is a nonexpansive, (iii)pseudocontractive if (iv)-strictly pseudocontractive if there exists a constant such that it is obvious that is a nonexpansive if and only if is a -strictly pseudocontractive, (v)-strongly monotone if there exists a constant such that (vi)-inverse-strongly monotone (or -cocoercive) if there exists a constant such that if , then is called that firmly nonexpansive; it is obvious that any -inverse-strongly monotone mapping is monotone and (1/)-Lipschitz continuous, (vii)relaxed -cocoercive if there exists a constant such that (viii)relaxed -cocoercive if there exists two constants such that it is obvious that any the -strongly monotonicity implies to the relaxed -cocoercivity.

Recall that a set-valued mapping is called monotone if for all and imply . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mappings.

The existence common fixed points for a finite family of nonexpansive mappings has been considered by many authers (see [25] and the references therein).

In this paper, we study the mapping defined by where is nonnegative real sequence in , for all , form a family of infinitely nonexpansive mappings of into itself. It is obvious that is nonexpansive from into itself, such a mapping is called a -mapping generated by and .

A typical problem is to minimize a quadratic function over the set of fixed points of a nonexpansive mapping in a real Hilbert space : where is a bounded linear operator on , is the fixed point set of a nonexpansive mapping on and is a given point in . Recall that be a strongly positive bounded linear operator on if there exists a constant such that

Marino and Xu [6] introduced the following general iterative scheme based on the viscosity approximation method introduced by Moudafi [7]: where is a strongly positive bounded linear operator on , is a contraction on and is a nonexpansive on . They proved that under those conditions are corrected, if , then the sequence generated by (1.15) converges strongly to the unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ).

The so-called the system of generalized quasivariational inclusions problem is to find such that where , are nonlinear mappings and for each . As special cases of problem (1.18), we have the following. (1)If , and , then problem (1.18) is reduced to find such that (2)If , then problem (1.18) is reduced to find such that which called that the system of quasivariational inclusions problem. (3)If and , then problem (1.20) is reduced to find such that (4)If , then problem (1.21) is reduced to find such that We denote by the set of solutions of variational inclusion of the problem (1.22). (5)If , where is a proper convex lower semicontinuous function and is the subdifferential of , then problem (1.22) is equivalent to find such that which is said to be the mixed quasivariational inequality (see, e.g., [8, 9] for more details). (6)If is the indicator function of , then problem (1.23) is equivalent to the classical variational inequality problem, denoted by , to find such that

Iiduka and Takahashi [10] introduced iterative scheme for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality (1.24) as the following theorem.

Theorem IT. Let be a closed convex subset of a real Hilbert space . Let be an -inverse-strongly monotone mapping of into and let be a nonexpansive mapping of into itself such that . Suppose that and is the sequence defined by where and for some satisfying the following conditions: (C1) and ;(C2) and .Then converges strongly to .

Definition 1.1 (see [11]). Let be a multivalued maximal monotone mapping. Then the single-valued mapping defined by , for all , is called the resolvent operator associated with , where is any positive number and is the identity mapping.

Recently, Zhang et al. [11] considered the problem (1.22). To be more precise, they proved the following theorem.

Theorem ZLC. Let be a real Hilbert space, be an -inverse-strongly monotone mapping, be a maximal monotone mapping, and be a nonexpansive mapping. Suppose that the set , where is the set of solutions of variational inclusion (1.22). Suppose that and is the sequence defined by for all , where and satisfying the following conditions: (C1) and ; (C2). Then converges strongly to .

Very recently, Li and Wu [12] introduced an iterative scheme: for all , where , is a strongly positive bounded linear operator on , is a contraction on and is a mapping on defined by such that is a -strictly pseudocontractive mapping on with a fixed point. They proved that under missing condition of , it should be by those Lemma 1.6, others are corrected, if , then the sequence generated by (1.27) converges strongly to of the variational inequality which is the optimality condition for the minimization problem: where is a potential function for .

Inspired and motivated by the works mentioned above, in this paper, we introduce an iterative scheme (2.2) below by the viscosity approximation to find the set of solutions of the generalized system of relaxed cocoercive quasivariational inclusions and the set of common fixed points of an infinite family of strictly pseudocontractive mappings problems in Hilbert spaces. We suggest and analyze an iterative scheme under some appropriate conditions imposed on the parameters, we prove that another strong convergence theorem for the above two sets is obtained. The results presented in this paper improve and extend the main results of Li and Wu [12] and many others.

We collect the following lemmas which be used in the proof for the main results in the next section.

Lemma 1.2 (see [6]). Let be a Hilbert space, be a nonempty closed convex subset of , be a contraction with coefficient , and be a strongly positive linear bounded operator with coefficient . Then, (1)if , then ; (2)if , then .

Lemma 1.3 (see [13]). Let and be bounded sequences in a Banach space E and be a sequence in which satisfies the following condition: Suppose that and . Then .

Lemma 1.4 (see [14]). Assume is a sequence of nonnegative real numbers such that where is a sequence in and is a sequence in such that (1); (2) or . Then .

Lemma 1.5 (see [15]). Let be a nonempty closed convex subset of a Hilbert space , define mapping as (1.12), let be a family of infinitely nonexpansive mappings with , and let be a sequence such that , for all . Then (1) is nonexpansive and for each ; (2)for each and for each positive integer , exists; (3)the mapping define by is a nonexpansive mapping satisfying and it is called the -mapping generated by and

Lemma 1.6 (see [11]). The resolvent operator associated with is single-valued and nonexpansive for all .

Lemma 1.7 (see [11]). is a solution of variational inclusion (1.22) if and only if , for all , that is,

Lemma 1.8. For any , where , we have is a solution of problem (1.18) if and only if is a fixed point of the mapping D defined by

Proof. Observe from (1.18) that

Lemma 1.9 (see [16]). Let be a closed convex subset of a strictly convex Banach space . Let and be two nonexpansive mappings on . Suppose that is nonempty. Then a mapping on C defined by , where , for is well defined and nonexpansive and holds.

Lemma 1.10 (see [17]). Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be a nonexpansive mapping. Then is demiclosed at zero.

Lemma 1.11 (see [18]). Let be a nonempty closed convex subset of a real Hilbert space and be a -strict pseudocontraction. Define by for each . Then, as , S is a nonexpansive such that .

2. Main Results

Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, for each Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of -strictly pseudocontractive mappings with a fixed point such that for all . Define , where , for all , and let be a -mapping generated by and such that for some .

Define sequence of mappings and mapping as follows: for all , where , , , , , for each and .

Under some appropriate imposed on the parameters and , we also know that , and, we also have that and are nonexpansive for each (see argument in the proof of Theorem 2.1 below). Observe that is a nonexpansive, and so is a contraction with coefficient . Therefore, by Banach contraction principle guarantees that has a unique fixed point in .

By the idea above, we obtain an iteration scheme by the viscosity approximation for solving the generalized system of relaxed cocoercive quasivariational inclusions and fixed point problems of an infinite family of strictly pseudocontractive mappings as the following theorem.

Theorem 2.1. Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, for each Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of -strictly pseudocontractive mappings with a fixed point such that for all . Define , where , for all , and let be a -mapping generated by and such that for some . Assume that and where defined by . For , suppose that be generated iteratively by for all , where ,,, , , , , , for each , satisfying the following conditions: ; ; and . Then the sequence converges strongly to where is a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ) and is a solution of problem (1.18).

Proof. From (C1) and (C2), we have and . Thus, we may assume without loss of generality that for all . For each , since where and , we have For any , it follows by the relaxed -cocoercivity and -Lipschitz continuity of and the relaxed -cocoercivity and -Lipschitz continuity of that which implies that is a nonexpansive. Since, is a linear bounded self-adjoint operator, we have Observe that Therefore, we obtain is positive. Thus, by the strong positively of , we get
Pick Then, we have And, let . Therefore, from (2.11), we have By the nonexpansivity of and , we have And, we have Therefore, by (2.13) and (2.14), we have
Let . Since , where and be a family of -strict pseudocontraction. By Lemma 1.11, we have is a nonexpansive and . Therefore, by Lemma 1.5(1), we get , which implies that . It follows by (2.15) and the nonexpansivity of that From (2.16), by the contraction of and the nonexpansivity of , we have It follows from induction that for all . Hence, is bounded, and so are , , and .
Next, we prove that and as . By the nonexpansivity of and , we have Similarly, we have Therefore, from (2.19) and (2.20), we have By the nonexpansivity of and , we have for some constant such that . Therefore, from (2.22), by the nonexpansivity of , we have Since combining (2.21), (2.23), and (2.24), we have
Let . Then we have Since combining (2.25) and (2.27), we have Therefore, by (C1), (C2) and , we get From (2.26) and (2.29), by (C2) and Lemma 1.3, we obtain From (2.26), by (2.30), we obtain Since therefore, From (2.31), by (C1) and (C2), we obtain
For all , by Lemma 1.2(2), the nonexpansivity of and the contraction of , we have Therefore, is a contraction with coefficient , by Banach contraction principle guarantees that has a unique fixed point, say , that is, .
Next, we claim that To show this inequality, we choose a subsequence of such that Since, is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that as .
Next, we prove that . Define sequence of mappings and mapping by Thus, by Lemma 1.5(3) and (C3), we have where . Since, and are nonexpansive and by Lemma 1.5(3), . Therefore, by Lemma 1.9, we get is a nonexpansive and . Since, , where . Thus, by Lemma 1.11, we obtain From (2.34), we have as Thus, from (2.38), we get as It follows from and by Lemma 1.10 that , that is . Therefore, from (2.37), we obtain
Next, we prove that as . Since , we have , and the same as in (2.16), we have It follows by the contraction of and the nonexpansivity of that where , and By (2.34), (2.41), (C1) and (C3), we can found that and . By Lemma 1.4, we obtain converges strongly to . This proof is completed.

Remarks 2.2. Theorem 2.1 improve and extend to the main results of Li and Wu [12] for solving the generalized system of relaxed cocoercive quasivariational inclusions and fixed points problems of an infinite family of strictly pseudocontractive mappings.

3. Applications

Theorem 3.1. Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively. Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of -strictly pseudocontractive mappings with a fixed point such that for all . Define , where , for all , and let be a -mapping generated by and such that for some . Assume that and where defined by . For , suppose that be generated iteratively by for all , where ,,, , , , for each , satisfying the following conditions: (C1);(C2); (C3) and . Then the sequence converges strongly to where is a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ) and is a solution of problem (1.19).

Proof. It is concluded obviously, from Theorem 2.1 by putting and .

Theorem 3.2. Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively. Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of nonexpansive mappings. Define , where , for all , and let be a -mapping generated by and such that for some . Assume that and where defined by . For , suppose that be generated iteratively by for all , where , , , , , , for each , satisfying the following conditions: (C1); (C2); (C3) and . Then the sequence converges strongly to where is a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ) and is a solution of problem (1.19).

Proof. It is concluded obviously, from Theorem 3.1 by putting for all .

Theorem 3.3. Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively, for each Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of -strictly pseudocontractive mappings with a fixed point such that for all . Define , where , for all , and let be a -mapping generated by and such that for some . Assume that and where defined by such that for each . For , suppose that be generated iteratively by for all , where ,,, , , for each , satisfying the following conditions: (C1); (C2); (C3) and . Then the sequence converges strongly to where is a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ) and is a solution of problem

Proof. It is concluded obviously, from Theorem 2.1 by putting for each .

Theorem 3.4. Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively. Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of -strictly pseudocontractive mappings with a fixed point such that for all . Define , where , for all , and let be a -mapping generated by and such that for some . Assume that and where defined by such that for each . For , suppose that be generated iteratively by for all , where ,,, , , for each , satisfying the following conditions: (C1); (C2); (C3) and . Then the sequence converges strongly to where is a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ) and is a solution of problem

Proof. It is concluded obviously, from Theorem 3.3 by putting and .

Theorem 3.5. Let be a real Hilbert space, be a maximal monotone mapping, be a relaxed -cocoercive and -Lipschitz continuous mappings, respectively. Let be a strongly positive linear bounded self-adjoint operator mapping with coefficient such that and be a contraction mapping with coefficient . Let be a family of nonexpansive mappings. Define , where , for all , and let be a -mapping generated by and such that for some . Assume that and where defined by such that for each . For , suppose that be generated iteratively by for all , where ,,, , , for each , satisfying the following conditions: (C1); (C2); (C3) and . Then the sequence converges strongly to where is a unique solution of the variational inequality which is the optimality condition for the minimization problem where is a potential function for (i.e., for ) and is a solution of problem

Proof. It is concluded obviously, from Theorem 3.4 by putting for all .