1 Introduction and preliminaries

Throughout this paper, and will represent the real Banach space, a nonempty subset of , the topological duala space of , uniformly convex, uniformly smooth Banach spaces and the set of real numbers, respectively. Let q > 1 be a constant and define the mapping by

(1.1)

Then, is called generalized duality map on . In particular, is called normalized duality mapping and it is often represented by . It has been established (see, for instance, [1]) that if and that if is strictly convex, then is single-valued, which is denoted by j_q, but it becomes an identity operator in Hilbert spaces. For convenience, we shall use J in place of where necessary.

Definition 1.1. Let be Banach space. Then

1. is called UCB if given ε with , the inequalities and entail that we can find such that .
2. is called smooth if for each , we can find a unique function such that and .
3. is called q-USB (q > 1) if we can find a fixed constant such that , ( denotes the set of positive real numbers). It is not difficult to see that every q-USB is uniformly smooth.

Definition 1.2. Let be a Banach space. Then, a function is called modulus of smoothness of if the following identity holds:

is called uniformly smooth if

It has been established that if is a USB then it is a smooth Banach space.

Suppose that is a nonlinear mapping. Then, we refer to and as the set of fixed point of , the the set of positive integers and the set of real numbers, respectively. When is a sequence in , strong (respectively weak) convergence to a point will be represented with (resp. ).

Definition 1.3. is called -enriched strictly pseudocontractive mapping (ESPM) (see [2,3]) if , and such that

or equivalently,

(1.2)

where . Set in (1.2). Then, by Proposition 1.1 (3) , it is easy to see that

(1.3)

where is an average operator and I denotes the identity mapping, and is called -strictly pseudocontractive. The inequality (1.3) could be written equivalently as

(1.4)

Remark 1.1. Set , then (1.2) becomes

(1.5)

which defines the class of mappings known as -enriched non-expansive mapping (-ENM), which was first considered by Berinde [4,5]. This class of mappings contains one of the initial classes of mappings (nonexpansive mappings) whose fixed points were evaluated using geometric properties instead of the compactness conditions. Aside generalizing the class of contraction mappings (and its intimate relationship with the monotonicity techniques), nonexpansive mapping practically stands as transition operators for initial value problems of differential inclusion, accretive operators, monotone operators, equilibrium problems, and variational inequality problems. Recently, several extensions and generalizations of this important class of mappings have been examined by different researchers; see, for instance, [6–8] and the references contained in them.

Example 1.1. (See [3]) Let be equipped with the Euclidean norm and

Define the mapping by Let . Then, for all , we have

(1.6)

Also, by setting , we obtain that

(1.7)

(1.6) and (1.7) imply

Thus, the mapping is a -enriched strictly pseudocontractive mapping. Further, observe that (0,0) is a unique fixed point of .

In the setup of a real Hilbert space, inequality (1.3) is written as

(1.8)

where is described in inequality (1.3). Set in (1.8), then we have the class of pseudocontractions.

It is worth noting that the class of -ESPM was initiated by Berinde [2], as a generalization of the class of -strictly pseudocontractions (Recall that is known as -strictly pseudocontraction if for all ). In [2], Berinde proved that every -enriched strictly pseudocontractive mapping defined on a bounded, closed and convex subset of a real Hilbert space has a fixed point.

Let be a Banach space, and . A mapping is sunny (see [10]) as long as , and , whenever . P is called a retraction if and the sunny nonexpansive retraction (SNR) from onto D if P is a retraction from onto D.

Let be a nonlinear operator. Then, A is called an accretive if we can find such that

A is called α-inverse strongly accretive if we can find and such that

Note that if is a -strictly pseudocontractive mapping, then is -inverse strongly accretive.

Let and be described as above. Then, the variational inequality in is to search for an that assures the inequality

(1.9)

for some . Problem (1.9) was studied by Aoyama et al. [11]. The set of solutions of the problem (1.9) in will be represented with , that is,

(1.10)

Many practically oriented problems in pure and applied sciences, physics and economics can be recast in the form of (1.9); see, for example, [12–14]. To solve the problem (1.9), several researchers have attempted to use the following methods. Let be a nonexpansive mapping with a fixed point and the sequence be chosen such that and . Then the normal Mann iterative sequence , first studied by Mann [15], is given by

(1.11)

The above sequence is known to admit weak convergence. In a quest to obtain strong convergence, Halpern [16], in 1967, initiated the following technique:

(1.12)

where is a nonexpansive mapping with a fixed point and . He noted that the conditions and are necessary in order to establish strong convergence of to the fixed point of . Subsequently, several authors have modified the iteration technique (1.12) in an attempt to establish strong convergence results; see, for example, [17] and the references therein.

In [18], Zhou obtained strong convergence results using modified Mann-Halpern iteration technique for -strictly pseudocontractive mappings in a real 2-uniformly smooth Banach space.

In [11], Aoyama et al. studied Mann-type recursive sequence and showed that such a method admits weak convergence results for the class of nonexpansive mappings. In addition, they proved weak convergence theorem for finding the solution of the variational inequality problem (1.9) in a 2-uniformly smooth Banach space.

In [26], Karntunykarn and Suantai gave the following definition.

Definition 1.4. Let be a real Banach space, and be a finite family of mapping, where . Let for each , where and . The mapping given by

(1.13)

is known as the -mapping generated from and

If , for every , then (1.13) reduces to the class of K-mapping which was first studied in [19].

In [20], by means of K-mapping defined in a real 2-uniformly smooth Banach space, Karntunykarn and Suantai constructed the Mann-type iterative method for finding a common element of the sets of fixed points of an -strictly pseudocontractive mapping and nonexpansive mapping, and the set of solutions of finite family of variational inequality problems.

Recently, Karntunykarn [10] introduced the mapping below as a generalization of (1.13).

Definition 1.5. Let be a real Banach space, and be finite families of mappings, where . Let for each , where and . The mapping defined by

(1.14)

is known as -mapping developed from and .

In addition, he established that mapping is nonexpansive and further showed that the Mann-Halpern-type algorithm can assure strong convergence results to the common element of the fixed point sets of a finite family of -strictly pseudocontractive mappings and nonexpansive mappings and the two sets of variational inequality problems in a 2-uniformly smooth and uniformly convex Banach space.

After reviewing the above papers, it becomes pertinent to ask the questions below.

Question 1.1.

1. Is it possible to construct a mapping that could obtain a common element of the set of solutions of the variational inequality problem and fixed point problems of a finite family of -ESPM and a finite family of -ENM?
2. Can the results established in [10] be extended to the classes of β-enriched non-expansive mappings and -enriched strictly pseudocontractive mappings?

Following the results in [10] , the purpose of this paper is to study the Mann-Halpern type method for the classes of a finite family of -ENM and -ESPM and variational inequality problems under a more general setting.

The rest of the paper is structured as follows: we present basic definitions and lemmas in Sect 2; in Sect 3, we present the algorithm for obtaining an answer to Question 1.1 and further analyze the convergence of the suggested method.

We need the following known results in the sequel. Meanwhile, and USB are still as described in Sect Sect 1.

Lemma 1.1. [21] Let be a real 2-USB and UCB with the best smooth constant B², then the inequality below holds:

Lemma 1.2. [22] Let , where is a UCB. Then, we can find a continuous and strictly increasing and convex function , such that

for all , and with .

Lemma 1.3. [11] Let be a smooth Banach space and be closed and convex. Let be an accretive operator and be a sunny non-expansive retraction from onto . Then, for all ,

Lemma 1.4. [21] Let and be a uniformly convex. Then, we can find a convex, strictly increasing and continuous function , with such that and for any , the following inequality holds:

Lemma 1.5. [23] Let be a real uniformly smooth Banach space, be closed and convex and for , let be an -enriched non-expansive mapping with nonempty fixed point . If is a bounded sequence such that

. Then, we can find a unique sunny non-expansive retraction such that

for any .

Lemma 1.6. [24] Let be a sequence in a set of nonnegative real numbers satisfying the identity

where is in (0,1) and is a sequence such that

1. (a) ,
2. (b) or .

Then, .

The following proposition provides some fundamental properties of duality mapping:

Proposition 1.1. [25] Let be a real Banach space. For , the duality has the following fundamental properties:

1. 1. for all and ;
2. 2. ;
3. 3. ;
4. 4. ;
5. 5. J_q is bounded, that is, for any bounded subset , J_q(A) is a bounded subset in ;
6. 6. J_q can be equivalently defined as the subdifferential of the functional , that is,

The authors in [27–33] worked on several iterative schemes that played an important role in variational inequality problems.

2 Main results

In this section, the main results of the paper are proved. In the sequel, we give the following definitions.

Definition 2.1. Suppose that is a real q-uniformly smooth Banach space for q > 1. A mapping Λ whose domain and range in are denoted by and ,respectively, is referred to as -ESPM in the thought of Browder and Petryshyn [9] if there exist , and such that for all , the following inequality holds:

(2.1)

Remark 2.1. Some immediate consequences of (2.1) are:

1. If q = 2, then (2.1) reduces to
   a notion studied in [2,3].
2. If , then (2.1) reduces to
   a notion examined in [32].
3. If is a real Hilbert space, then (2.1) reduces to
   a notion investigated in [2].
4. If , then (2.1) reduces to
   a notion considered in [6,30].
5. If , then (2.1) reduces to
   a notion studied in [4,5].

Set . Then, . Consequently, (2.1) becomes

which, by Proposition 1.1 (3) and on simplifying, yields

(2.2)

where , I is the identity mapping on . It is not hard to see that inequality (2.2) is equivalent to

(2.3)

Observe that the average operator in both (2.2) and (2.3) is strictly pseudocontractive.

Definition 2.2. Let be a real Banach space, and be finite families of mappings. For each , let , where and . The mapping is defined as follows:

where and , . This mapping is known as -enriched -mapping developed by and

Remark 2.2. If , then and -enriched -mapping reduces to -mapping. Thus, the class of -enriched -mapping properly includes the classes of S-mappings, -mappings and -mappings.

In the course of proving our main results, the following assumptions are used.

Assumption

1. is a 2-uniformly smooth and uniformly convex Banach space and is closed and convex;
2. is a finite family of -enriched strictly pseudocontractive mappings (ESPM) and is a finite family of -enriched nonexpansive mappings (ENM) with and with , where B is the 2-uniformly smooth constant of ;
3. , where , and for all ;
4. be an -enriched -mapping developed by the sequences and ;
5. are - and ρ-inverse strongly accretive mappings of onto , respectively; is a sunny non-expansive retraction from onto .

Assumption

and are sequences in [0,1] with and satisfying the conditions below:

1. (a) ;
2. (b) , for some ;
3. (c) ;
4. (d) ;
5. (e) and .

Now, we prove the following result by using -enriched -mapping in a 2-USB and UCB.

Theorem 2.1. Let and be as described in Assumption . Then, and -enriched -mapping is nonexpansive.

Proof: Let , and . Then, we have the following estimates: Set . Since

and , it follows that

(2.4)(2.5)

It therefore follows from (2.5) that

(2.6)

for every . Also, since for , we obtain, using (2.5), that

it follows that

(2.7)

Using (2.7), we obtain that ; that is, . Again, from the definition of , we obtain that

(2.8)

(2.5) and (2.8) imply that

The last inequality implies that

(2.9)

Now, since , it follows from the properties of and that

hence .

Using Definition 2.2 and the fact that

we obtain

(2.10)

and

(2.11)

From (2.5), (2.10), (2.11) and the uniform convexity of , we obtain the following:

(2.12)

(2.12) implies that

(2.13)

Assume that , then we have . By the properties of g, we have

(2.14)

This is a contradiction. Thus, . Using (2.10), we get .

From Definition 2.2, we obtain as follows:

Using the same approach as in above, we obtain

By continuing in this manner, we can conclude that

(2.15)

for . Now, since using (2.5) and (2.15), we obtain

it follows that

(2.16)

From (2.15) and the definition of in Definition 2.2, we get

which implies that

Considering the above information, we have the following:

(2.17)

Now, since

and , it follows that

Thus, from (2.16), we get

(2.18)

Also, from (2.17) and (2.18), we get

(2.19)

Consequently, It is not difficult to see that

.

By applying (2.5), it is not difficult to see that the mapping is non-expansive. □

We now prove the following important lemmas to obtain the next results.

Lemma 2.1. Let be a strictly convex Banach space and . Let be two -enriched non-expansive mapping (ENM) from into itself with . Define a mapping by

(2.20)

where is a constant. Then, W is an ENM and .

Proof: For every and (2.20), we obtain

Observe that whenever . Hence, W is an ENM and . □

Lemma 2.2. Let be a strictly convex Banach space and . Let be three -enriched non-expansive mappings from into itself with . Define a mapping W by

where are constants with . Then, W is an ENM and .

Proof: Using the definition of the mapping W, we get

(2.21)

where . It therefore follows from Lemma 2.1 that and U₁ is ENM. Also, (2.21) and Lemma 2.1 give the conclusion that and W is ENM. Thus, . □

Theorem 2.2. Let and be as described in Assumption . Let be a sequence developed by as

(2.22)

where with and satisfying the requirements of Assumption Then, converges strongly to , where is the sunny non-expansive retraction of onto .

Proof: Since and belong to the class of nonexpansive mappings (see [10] for details), we move on to show that the sequences are bounded. To achieve this, let . Then, using Lemma 1.3, we obtain that . Using (3.1), we get

Using inductional hypothesis, we get

Hence, the sequence is bounded. As a consequence, the sequences

are also bounded.

Next, we prove that . Using (3.1), we obtain the following estimates;

(2.23)

(2.23) and Lemma 1.6 imply that

(2.24)

Next, we prove that

(2.25)

Using (3.1), we obtain

(2.26)

where and .

Since (2.26) and Lemma 1.2 imply

it follows that

(2.27)

Observe that

so that

(2.28)

(2.27) and (2.28) imply that

(2.29)

From (2.24), (2.29) and condition (a), we have

which by employing properties of g yields

(2.30)

By using similar argument as in (2.30), we obtain

(2.31)

Define

(2.32)

for all with . Lemma 2.2 implies that while Lemma 1.3 and Theorem 2.1 imply that . Using (3.1), we obtain

which by (2.31) gives

(2.33)

(2.33) and Lemma 1.5 imply that

(2.34)

where .

Lastly, we show that the sequence converges strongly to .

(2.35)

(2.35), condition (a) and Lemma 1.6 imply that , and the proof is completed □

3 Application

The following strong convergence theorems could be obtained from our main results in the setup of real Banach space. In the sequel, we need the following lemma which is an immediate consequence of Theorem 2.1 and Definition 1.4.

Lemma 3.1. Let and be as described in Assumption . Let D be an - enriched -mapping developed from the sequences and . Then, and D is nonexpansive.

Theorem 3.1. Let and be as described in Assumption . Let be a sequence developed by as

(3.1)

where with and satisfying the requirements of Assumption Then, converges strongly to , where is the sunny non-expansive retraction of onto .

Proof: Set in Theorem 2.2. Then, the result follows immediately from Lemma 3.1 and Theorem 2.2. □

Theorem 3.2. Let and be as described in Assumption . Define a mapping by , where and B is the 2-uniformly smooth constant of , for all and for all . Let be an - enriched -mapping developed from the sequences and . Let be a sequence developed by as

(3.2)

where with and satisfying the requirements of Assumption Then, converges strongly to , where is the sunny non-expansive retraction of onto .

Proof: It is easy to see, by using similar argument as in the one used in [10] in an attempt to show that is non-expansive, that is non-expansive mapping. Since from Lemma 1.3 , for all , the conclusion of the proof follows directly from Theorem 2.2. □

4 Conclusions

In this study, the idea of -enriched mapping is introduced in the setup of a real Banach spaces which generalizes several well-known and vital mappings in the literature. We demonstrated the existence of fixed points of -enriched mapping in the setting of 2-uniformly smooth and uniformly convex Banach space. Furthermore, we obtained a common element of finite families of -enriched non-expansive mappings, and -enriched strictly psuedocontractive mappings and variational inequality problems by using newly defined mapping and Mann-Halpern type iterative method. Several previously known results are obtained by our main results. The results of this paper gave an affirmative answer to Question 1.1.