#### Abstract

Let be a real Banach space, and a closed convex nonempty subset of . Let be total asymptotically nonexpansive mappings. A simple iterative sequence is constructed in and necessary and sufficient conditions for this sequence to converge to a common fixed point of are given. Furthermore, in the case that is a uniformly convex real Banach space, strong convergence of the sequence to a common fixed point of the family is proved. Our recursion formula is much simpler and much more applicable than those recently announced by several authors for the same problem.

#### 1. Introduction

Let be a nonempty subset of a normed real linear space A mapping is said to be *nonexpansive* if for all

The mapping is called *asymptotically nonexpansive* if there exists a sequence with such that for all

The mapping is called *uniformly **-Lipschitzian* if there exists a constant such that for all

The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [1] as a generalization of the class of nonexpansive mappings. They proved that if is a nonempty closed convex bounded subset of a uniformly convex real Banach space and is an asymptotically nonexpansive self-mapping of then has a fixed point.

A mapping is said to be *asymptotically nonexpansive in the intermediate sense* (see, e.g., [2]) if it is continuous and the following inequality holds:

Observe that if we define

then as and (1.3) reduces to

The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [2]. It is known [3] that if is a nonempty closed convex bounded subset of a uniformly convex real Banach space and is a self-mapping of which is asymptotically nonexpansive in the intermediate sense, then has a fixed point. It is worth mentioning that the class of mappings which are asymptotically nonexpansive in the intermediate sense contains properly the class of asymptotically nonexpansive mappings (see, e.g., [4]).

Sahu [5], introduced the class of nearly Lipschitzian mappings. Let be a nonempty subset of a normed space and let be a sequence in such that A mapping is called *nearly Lipschitzian* with respect to if for each there exists such that

Define

Observe that for any sequence satisfying (1.6), and that

is called the *nearly Lipschitz constant.* A nearly Lipschitzian mapping is said to be

*nearly contraction*if for all (ii)

*nearly nonexpansive*if for all (iii)

*nearly asymptotically nonexpansive*if for all and (iv)

*nearly uniform*

*-Lipschitzian*if for all (v)

*nearly uniform*

*-contraction*if for all

*Example 1.1. *Let Define by
It is obvious that is not continuous, and thus, not Lipschitz. However, is nearly nonexpansive. In fact, for a real sequence with and as we have
This is because

*Remark 1.2. *If is a bounded domain of an asymptotically nonexpansive mapping then is nearly nonexpansive. In fact, for all and we have
Furthermore, we easily observe that every nearly nonexpansive mapping is nearly asymptotically nonexpansive with

*Remark 1.3. *If is a bounded domain of a nearly asymptotically nonexpansive mapping , then is asymptotically nonexpansive in the intermediate sense. To see this, let be a nearly asymptotically nonexpansive mapping. Then,
which implies that
Hence,

We observe from Remarks 1.2 and 1.3 that the classes of nearly nonexpansive mappings and nearly asymptotically nonexpansive mappings are intermediate classes between the class of asymptotically nonexpansive mappings and that of asymptotically nonexpansive in the intermediate sense mappings.

The main tool for approximation of fixed points of generalizations of nonexpansive mappings remains *iterative technique.* Several authors have studied approximation of fixed points of generalizations of nonexpansive mappings using Mann and Ishikawa iterative methods (see, e.g., [6–19]).

Bose [20] proved that if is a nonempty closed convex bounded subset of a uniformly convex real Banach space satisfying Opial's condition [21] (i.e., for all sequences in such that converges weakly to some the inequality holds for all in ) and is an *asymptotically nonexpansive mapping,* then the sequence converges *weakly* to a fixed point of provided that is *asymptotically regular at * that is, the limit

holds. Passty [13] and also Xu and Noor [22] showed that the requirement of Opial's condition can be replaced by the Fréchet differentiability of the space norm. Furthermore, Tan and Xu [23, 24] established that the asymptotic regularity of at a point can be weakened to the so-called *weakly asymptotic regularity of ** at * defined as follows: is weakly asymptotic regular at if

holds, where denotes the weak limit.

In [17, 18], Schu introduced a modified Mann iteration scheme for approximation of fixed points of asymptotically nonexpansive self-mappings defined on a nonempty closed convex and bounded subset of a uniformly convex real Banach space He proved that the iterative sequence generated by

converges *weakly* to some fixed point of if Opial's condition holds, for all is a real sequence such that for some positive constants and Neither condition (1.15) nor condition (1.16) is required with Schu's scheme. Schu's result, however, does not apply, for instance, to spaces with because none of these spaces satisfies Opial's condition.

Rhoades [15] obtained a *strong* convergence theorem for asymptotically nonexpansive mappings in uniformly convex real Banach spaces using the modified Ishikawa-type iteration method. Osilike and Aniagbosor proved in [12] that the results of [15, 17, 18] remain true without the boundedness requirement imposed on provided that Tan and Xu [25] extended the theorem of Schu [18] to uniformly convex Banach space with a Fréchet differentiable norm without assuming that the space satisfies Opial's condition. Thus, their result applies to spaces with

Chang et al*.* [26] established *weak* convergence theorems for asymptotically nonexpansive mappings in Banach spaces without assuming any of the following conditions: (i) satisfies the Opial's condition; (ii) T is asymptotically regular or weakly asymptotically regular; (iii) is bounded. Their results improve and generalize the corresponding results of Bose [20], Górnicki [27], Passty [13], Schu [18],Tan and Xu [23–25], Xu and Noor [22], and many others.

G. E. Kim and T. H. Kim [4] studied the strong convergence of the Mann and Ishikawa-type iteration methods *with errors* for mappings which are asymptotically nonexpansive in the intermediate sense in real Banach spaces.

In all the above papers, the mapping remains *a self-mapping* of nonempty closed convex subset of a uniformly convex real Banach space If, however, the domain of is a *proper* subset of then the Mann and Ishikawa-type iterative processes and Schu's modifications of type (1.17) may fail to be well defined.

Chidume et al. [28] proved convergence theorems for asymptotically nonexpansive *nonself-mappings* in Banach spaces and extended the corresponding results of [12, 15, 17, 18, 26].

Alber et al. [29] introduced a more general class of asymptotically nonexpansive mappings called *total asymptotically nonexpansive mappings* and studied methods of approximation of fixed points of mappings belonging to this class.

*Definition 1.4. *A mapping is said to be *total asymptotically nonexpansive* if there exist nonnegative real sequences and with as and strictly increasing continuous function with such that for all

*Remark 1.5. *If then (1.18) reduces to
In addition, if for all then total asymptotically nonexpansive mappings coincide with asymptotically nonexpansive mappings. If and for all we obtain from (1.18) the class of mappings that includes the class of nonexpansive mappings. If and where for all then (1.18) reduces to (1.5) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.

The idea of Definition 1.4 is to unify various definitions of classes of mappings associated with the class of asymptotically nonexpansive mappings and to prove a general convergence theorems applicable to all these classes of nonlinear mappings.

Another class of nonlinear mappings introduced as a further generalization of nonexpansive mappings with nonempty fixed point sets is the class of asymptotically quasi-nonexpansive mappings which properly contains the class of asymptotically nonexpansive operators with nonempty fixed point sets (see, e.g., [8, 16, 30–33]).

A mapping is said to be *quasi-nonexpansive* if and
is called *asymptotically quasi-nonexpansive* if and there exists a sequence with such that for all and

is said to be *asymptotically quasi-nonexpansive in intermediate sense* if it is continuous and

*Remark 1.6. *Observe that if we define
then as and (1.22) reduces to

Existence theorems for common fixed points of certain families of nonlinear mappings have been established by various authors (see, e.g., [2, 34–37]).

Within the past 30 years or so, research on iterative approximation of common fixed points of generalizations of nonlinear nonexpansive mappings surged. Considerable research efforts have been devoted to developing iterative methods for approximating common fixed points (when they exist) of finite families of this class of mappings (see, e.g., [33, 38–46]).

In [16], Shahzad and Udomene established necessary and sufficient conditions for convergence of Ishikawa-type iteration sequences involving two asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings in arbitrary real Banach spaces. They also established a sufficient condition for the convergence of the Ishikawa-type iteration sequences involving two uniformly continuous asymptotically quasi-nonexpansive mappings to a common fixed point of the mappings in real uniformly convex Banach spaces.

Recently, Chidume and Ofoedu [47] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces. More precisely, they proved the following theorems.

*Theorem CO1*

Let be a real Banach space, let be a nonempty closed convex subset of , and be total asymptotically nonexpansive mappings with sequences such that Let be given by
Suppose and suppose that there exist such that for all Then the sequence is bounded and exists, . Moreover, the sequence converges strongly to a common fixed point of if and only if where

*Theorem CO2*

Let be a uniformly convex real Banach space, let be a nonempty closed convex subset of and be uniformly continuous total asymptotically nonexpansive mappings with sequences such that and Let for some From arbitrary define the sequence by (1.25). Suppose that there exist such that whenever and that one of is compact, then converges strongly to some

It is our purpose in this paper to construct *a new iterative sequence* much simpler than (1.25) for approximation of common fixed points of finite families of total asymptotically nonexpansive mappings and give necessary and sufficient conditions for the convergence of the scheme to common fixed points of the mappings in arbitrary real Banach spaces. A sufficient condition for convergence of the iteration process to a common fixed point of mappings under our setting is also established in uniformly convex real Banach spaces. Our theorems unify, extend and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced for the approxima tion of common fixed points of finite families of several classes of nonlinear mappings. Our iteration process is also of independent interest.

#### 2. Preliminary

In the sequel, we shall need the following lemmas.

Lemma 2.1. *Let , , and be sequences of nonnegative real numbers such that
**
Suppose that and Then is bounded and exists. Moreover, if in addition, then *

Lemma 2.2 (Zeidler [48, pages 484-485]). *Let be a uniformly convex real Banach space and . Suppose that and are two sequences of such that
**
hold for some , then .*

#### 3. Main Results

Let be a nonempty closed convex subset of a real normed space Let be total asymptotically nonexpansive mappings. We define the iterative sequence by

where are sequences in such that .

We now state and prove our main theorems.

Theorem 3.1. *Let be a real Banach space, let be a nonempty closed convex subset of and let be total asymptotically nonexpansive mappings with sequences such that Let be given by (3.1). Suppose and suppose that there exist such that for all Then the sequence is bounded and exists, .*

*Proof. *Let Then we have from (3.1) that
Since is an increasing function, it follows that whenever and (by hypothesis) if In either case, we have
for some Thus,
for some constant Hence,
where and Observe that and So, from (3.5) and by Lemma 2.1 we obtain that the sequence is bounded and that exists. This completes the proof.

##### 3.1. Necessary and Sufficient Conditions for Convergence in Real Banach Spaces

Theorem 3.2. *Let be a real Banach space, let be a nonempty closed convex subset of and let be continuous total asymptotically nonexpansive mappings with sequences such that Let be given by (3.1). Suppose and suppose that there exist such that for all Then the sequence converges strongly to a common fixed point of if and only if where *

*Proof. *It suffices to show that implies that converges to a common fixed point of *Necessity*

Since (3.5) holds for all we obtain from it that
Lemma 2.1 then implies that exists. But, Hence,*Sufficiency*

Next, we first show that is a Cauchy sequence in For all integer we obtain from inequality (3.5) that
so that for all integers and all
We therefore have that
for some constant Taking infimum over in (3.9) gives
Now, since and given there exists an integer such that for all and So for all integers we obtain from (3.10) that
Hence, is a Cauchy sequence in and since is complete there exists such that as We now show that is a common fixed point of that is, we show that Suppose for contradiction that (where denotes the complement of ). Since is a closed subset of (recall each is continuous), we have that But, for all we have
This implies
so that as we obtain which contradicts Thus, is a common fixed point of This completes the proof.

*Remark 3.3. *If are asymptotically nonexpansive mappings, then for all and so that the assumption that there exist such that for all in the above theorems is no longer needed.

Thus, we have the following corollary.

Corollary 3.4. *Let be a real Banach space, let be a nonempty closed convex subset of and let be continuous asymptotically nonexpansive mappings with sequences such that Let be given by (3.1). Suppose Then the sequence is bounded and exists, Moreover, converges strongly to a common fixed point of if and only if *

##### 3.2. Convergence Theorem in Real Uniformly Convex Banach Spaces

Theorem 3.5. *Let be a uniformly convex real Banach space, be a nonempty closed convex subset of and be uniformly continuous total asymptotically nonexpansive mappings with sequences such that and From arbitrary define the sequence by (3.1). Suppose that there exist such that whenever Then .*

*Proof. *Let Then, by Theorem 3.1, exists. Let If then by continuity of we are done. Now suppose We show that We observe that
so that taking on both sides of this inequality, we obtain
Let be such that as and define
Then,
for some Thus,
Furthermore,
This implies that
But,
So,
Hence, by Lemma 2.2, we obtain
This completes the proof.

Theorem 3.6. *Let be a uniformly convex real Banach space, let be a nonempty closed convex subset of and let be uniformly continuous total asymptotically nonexpansive mappings with sequences such that and From arbitrary define the sequence by (1.24). Suppose that there exist such that whenever and that one of is compact, then converges strongly some *

*Proof. *We obtain from Theorem 3.5 that
Using the recursion formula (3.1), we observe that
It then follows from (3.24) and (3.25) that
Without loss of generality, let be compact. Since is continuous and compact, it is completely continuous. Thus, there exists a subsequence of such that as for some Thus as and from (3.24), we have that Also from (3.24) as Thus, as Now, since from (3.26), as it follows that as Next, we show that Observe that
Taking limit as and using the fact that are uniformly continuous we have that and so But by Theorem 3.1, exists, Hence, converges strongly to This completes the proof.

In view of Remark , the following corollary is now obvious.

Corollary 3.7. *Let be a uniformly convex real Banach space, let be a nonempty closed convex subset of and let be asymptotically nonexpansive mappings with sequences such that ; and that one of is compact. From arbitrary define the sequence by (3.1). Then converges strongly to to some *

*Remark 3.8. *Observe that the theorems of this paper remain true for mappings satisfying (1.5) provided that In this case, the requirement that there exist such that for all is not needed.

*Remark 3.9. *A prototype for satisfying the conditions of our theorems is Prototypes for the sequences in this paper are the following:

*Remark 3.10. *Addition of *bounded* (or the so called mean) error terms to the iteration process studied in this paper leads to no further generalization.

*Definition 3.11. *A mapping is said to be *total asymptotically quasi-nonexpansive* if and there exist nonnegative real sequences and with as and strictly increasing continuous function with such that for all

*Remark 3.12. *If then (3.29) reduces to
In addition, if for all then total asymptotically quasi-nonexpansive mappings coincide with asymptotically quasi-nonexpansive mappings studied by various authors. If and for all we obtain from (3.30) the class of quasi-nonexpansive mappings. Observe that the class of total asymptotically nonexpansive mappings with nonempty fixed point sets belongs to the class of total asymptotically quasi-nonexpansive mappings. Moreover, if and then (3.29) reduces to (1.24).

It is trivial to observe that all the theorems of this paper carry over to the class of total asymptotically quasi-nonexpansive mappings with little or no modifications.

A subset of a real normed linear space is said to be a retract of if there exists a continuous map such that for all It is well known (see, e.g., [28]) that every closed convex nonempty subset of a uniformly convex Banach space is a retract. A map is said to be a retraction if It follows that if a map ia a retraction, then for all in the range of The mapping is called a sunny nonexpansive retraction if for all and

*Definition 3.13. *Let be a nonempty closed and convex subset of Let be the nonexpansive retraction of onto A *nonself* map is said to be *total asymptotically nonexpansive* if there exist sequences in with as and a strictly increasing continuous function with such that for all
Let be total asymptotically nonexpansive *nonself* maps; assuming existence of common fixed points of these operators, our theorems and method of proof easily carry over to this class of mappings using the iterative sequence defined by
instead of (3.1) provided that the well definedness of as a sunny nonexpansive retraction is guaranteed.

*Remark 3.14. *It is clear that the recursion formula (3.1) introduced and studied in this paper is much simpler than the recursion formulas (1.25) studied earlier for this problem.

*Remark 3.15. *Our theorems unify, extend, and generalize the corresponding results of Alber et al. [29], Sahu [5], Shahzad and Udomene [16], and a host of other results recently announced (see, e.g., [8, 16, 22, 28, 30, 39, 40, 44, 47, 49–58]) for the approximation of common fixed points of finite families of several classes of nonlinear mappings.

#### Acknowledgment

This author's research was supported by the Japanese Mori Fellowship of UNESCO at The Abdus Salam International Centre for Theoretical Physics, Trieste, Italy.