A Hybrid Iterative Scheme for Variational Inequality Problems for Finite Families of Relatively Weak Quasi-Nonexpansive Mappings

Let be a Banach space and let be a nonempty, closed and convex subset of . Let be an operator. The classical variational inequality problem [1] for is to find such that

(1.1)

where denotes the dual space of and the generalized duality pairing between and . The set of all solutions of (1.1) is denoted by . Such a problem is connected with the convex minimization problem, the complementarity, the problem of finding a point satisfying , and so on. First, we recall that a mapping is said to be

(i)monotone if , for all

(ii) -inverse-strongly monotone if there exists a positive real number such that

(1.2)

Let be the normalized duality mapping from into given by

(1.3)

It is well known that if is uniformly convex, then is uniformly continuous on bounded subsets of . Some properties of the duality mapping are given in [2–4].

Recall that a mappings is said to be nonexpansive if

(1.4)

If is a nonempty closed convex subset of a Hilbert space and is the metric projection of onto , then is a nonexpansive mapping. This fact actually characterizes Hilbert spaces and, consequently, it is not available in more general Banach spaces. In this connection, Alber [5] recently introduced a generalized projection operator in a Banach space which is an analogue of the metric projection in Hilbert spaces.

Consider the functional defined by

(1.5)

for all , where is the normalized duality mapping from to . Observe that, in a Hilbert space , (1.5) reduces to = for all . The generalized projection is a mapping that assigns to an arbitrary point the minimum point of the functional , that is, = , where is the solution to the minimization problem:

(1.6)

The existence and uniqueness of the operator follows from the properties of the functional and strict monotonicity of the mapping (see, e.g., [3, 5–7]). In Hilbert spaces, = . It is obvious from the definition of the function that

(1) for all ,

(2) for all ,

(3) for all ,

(4)If is a reflexive, strictly convex and smooth Banach space, then, for all ,

(1.7)

For more detail see [2, 3]. Let be a closed convex subset of , and let be a mapping from into itself. We denote by the set of fixed point of . A point in is said to be an asymptotic fixed point of [8] if contains a sequence which converges weakly to such that . The set of asymptotic fixed points of will be denoted by . A mapping from into itself is called relatively nonexpansive [7, 9, 10] if = and for all and . The asymptotic behavior of relatively nonexpansive mappings were studied in [7, 9]. A point in is said to be a strong asymptotic fixed point of if contains a sequence which converges strongly to such that . The set of strong asymptotic fixed points of will be denoted by . A mapping from into itself is called relatively weak nonexpansive if and for all and . If is a smooth strictly convex and reflexive Banach space, and is a continuous monotone mapping with , then it is proved in [11] that , for is relatively weak nonexpansive. is called relatively weak quasi-nonexpansive if and for all and .

Remark 1.1.

The class of relatively weak quasi-nonexpansive mappings is more general than the class of relatively weak nonexpansive mappings [7, 9, 12–14] which requires the strong restriction = .

Remark 1.2.

If is relatively weak quasi-nonexpansive, then using the definition of (i.e., the same argument as in the proof of [12, page 260]) one can show that is closed and convex. It is obvious that relatively nonexpansive mapping is relatively weak nonexpansive mapping. In fact, for any mapping we have . Therefore, if is a relatively nonexpansive mapping, then = = .

Iiduka and Takahashi [15] introduced the following algorithm for finding a solution of the variational inequality for an -inverse-strongly monotone mapping with for all and in a -uniformly convex and uniformly smooth Banach space . For an initial point , define a sequence by

(1.8)

where is the duality mapping on , and is the generalized projection of onto . Assume that for some with where is the 2-uniformly convexity constant of . They proved that if is weakly sequentially continuous, then the sequence converges weakly to some element in where = .

The problem of finding a common element of the set of the variational inequalities for monotone mappings in the framework of Hilbert spaces and Banach spaces has been intensively studied by many authors; see, for instance, [16–18] and the references cited therein.

On the other hand, in 2001, Xu and Ori [19] introduced the following implicit iterative process for a finite family of nonexpansive mappings , with a real sequence in , and an initial point :

(1.9)

which can be rewritten in the following compact form:

(1.10)

where (here the function takes values in . They obtained the following result in a real Hilbert space.

Theorem XO

Let be a real Hilbert space, a nonempty closed convex subset of , and let be a finite family of nonexpansive self-mappings on such that . Let be a sequence defined by (1.10). If is chosen so that , as , then converges weakly to a common fixed point of the family of .

On the other hand, Halpern [20] considered the following explicit iteration:

(1.11)

where is a nonexpansive mapping and is a fixed point. He proved the strong convergence of to a fixed point of provided that = , where .

Very recently, Qin et al. [21] proposed the following modification of the Halpern iteration for a single relatively quasi-nonexpansive mapping in a real Banach space. More precisely, they proved the following theorem.

Theorem 1 QCKZ.

Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space and a closed and quasi- -nonexpansive mapping such that . Let be a sequence generated by the following manner:

(1.12)

Assume that satisfies the restriction: , then converges strongly to .

Motivated and inspired by the above results, Cai and Hu [22] introduced the hybrid projection algorithm to modify the iterative processes (1.10), (1.11), and (1.12) to have strong convergence for a finite family of relatively weak quasi-nonexpansive mappings in Banach spaces. More precisely, they obtained the following theorem.

Theorem CH

Let be a nonempty and closed convex subset of a uniformly convex and uniformly smooth Banach space and be finite family of closed relatively weak quasi-nonexpansive mappings of into itself with . Assume that is uniformly continuous for all . Let be a sequence generated by the following algorithm:

(1.13)

Assume that and are the sequences in satisfying and . Then converges strongly to , where is the generalized projection from onto .

Motivated and inspired by Iiduka and Takahashi [15], Xu and Ori [19], Qin et al.[21], and Cai and Hu [22], we introduce a new hybrid projection algorithm basing on the shrinking projection method for two finite families of closed relatively weak quasi-nonexpansive mappings to have strong convergence theorems for approximating the common element of the set of common fixed points of two finite families of such mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator in the framework of Banach spaces. Our results improve and extend the corresponding results announced by recent results.

A Banach space is said to be strictly convex if for all with and . It is also said to be uniformly convex if for any two sequences in such that and = . Let be the unit sphere of . Then the Banach space is said to be smooth provided

(2.1)

exists for each . It is also said to be uniformly smooth if the limit is attained uniformly for . It is well know that if is smooth, then the duality mapping is single valued. It is also known that if is uniformly smooth, then is uniformly norm-to-norm continuous on each bounded subset of . Some properties of the duality mapping have been given in [3, 23–25]. A Banach space is said to have Kadec-Klee property if a sequence of satisfying that and , then . It is known that if is uniformly convex, then has the Kadec-Klee property; see [3, 23, 25] for more details.

We define the function which is called the modulus of convexity of as following

(2.2)

Then is said to be -uniformly convex if there exists a constant such that constant for all . Constant is called the -uniformly convexity constant of . A 2-uniformly convex Banach space is uniformly convex, see [26, 27] for more details. We know the following lemma of 2-uniformly convex Banach spaces.

Lemma 2.1 (see [28, 29]).

Let be a -uniformly convex Banach, then for all from any bounded set of and ,

(2.3)

where is the -uniformly convexity constant of .

Now we present some definitions and lemmas which will be applied in the proof of the main result in the next section.

Lemma 2.2 (Kamimura and Takahashi [30]).

Let be a uniformly convex and smooth Banach space and let , be two sequences of such that either or is bounded. If , then .

Lemma 2.3 (Alber [5]).

Let be a nonempty closed convex subset of a smooth Banach space and . Then, if and only if for any .

Lemma 2.4 (Alber [5]).

Let be a reflexive, strictly convex and smooth Banach space, let be a nonempty closed convex subset of and let . Then

(2.4)

for all .

Let be a reflexive strictly convex, smooth and uniformly Banach space and the duality mapping from to . Then is also single-valued, one to one, surjective, and it is the duality mapping from to . We need the following mapping which studied in Alber [5]:

(2.5)

for all and . Obviously, . We know the following lemma.

Lemma 2.5 (Kamimura and Takahashi [30]).

Let be a reflexive, strictly convex and smooth Banach space, and let be as in (2.5). Then

(2.6)

for all and .

Lemma 2.6 ([31, Lemma ]).

Let be a uniformly convex Banach space and be a closed ball of . Then there exists a continuous strictly increasing convex function with such that

(2.7)

for all and with .

An operator of into is said to be hemicontinuous if for all , the mapping of into defined by is continuous with respect to the topology of . We denote by the normal cone for at a point , that is

(2.8)

Lemma 2.7 (see [32]).

Let be a nonempty closed convex subset of a Banach space and a monotone, hemicontinuous operator of into . Let be an operator defined as follows:

(2.9)

Then is maximal monotone and = .

In this section, we prove strong convergence theorem which is our main result.

Theorem 3.1.

Let be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space , let be an -inverse-strongly monotone mapping of into with for all and . Let and be two finite families of closed relatively weak quasi-nonexpansive mappings from into itself with , where . Assume that and are uniformly continuous for all . Let be a sequence generated by the following algolithm:

(3.1)

where , and is the normalized duality mapping on . Assume that , and are the sequences in satisfying the restrictions:

(C1) ;

(C2) for some with , where is the 2-uniformly convexity constant of ;

(C3) and if one of the following conditions is satisfied

(a) and and

(b) and .

Then converges strongly to , where is the generalized projection from onto .

Proof.

By the same method as in the proof of Cai and Hu [22], we can show that is closed and convex. Next, we show for all . In fact, is obvious. Suppose for some . Then, for all , we know from Lemma 2.5 that

(3.2)

Since and is -inverse-strongly monotone, we have

(3.3)

Therefore, from Lemma 2.1 and the assumption that for all and , we obtain that

(3.4)

Substituting (3.3) and (3.4) into (3.2) and using the condition that , we get

(3.5)

Using (3.5) and the convexity of , for each , we obtain

(3.6)

It follows from (3.6) that

(3.7)

So, . Then by induction, for all and hence the sequence generated by (3.1) is well defined. Next, we show that is a convergent sequence in . From , we have

(3.8)

It follows from for all that

(3.9)

From Lemma 2.4, we have

(3.10)

for each and for all . Therefore, the sequence is bounded. Furthermore, since = and = , we have

(3.11)

This implies that is nondecreasing and hence exists. Similarly, by Lemma 2.4, we have, for any positive integer , that

(3.12)

The existence of implies that as . From Lemma 2.2, we have

(3.13)

Hence, is a Cauchy sequence. Therefore, there exists a point such that as

Now, we will show that .

1. (I)

   We first show that . Indeed, taking in (3.12), we have

    

(3.14)

It follows from Lemma 2.2 that

(3.15)

This implies that

(3.16)

The property of the function implies that

(3.17)

Since , we obtain

(3.18)

It follows from the condition (3.14) and (3.17) that

(3.19)

From Lemma 2.2, we have

(3.20)

Combining (3.15) and (3.20), we have

(3.21)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(3.22)

On the other hand, noticing

(3.23)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(3.24)

Using (3.15), (3.20), and (3.24) that

(3.25)

Taking the constant , we have, from Lemma 2.6, that there exists a continuous strictly increasing convex function satisfying the inequality (2.7) and .

Case 1.

Assume that (a) holds. Applying (2.7) and (3.5), we can calculate

(3.26)

This implies that

(3.27)

We observe that

(3.28)

It follows from (3.15), (3.22), (3.23) and (3.25) that

(3.29)

From and (3.27), we get

(3.30)

By the property of function , we obtain that

(3.31)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(3.32)

From (3.15) and (3.32), we have

(3.33)

Noticing that

(3.34)

for all . By the uniformly continuity of , (3.16) and (3.33), we obtain

(3.35)

Thus

(3.36)

From the closeness of , we get . Therefore . In the same manner, we can apply the condition to conclude that

(3.37)

Again, by (C2) and (3.26), we have

(3.38)

It follows from (3.29) and that

(3.39)

Since , we have

(3.40)

From Lemmas 2.4, 2.5, and (3.4), we have

(3.41)

It follows from (3.40) that

(3.42)

Lemma 2.2 implies that

(3.43)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(3.44)

Combining (3.37) and (3.43), we also obtain

(3.45)

Moreover

(3.46)

By (3.43), (3.15), we have

(3.47)

This implies that

(3.48)

Noticing that

(3.49)

for all . Since is uniformly continuous, we can show that . From the closeness of , we get . Therefore . Hence .

Case 2.

Assume that (b) holds. Using the inequalities (2.7) and (3.5), we obtain

(3.50)

This implies that

(3.51)

It follows from (3.21), (3.24) and the condition that

(3.52)

By the property of function , we obtain that

(3.53)

Since is uniformly norm-to-norm continuous on any bounded sets, we have

(3.54)

On the other hand, we can calculate

(3.55)

Observe that

(3.56)

It follows from (3.53) and (3.54) that

(3.57)

Applying and (3.57) and the fact that is bounded to (3.55), we obtain

(3.58)

From Lemma 2.2, one obtains

(3.59)

We observe that

(3.60)

This together with (3.25) and (3.59), we obtain

(3.61)

Noticing that

(3.62)

for all . By the uniformly continuity of , (3.16) and (3.61), we obtain

(3.63)

Thus

(3.64)

From the closeness of , we get . Therefore . By the same proof as in Case 1, we obtain that

(3.65)

Hence as for each and

(3.66)

Combining (3.54), (3.61), and (3.65), we also have

(3.67)

Moreover

(3.68)

By (3.43), (3.15), we have

(3.69)

This implies that

(3.70)

Noticing that

(3.71)

for all . Since is uniformly continuous, we can show that . From the closeness of , we get . Therefore . Hence .

1. (II)

   We next show that .

    

Let be an operator defined by:

(3.72)

By Lemma 2.7, is maximal monotone and . Let , since , we have . From , we get

(3.73)

Since is -inverse-strong monotone, we have

(3.74)

On other hand, from and Lemma 2.3, we have , and hence

(3.75)

Because is constricted, it holds from (3.74) and (3.75) that

(3.76)

for all . By taking the limit as in (3.76) and from (3.43) and (3.44), we have as . By the maximality of we obtain and hence . Hence we conclude that

(3.77)

Finally, we show that . Indeed, taking the limit as in (3.9), we obtain

(3.78)

and hence by Lemma 2.3. This complete the proof.

Remark 3.2.

Theorem 3.1 improves and extends main results of Iiduka and Takahashi [15], Xu and Ori [19], Qin et al. [21], and Cai and Hu [22] because it can be applied to solving the problem of finding the common element of the set of common fixed points of two families of relatively weak quasi-nonexpansive mappings and the set of solutions of the variational inequality for an inverse-strongly monotone operator.

Strong convergence theorem for approximating a common fixed point of two finite families of closed relatively weak quasi-nonexpansive mappings in Banach spaces may not require that is 2-uniformly convex. In fact, we have the following theorem.

Corollary 3.3.

Let be a nonempty, closed, and convex subset of a uniformly convex and uniformly smooth Banach space . Let and be two finite families of closed relatively weak quasi-nonexpansive mappings from into itself with , where . Assume that and are uniformly continuous for all . Let be a sequence generated by the following algolithm:

(3.79)

where , and is the normalized duality mapping on . Assume that , and are the sequences in satisfying the following restrictions:

(C1)  ;

(C2) and if one of the following conditions is satisfied

(a) and and

(b) and .

Then converges strongly to , where is the generalized projection from onto .

Proof.

Put in Theorem 3.1. Then, we get that . Thus, the method of the proof of Theorem 3.1 gives the required assertion without the requirement that is 2-uniformly convex.

Remark 3.4.

Corollary 3.3 improves Theorem 3.1 of Cai and Hu [22] from a finite family of of relatively weak quasi-nonexpansive mappings to two finite families of relatively weak quasi-nonexpansive mappings.

If , a Hilbert space, then is -uniformly convex (we can choose ) and uniformly smooth real Banach space and closed relatively weak quasi-nonexpansive map reduces to closed weak quasi-nonexpansive map. Furthermore, , identity operator on and , projection mapping from into . Thus, the following corollaries hold.

Corollary 3.5.

Let be a nonempty, closed and convex subset of a Hilbert space . Let and be two finite families of closed weak quasi-nonexpansive mappings from into itself with , where with for all and . Assume that and are uniformly continuous for all . Let be a sequence generated by the following algorithm:

(3.80)

where , and is the normalized duality mapping on . Assume that , , , , and are the sequences in satisfying the restrictions:

(C1)  ;

(C2)  for some with , where is the 2-uniformly convexity constant of ;

(C3)  and if one of the following conditions is satisfied

(a) and and

(b) and .

Then converges strongly to , where is the metric projection from onto .

Let be a nonempty closed convex cone in , and let be an operator from into . We define its polar in to be the set

(3.81)

Then an element in is called a solution of the complementarity problem if

(3.82)

The set of all solutions of the complementarity problem is denoted by . Several problem arising in different fields, such as mathematical programming, game theory, mechanics, and geometry, are to find solutions of the complementarity problems.

Theorem 3.6.

Let be a nonempty, closed and convex subset of a 2-uniformly convex and uniformly smooth Banach space , let be an -inverse-strongly monotone mapping of into with for all and . Let and be two finite families of closed relatively weak quasi-nonexpansive mappings from into itself with , where . Assume that and are uniformly continuous for all . Let be a sequence generated by the following algorithm:

(3.83)

where = = , and is the normalized duality mapping on . Assume that , and are the sequences in satisfying the restrictions:

(C1) ;

(C2) for some with , where is the 2-uniformly convexity constant of ;

(C3) and if one of the following conditions is satisfied

(a) and and

(b) and .

Then converges strongly to , where is the generalized projection from onto .

Proof.

From [25, Lemma ], we have . From Theorem 3.1, we can obtain the desired conclusion easily.