# Strong convergence theorems for equilibrium problems and fixed point problems: A new iterative method, some comments and applications

- Zhenhua He
^{1}and - Wei-Shih Du
^{2}Email author

**2011**:33

https://doi.org/10.1186/1687-1812-2011-33

© He and Du; licensee Springer. 2011

**Received: **2 April 2011

**Accepted: **12 August 2011

**Published: **12 August 2011

## Abstract

In this paper, we introduce a new approach method to find a common element in the intersection of the set of the solutions of a finite family of equilibrium problems and the set of fixed points of a nonexpansive mapping in a real Hilbert space. Under appropriate conditions, some strong convergence theorems are established. The results obtained in this paper are new, and a few examples illustrating these results are given. Finally, we point out that some 'so-called' mixed equilibrium problems and generalized equilibrium problems in the literature are still usual equilibrium problems.

**2010 Mathematics Subject Classification**: 47H09; 47H10, 47J25.

## Keywords

## 1 Introduction and preliminaries

Throughout this paper, we assume that *H* is a real Hilbert space with zero vector *θ*, whose inner product and norm are denoted by 〈·, ·〉 and || · ||, respectively. The symbols ℕ and ℝ are used to denote the sets of positive integers and real numbers, respectively. Let *K* be a nonempty closed convex subset of *H* and *T* : *K* → *H* be a mapping. In this paper, the set of fixed points of *T* is denoted by *F*(*T*). We use symbols → and ⇀ to denote strong and weak convergence, respectively.

*P*

_{ K }is called the

*metric projection*from

*H*onto

*K*. It is well known that

*P*

_{ K }satisfies

*f*be a bi-function from

*K*×

*K*into ℝ. The classical equilibrium problem is to find

*x*∈

*K*such that

Let *EP*(*f*) denote the set of all solutions of the problem (1.1). Since several problems in physics, optimization, and economics reduce to find a solution of (1.1) (see, e.g., [1, 2]), some authors had proposed some methods to find the solution of equilibrium problem (1.1); for instance, see [1–4]. We know that a mapping *S* is said to be nonexpansive mapping if for all *x*, *y* ∈ *K*, ||*Sx* - *Sy*|| ≤ ||*x* - *y*||. Recently, some authors used iterative method including composite iterative, CQ iterative, viscosity iterative etc. to find a common element in the intersection of *EP*(*f*) and *F*(*S*); see, e.g., [5–11].

*I*be an index set. For each

*i*∈

*I*, let

*f*

_{ i }be a bi-function from

*K*×

*K*into ℝ. The system of equilibrium problem is to find

*x*∈

*K*such that

We know that is the set of all solutions of the system of equilibrium problem (1.2).

*i*∈

*I*, if

*f*

_{ i }(

*x*,

*y*) = 〈

*A*

_{ i }

*x*,

*y*-

*x*〉, where

*A*

_{ i }:

*K*→

*K*is a nonlinear operator, then the problem (1.2) becomes the following system of variational inequality problem:

It is obvious that the problem (1.3) is a special case of the problem (1.2).

The following Lemmas are crucial to our main results.

**Lemma 1.1 (Demicloseness principle**[12]) *Let H be a real Hilbert space and K a closed convex subset of H*. *S* : *K* → *H is a nonexpansive mapping. Then the mapping I - S is demiclosed on K, where I is the identity mapping, i.e., x*_{
n
} ⇀ *x in K and* (*I* - *S*)*x*_{
n
} → *y implies that ×* ∈ *K and* (*I* - *S*)*x* = *y*.

**Lemma 1.2**[13]* Let* {*x*_{
n
} }*and* {*y*_{
n
} } *be bounded sequences in a Banach space E and let* {*β*_{
n
} } *be a sequence in [0,1] with* 0 < lim inf_{n→∞}*β*_{
n
}≤ lim sup_{n→∞}*β*_{
n
}< 1. *Suppose x*_{n+1}= *β*_{
n
}*y*_{
n
} + (1 - *β*_{
n
} )*x*_{
n
} *for all integers n* ≥ 0 *and* lim sup_{n→∞}(||*y*_{n+1}- *y*_{
n
} || - ||*x*_{n+1}- *x*_{
n
} ||) ≤ 0, *then* lim_{n→∞}||*y*_{
n
} - *x*_{
n
} || = 0.

**Lemma 1.3**[5]

*Let H be a real Hilbert space. Then the following hold*.

- (a)
||

*x*+*y*||^{2}≤ ||*y*||^{2}+ 2〈*x*,*x*+*y*〉*for all x*,*y*∈*H;* - (b)
||

*αx*+ (1 -*α*)*y*||^{2}=*α*||*x*||^{2}+ (1 - α) ||*y*||^{2}-*α*(1 -*α*) ||*x*-*y*||^{2}*for all**x*,*y*∈*H and α*∈ ℝ*;* - (c)
||

*x*-*y*||^{2}= ||*x*||^{2}+ ||*y*||^{2}- 2 〈*x*,*y*〉*for all x*,*y*∈*H*.

**Lemma 1.4**. [14]

*Let*{

*a*

_{ n }}

*be a sequence of nonnegative real numbers satisfying the following relation:*

**Lemma 1.5**[1]* Let K be a nonempty closed convex subset of H and F be a bi-function of K × K into* ℝ *satisfying the following conditions*.

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

(A2) *F is monotone, that is, F*(*x*, *y*) + *F*(*y*, *x*) ≤ 0 *for all x*, *y* ∈ *K;*

(A4) *for each ×* ∈ *K*, *y* → *F*(*x*, *y*) *is convex and lower semi-continuous*.

**Lemma 1.6**[3]

*Let K be a nonempty closed convex subset of H and let F be a bi-function of K × K into*R

*satisfying (A1)*-

*(A4). For r >*0

*and ×*∈

*H, define a mapping T*

_{ r }:

*H*→

*K as follows:*

## 2 Main results and their applications

*I*= {1, 2,...,

*k*} be a finite index set, where

*k*∈ ℕ. For each

*i*∈

*I*, let

*f*

_{ i }be a bi-functions from

*K*×

*K*into ℝ satisfying the conditions (A1)-(A4). Denote by

For each (*i*, *n*) ∈ *I ×* ℕ, applying Lemmas 1.5 and 1.6,
is a firmly nonexpansive single-valued mapping such that
is closed and convex. For each *i* ∈ *I*, let
, *n* ∈ ℕ.

First, let us consider the following example.

*Example A*Let

*f*

_{ i }: [-1, 0]

*×*[-1,0]

*→*ℝ be defined by

*f*

_{ i }(

*x*,

*y*) = (1+

*x*

^{2i})(

*x - y*),

*i*= 1, 2, 3. It is easy to see that for any

*i*∈ {1, 2, 3},

*f*

_{ i }(

*x*,

*y*) satisfies the conditions (A1)-(A4) and . Let

*Sx*=

*x*

^{3}and , ∀

*x*∈ [-1, 0] Then

*g*is a -contraction from

*K*into itself and

*S*:

*K → K*is a nonexpansive mapping with . Let

*λ*∈ (0, 1), {

*r*

_{ n }} ⊂ [1, + ∞) and {

*α*

_{ n }} ⊂ (0,1) satisfy the conditions (i) lim

_{n→∞}

*α*

_{ n }= 0, and (ii) , or equivalently, ; e.g., let , {

*α*

_{ n }} ⊂ (0, 1) and {

*r*

_{ n }} ⊂ [1, + ∞) be given by

Then the sequences {*x*_{
n
} } and
, *i* = 1, 2, 3, defined by (2.1) all strongly converge to 0.

*Proof*

- (a)
By Lemmas 1.5 and 1.6, (2.1) is well defined.

- (b)

*z*= 0 is a solution of the problem . On the other hand, there does not exist

*z*∈ [-1, 0) such that

*z - y*≤ 0 and . So

*z*= 0 is the unique solution of the problem .

- (c)

*x*

_{ n }} ⊂ [-1, 0], so, by (b), for all

*n*∈ ℕ. We need to prove

*x*

_{ n }

*→*0 as

*n*→ ∞. Since

*z*

_{ n }= 0 for all

*n*∈ ℕ, we have

*y*

_{ n }= (1 -

*λ*)

*x*

_{ n }and

Hence {|*x*_{
n
}*|*} is a strictly deceasing sequence and *|x*_{
n
}*|* ≥ 0 for all *n* ∈ ℕ. So
exists.

which implies
. Therefore {*x*_{
n
} } strongly converges to 0. □

In this paper, motivated by the preceding *Example A*, we introduce a new iterative algorithm for the problem of finding a common element in the set of solutions to the system of equilibrium problem and the set of fixed points of a nonexpansive mapping. The following new strong convergence theorem is established in the framework of a real Hilbert space *H*.

**Theorem 2.1**

*Let K be a nonempty closed convex subset of a real Hilbert space H and I*= {1, 2,...,

*k*}

*be a finite index set. For each i*∈

*I, let f*

_{ i }

*be a bi-function from K × K into*ℝ

*satisfying (A1)*-

*(A4). Let S*:

*K → K be a nonexpansive mapping with*.

*Let λ*,

*ρ*∈ (0, 1)

*and g*:

*K → K is a ρ-contraction. Let*{

*x*

_{ n }}

*be a sequence generated in the following manner:*

*If the above control coefficient sequences* {*α*_{
n
} } ⊂ (0, 1) *and* {*r*_{
n
} } ⊂ (0, +∞) *satisfy the following restrictions:*

*then the sequences* {*x*_{
n
} } *and*
, *for all i* ∈ *I, converge strongly to an element c* = *P*_{Ω}*g*(*c*) ∈ Ω. *The following conclusion is immediately drawn from Theorem 2.1*.

**Corollary 2.1**

*Let K be a nonempty closed convex subset of a real Hilbert space H. Let f be a bi-function from K × K into*ℝ

*satisfying (A1)-(A4) and S*:

*K*→

*K be a nonexpansive mapping with*Ω =

*EP*(

*f*)

*∩F*(

*S*) ≠ ∅.

*Let λ, ρ*∈ (0,1)

*and g*:

*K → K is a ρ-contraction. Let*{

*x*

_{ n }}

*be a sequence generated in the following manner:*

*If the above control coefficient sequences* {*α*_{
n
} } ⊂ (0, 1) *and* {*r*_{
n
} } ⊂ (0, +∞) *satisfy all the restrictions in Theorem 2.1, then the sequences* {*x*_{
n
} } *and* {*u*_{
n
} } *converge strongly to an element c* = *P*_{Ω}*g*(*c*) ∈ Ω, *respectively*.

*If f*_{
i
} (*x, y*) ≡ 0 *for all* (*x*, *y*) ∈ *K* × *K in Theorem 2.1 and all i* ∈ *I, then, from the algorithm* (*D*_{
H
} ), *we obtain*
, ∀ *i* ∈ *I. So we have the following result*.

**Corollary 2.2**

*Let K be a nonempty closed convex subset of a real Hilbert space H. Let S*:

*K*→

*K be a nonexpansive mapping with F*(

*S*) ≠ ∅.

*Let λ, ρ*∈ (0, 1)

*and g*:

*K*→

*K is a ρ-contraction. Let*{

*x*

_{ n }}

*be a sequence generated in the following manner:*

*If the above control coefficient sequences* {*α*_{
n
} } ⊂ (0, 1) *satisfy*
,
*and*
, *then the sequences* {*x*_{
n
} } *converge strongly to an element c* = *P*_{Ω}*g*(*c*) ∈ *F* (*S*).

As some interesting and important applications of Theorem 2.1 for optimization problems and fixed point problems, we have the following.

*Application (I) of Theorem 2.1*We will give an iterative algorithm for the following optimization problem with a nonempty common solution set:

where *h*_{
i
} (*x*), *i* ∈ {1, 2,..., *k*}, are convex and lower semi-continuous functions defined on a closed convex subset *K* of a Hilbert space *H* (for example, *h*_{
i
} (*x*) = *x*^{
i
} , *x* ∈ *K* := [0, 1], *i* ∈ {1, 2,..., *k*}).

*f*

_{ i }(

*x*,

*y*) =

*h*

_{ i }(

*y*) -

*h*

_{ i }(

*x*),

*i*∈ {1, 2,...,

*k*}, then is the common solution set of the problem (

*OP*), where denote the common solution set of the following equilibrium:

*i*∈ {1, 2,...,

*k*}, it is obvious that the

*f*

_{ i }(

*x*,

*y*) satisfies the conditions (A1)-(A4). Let

*S*=

*I*(identity mapping), then from (

*D*

_{ H }), we have the following algorithm

where *x*_{1} ∈ *K*, *λ* ∈ (0, 1), *g* : *K → K* is a *ρ*-contraction. From Theorem 2.1, we know that {*x*_{
n
} } and
, *i* ∈{1,2,..., *k*}, generated by (2.5), strongly converge to an element of
if the coefficients {*α*_{
n
} } and {*r*_{
n
} } satisfy the conditions of Theorem 2.1.

*Application (II) of Theorem 2.1* Let *H*, *K*, *I*, *λ*, *ρ*, *g* be the same as Theorem 2.1. Let *A*_{1}, *A*_{2},..., *A*_{
k
} : *K* → *K* be *k* nonlinear mappings with
. For any *i* ∈ *I*, put *f*_{
i
} (*x*, *y*) = 〈*x - A*_{
i
}*x*, *y - x*〉, ∀ *x*, *y* ∈ *K*. Since
, we have
. Let *S* = *I* (identity mapping) in the algorithm (*D*_{
H
} ). Then the sequences {*x*_{
n
} } and
, defined by the algorithm (*D*_{
H
} ), converge strongly to a common fixed point of {*A*_{1}, *A*_{2},..., *A*_{
k
} }, respectively.

The following result is important in this paper.

**Lemma 2.1**

*Let H be a real Hilbert space. Then for any x*

_{1},

*x*

_{2},...

*x*

_{ k }∈

*H and a*

_{1},

*a*

_{2},...,

*a*

_{ k }∈ [0,1]

*with*,

*k*∈ ℕ,

*we have*

*Proof*It is obvious that (2.6) is true if

*a*

_{ j }= 1 for some

*j*, so it suffices to show that (2.6) is true for

*a*

_{ j }≠ 1 for all

*j*. The proof is by mathematic induction on

*k*. Clearly, (2.6) is true for

*k*= 1. Let

*x*

_{1},

*x*

_{2}∈

*H*and

*a*

_{1},

*a*

_{2}∈ [0,1] with

*a*

_{1}+

*a*

_{2}= 1. By Lemma 1.3, we obtain

*k*= 2. Suppose that (2.6) is true for

*k*=

*l*∈ ℕ. Let

*x*

_{1},

*x*

_{2},...,

*x*

_{ l },

*x*

_{l+1}∈

*H*and

*a*

_{1},

*a*

_{2},...,

*a*

_{ l },

*a*

_{l+1}∈ [0, 1) with . Let . Then applying the induction hypothesis we have

Hence, the equality (2.6) is also true for *k* = *l* + 1. This completes the induction. □

## 3 Proof of Theorem 2.1

We will proceed with the following steps.

**Step 1**: There exists a unique *c* ∈ Ω ⊂ *H* such that *P*_{Ω}*g*(*c*) = *c*.

Since *P*_{Ω}*g* is a *ρ*-contraction on *H*, Banach contraction principle ensures that there exists a unique *c* ∈ *H* such that *c* = *P*_{Ω}*g*(*c*) ∈ Ω.

**Step 2**: We prove that the sequences {*x*_{
n
} }, {*y*_{
n
} }, {*z*_{
n
} } and
, ∀*i* ∈ *I*, are all bounded.

which shows that {*x*_{
n
} } is bounded. Also, we know that {*y*_{
n
} }, {*z*_{
n
} } and
, ∀*i* ∈ *I*, are all

bounded.

**Step 3**: We prove lim_{n→∞}||*x*_{n+1}- *x*_{
n
} || = 0.

**Step 5**: Prove lim sup_{n→∞}〈*g*(*c*) - *q*, *x*_{
n
} - *c*〉 ≤ 0.

Since
is bounded, there exists a subsequence of
which is still denoted by
such that
as *ℓ* → ∞. Notice that for each *i* ∈ *I*,
by (3.17), so we also have
as *ℓ* → ∞, ∀ *i* ∈ *I*.

We want to show *z* ∈ Ω. First, we show that *z* ∈ *F*(*S*). In fact, since
and
as *ℓ* → ∞, by Lemma 1.1, we have (*I* - *S*)*z* = *θ* or, equivalently, *z* ∈ *F*(*S*).

*y*∈

*K*be given. Put

*y*

_{ t }=

*ty*+ (1 -

*t*)

*z, t*∈ (0, 1). Then

*y*

_{ t }∈

*K*and

*f*

_{ i }(

*y*

_{ t },

*z*) ≤ 0 for all

*i*∈

*I*. By (A1) and (A4), we get

**Step 6**: Finally, we prove {*x*_{
n
} } and
, for all *i* ∈ *I*, converge strongly to *c* = *P*_{Ω}*g*(*c*) ∈ Ω.

_{n→∞}

*a*

_{ n }= 0 which implies

or equivalence, {*x*_{
n
} } strongly converges to *c*. By (3.17), we can prove that for any *i* ∈ *I*,
strongly converges to *c*. The proof of Theorem 2.1 is completed. □

## 4 Further remarks

Let *K* be a nonempty closed convex subset of *H* and *f* be a bi-function of *K* × *K* into ℝ.

*Remark 4.1*Recently, some authors introduced the following mixed equilibrium problem (MEP, for short) (see [15–17] and references therein) and generalized equilibrium problem (GEP, for short) (see [18–20] and references therein):

- (a)

where *A* : *C* → *H* is a nonlinear operator.

In [15–17], the authors gave some iterative methods for finding the solution of MEP when the bi-function *f*(*x*, *y*) admits the conditions (A1)-(A4) and the real-valued function *φ* satisfies the following condition:

(A5) *φ* : *C* → ℝ is a proper lower semi-continuous and convex function.

However, in this case, we argue that the problem MEP is still the equilibrium problem (1.1). In fact, if we put *f*_{1}(*x*, *y*) = *f*(*x*, *y*), *f*_{2}(*x*, *y*) = *φ*(*y*) - *φ*(*x*) and *F*(*x*, *y*) = *f*_{1}(*x*, *y*) + *f*_{2}(*x*, *y*) for each (*x*, *y*) ∈ *C* × *C*, then *f*_{1}(*x*, *y*) satisfies the conditions (A1)-(A4), *f*_{2}(*x*, *y*) satisfies the condition (A5) and the function *φ* must satisfy the conditions (A1)-(A4). This shows that for each (*x*, *y*) ∈ *C* × *C*, *F*(*x*, *y*) satisfies the conditions (A1)-(A4). So, when we study the solution of MEP, we only need to study the solution of the equilibrium (1.1). This also shows that some "so-called" mixed equilibrium problem studied in [15–17] is still the equilibrium problem (1.1).

*Remark 4*.

*2*Let us recall some well-known definitions. A mapping

*T*:

*C*→

*C*is said to be

- (1)
*v*-*expansive*if ||*Tx*-*Ty*|| ≥*v*||*x*-*y*|| for all*x*,*y*∈*C*. In particular, if*v*= 1, then*T*is called*expansive*. - (2)

*v*-strongly monotone mapping is

*v*-expansive.

- (3)
- (4)
*L-Lipschitz continuous*if ||*Tx*-*Ty*|| ≤*L*||*x*-*y*|| for all*x*,*y*∈*C*. In particular, if*L*= 1, then*T*is called*nonexpansive*.

It is easy to see that a *u*-inverse strongly monotone operator is
-Lipschitz continuous.

For the problem GEP, if the nonlinear operator *A* : *C* → *H* is a *u*-inverse strongly monotone operator and the bi-function *f*(*x*, *y*) admits the conditions (A1)-(A4), we argue that the problem GEP is still the problem (1.1) and so it is indeed not a generalization. In fact, if *A* is a *u*-inverse strongly monotone operator from *C* into *H*, then *A* is a continuous operator. So, we obtain easily that the function (*x*, *y*) → *<Ax*, *y* - *x*〉, ∀*x*, *y* ∈ *C*, satisfies the conditions (A1)-(A4). Hence, if we put *F*(*x*, *y*) = *f*(*x*, *y*) + 〈*Ax*, *y* - *x*〉 ≥ 0, then the problem GEP studied in [18–20] is still the problem (1.1).

## 5 Conclusion

## Declarations

### Acknowledgements

Zhenhua He was supported by the Natural Science Foundation of Yunnan Province (2010ZC152) and the Scientific Research Foundation from Yunnan Province Education Committee (08Y0338); Wei-Shih Du was supported by the National Science Council of the Republic of China.

## Authors’ Affiliations

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