Strong Convergence for Generalized Equilibrium Problems, Fixed Point Problems and Relaxed Cocoercive Variational Inequalities

Variational inequalities introduced by Stampacchia [1] in the early sixties have had a great impact and influence in the development of almost all branches of pure and applied sciences. It is well known that the variational inequalities are equivalent to the fixed point problems. This alternative equivalent formulation has been used to suggest and analyze in variational inequalities. In particular, the solution of the variational inequalities can be computed using the iterative projection methods. It is well known that the convergence of a projection method requires the operator to be strongly monotone and Lipschitz continuous. Gabay [2] has shown that the convergence of a projection method can be proved for cocoercive operators. Note that cocoercivity is a weaker condition than strong monotonicity.

Equilibrium problem theory provides a novel and unified treatment of a wide class of problems which arise in economics, finance, image reconstruction, ecology, transportation, network, elasticity, and optimization which has been extended and generalized in many directions using novel and innovative technique; see [3, 4]. Related to the equilibrium problems, we also have the problem of finding the fixed points of the nonexpansive mappings. It is natural to construct a unified approach for these problems. In this direction, several authors have introduced some iterative schemes for finding a common element of a set of the solutions of the equilibrium problems and a set of the fixed points of infinitely (finitely) many nonexpansive mappings; see [5–7] and the references therein. In this paper, we suggest and analyze a new iterative method for finding a common element of a set of the solutions of generalized equilibrium problems and a set of fixed points of an infinite family of nonexpansive mappings and the set solution of the variational inequality problems for a relaxed -cocoercive mapping in a real Hilbert space.

Let be a real Hilbert space and let be a nonempty closed convex subset of and is the metric projection of onto Recall that a mapping is contraction on if there exists a constant such that for all A mapping of into itself is called nonexpansive if for all We denote by the set of fixed points of , that is, . If is nonempty, bounded, closed, and convex and is a nonexpansive mapping of into itself, then is nonempty; see, for example, [8]. We recalled some definitions as follows.

Definition 1.1.

Let be a mapping. Then one has the following.

(1) is calledmonotone if for all

(2) is called -strongly monotone if there exists a positive real number such that

(1.1)

(3) is called -Lipschitz continuous if there exists a positive real number such that

(1.2)

(4) is called -inverse-strongly monotone, [9, 10] if there exists a positive real number such that

(1.3)

If we say that is firmly nonexpansive. It is obvious that any -inverse-strongly monotone mapping is monotone and -Lipschitz continuous.

(5) is calledrelaxed -cocoercive if there exists a positive real number such that

(1.4)

For , is -strongly monotone. This class of maps is more general than the class of strongly monotone maps. It is easy to see that we have the following implication: -strongly monotonicity relaxed -cocoercivity.

(6)A set-valued mapping is called monotone if for all , and imply . A monotone mapping is maximal if the graph of of is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping is maximal if and only if for , for every implies .

Let be a monotone mapping of into and let be the normal cone to at , that is,

(1.5)

Define

(1.6)

Then is the maximal monotone and if and only if ; see [11, 12]

In addition, let be a inverse-strongly monotone mapping. Let be a bifunction of into , where is the set of real numbers. The generalized equilibrium problem for is to find such that

(1.7)

The set of such is denoted by that is,

(1.8)

Special Cases

( ) If (:the zero mapping), then the problem (1.7) is reduced to the equilibrium problem:

(1.9)

The set of solutions of (1.9) is denoted by that is,

(1.10)

( ) If , the problem (1.7) is reduced to the variational inequality problem:

(1.11)

The set of solutions of (1.11) is denoted by , that is,

(1.12)

The generalized equilibrium problem (1.7) is very general in the sense that it includes, as special case, some optimization, variational inequalities, minimax problems, the Nash equilibrium problem in noncooperative games, economics, and others (see, e.g., [4, 13]). Some methods have been proposed to solve the equilibrium problem and the generalized equilibrium problem; see, for instance, [5, 14–28]. Recently, Combettes and Hirstoaga [29] introduced an iterative scheme of finding the best approximation to the initial data when is nonempty and proved a strong convergence theorem. Very recently, Moudafi [24] introduced an itertive method for finding an element of , where is an inverse-strongly monotone mapping and then proved a weak convergence theorem.

For finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of variational inequality problem for an -inverse-strongly monotone, Takahashi and Toyoda [30] introduced the following iterative scheme:

(1.13)

where is an -inverse-strongly monotone mapping, is a sequence in (0, 1), and is a sequence in . They showed that if is nonempty, then the sequence generated by (1.13) converges weakly to some . On the other hand, Shang et al. [31] introduced a new iterative process for finding a common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for a relaxed -cocoercive mapping in a real Hilbert space. Let be a nonexpansive mapping. Starting with arbitrary initial defined sequences recursively by

(1.14)

They proved that under certain appropriate conditions imposed on , , and , the sequence converges strongly to , where

In 2008, S. Takahashi and W. Takahashi [27] introduced the following iterative scheme for finding an element of under some mild conditions. 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 and let , , and by sequences generated by

(1.15)

where , and satisfy some parameters controlling conditions. They proved that the sequence defined by (1.15) converges strongly to a common element of .

On the other hand, iterative methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [32–35] and the references therein. Convex minimization problems have a great impact and influence in the development of almost all branches of pure and applied sciences.

A typical problem is to minimize a quadratic function over the set of the fixed points a nonexpansive mapping in a real Hilbert space :

(1.16)

where is the fixed point set of a nonexpansive mapping on and is a given point in . Assume that is a strongly positive bounded linear operator on ; that is, there exists a constant such that

(1.17)

In 2006, Marino and Xu [36] considered the following iterative method:

(1.18)

They proved that if the sequence of parameters satisfies appropriate conditions, then the sequence generated by (1.18) converges strongly to the unique of the variational inequality

(1.19)

which is the optimality condition for the minimization problem

(1.20)

where is a potential function for (i.e., for ).

In 2008, Qin et al. [26] proposed the following iterative algorithm:

(1.21)

where is a strongly positive linear bounded operator and is a relaxed cocoercive mapping of into . They prove that if the sequences , and of parameters satisfy appropriate condition, then the sequences and both converge to the unique solution of the variational inequality

(1.22)

which is the optimality condition for the minimization problem

(1.23)

where is a potential function for (i.e., for ).

Furthermore, for finding approximate common fixed points of an infinite family of nonexpansive mappings under very mild conditions on the parameters, we need the following definition.

Definition 1.2 (see [37]).

Let be a sequence of nonexpansive mappings of into itself and let be a sequence of nonnegative numbers in . For each , define a mapping of into itself as follows:

(1.24)

Such a mappings is called the -mapping generated by and . It is obvious that is nonexpansive, and if then .

On the other hand, Yao et al. [38] introduced and considered an iterative scheme for finding a common element of the set of solutions of the equilibrium problem and the set of common fixed points of an infinite family of nonexpansive mappings on . Starting with an arbitrary initial , define sequences and recursively by

(1.25)

where is a sequence in . It is proved [38] that under certain appropriate conditions imposed on and , the sequence generated by (1.25) converges strongly to . Very recently, Qin et al. [6] introduced an iterative scheme for finding a common fixed points of a finite family of nonexpansive mappings, the set of solutions of the variational inequality problem for a relaxed cocoercive mapping, and the set of solutions of the equilibrium problems in a real Hilbert space. Starting with an arbitrary initial , define sequences and recursively by

(1.26)

where is a relaxed -cocoercive mapping and is a strongly positive linear bounded operator. They proved that under certain appropriate conditions imposed on and , the sequences and generated by (1.26) converge strongly to some point , which is a unique solution of the variation inequality:

(1.27)

and is also the optimality for some minimization problems.

In this paper, motivated by iterative schemes considered in (1.15), (1.25), and (1.26) we will introduce a new iterative process (3.4) below for finding a common element of the set of fixed points of an infinite family of nonexpansive mappings, the set of solutions of the generalized equilibrium problem, and the set of solutions of variational inequality problem for a relaxed -cocoercive mapping in a real Hilbert space. The results obtained in this paper improve and extend the recent ones announced by Yao et al. [38], S. Takahashi and W. Takahashi [27], and Qin et al. [6] and many others.

Let be a real Hilbert space with inner product and norm . Let be a nonempty closed convex subset of We denote weak convergence and strong convergence by notations and , respectively. Recall that the (nearest point) projection from to assigns each the unique point in satisfying the property

(2.1)

The following characterizes the projection .

We need some facts tools in a real Hilbert space which are listed as follows.

Lemma 2.1.

For any ,

(2.2)

It is well known that is a firmly nonexpansive mapping of onto and satisfies

(2.3)

Moreover, is characterized by the following properties: and for all

(2.4)

Lemma 2.2 (see [39]).

Let be a Hilbert space, let be a nonempty closed convex subset of and let be a mapping of into . Let . Then for ,

(2.5)

where is the metric projection of onto .

It is clear from Lemma 2.2 that variational inequality and fixed point problem are equivalent. This alternative equivalent formulation has played a significant role in the studies of the variational inequalities and related optimization problems.

Lemma 2.3 (see [40]).

Each Hilbert space satisfies Opials condition; that is, for any sequence with , the inequality

(2.6)

holds for each with .

Lemma 2.4 (see [36]).

Assume that is a strongly positive linear bounded operator on with coefficient and . Then .

For solving the equilibrium problem for a bifunction , let us assume that satisfies the following conditions:

, for all

is monotone, that is, , for all

, for all

for each is convex and lower semicontinuous.

The following lemma appears implicitly in [4].

Lemma 2.5 (see [4]).

Let be a nonempty closed convex subset of and let be a bifunction of into satisfying (A1)–(A4). Let and . Then, there exists such that

(2.7)

The following lemma was also given in [5].

Lemma 2.6 (see [5]).

Assume that satisfies (A1)–(A4). For and , define a mapping as follows:

(2.8)

for all . Then, the following holds:

(1) is single-valued;

(2) is firmly nonexpansive, that is, for any

(2.9)

(3)

(4) is closed and convex.

Remark 2.7.

Replacing with in (2.7), then there exists , such that

(2.10)

Lemma 2.8 (see [41]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, and let be real numbers such that for every . Then, for every and , the limit exists.

Using Lemma 2.8, one can define a mapping of into itself as follows:

(2.11)

for every . Such a is called the -mapping generated by and . Throughout this paper, we will assume that for every . Then, we have the following results.

Lemma 2.9 (see [41]).

Let be a nonempty closed convex subset of a real Hilbert space . Let be nonexpansive mappings of into itself such that is nonempty, let be real numbers such that for every . Then, .

Lemma 2.10 (see [7]).

If is a bounded sequence in , then .

Lemma 2.11 (see [42]).

Let and be bounded sequences in a Banach space and let be a sequence in with Suppose for all integers and Then,

Lemma 2.12.

Let be a real Hilbert space. Then the following inequality holds:

(1) ,

(2)

for all .

Lemma 2.13 (see [43]).

Assume that is a sequence of nonnegative real numbers such that

(2.12)

where is a sequence in and is a sequence in such that

(1) ,

(2) or

Then

In this section, we prove a strong convergence theorem of a new iterative method (3.4) for an infinite family of nonexpansive mappings and relaxed -cocoercive mappings in a real Hilbert space.

We first prove the following lemmas.

Lemma 3.1.

Let be a real Hilbert space, let be a nonempty closed convex subset of , and let be -inverse-strongly monotone. It , then is a nonexpansive mapping in .

Proof.

For all and , we have

(3.1)

So, is a nonexpansive mapping of into .

Lemma 3.2.

Let be a real Hilbert space, let be a nonempty closed convex subset of and let be a relaxed -cocoercive and -Lipschitz continuous. If , , then is a nonexpansive mapping in .

Proof.

For any and , .

Putting , we obtain

(3.2)

that is, . It follows that

(3.3)

for all . Thus .

So, is a nonexpansive mapping of into .

Now, we prove the following main theorem.

Theorem 3.3.

Let be a nonempty closed convex subset of a real Hilbert space , and let be a bifunction satisfying (A1)–(A4). Let

(1) be an infinite family of nonexpansive mappings of into ;

(2) be an -inverse strongly monotone mappings of into ;

(3) be relaxed -cocoercive and -Lipschitz continuous mappings of into .

Assume that . Let be a contraction mapping with and let be a strongly positive linear bounded operator on with coefficient and . Let , , and be sequences generated by

(3.4)

where is the sequence generated by (1.24) and , , and are sequences in satisfy the following conditions:

and

,

, , for some with , ,

for some with .

Then, and converge strongly to a point , where , which solves the variational inequality

(3.5)

which is the optimality condition fot the minimization problem

(3.6)

where is a potential function for (i.e., for ).

Proof.

Since by the condition (C1) and , we may assume, without loss of generality, that . Since is a strongly positive bounded linear operator on , then

(3.7)

Observe that

(3.8)

that is to say is positive. It follows that

(3.9)

We will divide the proof of Theorem 3.3 into six steps.

Step 1.

We prove that there exists such that .

Let . Note that is a contraction mapping of into itself with coefficient . Then, we have

(3.10)

Therefore, is a contraction mapping of into itself. Therefore by the Banach Contraction Mapping Principle guarantee that has a unique fixed point, say . That is, .

Step 2.

We prove that is bounded.

Since

(3.11)

we obtain

(3.12)

From Lemma 2.6, we have for all .

For any , it follows from that

(3.13)

So, we have

(3.14)

By Lemma 2.6 again, we have for all . If follows that

(3.15)

If we applied Lemma 3.2, we get and are nonexpansive. Since and is a nonexpansive, we have , and we have

(3.16)

It follows that

(3.17)

which yields that

(3.18)

This in turn implies that

(3.19)

Therefore, is bounded. We also obtain that , ,   , , ,   , , , and are all bounded.

Step 3.

We claim that and .

From Lemma 2.6, we have and . Let , we get , , and so

(3.20)

(3.21)

Putting in (3.20) and in (3.21), we have

(3.22)

So, from the monotonicity of , we get

(3.23)

and hence

(3.24)

Without loss of generality, let us assume that there exists a real number such that for all Then, we have

(3.25)

and hence

(3.26)

where .

Put , and . Since , and are nonexpansive, then we have the following some estimates:

(3.27)

Similarly, we can prove that

(3.28)

(3.29)

Since and are nonexpansive, we deduce that, for each ,

(3.30)

where is a constant such that for all

Similarly, we can obtain that there exist nonnegative numbers , such that

(3.31)

and so are

(3.32)

Observing that

(3.33)

we obtain

(3.34)

which yields that

(3.35)

Substitution of (3.27) and (3.30) into (3.35) yields that

(3.36)

where is an appropriate constant such that .

Observing that

(3.37)

we obtain

(3.38)

which yields that

(3.39)

Substitution of (3.28) and (3.32) into (3.39) yields that

(3.40)

where is an appropriate constant such that .

Substituting (3.26) and (3.36) into (3.40), we obtain

(3.41)

where is an appropriate constant such that .

Substituting (3.41) into (3.29), we obtain

(3.42)

where is an appropriate constant such that .

Define

(3.43)

Observe that from the definition , we obtain

(3.44)

It follows from (3.32), (3.42), and (3.44) that

(3.45)

where is an appropriate constant such that .

It follows from conditions (C1), (C2), (C3), (C4), (C5), and for all

(3.46)

Hence, by Lemma 2.11, we obtain

(3.47)

It follows that

(3.48)

Applying (3.48) and conditions in Theorem 3.3 to (3.26), (3.41), and (3.42), we obtain that

(3.49)

From (3.49), (C2), (C5), and for all , we also have

(3.50)

Since , we have

(3.51)

that is,

(3.52)

By (C1), (C3), and (3.48) it follows that

(3.53)

Step 4.

We claim that the following statements hold:

(i) ;

(ii) ;

(iii) .

Since is relaxed -cocoercive and -Lipschitz continuous mappings, by the assumptions imposed on for any , we have

(3.54)

Similarly, we have

(3.55)

Observe that

(3.56)

where

(3.57)

It follows from condition (C1) that

(3.58)

Substituting (3.54) into (3.56), and using condition (C6), we have

(3.59)

It follows that

(3.60)

Since as and (3.48), we obtain

(3.61)

Note that

(3.62)

(3.63)

Using (3.56) again, we have

(3.64)

Substituting (3.62) into (3.64) and using condition (C2) and (C6), we have

(3.65)

It follows that

(3.66)

Since as and (3.48), we obtain

(3.67)

In a similar way, we can prove

(3.68)

By (2.3), we also have

(3.69)

which yields that

(3.70)

Substituting (3.70) into (3.56), we have

(3.71)

It follows that

(3.72)

Applying , and as to the last inequality, we have

(3.73)

On the other hand, we have

(3.74)

which yields that

(3.75)

Similarly, we can prove

(3.76)

(3.77)

Substituting (3.75) into (3.56), we have

(3.78)

which yields that

(3.79)

Applying (3.48) and (3.61) to the last inequality, we have

(3.80)

Using (3.64) again, we have

(3.81)

which implies that

(3.82)

From (3.48) and (3.67), we obtain

(3.83)

By using the same argument, we can prove that

(3.84)

Note that

(3.85)

Since and as , respectively, we also have

(3.86)

On the other hand, we observe

(3.87)

Applying (3.73), (3.83), (3.84), and (3.86), we have

(3.88)

On the other hand, we have

(3.89)

Substituting (3.89) into (3.64) and using conditions (C2) and (C7), we have

(3.90)

This implies that

(3.91)

In view of the restrictions (C2) and (C7), we obtain that

(3.92)

Let . Since and is firmly nonexpansive (Lemma 2.6), then we obtain

(3.93)

So, we obtain

(3.94)

Therefore, we have

(3.95)

It follows that

(3.96)

Using , as , (3.48), (3.88), and (3.92), we obtain

(3.97)

Since , we obtain

(3.98)

Note that

(3.99)

and thus from (3.88) and (3.97), we have

(3.100)

Observe that

(3.101)

Applying (3.53) and (3.100), we obtain

(3.102)

Let be the mapping defined by (2.11). Since is bounded, applying Lemma 2.10 and (3.102), we have

(3.103)

Step 5.

We claim that where is the unique solution of the variational inequality for all

Since is a unique solution of the variational inequality (3.5), to show this inequality, we choose a subsequence of such that

(3.104)

Since is bounded, there exists a subsequence of which converges weakly to . Without loss of generality, we can assume that From we obtain . Next, We show that , where .

1. (a)

   First, we prove .

    

Since , we know that

(3.105)

From (A2), we also have

(3.106)

Replacing by , we have

(3.107)

For any with and let Since and we have So, from (3.107) we have

(3.108)

Since is Lipschitz continuous, from (3.97), we have as .

Further, from the monotonicity of , we get that

(3.109)

It follows from (A4) and (3.108) that

(3.110)

From (A1), (A4), and (3.110), we also have

(3.111)

and hence

(3.112)

Letting in the above inequality, we have, for each ,

(3.113)

Thus

1. (b)

   Next, we show that

    

By Lemma 2.9, we have . Assume Since we know that and and it follows by the Opial's condition (Lemma 2.3) that

(3.114)

that is a contradiction. Thus, we have .

1. (c)
   Finally, Now we prove that Define,

   

   (3.115)
    

Since is relaxed -cocoercive and condition (C6), we have

(3.116)

which yields that is monotone. Then, is maximal monotone. Let Since and we have On the other hand, from we have

(3.117)

and hence

(3.118)

Therefore, we have

(3.119)

which implies that

(3.120)

Since is maximal monotone, we have and hence That is, , where . Since , it follows that

(3.121)

On the other hand, we have

(3.122)

From (3.53) and (3.121), we obtain that

(3.123)

Step 6.

Finally, we show that and converge strongly to .

Indeed, from (3.4) and Lemma 2.4, we obtain

(3.124)

Since , and are bounded, we can take a constant such that

(3.125)

for all . It then follows that

(3.126)

where

(3.127)

Using (C1), (3.121), and (3.123), we get , and . Applying Lemma 2.13 to (3.126), we conclude that in norm. Finally, noticing we also conclude that in norm. This completes the proof.

Corollary 3.4.

Let be a nonempty closed convex subset of a real Hilbert space . Let be a bifunction satisfying (A1)–(A4), let be relaxed -cocoercive and -Lipschitz continuous mappings, and let be an infinite family of nonexpansive mappings of into itself such that . Let be a contraction mapping of into itself with . Let , , and be sequences generated by

(3.128)

where is the sequence generated by (1.24) and , , , and are sequences in and is a real sequence in satisfying the following conditions:

,

, , for some with , .

Then, and converge strongly to a point , where .

Proof.

Put , , (:the zero mapping) and in Theorem 3.3. Then and for any , we see that

(3.129)

Let be a sequence satisfying the restriction: , where . Then we can obtain the desired conclusion easily from Theorem 3.3.

Corollary 3.5.

Let be a nonempty closed convex subset of a real Hilbert space . Let be an infinite family of nonexpansive mappings of into itself and let be relaxed -cocoercive and -Lipschitz continuous mappings such that . Let be a contraction mapping with and let be a strongly positive linear bounded operator on with coefficient and . Let , and be sequences generated by

(3.130)

where is the sequence generated by (1.24) and , , and are sequences in satisfying the following conditions:

,

, , for some with , .

Then, converges strongly to a point , where , which solves the variational inequality

(3.131)

which is the optimality condition fot the minimization problem

(3.132)

where is a potential function for (i.e., for ).

Proof.

Put , for all and for all in Theorem 3.3. Then, we have . So, by Theorem 3.3, we can conclude the desired conclusion easily.