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# Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions

*Fixed Point Theory and Applications*
**volume 2011**, Article number: 23 (2011)

## Abstract

In this article, we introduce a new mapping generated by an infinite family of *κ*_{
i
}*-* strict pseudo-contractions and a sequence of positive real numbers. By using this mapping, we consider an iterative method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Strong convergence theorem of the purposed iteration is established in the framework of Hilbert spaces.

## 1 Introduction

Let *C* be a closed convex subset of a real Hilbert space *H*, and let *G* : *C* × *C* → ℝ be a bifunction. We know that the equilibrium problem for a bifunction *G* is to find *x* ∈ *C* such that

The set of solutions of (1.1) is denoted by *EP*(*G*). Given a mapping *T* : *C* → *H*, let *G*(*x*, *y*) = 〈*Tx*, *y* - *x*〉 for all *x*, *y* ∈. Then, *z* ∈ *EP*(*G*) if and only if 〈*Tz*, *y* - *z*〉 ≥ 0 for all *y* ∈ *C*, i.e., *z* is a solution of the variational inequality. Let *A* : *C* → *H* be a nonlinear mapping. The variational inequality problem is to find a *u* ∈ *C* such that

for all *v* ∈ *C*. The set of solutions of the variational inequality is denoted by *V I*(*C*, *A*). Now, we consider the following generalized equilibrium problem:

The set of such *z* ∈ *C* is denoted by *EP*(*G*, *A*), i.e.,

In the case of *A* ≡ 0, *EP*(*G*, *A*) is denoted by *EP*(*G*). In the case of *G* ≡ 0, *EP*(*G*, *A*) is also denoted by *V I*(*C*, *A*). Numerous problems in physics, optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, and economics reduce to find a solution of (1.3) (see, for instance, [1]-[3]).

A mapping *A* of *C* into *H* is called *inverse-strongly monotone* (see [4]), if there exists a positive real number *α* such that

for all *x*, *y* ∈ *C*.

A mapping *T* with domain *D*(*T*) and range *R*(*T*) is called *nonexpansive* if

for all *x*, *y* ∈ *D*(*T*) and *T* is said to be *κ*-*strict pseudo-contration* if there exist *κ* ∈ [0, 1) such that

We know that the class of *κ*-strict pseudo-contractions includes class of nonexpansive mappings. If *κ* = 1, then *T* is said to be *pseudo-contractive*. *T* is *strong pseudo-contraction* if there exists a positive constant *λ* ∈ (0, 1) such that *T* + *λI* is pseudo-contractive. In a real Hilbert space *H* (1.5) is equivalent to

*T* is pseudo-contractive if and only if

Then, *T* is strongly pseudo-contractive, if there exists a positive constant *λ* ∈ (0, 1) such that

The class of *κ*-strict pseudo-contractions fall into the one between classes of nonexpansive mappings and pseudo-contractions, and the class of strong pseudo-contractions is independent of the class of *κ*-strict pseudo-contractions.

We denote by *F*(*T*) the set of fixed points of *T*. If *C* ⊂ *H* is bounded, closed and convex and *T* is a nonexpansive mapping of C into itself, then *F*(*T*) is nonempty; for instance, see [5]. Recently, Tada and Takahashi [6] and Takahashi and Takahashi [7] considered iterative methods for finding an element of *EP*(*G*) ∩ *F*(*T*). Browder and Petryshyn [8] showed that if a *κ*-strict pseudo-contraction T has a fixed point in C, then starting with an initial *x*_{0} ∈ *C*, the sequence {*x*_{
n
}} generated by the recursive formula:

where *α* is a constant such that 0 *< α <* 1, converges weakly to a fixed point of *T*. Marino and Xu [9] extended Browder and Petryshyn's above mentioned result by proving that the sequence {*x*_{
n
}} generated by the following Manns algorithm [10]:

converges weakly to a fixed point of *T* provided the control sequence satisfies the conditions that *κ < α*_{
n
} *<* 1 for all *n* and .

Recently, in 2009, Qin et al. [11] introduced a general iterative method for finding a common element of *EP*(*F*, *T*), *F*(*S*), and *F*(*D*). They defined {*x*_{
n
}} as follows:

where the mapping *D* : *C* → *C* is defined by *D*(*x*) = *P*_{
C
}(*P*_{
C
}(*x* - *ηBx*) - *λAP*_{
C
}(*x* - *ηBx*)), *S*_{
k
} is the mapping defined by *S*_{
k
}*x* = *kx* + (1 - *k*)*Sx*, ∀*x* ∈ *C*, *S* : *C* → *C* is a *κ*-strict pseudo-contraction, and *A*, *B* : *C* ∈ *H* are a-inverse-strongly monotone mapping and b-inverse-strongly monotone mappings, respectively. Under suitable conditions, they proved strong convergence of {*x*_{
n
}} defined by (1.9) to *z* = *P*_{EP(F, T)∩F(S) ∩F(D)}*u*.

Let *C* be a nonempty convex subset of a real Hilbert space. Let *T*_{
i
}, *i* = 1, 2, ... be mappings of *C* into itself. For each *j* = 1, 2, ..., let where *I* = [0, 1] and . For every *n* ∈ ℕ, we define the mapping *S*_{
n
} : *C* → *C* as follows:

This mapping is called *S-mapping* generated by *T*_{
n
}, ..., *T*_{1} and *α*_{
n
}, *α*_{n-1}, ..., *α*_{1}.

**Question**. How can we define an iterative method for finding an element in ?

In this article, motivated by Qin et al. [11], by using *S*-mapping, we introduce a new iteration method for finding a common element of the set of a generalized equilibrium problem of the set of solution to a system of variational inequalities, and of the set of fixed points of an infinite family of strict pseudo-contractions. Our iteration scheme is define as follows.

For *u*, *x*_{1} ∈ *C*, let {*x*_{
n
}} be generated by

For *i* = 1, 2, ..., *N*, let *F*_{
i
} : *C* × *C* → ℝ be bifunction, *A*_{
i
} : *C* → *H* be *α*_{
i
}-inverse strongly monotone and let *G*_{
i
} : *C* → *C* be defined by *G*_{
i
}(*y*) = *P*_{
C
}(*I* - *λ*_{
i
}*A*_{
i
})*y*, ∀*y* ∈ *C* with (0, 1] ⊂ (0, 2 *α*_{
i
} ) such that , where *B* is the *K*-mapping generated by *G*_{1}, *G*_{2}, ..., *G*_{
N
} and *β*_{1}, *β*_{2}, ..., *β*_{
N
} .

We prove a strong convergence theorem of purposed iterative sequence {*x*_{
n
}} to a point and *z* is a solution of (1.10)

## 2 Preliminaries

In this section, we collect and provide some useful lemmas that will be used for our main result in the next section.

Let *C* be a closed convex subset of a real Hilbert space *H*, and let *P*_{
C
} be the metric projection of *H* onto *C* i.e., so that for *x* ∈ *H*, *P*_{
C
}*x* satisfies the property:

The following characterizes the projection *P*_{
C
}.

**Lemma 2.1** [5]. *Given x* ∈ *H and y* ∈ *C. Then, P*_{
C
}*x* = *y if and only if there holds the inequality*

**Lemma 2.2** [12]. *Let* {*s*_{
n
}} *be a sequence of nonnegative real number satisfying*

*where* {*α*_{
n
}}, {*β*_{
n
}} *satisfy the conditions*

*Then* lim_{n→∞}*s*_{
n
} = 0.

**Lemma 2.3** [13]. *Let C be a closed convex subset of a strictly convex Banach space E. Let* {*T*_{
n
} : *n* ∈ ℕ} *be a sequence of nonexpansive mappings on C. Suppose* *is nonempty. Let* {*λ*_{
n
}} *be a sequence of positive numbers with* . *Then, a mapping S on C defined by*

*for x* ∈ *C is well defined, nonexpansive and* *hold*.

**Lemma 2.4** [14]. *Let E be a uniformly convex Banach space, C be a nonempty closed convex subset of E and S* : *C* → *C be a nonexpansive mapping. Then, I* - *S is demi-closed at zero*.

**Lemma 2.5** [15]. *Let* {*x*_{
n
}} *and* {*z*_{
n
}} *be bounded sequences in a Banach space X and let* {*β*_{
n
}} *be a sequence in 0*[1] *with* 0 *<* lim inf_{n→∞}*β*_{
n
} ≤ lim sup_{n→∞}*β*_{
n
} *<* 1.

*Suppose*

*for all integer n* ≥ 0 *and*

*Then* lim_{n→∞}||*x*_{
n
} - *z*_{
n
}|| = 0.

For solving the equilibrium problem for a bifunction *F* : *C* × *C* → ℝ, let us assume that *F* satisfies the following conditions:

(*A* 1) *F*(*x*, *x*) = 0 ∀*x* ∈ *C*;

(*A* 2) *F* is monotone, i.e. *F*(*x*, *y*) + *F*(*y*, *x*) ≤ 0, ∀*x*, *y* ∈ *C*;

(*A* 3) ∀*x*, *y*, *z* ∈ *C*,

(*A* 4) ∀*x* ∈ *C*, *y* ↦ *F*(*x*, *y*) is convex and lower semicontinuous.

The following lemma appears implicitly in [1].

**Lemma 2.6** [1]. *Let C be a nonempty closed convex subset of H, and let F be a bifunction of C* × *C into* ℝ *satisfying* (*A* 1) - (*A* 4). *Let r >* 0 *and x* ∈ *H. Then, there exists z* ∈ *C such that*

*for all x* ∈ *C*.

**Lemma 2.7** [16]. *Assume that F* : *C* × *C* → ℝ *satisfies* (*A* 1) - (*A* 4). *For r >* 0 *and x* ∈ *H, define a mapping T*_{
r
} : *H* → *C as follows*.

*for all z* ∈ *H. Then, the following hold*.

*(1) T*_{
r
} *is single-valued*,

*(2) T*
_{
r
}
*is firmly nonexpansive i.e*

*(3) F*(*T*_{
r
}) = *EP* (*F* );

*(4) EP*(*F*) *is closed and convex*.

**Definition 2.1** [17]. *Let C be a nonempty convex subset of real Banach space. Let* *be a finite family of nonexpanxive mappings of C into itself, and let λ*_{1}, ..., *λ*_{
N
} *be real numbers such that* 0 ≤ *λ*_{
i
} ≤ 1 *for every i* = 1, ..., *N . We define a mapping K* : *C* → *C as follows*.

Such a mapping *K* is called the *K-mapping* generated by *T*_{1}, ..., *T*_{
N
} and *λ*_{1}, ..., *λ*_{
N
} .

**Lemma 2.8** [17]. *Let C be a nonempty closed convex subset of a strictly convex Banach space. Let* *be a finite family of nonexpanxive mappings of C into itself with* *and let λ*_{1}, ..., *λ*_{
N
} *be real numbers such that* 0 *< λ*_{
i
} *<* 1 *for every i* = 1, ..., *N* - 1 *and* 0 *< λ*_{
N
} ≤ 1. *Let K be the K-mapping generated by T*_{1}, ..., *T*_{
N
} *and λ*_{1}, ..., *λ*_{
N
} *. Then* .

**Lemma 2.9** [9]. *Let C be a nonempty closed convex subset of a real Hilbert space H and S* : *C* → *C be a self-mapping of C. If S is a κ-strict pseudo-contraction mapping, then S satisfies the Lipschitz condition*.

**Lemma 2.10**. *Let C be a nonempty closed convex subset of a real Hilbert space. Let* *be κ*_{
i
}*-strict pseudo-contraction mappings of C into itself with* *and κ* = sup_{
i
} *κ*_{
i
} *and let* , *where I* = [0, 1], ,, *and* *for all j* = 1, 2, .... *For every n* ∈ ℕ, *let S*_{
n
} *be S-mapping generated by T*_{
n
}, ..., *T*_{1} *and α*_{
n
}, *α*_{n-1}, ..., *α*_{1}. *Then, for every x* ∈ *C and k* ∈ ℕ, lim_{n→∞}*U*_{
n
},_{
k
}*x exists*.

*Proof*. Let *x* ∈ *C* and . Fix *k* ∈ ℕ, then for every *n* ∈ ℕ with *n* ≥ *k*,

we have

It follows that

where and

For any *k*, *n*, *p* ∈ ℕ, *p >* 0, *n* ≥ *k*, we have

Since *a* ∈ (0, 1), we have lim_{n→∞}*a*^{n} = 0. From (2.5), we have that {*U*_{
n
},_{
k
}*x*} is a Cauchy sequence. Hence lim _{n→∞}*U*_{n,k}*x* exists. □

For every *k* ∈ ℕ and *x* ∈ *C*, we define mapping *U*_{∞,k}and *S* : *C* ∈ *C* as follows:

and

Such a mapping *S* is called *S*-mapping generated by *T*_{
n
}, *T*_{n-1}, ... and *α*_{
n
}, *α*_{n- 1}, ...

*Remark* 2.11. For each *n* ∈ ℕ, *S*_{
n
} is nonexpansive and lim_{n→∞}*sup*_{x∈D}||*S*_{
n
}*x* - *Sx*|| = 0 for every bounded subset *D* of *C*. To show this, let *x*, *y* ∈ *C* and *D* be a bounded subset of *C*. Then, we have

Then, we have that *S* : *C* → *C* is also nonexpansive indeed, observe that for each *x*, *y* ∈ *C*

By (2.8), we have

This implies that for *m > n* and *x* ∈ *D*,

By letting *m* → ∞, for any *x* ∈ *D*, we have

It follows that

**Lemma 2.12**. *Let C be a nonempty closed convex subset of a real Hilbert space. Let* *be κ*_{
i
}*-strict pseudo-contraction mappings of C into itself with* *and κ* = sup_{i∈}*κ*_{
i
} *and let* , *where I* = [0, 1], , *and* *for all j* = 1, .... *For every n* ∈ ℕ, *let S*_{
n
} *and S be S-mappings generated by T*_{
n
}, ..., *T*_{1} *and α*_{
n
}, *α*_{n-1}, ..., *α*_{1} *and T*_{
n
}, *T*_{n-1}, ..., *and α*_{
n
}, *α*_{n-1}, ..., *respectively. Then* .

*Proof*. It is evident that . For every *n*, *k* ∈ ℕ, *with n* ≥ *k*, let *x*_{0} ∈ *F* (*S*) and , we have

For *k* ∈ ℕ and (2.12), we have

as *n* → ∞. This implies that *U*_{∞},_{
k
}*x*_{0} = *x*_{0}, ∀*k* ∈ ℕ.

Again by (2.12), we have

as *n* → ∞. Hence

From *U*_{∞,k}*x*_{0} = *x*_{0}, ∀*k* ∈ ℕ, and (2.15), we obtain that *T*_{
k
}*x*_{0} = *x*_{0}, ∀*k* ∈ ℕ. This implies that . □

**Lemma 2.13**. *Let C be a closed convex subset of Hilbert space H. Let A*_{
i
} : *C* → *H be mappings and let G*_{
i
} : *C* → *C be defined by G*_{
i
}(*y*) = *P*_{
C
}(*I* - *λ*_{
i
}*A*_{
i
})*y with λ*_{
i
} *>* 0, ∀_{
i
} = 1, 2, ... *N. Then* *if and only if* .

*Proof*. For given , we have *x** ∈ *VI*(*C*, *A*_{
i
}), ∀_{
i
} = 1, 2, ..., *N*. Since 〈*A*_{
i
}*x**, *x* - *x**〉 ≥ 0, we have 〈*λ*_{
i
}*A*_{
i
}*x**, *x* - *x**〉 ≥ 0, ∀*λ*_{
i
} *>* 0, *i* = 1, 2, ..., *N*. It follows that

Hence, *x** = *P*_{
C
}(*I* - *λ*_{
i
}*A*_{
i
})*x** = *G*_{
i
}(*x**), ∀*x* ∈ *C*, *i* = 1, 2, ..., *N*. Therefore, we have . For the converse, let ; then, we have for every *i* = 1, ..., *N*, *x** = *G*_{
i
}(*x**) = *P*_{
C
}(*I* - *λ*_{
i
}*A*_{
i
})*x**, ∀*λ*_{
i
} *>* 0, *i* = 1, 2, ..., *N*. It implies that

Hence, 〈*A*_{
i
}*x**, *x - x**〉 ≥ 0, ∀*x* ∈ *C*, so *x** ∈ *VI*(*C*, *A*_{
i
}), ∀*i* = 1, 2, ..., *N*. Hence, .

□

## 3 Main results

**Theorem 3.1**. *Let C be a closed convex subset of Hilbert space H. For every i* = 1, 2, ..., *N*, *let F*_{
i
} : *C* × *C* → ℝ *be a bifunction satisfying* (*A*_{1}) - (*A*_{4}), *let A*_{
i
} : *C* → *H be α*_{
i
}*-inverse strongly monotone and let G*_{
i
} : *C* → *C be defined by G*_{
i
}(*y*) = *P*_{
C
}(*I* - *λ*_{
i
}*A*_{
i
})*y*, ∀*y* ∈ *C with λ*_{
i
} ∈ (0, 1] ⊂ (0, 2*α*_{
i
}). *Let B* : *C* → *C be the K-mapping generated by G*_{1}, *G*_{2}, ..., *G*_{
N
} *and β*_{1}, *β*_{2}, ..., *β*_{
N
} *where β*_{
i
} ∈ (0, 1), ∀*i* = 1, 2, 3, ..., *N -* 1, *β*_{
N
} ∈ (0, 1] *and let* *be κ*_{
i
}*-strict pseudo-contraction mappings of C into itself with κ =* sup_{
i
}*κ*_{
i
} *and let* *, where I* = [0, 1], , , *and* *for all j* = 1, 2, ... . *For every n* ∈ ℕ, *let S*_{
n
} *and S are S-mapping generated by T*_{
n
}, ..., *T*_{1} *and ρ*_{
n
}, *ρ*_{n - 1}, ..., *ρ*_{1} *and T*_{
n
}, *T*_{n- 1}, ..., *and ρ*_{
n
}, *ρ*_{n - 1}, ..., *respectively. Assume that* . *For every n* ∈ ℕ, *i* = 1, 2, ..., *N, let* {*x*_{
n
}} *and* *be generated by x*_{1}, *u* ∈ *C and*

*where* {*α*_{
n
}}, {*β*_{
n
}}, {*γ*_{
n
}}, {*a*_{
n
}}, {*b*_{
n
}}, {*c*_{
n
}} ⊂ (0, 1), , *and* , *satisfy the following conditions:*

(*i*) *and* ,

(*ii*) ,

(*iii*) , , , *with a*, *b*, *c* ∈ (0, 1).

*Then, the sequence* {*x*_{
n
}}, {*y*_{
n
}}, , ∀*i* = 1, 2, ..., *N*, *converge strongly to* *and z is a solution of (1.10)*.

*Proof*. First, we show that (*I* - *λ*_{
i
}*A*_{
i
}) is nonexpansive mapping for every *i* = 1, 2, ..., *N*. For *x*, *y* ∈ *C*, we have

Thus, (*I - λ*_{
i
}*A*_{
i
}) is nonexpansive, and so are *B* and *G*_{
i
}, for all *i* = 1, 2, ..., *N*.

Now, we shall divide our proof into five steps.

**Step 1**. We shall show that the sequence {*x*_{
n
}} is bounded. Since

we have

By Lemma 2.7, we have .

Let . Then *F*(*z*, *y*) + 〈*y - z*, *A*_{
i
}*z*〉 ≥ 0 ∀*y* ∈ *C*, so we have

Again by Lemma 2.7, we have , ∀*i* = 1, 2, ..., *N*. Since *B* is *K*-mapping generated by *G*_{1}, *G*_{2}, ..., *G*_{
N
} and *β*_{1}, *β*_{2}, ..., *β*_{
N
} and . By Lemma 2.8, we have . Since , we have *z* ∈ *F*(*B*). Setting *e*_{
n
} = *a*_{
n
}*S*_{
n
}*x*_{
n
} + *b*_{
n
}*Bx*_{
n
} + *c*_{
n
}*y*_{
n
}, ∀*n* ∈ ℕ, we have

By induction, we can prove that {*x*_{
n
}} is bounded, and so are , {*y*_{
n
}}, {*Bx*_{
n
}} {*S*_{
n
}*x*_{
n
}}, {*e*_{
n
}}.

**Step 2**. We will show that lim_{n→∞}||*x*_{n+1}- *x*_{
n
}|| = 0. Let , and then we have

From definition of *d*_{
n
}, we have

By definition of *e*_{
n
}, we have

By (3.6) and (3.7), we have

It follows that

From Remark 2.11 and conditions (i)-(iii), we have

From (3.5), (3.12) and Lemma 2.5, we have

We can rewrite (3.5) as

By (3.13) and (3.14), we have

**Step. 3**. Show that lim_{n→∞}||*x*_{
n
} *- e*_{
n
}|| = 0. From (3.1), we have

It implies that

By conditions (i), (ii), and (3.15), we have

**Step. 4**. We show that lim sup_{n→∞}〈*u* - *z*, *x*_{
n
} *- z*〉 ≤ 0, where . Let be a subsequence of {*x*_{
n
}} such that

Without loss of generality, we may assume that converges weakly to some *q* in *H*. Next, we will show that

First, we define a mapping *A* : *C* → *C* by

Since , we have . By Lemma 2.3, we have .

Next, we define *Q* : *C* → *C* by

Again, by Lemma 2.3, we have

By (3.19), we have

By condition (iii), (3.20), and (2.11), we have

Since

by (3.16) and (3.21), we have

From, (3.22), we have

By Lemma 2.4, we obtain that

From (3.17)

**Step. 5**. Finally, we show that lim_{n→∞}*x*_{
n
} = *z*, where .

By nonexpansiveness of *S*_{
n
} and *B*, we can show that ||*e*_{
n
} *- z*|| ≤ ||*x*_{
n
} *- z*||. Then,

It follows that

From Step 4, (3.26), and Lemma 2.2, we have lim_{n→∞} *x*_{
n
} = *z*, where . The proof is complete. □

## 4 Applications

From Theorem 3.1, we obtain the following strong convergence theorems in a real

Hilbert space:

**Theorem 4.1**. *Let C be a closed convex subset of Hilbert space H. For every i* = 1, 2, ..., *N*, *let F*_{
i
} : *C* × *C* → ℝ *be a bifunction satisfying* (*A*_{1}) - (*A*_{4}) *and let* *be κ*_{
i
}*-strict pseudo-contraction mappings of C into itself with κ* = sup_{
i
}*κ*_{
i
} *and let* , *where I* = [0, 1], , , *and* *for all j* = 2, ... .. *For every n* ∈ ℕ, *let S*_{
n
} *and S are S-mappings generated by T*_{
n
}, ..., *T*_{1} *and ρ*_{
n
}, *ρ*_{n - 1}, ..., *ρ*_{1} *and T*_{
n
}, *T*_{n- 1}, ..., *and ρ*_{
n
}, *ρ*_{n- 1}, ..., *respectively. Assume that* . *For every n*∈ ℕ, *i* = 1, 2, ..., *N, let* {*x*_{
n
}} *and* *be generated by x*_{1}, *u* ∈ *C and*

*where* {*α*_{
n
}}, {*β*_{
n
}}, {*γ*_{
n
}}, {*a*_{
n
}}, {*b*_{
n
}}, {*c*_{
n
}} ⊂ (0, 1), , *and* , *satisfy the following conditions:*

(*i*) *and* ,

(*ii*) ,

(*iii*) , , , *with a*, *b*, *c* ∈ (0, 1),

*Then, the sequence* {*x*_{
n
}}, {*y*_{
n
}}, , ∀*i* = 1, 2, ..., *N*, *converge strongly to* , *and z is solution of (1.10)*

*Proof*. From Theorem 3.1, let *A*_{
i
} ≡ 0; then we have *G*_{
i
}(*y*) = *P*_{
Cy
} = *y* ∀*y* ∈ *C*. Then, we get *Bx*_{
n
} = *x*_{
n
} ∀*n* ∈ ℕ. Then, from Theorem 3.1, we obtain the desired conclusion. □

Next theorem is derived from Theorem 3.1, and we modify the result of [11] as follows:

**Theorem 4.2**. *Let C be a closed convex subset of Hilbert space H and let F* : *C* × *C* → ℝ *be a bifunction satisfying* (*A*_{1})*-*(*A*_{4}), *let A* : *C* → *H be α-inverse strongly monotone mapping, and let T be κ-strict pseudo-contraction mappings of C into itself. Define a mapping T*_{
κ
} *by T*_{
κ
}*x* = *κx* + (1 - *κ*)*Tx*, ∀*x* ∈ *C*. *Assume that* . *For every n* ∈ ℕ, *let* {*x*_{
n
}} *and* {*v*_{
n
}} *be generated by x*_{1}, *u* ∈ *C and*

*where* {*α*_{
n
}}, {*β*_{
n
}}, {*γ*_{
n
}}, {*a*, *b*, *c*} ⊂ (0, 1), *α*_{
n
} + *β*_{
n
} + *γ*_{
n
} = *a* + *b* + *c* = 1, *and* {*r*, *λ*} ⊂ (*ς*, *τ*) ⊂ (0, 2*α*) *satisfy the following conditions:*

(*i*) *and* ,

(*ii*) ,

*Then, the sequence* {*x*_{
n
}} *and* {*v*_{
n
}} *converge strongly to* .

*Proof*. From Theorem 3.1, choose *N* = 1 and let *A*_{1} = *A*, *λ*_{1} = *λ*. Then, we have *B*(*y*) = *G*_{1}(*y*) = *P*_{
C
}(*I* - *λA*)*y*, ∀*y* ∈ *C*. Choose , *a* = *a*_{
n
}, *b* = *b*_{
n
}, *c* = *c*_{
n
} for all *n* ∈ ℕ, and let *T*_{
κ
} ≡ *S*_{1} : *C* → *C* be *S*-mapping generated by *T*_{1} and *ρ*_{1} with *T*_{1} = *T* and , and then we obtain the desired result from Theorem 3.1 □

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## Acknowledgements

The authors would like to thank Professor Dr. Suthep Suantai for his valuable suggestion in the preparation and improvement of this article.

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Kangtunyakarn, A. Strong convergence theorem for a generalized equilibrium problem and system of variational inequalities problem and infinite family of strict pseudo-contractions.
*Fixed Point Theory Appl* **2011, **23 (2011). https://doi.org/10.1186/1687-1812-2011-23

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DOI: https://doi.org/10.1186/1687-1812-2011-23

### Keywords

- nonexpansive mappings
- strongly positive operator
- generalized equilibrium problem
- strict pseudo-contraction
- fixed point