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# Strong convergence theorems for system of equilibrium problems and asymptotically strict pseudocontractions in the intermediate sense

- Peichao Duan
^{1}Email author and - Jing Zhao
^{1}

**2011**:13

https://doi.org/10.1186/1687-1812-2011-13

© Duan and Zhao; licensee Springer. 2011

**Received:**22 January 2011**Accepted:**5 July 2011**Published:**5 July 2011

## Abstract

Let
be *N* uniformly continuous asymptotically *λ*_{
i
} -strict pseudocontractions in the intermediate sense defined on a nonempty closed convex subset *C* of a real Hilbert space *H*. Consider the problem of finding a common element of the fixed point set of these mappings and the solution set of a system of equilibrium problems by using hybrid method. In this paper, we propose new iterative schemes for solving this problem and prove these schemes converge strongly.

**MSC:** 47H05; 47H09; 47H10.

## Keywords

- asymptotically strict pseudocontraction in the intermediate sense
- system of equilibrium problem
- hybrid method
- fixed point

## 1. Introduction

Let *H* be a real Hilbert space and let *C* be a nonempty closed convex subset of *H*.

*S*:

*C*→

*C*is a self mapping of

*C*. We denote the set of fixed points of

*S*by

*F*(

*S*) (i.e.,

*F*(

*S*) = {

*x*∈

*C*:

*Sx*=

*x*}). Recall the following concepts.

- (1)
- (2)
- (3)
- (4)
*S*is asymptotically nonexpansive in the intermediate sense [1] provided*S*is continuous and the following inequality holds: - (5)
*S*is asymptotically*λ*-strict pseudocontractive mapping [2] with sequence {*γ*_{ n }} if there exists a constant*λ*∈ [0, 1) and a sequence {*γ*_{ n }} in [0, ∞) with lim_{n→∞}*γ*_{ n }= 0 such thatfor all

*x*,*y*∈*C*and*n*∈ ℕ. - (6)

for all *x*, *y* ∈ *C* and *n* ∈ ℕ.

for all *x*, *y* ∈ *C* and *n* ∈ ℕ.

When *c*_{
n
} = 0 for all *n* ∈ *N* in (1.2), then *S* is an asymptotically *λ*-strict pseudocontractive mapping with sequence {*γ*_{
n
} }. We note that *S* is not necessarily uniformly *L*-Lipschitzian (see [4]), more examples can also be seen in [3].

*F*

_{ k }} be a countable family of bifunctions from

*C*×

*C*to ℝ, where ℝ is the set of real numbers. Combettes and Hirstoaga [5] considered the following system of equilibrium problems:

The solution set of (1.4) is denoted by *EP*(*F*).

The problem (1.3) is very general in the sense that it includes, as special cases, optimization problems, variational inequalities, minimax problems, Nash equilibrium problem in noncooperative games and others; see, for instance, [6, 7] and the references therein. Some methods have been proposed to solve the equilibrium problem (1.3), related work can also be found in [8–11].

For solving the equilibrium problem, let us assume that the bifunction *F* satisfies the following conditions:

(A1) *F*(*x, x*) = 0 for all *x* ∈ *C*;

(A2) *F* is monotone, i.e.*F*(*x, y*) + *F*(*y, x*) ≤ 0 for any *x*, *y* ∈ *C*;

(A3) for each *x*, *y*, *z* ∈ *C*, lim sup_{t→0}*F*(*tz* + (1 - *t*)*x*, *y*) ≤ *F*(*x*, *y*);

(A4) *F*(*x*,·) is convex and lower semicontionuous for each *x* ∈ *C*.

Recall Mann's iteration algorithm was introduced by Mann [12]. Since then, the construction of fixed points for nonexpansive mappings and asymptotically strict pseudocontractions via Mann' iteration algorithm has been extensively investigated by many authors (see, e.g., [2, 6]).

where *α*_{
n
} is a real sequence in (0, 1) which satisfies certain control conditions.

*λ*

_{ i }-strict pseudocontractions. Let

*x*

_{0}∈

*C*and be a sequence in (0, 1). The sequence {

*x*

_{ n }} by the following way:

*λ*

_{ i }-strict pseudocontractions {

*S*

_{1},

*S*

_{2},...,

*S*

_{ N }}. Since, for each

*n*≥ 1, it can be written as

*n*= (

*h*- 1)

*N*+

*i*, where

*i*=

*i*(

*n*) ∈ {1, 2,...,

*N*},

*h*=

*h*(

*n*) ≥ 1 is a positive integer and

*h*(

*n*) → ∞, as

*n*→ ∞. We can rewrite the above table in the following compact form:

Recently, Sahu et al. [4] introduced new iterative schemes for asymptotically strict pseudocontractive mappings in the intermediate sense. To be more precise, they proved the following theorem.

**Theorem 1.1**.

*Let C be a nonempty closed convex subset of a real Hilbert space H and T: C*→

*C a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence γ*

_{ n }

*such that F*(

*T*)

*is nonempty and bounded. Let*

*α*

_{ n }

*be a sequence in*[0, 1]

*such that*0 <

*δ*≤

*α*

_{ n }≤ 1 -

*κ*

*for all n*∈

*N. Let*{

*x*

_{ n }} ⊂

*C*

*be sequences generated by the following*

*(CQ) algorithm:*

*where* *θ*_{
n
} = *c*_{
n
} + *γ*_{
n
} Δ _{
n
}*and* Δ _{
n
} = *sup* {||*x*_{
n
} - *z*||: *z* ∈ *F*(*T*)} < ∞. *Then*, {*x*_{
n
} } *converges strongly to P*_{F(T)}(*u*).

Very recently, Hu and Cai [3] further considered the asymptotically strict pseudocontractive mappings in the intermediate sense concerning equilibrium problem. They obtained the following result in a real Hilbert space.

**Theorem 1.2**.

*Let C be a nonempty closed convex subset of a real Hilbert space H and*

*N*≥ 1

*be an integer*,

*ϕ*:

*C*→

*C be a bifunction satisfying*

*(A1)-(A4) and A*:

*C*→

*H be an*

*α*-

*inverse-strongly monotone mapping. Let for each*1 ≤

*i*≤

*N, T*

_{ i }:

*C*→

*C be a uniformly continuous k*

_{ i }-

*strictly asymptotically pseudocontractive mapping in the intermediate sense for some*0 ≤

*k*

_{ i }< 1

*with sequences*{

*γ*

_{ n,i }} ⊂ [0, ∞)

*such that*lim

_{n→∞}

*γ*

_{ n,i }= 0

*and*{

*c*

_{ n,i }} ⊂ [0, ∞)

*such that*lim

_{n→∞}

*c*

_{ n,i }= 0.

*Let k*= max{

*k*

_{ i }: 1 ≤

*i*≤

*N*},

*γ*

_{ n }=

*max*{

*γ*

_{ n,i }: 1 ≤

*i*≤

*N*}

*and c*

_{ n }= max{

*c*

_{ n,i }: 1 ≤

*i*≤

*N*}.

*Assume that*

*is nonempty and bounded. Let*{

*α*

_{ n }}

*and*{

*β*

_{ n }}

*be sequences in*[0, 1]

*such that*0 <

*a*≤

*α*

_{ n }≤ 1, 0 <

*δ*≤

*β*

_{ n }≤ 1 -

*k for all n*∈

*N and*0 <

*b*≤

*r*

_{ n }≤

*c*< 2

*α*.

*Let*{

*x*

_{ n }}

*and*{

*u*

_{ n }}

*be sequences generated by the following algorithm:*

*where*
*, as n* → ∞*, where* *ρ*_{
n
} = *sup*{||*x*_{
n
} - *v*||: *v* ∈ *F*} < ∞. *Then*, {*x*_{
n
} } *converges strongly to P*_{F(T)}*x*_{0}.

Motivated by Hu and Cai [3], Sahu et al. [4], and Duan [8], the main purpose of this paper is to introduce a new iterative process for finding a common element of the fixed point set of a finite family of asymptotically *λ*_{
i
} -strict pseudocontractions and the solution set of the problem (1.3). Using the hybrid method, we obtain strong convergence theorems that extend and improve the corresponding results [3, 4, 13, 14].

- 1.
⇀ for the weak convergence and → for the strong convergence.

- 2.
denotes the weak

*ω*-limit set of {*x*_{ n }}.

## 2. Preliminaries

We need some facts and tools in a real Hilbert space *H* which are listed below.

**Lemma 2.1**. *Let H be a real Hilbert space. Then, the following identities hold*.

*(i)* ||*x* - *y*||^{2} = ||*x*||^{2} - ||*y*||^{2} - 2〈*x* - *y, y*〉, ∀*x, y* ∈ *H*.

*(ii)* ||*tx* +(1 - *t*)*y*||^{2} = *t*||*x*||^{2}+(1 - *t*)||*y*||^{2} - *t*(1 - *t*)||*x* - *y*||^{2}, ∀*t* ∈ [0, 1], ∀*x, y* ∈ *H*.

**Lemma 2.2**. ([10])

*Let H be a real Hilbert space. Given a nonempty closed convex subset C*⊂

*H and points x, y, z*∈

*H and given also a real number a*∈ ℝ

*, the set*

*is convex (and closed)*.

**Lemma 2.3**. ([15])

*Let C be a nonempty, closed and convex subset of H. Let*{

*x*

_{ n }}

*be a sequence in H and u*∈

*H. Let q*=

*P*

_{ C }

*u. Suppose that*{

*x*

_{ n }}

*is such that*

*ω*

_{ w }(

*x*

_{ n }) ⊂

*C and satisfies the following condition*

*Then*, *x*_{
n
} → *q*.

**Lemma 2.4**. ([4]) *Let C be a nonempty closed convex subset of a real Hilbert space H and T* : *C* → *C a continuous asymptotically* *κ*-*strict pseudocontractive mapping in the intermediate sense. Then I* - *T is demiclosed at zero in the sense that if* {*x*_{
n
} } *is a sequence in C such that x*_{
n
} ⇀ *x* ∈ *C and* lim sup_{m→∞}lim sup_{n→∞}||*x*_{
n
} - *T*^{
m
}*x*_{
n
} || = 0*, then* (*I* - *T*)*x* = 0.

**Lemma 2.5**. ([4])

*Let C be a nonempty subset of a Hilbert space H and T*:

*C*→

*C an asymptotically κ*-

*strict pseudocontractive mapping in the intermediate sense with sequence*{

*γ*

_{ n }}

*. Then*

*for all x, y* ∈ *C and n* ∈ *N*.

**Lemma 2.6**. ([6])

*Let C be a nonempty closed convex subset of H, let F be bifunction from C*×

*C to*ℝ

*satisfying (A1)-(A4) and let r*> 0

*and x*∈

*H. Then there exists z*∈

*C such that*

## 3. Main result

**Theorem 3.1**.

*Let C be a nonempty closed convex subset of a real Hilbert space H and N*≥ 1

*be an integer, let*

*F*

_{ k }

*, k*∈ {1, 2, ...

*M*}

*, be a bifunction from*

*C*×

*C to*ℝ

*which satisfies conditions (A1)-(A4). Let, for each*1 ≤

*i*≤

*N, S*

_{ i }:

*C*→

*C*

*be a uniformly continuous asymptotically*

*λ*

_{ i }-

*strict pseudocontractive mapping in the intermediate sense for some*0 ≤

*λ*

_{ i }< 1

*with sequences*{

*γ*

_{ n,i }} ⊂ [0, ∞)

*such that*lim

_{n→∞}

*γ*

_{ n,i }= 0

*and*{

*c*

_{ n,i }} ⊂ [0, ∞)

*such that*lim

_{n→∞}

*c*

_{ n,i }= 0.

*Let*

*λ*= max{

*λ*

_{ i }: 1 ≤

*i*≤

*N*},

*γ*

_{ n }= max{

*γ*

_{ n,i }: 1 ≤

*i*≤

*N*}

*and*

*c*

_{ n }= max{

*c*

_{ n,i }: 1 ≤

*i*≤

*N*}

*. Assume that*

*is nonempty and bounded. Let*{

*α*

_{ n }}

*and*{

*β*

_{ n }}

*be sequences in*[0, 1]

*such that*0 <

*a*≤

*α*

_{ n }≤ 1, 0 <

*δ*≤

*β*

_{ n }≤ 1 -

*λ*

*for all n*∈ ℕ

*and*{

*r*

_{ k,n }} ⊂ (0, ∞)

*satisfies*lim inf

_{n→∞}

*r*

_{ k,n }> 0

*for all*

*k*∈ {1, 2, ...

*M*}

*. Let*{

*x*

_{ n }}

*and*{

*u*

_{ n }}

*be sequences generated by the following algorithm:*

*where*
*, as* *n* → ∞*, where ρ*_{
n
} = sup{||*x*_{
n
} - *v*|| : *v* ∈ Ω} < ∞. *Then* {*x*_{
n
} } *converges strongly to* *P*_{Ω}*x*_{1}.

*Proof*. Denote
for every *k* ∈ {1, 2,..., *M*} and
for all *n* ∈ ℕ. Therefore
. The proof is divided into six steps.

**Step 1**. The sequence {*x*_{
n
} } is well defined.

It is obvious that *C*_{
n
} is closed and *Q*_{
n
} is closed and convex for every *n* ∈ ℕ. From Lemma 2.2, we also get that *C*_{
n
} is convex.

It follows that *p* ∈ *C*_{
n
} for all *n* ∈ ℕ. Thus, Ω ⊂ *C*_{
n
} .

*Q*

_{ n }for all

*n*∈ ℕ by induction. For

*n*= 1, we have Ω ⊂

*C*=

*Q*

_{1}. Assume that Ω ⊂

*Q*

_{ n }for some

*n*≥ 1. Since , we obtain

As Ω ⊂ *C*_{
n
} ⋂ *Q*_{
n
} by induction assumption, the inequality holds, in particular, for all *z* ∈ Ω. This together with the definition of *Q*_{n+1}implies that Ω ⊂ *Q*_{n +1}.

Hence Ω ⊂ *Q*_{
n
} holds for all *n* ≥ 1. Thus Ω ⊂ *C*_{
n
} ⋂ *Q*_{
n
} and therefore the sequence {*x*_{
n
} } is well defined.

Since Ω is a nonempty closed convex subset of *H*, there exists a unique *q* ∈ Ω such that *q* = *P*_{Ω}*x*_{1}.

Since *q* ∈ Ω ⊂ *C*_{
n
} ⋂ *Q*_{
n
} , we get (3.6).

Therefore, {*x*_{
n
} } is bounded. So are {*u*_{
n
} } and {*y*_{
n
} }.

*Q*

_{ n }, we have , which together with the fact that

*x*

_{n+1}∈

*C*

_{ n }⋂

*Q*

_{ n }⊂

*Q*

_{ n }implies that

This shows that the sequence {||*x*_{
n
} - *x*_{1}||} is nondecreasing. Since {*x*_{
n
} } is bounded, the limit of {||*x*_{
n
} - *x*_{1}||} exists.

*p*∈ Ω, it follows from the firmly nonexpansivity of that for each

*k*∈ {1, 2,...,

*M*}, we have

**Step 4**. Show that ||*u*_{
n
} - *S*_{
i
}*u*_{
n
} || → 0, ||*x*_{
n
} - *S*_{
i
}*x*_{
n
} || → 0, as *n* → ∞; ∀*i* ∈ {1, 2,..., *N*}.

*n*≥

*N*, it can be written as

*n*= (

*h*(

*n*) - 1)

*N*+

*i*(

*n*), where

*i*(

*n*) ∈ {1, 2,...,

*N*}. Observe that

*a*≤

*α*

_{ n }≤ 1 and 0 <

*δ*≤

*β*

_{ n }≤ 1 -

*λ*, we obtain

It is obvious that the relations hold: *h*(*n*) = *h*(*n* - *N*) + 1, *i*(*n*) = *i*(*n* - *N*).

*S*

_{ i }, we obtain

We first show that
. To this end, we take *ω* ∈ *ω*_{
w
} (*x*_{
n
} ) and assume that
as *j* → ∞ for some subsequence
of *x*_{
n
} .

Note that *S*_{
i
} is uniformly continuous and (3.23), we see that
, for all *m* ∈ ℕ. So by Lemma 2.4, it follows that
and hence
.

*j*→ ∞ for each

*k*= 1, 2, ...,

*M*(especially, ). Together with (3.11) and (A4) we have, for each

*k*= 1, 2, ...,

*M*, that

*t*≤ 1 and

*y*∈

*C*, let

*y*

_{ t }=

*ty*+ (1 -

*t*)

*ω*. Since

*y*∈

*C*and

*ω*∈

*C*, we obtain that

*y*

_{ t }∈

*C*and hence

*F*

_{ k }(

*y*

_{ t },

*ω*) ≤ 0. So, we have

for all *y* ∈ *C* and *ω* ∈ *EP*(*F*_{
k
} ) for each *k* = 1, 2, ..., *M*, i.e.,
.

Hence (3.24) holds.

**Step 6**. Show that *x*_{
n
} → *q* = *P*_{Ω}*x*_{1}.

From (3.6), (3.24) and Lemma 2.3, we conclude that *x*_{
n
} → *q*, where *q* = *P*_{Ω}*x*_{1}. □

**Corollary 3.2**. *Let C be a nonempty closed convex subset of a real Hilbert space H and N* ≥ 1 *be an integer, let F be a bifunction from C* × *C to* ℝ *which satisfies conditions (A1)*-*(A4). Let, for each* 1 ≤ *i* ≤ *N*, *S*_{
i
} : *C* → *C be a uniformly continuous λ*_{
i
}*-strict asymptotically pseudocontractive mapping in the intermediate sense for some 0* ≤ λ _{
i
} *<* 1 *with sequences* {*γ*_{n,i}} ⊂ [0, ∞) *such that* lim_{n→∞}*γ*_{n,i}= 0 *and* {*c*_{n,i}} ⊂ [0, ∞) *such that* lim_{n→∞}*c*_{
n
} ,_{
i
}= 0*. Let λ = max*{*λ*_{
i
} : 1 ≤ *i* ≤ *N*}, *γ*_{
n
} *= max*{*γ*_{n,i}: 1 ≤ *i* ≤ *N*} *and c*_{
n
} *= max*{*c*_{n,i}: 1 ≤ *i* ≤ *N*}. *Assume that*
*is nonempty and bounded. Let* {*α*_{
n
} } *and* {*β*_{
n
} } *be sequences in* [0, 1] *such that 0 < a* ≤ *α*_{
n
} ≤ 1,0 < *δ* ≤ *β*_{
n
} ≤ 1 - *λ for all n* ∈ *N and* {*r*_{
n
} } ⊂ (0,∞) *satisfies* lim inf_{n→∞}*r*_{
n
} *> 0 for all k* ∈ {1, 2, ... *M*}.

*where*
, as *n* → ∞, *where* *ρ*_{
n
} = *sup*{||*x*_{
n
} - *v*|| : *v* ∈ Ω} < ∞. *Then* {*x*_{
n
} } *converges strongly to* *P*_{Ω}*x*_{1}.

*Proof*. Putting *M* = 1, we can draw the desired conclusion from Theorem 3.1.

□

*Remark* 3.3. Corollary 3.2 extends the theorem of Tada and Takahashi [14] from a nonexpansive mapping to a finite family of asymptotically *λ*_{
i
} -strict pseudocontractive mappings in the intermediate sense.

**Corollary 3.4**.

*Let C be a nonempty closed convex subset of a real Hilbert space H and N*≥ 1

*be an integer, let, for each*1 ≤

*i*≤

*N*,

*S*

_{ i }

*: C*→

*C be a uniformly continuous λ*

_{ i }

*-strict asymptotically pseudocontractive mapping in the intermediate sense for some*0 ≤ λ

_{ i }

*<*1

*with sequences*{

*γ*

_{n,i}} ⊂ [0, ∞)

*such that*lim

_{n→∞}

*γ*

_{n,i}= 0

*and*{

*c*

_{ n,i }} ⊂ [0, ∞)

*such that*lim

_{n→∞}

*c*

_{ n,i }= 0

*. Let λ= max*{

*λ*

_{ i }: 1 ≤

*i*≤

*N*},

*γ*

_{ n }

*= max*{

*γ*

_{n,i}: 1 ≤

*i*≤

*N*}

*and c*

_{ n }

*=*max{

*c*

_{ n,i }: 1 ≤

*i*≤

*N*}.

*Assume that*

*is nonempty and bounded. Let*{

*α*

_{ n }}

*and*{

*β*

_{ n }}

*be sequences in*[0, 1]

*such that*0

*< a*≤

*α*

_{ n }≤ 1, 0

*<δ*≤

*β*

_{ n }≤ 1 -

*λ for all n*∈ ℕ.

*Let*{

*x*

_{ n }}

*and*{

*u*

_{ n }}

*be sequences generated by the following algorithm*:

*where*
, *as n* → ∞, *where ρ*_{
n
} *= sup*{||*x*_{
n
} - *v*|| : *v* ∈ Ω} *<* ∞*. Then* {*x*_{
n
}*} converges strongly to P*_{Ω}*x*_{1}.

*Proof*. If *F*_{
k
} (*x*, *y*) = 0, α _{
n
} = 1 in Theorem 3.1, we can draw the conclusion easily. □

*Remark* 3.5. Corollary 3.4 extends the Theorem 4.1 of [4] and Theorem 2.2 of [13], respectively.

## 4. Numerical result

In this section, in order to demonstrate the effectiveness, realization and convergence of the algorithm in Theorem 3.1, we consider the following simple example ever appeared in the reference [4]:

*where* 0 < *k* < 1.

for all *x*, *y* ∈ *C*, *n* ∈ ℕ and some *K* > 0. Therefore, *T* is an asymptotically *k*-strict pseudocontractive mapping in the intermediate sense.

In the algorithm (3.1), set
. We apply it to find the fixed point of *T* of Example 4.1.

*C*

_{ n }⋂

*Q*

_{ n }is an closed interval. If we set [

*a*

_{ n },

*b*

_{ n }] :=

*C*

_{ n }⋂

*Q*

_{ n }, then the projection point

*x*

_{n+1}of

*x*

_{1}∈

*C*onto

*C*

_{ n }⋂

*Q*

_{ n }can be expressed as:

Since the conditions of Theorem 3.1 are satisfied in Example 4.1, the conclusion holds, i.e., *x*_{
n
} → 0 ∈ *F* (*T*).

*T*. Take the initial guess

*x*

_{1}= 1/2, 1/5 and 5/8, respectively. All the numerical results are given in Tables 1, 2 and 3. The corresponding graph appears in Figure 1a,b,c.

*x*_{1} = 0.5

n(iterative number) | x | Errors (n) |
---|---|---|

5 | 0.2471 | 2.471 × 10 |

20 | 0.0527 | 5.27 × 10 |

50 | 0.0028 | 2.8 × 10 |

93 | 0.0000 | 0 |

*x*_{1} = 0.2

n(iterative number) | x | Errors (n) |
---|---|---|

5 | 0.0998 | 9.98 × 10 |

20 | 0.0211 | 2.11 × 10 |

50 | 0.0011 | 1.1 × 10 |

83 | 0.0000 | 0 |

n(iterative number) | x | Errors (n) |
---|---|---|

5 | 0.2636 | 2.636 × 10 |

20 | 0.0562 | 5.62 × 10 |

50 | 0.0030 | 3.0 × 10 |

93 | 0.0000 | 0 |

## Declarations

### Acknowledgements

The authors would like to thank the reviewers for their good suggestions. This research is supported by Fundamental Research Funds for the Central Universities (ZXH2011C002).

## Authors’ Affiliations

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