## Fixed-Point Theory, Variational Inequalities, and Its Approximation Algorithms

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# On Common Solutions for Fixed-Point Problems of Two Infinite Families of Strictly Pseudocontractive Mappings and the System of Cocoercive Quasivariational Inclusions Problems in Hilbert Spaces

**Academic Editor:**Vittorio Colao

#### Abstract

This paper is concerned with a common element of the set of common fixed points for two infinite families of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. The strong convergence theorem for the above two sets is obtained by a novel general iterative scheme based on the viscosity approximation method, and applicability of the results has shown difference with the results of many others existing in the current literature.

#### 1. Introduction

Throughout this paper, we always assume that is a nonempty closed-convex subset of a real Hilbert space with inner product and norm denoted by and , respectively, and denotes the family of all the nonempty subsets of .

Let be a single-valued nonlinear mapping and a set-valued mapping. We consider the following * quasivariational inclusion problem*, which is to find a point such that
where is the zero vector in . The set of solutions of the problem (1.1) is denoted by . As special cases of the problem (1.1), we have the following. (i)If , where is a proper convex lower semicontinuous function such that is the set of real numbers, and is the subdifferential of , then the quasivariational inclusion problem (1.1) is equivalent to find such that
which is called the * mixed quasivariational inequality problem* (see [1]). (ii)If , where is the indicator function of , that is,
then the quasivariational inclusion (1.1) is equivalent to find such that
which is called * Hartman-Stampacchia variational inequality problem* (see [2â€“4]).

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 mapping of into itself, then is nonempty (see [5]). 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
and it is obvious that is a nonexpansive if and only if is a 0-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
and it is obvious that any -strongly monotonicity implies to the relaxed -cocoercivity.

The existence common fixed points for a finite family of nonexpansive mappings have been considered by many authors (see [6â€“9] and the references therein).

In this paper, we study the mapping defined by
where is nonnegative real sequence in , for all , from a family of infinitely nonexpansive mappings of into itself. It is obvious that is a nonexpansive of 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 is a strongly positive bounded linear operator on if there exists a constant such that

Marino and Xu [10] introduced the following iterative scheme based on the viscosity approximation method introduced by Moudafi [11]: where , is a strongly positive bounded linear operator on , is a contraction on , and is a nonexpansive on . They proved that under some appropriateness conditions imposed on the parameters, if , then the sequence generated by (1.19) 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 ).

Iiduka and Takahashi [12] introduced an 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 (1.4) as in the following theorem.

Theorem IT. *Let be a nonempty 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
**
for all , where and such that satisfying the following conditions: *

(C1)* and ,*(C2)* and ,**
then converges strongly to .*

*Definition 1.1 (see [13]). *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. [13] considered the problem (1.1). To be more precise, they proved the following theorem.

Theorem ZLC. *Let be a real Hilbert space, let be an -inverse-strongly monotone mapping, let be a maximal monotone mapping, and let be a nonexpansive mapping. Suppose that the set , where is the set of solutions of quasivariational inclusion (1.1). Suppose that and is the sequence defined by
**
for all , where and satisfying the following conditions:*

(C1)* and ,*(C2)*,**
then converges strongly to .*

Peng et al. [14] introduced an iterative scheme for all , where , is an -cocoercive mapping on H, is a contraction on , is a nonexpansive on , is a maximal monotone mapping of into , and is a bifunction from into .

We note that their iteration is well defined if we let , and the appropriateness of the control conditions and of their iteration should be and (see Theoremâ€‰â€‰3.1 in [14]). They proved that under some appropriateness imposed on the other parameters, if , then the sequences , , and generated by (1.24) converge strongly to of the variational inequality
where is the set of solutions of * equilibrium problem* defined by

Moreover, Plubtieng and Sriprad [15] introduced an iterative scheme for all , where , is a strongly bounded linear operator on , is an -cocoercive mapping on H, is a contraction on , is a nonexpansive on , is a maximal monotone mapping of into , and is a bifunction from into .

We note that the appropriateness of the control conditions and of their iteration should be and (see Theorem 3.2 in [15]). They proved that under some appropriateness imposed on the other parameters, if , then the sequences , , and generated by (1.27) converge strongly to .

On the other hand, Li and Wu [16] introduced an iterative scheme for finding a common element of the set of fixed points of a -strictly pseudocontractive mapping with a fixed point and the set of solutions of relaxed cocoercive quasivariational inclusions as follows: for all , where , is a strongly positive bounded linear operator on , is a contraction on , is a mapping on defined by for all , such that is a -strictly pseudocontractive mapping on with a fixed point, is relaxed cocoercive and Lipschitz continuous mappings on , and is a maximal monotone mapping of into .

They proved that under the missing condition of , which should be (see Theoremâ€‰â€‰2.1 in [16]) and some appropriateness imposed on the other parameters, if , then the sequence generated by (1.28) converges strongly to .

Very recently, Tianchai and Wangkeeree [17] introduced an implicit iterative scheme for finding a common element of the set of common fixed points of an infinite family of a -strictly pseudocontractive mapping and the set of solutions of the system of generalized relaxed cocoercive quasivariational inclusions as follows: for all , where , is a strongly positive bounded linear operator on , is a contraction on , is a -mapping on generated by and such that for all , is a -strictly pseudocontractive mapping on with a fixed point, is a maximal monotone mapping of into , and , are two mappings of relaxed cocoercive and Lipschitz continuous mappings on for each .

They proved that under some appropriateness imposed on the parameters, if such that the mapping defined by then the sequence generated by (1.29) converges strongly to .

In this paper, we introduce a novel general iterative scheme (1.32) below by the viscosity approximation method to find a common element of the set of common fixed points for two infinite families of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces. Firstly, we introduce a mapping , where is a -mapping generated by and for solving a common fixed point for two infinite families of strictly pseudocontractive mappings by iteration such that the mapping defined by for all , where and are two infinite families of and -strictly pseudocontractive mappings with a fixed point, respectively, and for some . It follows that a linear general iterative scheme of the mappings and is obtained as follows: for all , where , is a maximal monotone mapping, is a cocoercive mapping for each , is a contraction mapping, and are two mappings of the strongly positive linear bounded self-adjoint operator mappings.

As special cases of the iterative scheme (1.32), we have the following. (i)If for all , then (1.32) is reduced to the iterative scheme (ii)If , then (1.32) is reduced to the iterative scheme (iii)If for all , then (1.34) is reduced to the iterative scheme (iv)If for all , then (1.34) is reduced to the iterative scheme (v)If for all , then (1.36) is reduced to the iterative scheme (vi)If and , then (1.37) is reduced to the iterative scheme (vii)If for each and , then (1.32) is reduced to the iterative scheme

Furthermore, if for all , then the mapping in (1.31) is reduced to for all . It follows that the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for an infinite family of strictly pseudocontractive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.

It is well known that the class of strictly pseudocontractive mappings contains the class of nonexpansive mappings; it follows that if the mapping is defined as (1.31) and , then the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for two infinite families of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces, and if the mapping is defined as (1.40) and , then the iterative scheme (1.32) is reduced to find a common element of the set of common fixed points for an infinite family of nonexpansive mappings and the set of solutions of a system of cocoercive quasivariational inclusions problems in Hilbert spaces.

We suggest and analyze the iterative scheme (1.32) above under some appropriateness conditions imposed on the parameters, the strong convergence theorem for the above two sets is obtained, and applicability of the results has shown difference with the results of many others existing in the current literature.

#### 2. Preliminaries

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

Lemma 2.1. *Let be a nonempty closed-convex subset of a Hilbert space then the following inequalities hold:*

(1)*,*(2)*. *

Lemma 2.2 (see [10]). *Let be a Hilbert space, let be a contraction with coefficient , and let be a strongly positive linear bounded operator with coefficient , then*

(1)*if , then
*(2)*if , then . *

Lemma 2.3 (see [18]). *Assume that is a sequence of nonnegative real numbers such that
**
where is a sequence in and is a sequence in such that *

(1)* and ,*(2)* or ,**then .*

Lemma 2.4 (see [9]). *Let be a nonempty closed-convex subset of a Hilbert space , define mapping as (1.16), 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 defined by
is a nonexpansive mapping satisfying , and it is called the -mapping generated by and .*

Lemma 2.5 (see [13]). *The resolvent operator associated with is single-valued and nonexpansive for all .*

Lemma 2.6 (see [13]). * is a solution of quasivariational inclusion (1.1) if and only if , for all , that is,
*

Lemma 2.7 (see [19]). *Let be a nonempty closed-convex subset of a strictly convex Banach space . Let be a sequence of nonexpansive mappings on . Suppose that . Let be a sequence of positive real numbers such that , then a mapping on defined by
**
for , is well defined, nonexpansive, and holds.*

Lemma 2.8 (see [2]). *Let be a nonempty closed-convex subset of a Hilbert space and a nonexpansive mapping, then is demiclosed at zero. That is, whenever is a sequence in weakly converging to some and the sequence strongly converges to some , it follows that .*

Lemma 2.9 (see [20]). *Let be a nonempty closed-convex subset of a real Hilbert space and a -strict pseudocontraction. Define by for each , then, as , S is a nonexpansive such that .*

#### 3. Main Results

Lemma 3.1. *Let be a nonempty closed-convex subset of a real Hilbert space , and let be two mappings of and -strictly pseudocontractive mappings with a fixed point, respectively. Suppose that and define a mapping by
**
where such that , then is well defined, nonexpansive, and .*

*Proof. *Define the mappings as follows:
for all . By Lemma 2.9, we have and as nonexpansive such that and . Therefore, for all , we have
It follows from Lemma 2.7 that is well defined, nonexpansive, and .

Theorem 3.2. *Let be a real Hilbert space, let be a maximal monotone mapping, and let be a -cocoercive mapping for each . Let be two mappings of the strongly positive linear bounded self-adjoint operator mappings with coefficients such that and , respectively, and let be a contraction mapping with coefficient . Let and be two infinite families of and -strictly pseudocontractive mappings with a fixed point such that , respectively. Define a mapping by
**
for all , where such that . Let be a -mapping generated by and such that , for some . Assume that and . For , suppose that is generated iteratively by
**
for all , where , such that , , and for each satisfying the following conditions: *

(C1)*,*(C2)* and ,*(C3)* and ,*(C4)*, , and ,*(C5)* and ,**
then the sequences and converge strongly to where is a unique solution of the variational inequality
*

*Proof. *From , for all , (C1) and (C2), we have , as and . Thus, we may assume without loss of generality that for all . For any and for each , by the -cocoercivity of , we have
which implies that is a nonexpansive. Since and are two mappings of the linear bounded self-adjoint operators, we have
Observe that
Therefore, we obtain that is positive. Thus, by the strong positivity of and , we get

Define the sequences of mappings and as follows:
for all . Firstly, we prove that has a unique fixed point in . Note that for all , by (3.11), (C3), the nonexpansiveness of , and , we have
Therefore, is a nonexpansive. It follows from (3.10), (3.11), (3.12), the contraction of , and the linearity of and that
Hence, is a contraction with coefficient . Therefore, Banach contraction principle guarantees that has a unique fixed point in , and so the iteration (3.5) is well defined.

Next, we prove that is bounded. Pick . Therefore, by Lemma 2.6, we have
for each . By (3.14), the nonexpansiveness of , and , we have

Let . By (3.14), (C3), the nonexpansiveness of , and , we have
Since , where , and are two infinite families of and -strict pseudocontractions with a fixed point, respectively, such that ; therefore, by Lemma 3.1, we have that is a nonexpansive and for all . It follows from Lemma 2.4(1) that we get , which implies that . Hence, by (3.16) and the nonexpansiveness of , we have
By (3.10), (3.17), the contraction of , and the linearity of and , we have
It follows from induction that
for all . Hence, is bounded, and so are , , , , , , and .

Next, we prove that as . By (C3), the nonexpansiveness of , and , we have
By the nonexpansiveness of and , we have
for some constant such that . Therefore, from (3.21), by the nonexpansiveness of , we have
Since
combining (3.20), (3.22), and (3.23), we have
By the linearity of and , we have