Strong Convergence Theorems of a New General Iterative Process with Meir-Keeler Contractions for a Countable Family of -Strict Pseudocontractions in -Uniformly Smooth Banach Spaces

Abstract

We introduce a new iterative scheme with Meir-Keeler contractions for strict pseudocontractions in -uniformly smooth Banach spaces. We also discuss the strong convergence theorems for the new iterative scheme in -uniformly smooth Banach space. Our results improve and extend the corresponding results announced by many others.

1. Introduction

Throughout this paper, we denote by and a real Banach space and the dual space of , respectively. Let be a subset of , and lrt be a non-self-mapping of . We use to denote the set of fixed points of .

The norm of a Banach space is said to be Gâteaux differentiable if the limit

(1.1)

exists for all , on the unit sphere . If, for each , the limit (1.1) is uniformly attained for , then the norm of is said to be uniformly Gâteaux differentiable. The norm of is said to be Fréchet differentiable if, for each , the limit (1.1) is attained uniformly for . The norm of is said to be uniformly Fréchet differentiable (or uniformly smooth) if the limit (1.1) is attained uniformly for .

Let be the modulus of smoothness of defined by

(1.2)

A Banach space is said to be uniformly smooth if as . Let . A Banach space is said to be -uniformly smooth, if there exists a fixed constant such that . It is well known that is uniformly smooth if and only if the norm of is uniformly Fréchet differentiable. If is -uniformly smooth, then and is uniformly smooth, and hence the norm of is uniformly Fréchet differentiable, in particular, the norm of is Fréchet differentiable. Typical examples of both uniformly convex and uniformly smooth Banach spaces are , where . More precisely, is -uniformly smooth for every .

By a gauge we mean a continuous strictly increasing function defined such that and . We associate with a gauge a (generally multivalued) duality map defined by

(1.3)

In particular, the duality mapping with gauge function denoted by , is referred to the (generalized) duality mapping. The duality mapping with gauge function denoted by , is referred to the normalized duality mapping. Browder [1] initiated the study . Set for

(1.4)

Then it is known that is the subdifferential of the convex function at . It is well known that if is smooth, then is single valued, which is denoted by .

The duality mapping is said to be weakly sequentially continuous if the duality mapping is single valued and for any with , . Every space has a weakly sequentially continuous duality map with the gauge . Gossez and Lami Dozo [2] proved that a space with a weakly continuous duality mapping satisfies Opial's condition. Conversely, if a space satisfies Opial's condition and has a uniformly Gâteaux differentiable norm, then it has a weakly continuous duality mapping. We already know that in -uniformly smooth Banach space, there exists a constant such that

(1.5)

for all .

Recall that a mapping is said to be nonexpansive, if

(1.6)

is said to be a -strict pseudocontraction in the terminology of Browder and Petryshyn [3], if there exists a constant such that

(1.7)

for every , , and for some . It is clear that (1.7) is equivalent to the following:

(1.8)

The following famous theorem is referred to as the Banach contraction principle.

Theorem 1.1 (Banach [4]).

Let be a complete metric space and let be a contraction on , that is, there exists such that for all . Then has a unique fixed point.

Theorem 1.2 (Meir and Keeler [5]).

Let be a complete metric space and let be a Meir-Keeler contraction (MKC, for short) on , that is, for every , there exists such that implies for all . Then has a unique fixed point.

This theorem is one of generalizations of Theorem 1.1, because contractions are Meir-Keeler contractions.

In a smooth Banach space, we define an operator is strongly positive if there exists a constant with the property

(1.9)

where is the identity mapping and is the normalized duality mapping.

Attempts to modify the normal Mann's iteration method for nonexpansive mappings and -strictly pseudocontractions so that strong convergence is guaranteed have recently been made; see, for example, [6–11] and the references therein.

Kim and Xu [6] introduced the following iteration process:

(1.10)

where is a nonexpansive mapping of into itself is a given point. They proved the sequence defined by (1.10) converges strongly to a fixed point of , provided the control sequences and satisfy appropriate conditions.

Hu and Cai [12] introduced the following iteration process:

(1.11)

where is non-self--strictly pseudocontraction, is a contraction and is a strong positive linear bounded operator in Banach space. They have proved, under certain appropriate assumptions on the sequences , and , that defined by (1.11) converges strongly to a common fixed point of a finite family of -strictly pseudocontractions, which solves some variational inequality.

Question 1.

Can Theorem 3.1 of Zhou [8], Theorem 2.2 of Hu and Cai [12] and so on be extended from finite -strictly pseudocontraction to infinite -strictly pseudocontraction?

Question 2.

We know that the Meir-Keeler contraction (MKC, for short) is more general than the contraction. What happens if the contraction is replaced by the Meir-Keeler contraction?

The purpose of this paper is to give the affirmative answers to these questions mentioned above. In this paper we study a general iterative scheme as follows:

(1.12)

where is non-self -strictly pseudocontraction, is a MKC contraction and is a strong positive linear bounded operator in Banach space. Under certain appropriate assumptions on the sequences , , , and , that defined by (1.12) converges strongly to a common fixed point of an infinite family of -strictly pseudocontractions, which solves some variational inequality.

2. Preliminaries

In order to prove our main results, we need the following lemmas.

Lemma 2.1 (see [13]).

Let be bounded sequences in a Banach space and be a sequence in which satisfies the following condition: . Suppose that for all and . Then, .

Lemma 2.2 (see Xu [14]).

Assume that is a sequence of nonnegative real numbers such that , where is a sequence in (0, 1) and is a sequence in such that

(i),

(ii) or .

Then .

Lemma 2.3 (see [15] demiclosedness principle).

Let be a nonempty closed convex subset of a reflexive Banach space which satisfies Opial's condition, and suppose is nonexpansive. Then the mapping is demiclosed at zero, that is, , implies .

Lemma 2.4 (see [16, Lemmas 3.1, 3.3]).

Let be real smooth and strictly convex Banach space, and be a nonempty closed convex subset of which is also a sunny nonexpansive retraction of . Assume that is a nonexpansive mapping and is a sunny nonexpansive retraction of onto , then .

Lemma 2.5 (see [17, Lemma 2.2]).

Let be a nonempty convex subset of a real -uniformly smooth Banach space and be a -strict pseudocontraction. For , we define . Then, as , , is nonexpansive such that .

Lemma 2.6 (see [12, Remark 2.6]).

When is non-self-mapping, the Lemma 2.5 also holds.

Lemma 2.7 (see [12, Lemma 2.8]).

Assume that is a strongly positive linear bounded operator on a smooth Banach space with coefficient and . Then,

(2.1)

Lemma 2.8 (see [18, Lemma 2.3]).

Let be an MKC on a convex subset of a Banach space . Then for each , there exists such that

(2.2)

Lemma 2.9.

Let be a closed convex subset of a reflexive Banach space which admits a weakly sequentially continuous duality mapping from to . Let be a nonexpansive mapping with and be a MKC, is strongly positive linear bounded operator with coefficient . Assume that . Then the sequence define by converges strongly as to a fixed point of which solves the variational inequality:

(2.3)

Proof.

The definition of is well definition. Indeed, from the definition of MKC, we can see MKC is also a nonexpansive mapping. Consider a mapping on defined by

(2.4)

It is easy to see that is a contraction. Indeed, by Lemma 2.8, we have

(2.5)

Hence, has a unique fixed point, denoted by , which uniquely solves the fixed point equation

(2.6)

We next show the uniqueness of a solution of the variational inequality (2.3). Suppose both and are solutions to (2.3), not lost generality, we may assume there is a number such that . Then by Lemma 2.8, there is a number such that . From (2.3), we know

(2.7)

Adding up (2.7) gets

(2.8)

Noticing that

(2.9)

Therefore and the uniqueness is proved. Below, we use to denote the unique solution of (2.3).

We observe that is bounded. Indeed, we may assume, with no loss of generality, , for all , fixed , for each .

Case 1 ().

In this case, we can see easily that is bounded.

Case 2 ().

In this case, by Lemmas 2.7 and 2.8, there is a number such that

(2.10)

therefore, . This implies the is bounded.

To prove that as .

Since is bounded and is reflexive, there exists a subsequence of such that . By . We have , as . Since satisfies Opial's condition, it follows from Lemma 2.3 that . We claim

(2.11)

By contradiction, there is a number and a subsequence of such that . From Lemma 2.8, there is a number such that , we write

(2.12)

to derive that

(2.13)

It follows that

(2.14)

Therefore,

(2.15)

Using that the duality map is single valued and weakly sequentially continuous from to , by (2.15), we get that . It is a contradiction. Hence, we have .

We next prove that solves the variational inequality (2.3). Since

(2.16)

we derive that

(2.17)

Notice

(2.18)

It follows that, for ,

(2.19)

Now replacing in (2.19) with and letting , noticing for , we obtain . That is, is a solution of (2.3); Hence by uniqueness. In a summary, we have shown that each cluster point of (at ) equals , therefore, as .

Lemma 2.10 . (see, e.g., Mitrinović [19, page 63]).

Let . Then the following inequality holds:

(2.20)

for arbitrary positive real numbers , .

Lemma 2.11.

Let be a -uniformly smooth Banach space which admits a weakly sequentially continuous duality mapping from to and be a nonempty convex subset of . Assume that is a countable family of -strict pseudocontraction for some and such that . Assume that is a positive sequence such that . Then is a -strict pseudocontraction with and .

Proof.

Let

(2.21)

and . Then, is a -strict pseudocontraction with . Indeed, we can firstly see the case of .

(2.22)

which shows that is a -strict pseudocontraction with . By the same way, our proof method easily carries over to the general finite case.

Next, we prove the infinite case. From the definition of -strict pseudocontraction, we know

(2.23)

Hence, we can get

(2.24)

Taking , from (2.24), we have

(2.25)

Consquently, for all , if , and , then strongly converges. Let

(2.26)

we have

(2.27)

Hence,

(2.28)

So, we get is -strict pseudocontraction.

Finally, we show . Suppose that , it is sufficient to show that . Indeed, for , we have

(2.29)

where . Hence, for each , this means that .

3. Main Results

Lemma 3.1.

Let be a real -uniformly smooth, strictly convex Banach space and be a closed convex subset of such that . Let be also a sunny nonexpansive retraction of . Let be a MKC. Let be a strongly positive linear bounded operator with the coefficient such that and be -strictly pseudo-contractive non-self-mapping such that . Let . Let be a sequence of generated by (1.12) with the sequences , and in , assume for each , be an infinity sequence of positive number such that for all and . The following control conditions are satisfied

(i),

(ii), for some and for all ,

(iii), ,

(iv).

Then, .

Proof.

Write, for each , . By Lemma 2.11, each is a -strict pseudocontraction on and for all and the algorithm (1.12) can be rewritten as

(3.1)

The rest of the proof will now be split into two parts.

Step 1.

First, we show that sequences and are bounded. Define a mapping

(3.2)

Then, from the control condition (ii), Lemmas 2.5 and 2.6, we obtain is nonexpansive. Taking a point , by Lemma 2.4,   we can get . Hence, we have

(3.3)

From definition of MKC and Lemma 2.8, for each there is a number , if then ; If then . It follow (3.1)

(3.4)

By induction, we have

(3.5)

which gives that the sequence is bounded, so are and .

Step 2.

In this part, we shall claim that , as . From (3.1), we get

(3.6)

Define

(3.7)

where

(3.8)

It follows that

(3.9)

which yields that

(3.10)

Next, we estimate . Notice that

(3.11)

Substituting (3.11) into (3.10), we have

(3.12)

Hence, we have

(3.13)

Observing conditions (i), (iii), (iv), and the boundedness of , , , , it follows that

(3.14)

Thus by Lemma 2.1, we have .

From (3.7), we have

(3.15)

Therefore,

(3.16)

Theorem 3.2.

Let be a real -uniformly smooth, strictly convex Banach space which admits a weakly sequentially continuous duality mapping from to and be a closed convex subset of which be also a sunny nonexpansive retraction of such that . Let be a MKC. Let be a strongly positive linear bounded operator with the coefficient such that and be -strictly pseudo-contractive non-self-mapping such that . Let . Let be a sequence of generated by (1.12) with the sequences , and in , assume for each , for all and for all . They satisfy the conditions (i), (ii), (iii), (iv) of Lemma 3.1 and (v) and . Then converges strongly to , which also solves the following variational inequality

(3.17)

Proof.

From (3.1), we obtain

(3.18)

So , which together with the condition (i), (iv) and Lemma 3.1 implies

(3.19)

Define , then is a -strict pseudocontraction such that by Lemma 2.11, furthermore as for all . Defines by

(3.20)

Then, is nonexpansive with by Lemma 2.5. It follows from Lemma 2.4 that . Notice that

(3.21)

which combines with (3.19) yielding that

(3.22)

Next, we show that

(3.23)

where with being the fixed point of the contraction

(3.24)

To see this, we take a subsequence of such that

(3.25)

We may also assume that . Note that in virtue of Lemma 2.3 and (3.22). It follow from the Lemma 2.9 and is weak weakly sequentially continuous duality mapping that

(3.26)

Hence, we have

(3.27)

Finally, We show . By contradiction, there is a number such that

(3.28)

Case 1.

Fixed (), if for some such that , and for the other such that .

Let

(3.29)

From (3.23), we know . Hence, there is a number , when , we have . We extract a number stastifying , then we estimate .

(3.30)

which implies that

(3.31)

Hence, we have

(3.32)

In the same way, we can get

(3.33)

It contradict the .

Case 2.

Fixed (), if for all , from Lemma 2.8, there is a number , such that

(3.34)

It follow (3.1) that

(3.35)

which implies that

(3.36)

Apply Lemma 2.2 to (3.36) to conclude as . It contradict the . This completes the proof.

Corollary 3.3.

Let be a closed convex subset of a Hilbert space such that and with the coefficient . Let be a strongly positive linear bounded operator with the coefficient such that and be -strictly pseudo-contractive non-self-mapping such that . Let . Let be a sequence of generated by (1.12) with the sequences , , and in , assume for each , for all and for all . They satisfy the conditions (i), (ii), (iii), (iv) of Lemma 3.1 and (v) , and . Then converges strongly to , which also solves the following variational inequality

(3.37)

Remark 3.4.

We conclude the paper with the following observations.

(i)Theorem 3.2 improve and extends Theorem 3.1 of Zhang and Su [17], Theorem 1 of Yao et al. [11], and Theorem 2.2 of Cai and Hu [12]. Corollary 3.3 also improve and extend Theorem 2.1 of Choa et al. [20], Theorem 2.1 of Jung [21], Theorem 2.1 of Qin et al. [22] and includes those results as special cases. Especially, Our results extends above results form contractions to more general Meir-Keeler contraction (MKC, for short). Our iterative scheme studied in present paper can be viewed as a refinement and modification of the iterative methods in [12, 13, 17, 22]. On the other hand, our iterative schemes concern an infinite countable family of -strict pseudocontractions mappings, in this respect, they can be viewed as an another improvement.

(ii)The advantage of the results in this paper is that less restrictions on the parameters , , and are imposed. Our results unify many recent results including the results in [12, 17, 22].

(iii)It is worth noting that we obtained two strong convergence results concerning an infinite countable family of -strict pseudocontractions mappings. Our result is new and the proofs are simple and different from those in [11, 12, 17, 19–25].