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

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## 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.

## Keywords

nonexpansive mappings strongly positive operator generalized equilibrium problem strict pseudo-contraction fixed point## 1 Introduction

*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

*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

*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:

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*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*.

*T*with domain

*D*(

*T*) and range

*R*(

*T*) is called

*nonexpansive*if

*x*,

*y*∈

*D*(

*T*) and

*T*is said to be

*κ*-

*strict pseudo-contration*if there exist

*κ*∈ [0, 1) such that

*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

*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.

*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:

*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 Open image in new window satisfies the conditions that *κ < α*_{ n } *<* 1 for all *n* and Open image in new window .

*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*.

*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 Open image in new window where

*I*= [0, 1] and Open image in new window . 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 Open image in new window ?

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 *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 Open image in new window , where *B* is the *K*-mapping generated by *G*_{1}, *G*_{2}, ..., *G*_{ N } and *β*_{1}, *β*_{2}, ..., *β*_{ N } .

*x*

_{ n }} to a point Open image in new window 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.

*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 }.

*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*Open image in new window

*is nonempty. Let*{

*λ*

_{ n }}

*be a sequence of positive numbers with*Open image in new window .

*Then, a mapping S on C defined by*

*for x* ∈ *C is well defined, nonexpansive and* Open image in new window *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*

*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* 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*,

*(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*Open image in new window

*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* Open image in new window *be a finite family of nonexpanxive mappings of C into itself with* Open image in new window *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* Open image in new window .

**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* Open image in new window *be κ*_{ i }*-strict pseudo-contraction mappings of C into itself with* Open image in new window *and κ* = sup_{ i } *κ*_{ i } *and let* Open image in new window , *where I* = [0, 1], Open image in new window , Open image in new window , *and* Open image in new window *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 Open image in new window . Fix *k* ∈ ℕ, then for every *n* ∈ ℕ with *n* ≥ *k*,

where Open image in new window and Open image in new window

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. □

*k*∈ ℕ and

*x*∈

*C*, we define mapping

*U*

_{∞,k}and

*S*:

*C*∈

*C*as follows:

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

*S*:

*C*→

*C*is also nonexpansive indeed, observe that for each

*x*,

*y*∈

*C*

**Lemma 2.12**. *Let C be a nonempty closed convex subset of a real Hilbert space. Let* Open image in new window *be κ*_{ i }*-strict pseudo-contraction mappings of C into itself with* Open image in new window *and κ* = sup_{i∈}*κ*_{ i } *and let* Open image in new window , *where I* = [0, 1], Open image in new window , Open image in new window *and* Open image in new window *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* Open image in new window .

*Proof*. It is evident that Open image in new window . For every

*n*,

*k*∈ ℕ,

*with n*≥

*k*, let

*x*

_{0}∈

*F*(

*S*) and Open image in new window , we have

*k*∈ ℕ and (2.12), we have

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

*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 Open image in new window . □

**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* Open image in new window *if and only if* Open image in new window .

*Proof*. For given Open image in new window , 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

*x** =

*P*

_{ C }(

*I*-

*λ*

_{ i }

*A*

_{ i })

*x** =

*G*

_{ i }(

*x**), ∀

*x*∈

*C*,

*i*= 1, 2, ...,

*N*. Therefore, we have Open image in new window . For the converse, let Open image in new window ; 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, Open image in new window .

□

## 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*Open image in new window

*be κ*

_{ i }

*-strict pseudo-contraction mappings of C into itself with κ =*sup

_{ i }

*κ*

_{ i }

*and let*

*, where I*= [0, 1], Open image in new window , Open image in new window ,

*and*Open image in new window

*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*Open image in new window .

*For every n*∈ ℕ,

*i*= 1, 2, ...,

*N, let*{

*x*

_{ n }}

*and*Open image in new window

*be generated by x*

_{1},

*u*∈

*C and*

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

(*i*) Open image in new window *and* Open image in new window ,

(*ii*) Open image in new window ,

(*iii*) Open image in new window , Open image in new window , Open image in new window , *with a*, *b*, *c* ∈ (0, 1).

*Then, the sequence* {*x*_{ n }}, {*y*_{ n }}, Open image in new window , ∀*i* = 1, 2, ..., *N*, *converge strongly to* Open image in new window *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.

By Lemma 2.7, we have Open image in new window .

*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 Open image in new window . By Lemma 2.8, we have Open image in new window . Since Open image in new window , 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 Open image in new window , {*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 Open image in new window , and then we have

**Step. 4**. We show that lim sup

_{n→∞}〈

*u*-

*z*,

*x*

_{ n }

*- z*〉 ≤ 0, where Open image in new window . Let Open image in new window be a subsequence of {

*x*

_{ n }} such that

*q*in

*H*. Next, we will show that

Since Open image in new window , we have Open image in new window . By Lemma 2.3, we have Open image in new window .

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

*S*

_{ n }and

*B*, we can show that ||

*e*

_{ n }

*- z*|| ≤ ||

*x*

_{ n }

*- z*||. Then,

From Step 4, (3.26), and Lemma 2.2, we have lim_{n→∞} *x*_{ n } = *z*, where Open image in new window . 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*Open image in new window

*be κ*

_{ i }

*-strict pseudo-contraction mappings of C into itself with κ*= sup

_{ i }

*κ*

_{ i }

*and let*Open image in new window ,

*where I*= [0, 1], Open image in new window , Open image in new window ,

*and*Open image in new window

*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*Open image in new window .

*For every n*∈ ℕ,

*i*= 1, 2, ...,

*N, let*{

*x*

_{ n }}

*and*Open image in new window

*be generated by x*

_{1},

*u*∈

*C and*

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

(*i*) Open image in new window *and* Open image in new window ,

(*ii*) Open image in new window ,

(*iii*) Open image in new window , Open image in new window , Open image in new window , *with a*, *b*, *c* ∈ (0, 1),

*Then, the sequence* {*x*_{ n }}, {*y*_{ n }}, Open image in new window , ∀*i* = 1, 2, ..., *N*, *converge strongly to* Open image in new window , *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*Open image in new window .

*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*) Open image in new window *and* Open image in new window ,

(*ii*) Open image in new window ,

*Then, the sequence* {*x*_{ n }} *and* {*v*_{ n }} *converge strongly to* Open image in new window .

*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 Open image in new window , *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 Open image in new window , and then we obtain the desired result from Theorem 3.1 □

## Notes

### 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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