# On Unification of the Strong Convergence Theorems for a Finite Family of Total Asymptotically Nonexpansive Mappings in Banach Spaces

Authors: Farrukh Mukhamedov; Mansoor Saburov
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

We unify all known iterative methods by introducing a new explicit iterative scheme for approximation of common fixed points of finite families of total asymptoticallyI-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced by (C. E. Chidume and E. U. Ofoedu, 2009). We construct examples of total asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Note that no such kind of examples were known in the literature. We prove the strong convergence theorems for such iterative process to a common fixed point of the finite family of total asymptotically I-nonexpansive and total asymptotically nonexpansive mappings, defined on a nonempty closed-convex subset of uniformly convex Banach spaces. Moreover, our results extend and unify all known results.


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## Body

## 1. Introduction

LetK be a nonempty subset of a real normed linear space X, and let T:K→K be a mapping. Denote by F(T) the set of fixed points of T, that is, F(T)={x∈K:Tx=x}. Throughout this paper, we always assume that X is a real Banach space and F(T)≠∅. Now let us recall some known definitions.Definition 1.1.
A mappingT:K→K is said to be(i)
nonexpansive if∥Tx-Ty∥≤∥x-y∥ for all x,y∈K,(ii)
asymptotically nonexpansive if there exists a sequence{λn}⊂[1,∞) with limn→∞λn=1 such that ∥Tnx-Tny∥≤λn∥x-y∥ for all x,y∈K and n∈ℕ,(iii)
asymptotically nonexpansive in the intermediate sense, if it is continuous and the following inequality holds:(1.1)limsup⁡n→∞sup⁡x,y∈K(‖Tnx-Tny‖-‖x-y‖)≤0.Remark 1.2.
Observe that if we define(1.2)an:=sup⁡x,y∈K(‖Tnx-Tny‖-‖x-y‖),σn:=max⁡{0,an},
then σn→0 as n→∞, and (1.1) reduces to
(1.3)‖Tnx-Tny‖≤‖x-y‖+σn,∀x,y∈K,n≥1.In [1, 2], Browder studied the iterative construction for fixed points of nonexpansive mappings on closed and convex subsets of a Hilbert space. Note that for the past 30 years or so, the study of the iterative processes for the approximation of fixed points of nonexpansive mappings and fixed points of some of their generalizations have been flourishing areas of research for many mathematicians (see for more details [3, 4]).The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [5] as a generalization of the class of nonexpansive mappings. They proved that if K is a nonempty closed-convex bounded subset of a uniformly convex real Banach space and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.The class of mappings which are asymptotically nonexpansive in the intermediate sense was introduced by Bruck et al. [6]. It is known [7] that if K is a nonempty closed-convex bounded subset of a uniformly convex Banach space X and T:K→K is an asymptotically nonexpansive mapping in the intermediate sense, then T 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., [8]).The iterative approximation problems for nonexpansive mapping, asymptotically nonexpansive mapping, and asymptotically nonexpansive mapping in the intermediate sense were studied extensively in [5–20].There are many different types of concepts which generalize a notion of nonexpansive mapping. One of such concepts is a total asymptotically nonexpansive mapping [21], and second one is an asymptotically I-nonexpansive mapping [22]. Let us recall some notions.Definition 1.3.
LetK be a nonempty closed subset of a real normed linear space X.T:K→K is called a total asymptotically nonexpansive mapping if there exist nonnegative real sequence {μn} and {λn} with μn,λn→0 as n→∞ and strictly increasing continuous function ϕ:ℝ+→ℝ+ with ϕ(0)=0 such that for all x,y∈K,
(1.4)‖Tnx-Tny‖≤‖x-y‖+μnϕ(‖x-y‖)+λn,n≥1.Remark 1.4.
Ifϕ(ξ)=ξ, then (1.4) reduces to
(1.5)‖Tnx-Tny‖≤(1+μn)‖x-y‖+λn,n≥1.
In addition, if λn=0 for all n≥1, then total asymptotical nonexpansive mappings coincide with asymptotically nonexpansive mappings. If μn=0 and λn=0 for all n≥1, we obtain from (1.5) the class of mappings that includes the class of nonexpansive mappings. If μn=0 and λn=σn=max{0,an}, where an:=supx,y∈K(∥Tnx-Tny∥-∥x-y∥) for all n≥1, then (1.5) reduces to (1.3) which has been studied as mappings asymptotically nonexpansive in the intermediate sense.The idea of the definition of a total asymptotically nonexpansive mappings 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.Alber et al. [21] studied methods of approximation of fixed points of total asymptotically nonexpansive mappings.  C. E. Chidume and E. U. Ofoedu  [23] introduced an iterative scheme for approximation of a common fixed point of a finite family of total asymptotically nonexpansive mappings in Banach spaces. Recently,  C. E. Chidume and E. U. Ofoedu [24] constructed a new iterative sequence much simpler than other types of approximation of common fixed points of finite families of total asymptotically nonexpansive mappings.On the other hand, in [22] an asymptotically I-nonexpansive mapping was introduced.Definition 1.5.
LetT:K→K, I:K→K be two mappings of a nonempty subset K of a real normed linear space X, then T is said to be(i)
I-nonexpansive if ∥Tx-Ty∥≤∥Ix-Iy∥ for all x,y∈K,(ii)
asymptoticallyI-nonexpansive, if there exists a sequence {λn}⊂[1,∞) with limn→∞λn=1 such that ∥Tnx-Tny∥≤λn∥Inx-Iny∥ for all x,y∈K and n≥1.Best approximation properties ofI-nonexpansive mappings were investigated in [22, 25]. In [26], strong convergence of Mann iterations of I-nonexpansive mapping has been proved. In [27], the weak convergence of three-step Noor iterative scheme for an I-nonexpansive mapping in a Banach space has been established. In [28], the weakly convergence theorem for asymptotically I-nonexpansive mapping defined in Hilbert space was proved. Recently, in [29–31], the weak and strong convergence of explicit and implicit iteration process to a common fixed point of a finite family of asymptotically I-nonexpansive mappings have been studied.In this paper, we introduce a new type of concept of a generalization of nonexpansive mapping’s nation, which is a combination of Definitions1.3 and 1.5.Definition 1.6.
LetT:K→K, I:K→K be two mappings of a nonempty subset K of a real normed linear space X, then T is said to be a total asymptotically I-nonexpansive mapping if there exist nonnegative real sequences {μn} and {λn} with μn,λn→0 as n→∞ and the strictly increasing continuous function ϕ:ℝ+→ℝ+ with ϕ(0)=0 such that for all x,y∈K,
(1.6)‖Tnx-Tny‖≤‖Inx-Iny‖+μnϕ(‖Inx-Iny‖)+λn,n≥1.Now let us provide an example of a total asymptoticallyI-nonexpansive mapping, which is not asymptotically nonexpansive mapping.Example 1.7.
Let us consider the spaceℓ1, and let B1={x∈ℓ1:∥x∥1≤1}. Define a nonlinear operator T:ℓ1→ℓ1 by
(1.7)Tα(x1,x2,…,xn,…)=(0,α|x1|,αx2,…,αxn,…),α∈(0,1).
Let ∥x∥1≤1, then from
(1.8)‖Tα(x)‖1=α(‖x‖1-|x1|+|x1|)≤α(1-|x1|+|x1|)≤1,
one gets T(B1)⊂B1.
One can find that(1.9)Tαk(x1,x2,…,xn,…)=(0,…,0︸k,αk|x1|,αkx2,…,αkxn,…).
Hence,
(1.10)‖Tαk(x)-Tαk(y)‖1=αk(‖x-y‖1+||x1|-|y1||-|x1-y1|).
From x,y∈B1, we have
(1.11)||x1|-|y1||≤||x1|-|y1||≤‖x-y‖1.
So, it follows from (1.10) and (1.11) that
(1.12)‖Tαk(x)-Tαk(y)‖1≤αk(‖x-y‖1+‖x-y‖1)∀x,y∈B1,k∈N.
Now consider a new Banach spaceℝ×ℓ1 with a norm ∥𝕏∥=|x|+∥x∥1, where 𝕏=(x,x) and define a new mapping S:ℝ×ℓ1→ℝ×ℓ1 by
(1.13)S(x,x)=(x,Tα(x)).
Let K=[0,1]×B1, then it is clear that S(K)⊂K. One can see that Sk(x,x)=(x,Tαk(x)). Therefore, using (1.14), we obtain
(1.14)‖Sk(X)-Sk(Y)‖=|x-y|+‖Tαk(x)-Tαk(y)‖1≤|x-y|+αk(‖x-y‖1+‖x-y‖1)≤‖X-Y‖+αk(‖X-Y‖+‖X-Y‖).
We let ϕ(t)=t+t and μk=αk. It is clear that ϕ(0)=0 and ϕ is strictly increasing, and moreover, (1.14) implies
(1.15)‖Sk(X)-Sk(X)‖≤‖IX-IY‖+μkϕ(‖IX-IY‖),
that is S is a totally asymptotically I-nonexpansive mapping. Here, I is the identity mapping of ℝ×ℓ1.
Now we are going to show thatS is not asymptotically nonexpansive. Namely, we will establish that for any sequence of positive numbers {λn} with λn→0 and any k∈ℕ, one can find 𝕏0,𝕐0 such that
(1.16)‖Sk(X0)-Sk(Y0)‖>(1+λk)‖X0-Y0‖.
In fact, choose 𝕏0, 𝕐0 as follows:
(1.17)X0=(0,x0),Y0=(0,y0),
where
(1.18)x0=(x0,0,…,0,…),y0=(x04,0,…,0,…),0<x0<4α2k9(1+λk)2.
From (1.10), one finds that
(1.19)‖Sk(X0)-Sk(Y0)‖=αk|x0-x02|=αkx02,‖X0-Y0‖=3x04.
The last equalities with (1.18) imply that
(1.20)‖Sk(X0)-Sk(Y0)‖‖X0-Y0‖=2αk3x0>1+λk.
This yields the required assertion. Note that S has infinitely many fixed points in K, that is, Fix(S)={(x,0):x∈[0,1]}.Example 1.8.
Let us consider the Banach spaceℝ×ℓ1 defined as before, and let f be a mapping of a segment C⊂ℝ to itself, that is, f:C→C with f(0)=0 and
(1.21)|fn(x)-fn(y)|≤|x-y|+cn,cn>0,n∈N,
where cn→0. Note that such kind of functions do exist. One can take (see for more details [8]) C=[-1/π,1/π] and
(1.22)fκ(x)={κxsin⁡⁡1x,x≠0,κ∈(0,1).0,x=0.
Define a new mappingSf:C×B1→C×B1 by
(1.23)Sf(x,x)=(f(x),Tα(x)),
here T is defined as above (see (1.7)). Using the same argument as the above Example 1.7, we can establish that
(1.24)‖Sfk(X)-Sfk(X)‖≤‖X-Y‖+μkϕ(‖X-Y‖)+ck∀k∈N.
Moreover, such a mapping is not asymptotically nonexpansive. Note that the mapping Sfκ with the function fκ has a unique fixed point in C×B1.Remark 1.9.
To the best our knowledge, we should stress that the constructed examples are currently only unique examples of totaly asymptotically nonexpansive mappings which are not asymptotically nonexpansive. Before, no such examples were known in the literature.The aim of the present paper is unification of all known iterative methods by introducing a new iterative scheme for approximation of common fixed points of finite families of total asymptoticallyI-nonexpansive mappings. Note that such a scheme contains a particular case of the method introduced in [24] and allows us to construct more simpler methods than [23, 24].Namely, letK be a nonempty closed-convex subset of a real Banach space X and {Ti}i=1m: K→K be a finite family of total asymptotically Ii-nonexpansive mappings, that is,
(1.25)‖Tinx-Tiny‖≤‖Iinx-Iiny‖+μinϕi(‖Iinx-Iiny‖)+λin,
and {Ii}i=1m:K→K is a finite family of total asymptotically nonexpansive mappings, that is,
(1.26)‖Iinx-Iiny‖≤‖x-y‖+μ̃inφi(‖x-y‖)+λ̃in,
here ϕi,φi:ℝ+→ℝ+ are the strictly increasing continuous functions with ϕi(0)=φi(0)=0 for all i=1,m¯, and {μin}n=1∞,{λin}n=1∞,{μ̃in}n=1∞,{λ̃in}n=1∞ are nonnegative real sequences with μin,λin,μ̃in,λ̃in→0 as n→∞ for all i=1,m¯. Then for given sequences {αjn}n=1∞,{βjn}n=1∞ in (0,1), where j=0,m¯, we will consider the following explicit iterative process:(1.27)x0∈K,xn+1=α0nxn+∑i=1mαinTinyn,yn=β0nxn+∑i=1mβinIinxn,
such that ∑j=0mαjn=1 and ∑j=0mβjn=1.C. E. Chidume and E. U. Ofoedu [24] have considered only a particular case of the explicit iterative process (1.27), in which {Ii}i=1m is to be taken as the identity mappings. One of the main results of ([24], see Theorem 3.5, page 11) was correct, while the provided proof of that result was wrong. Since, in their proof, they used Lemma 2.3, but which actually is not applicable in that situation, the sequence {tn}n=1∞ tends to 0. As a counterexample, we can consider the following one: let x∈X,∥x∥=d>0, and let the sequences xn, yn, and tn be defined as follows:(1.28)xn=x,yn=-x,tn=1n,∀n∈N.
It is then clear that(1.29)lim⁡n→∞‖tnxn+(1-tn)yn‖=‖x‖lim⁡n→∞|1-2n|=d.
However,(1.30)lim⁡n→∞‖xn-yn‖=2d>0.In this paper, we shall provide a correct proof of Theorem 3.5 page 11 in [24]. As we already mentioned in Lemma 2.3 is not applicable the main result of [24]. Therefore, we first will generalize Lemma 2.3 to the case of finite number of sequences. Such a generalization gives us a possibility to prove the mentioned result. On other hand, the provided generalization presents an independent interest as well. Moreover, we extend and unify the main result of [24] for a finite family of total asymptoticallyIi-nonexpansive mappings {Ti}i=1m. Namely, we shall prove the strong convergence of the explicit iterative process (1.27) to a common fixed point of the finite family of total asymptoticallyIi-nonexpansive mappings {Ti}i=1m and the finite family of total asymptotically nonexpansive mappings {Ii}i=1m. Here, we stress that Lemmas 3.1 and 3.2 play a crucial role. All presented results here extend, generalize, unify, and improve the corresponding main results of [21, 24, 29–33].


## 2. Preliminaries

Throughout this paper, we always assume thatX is a real Banach space. The following lemmas play an important role in proving our main results.Lemma 2.1 (see [16]).
Let{an}, {bn}, and {cn} be three sequences of nonnegative real numbers with ∑n=1∞bn<∞, ∑n=1∞cn<∞. If the following condition is satisfied:(i)
an+1≤(1+bn)an+cn,n≥1, then the limit limn→∞an exists.Lemma 2.2 (see [34]).
LetX be a uniformly convex Banach space and t∈(0,1). Suppose that {xn},{yn} are two sequences in X such that
(2.1)lim⁡n→∞‖txn+(1-t)yn‖=d,limsup⁡n→∞‖xn‖≤d,limsup⁡n→∞‖yn‖≤d
hold some d≥0, then limn→∞∥xn-yn∥=0.Lemma 2.3 (see [14]).
LetX be a uniformly convex Banach space, and let b, c be two constants with 0<b<c<1. Suppose that {tn} is a sequence in [b,c] and {xn},{yn} are two sequences in X such that
(2.2)lim⁡n→∞‖tnxn+(1-tn)yn‖=d,limsup⁡n→∞‖xn‖≤d,limsup⁡n→∞‖yn‖≤d,
hold some d≥0, then limn→∞∥xn-yn∥=0.


## 3. Main Results

In this section, we shall prove our main results. To formulate ones, we need some auxiliary results.First we are going to generalize Lemmas2.2 and 2.3 for m number of sequences {zin}n=1∞ from the uniformly convex Banach space X, where i=1,m¯.Lemma 3.1.
LetX be a uniformly convex Banach space and αi∈(0,1),i=1,m¯ any constants with ∑i=1mαi=1. Suppose that {zin}n=1∞,i=1,m¯ are sequences in X such that
(3.1)lim⁡n→∞‖∑i=1mαizin‖=d,limsup⁡n→∞‖zin‖≤d,∀i=1,m¯
hold some d≥0, then limn→∞∥zin∥=d and limn→∞∥zin-zjn∥=0 for any i,j=1,m¯.Proof.
Let us first provelimn→∞∥zin∥=d for any i=1,m¯. Indeed, it follows from (3.1) that
(3.2)d=lim⁡n→∞‖∑k=1mαkzkn‖=liminf⁡n→∞‖∑k=1mαkzkn‖≤liminf⁡n→∞(∑k=1mαk‖zkn‖)≤αilim⁡inf⁡n→∞‖zin‖+∑k≠iαklimsup⁡n→∞‖zkn‖≤αilim⁡inf⁡n→∞‖zin‖+(1-αi)d.
We then get that liminfn→∞∥zin∥≥d, which means limn→∞∥zin∥=d.
Now we prove the statementlimn→∞∥zin-zjn∥=0 by means of mathematical induction with respect to m. For m=2, the statement immediately follows from Lemma 2.2. Assume that the statement is true, for m=k-1. Let us prove for m=k. To do this, denote
(3.3)tn=11-αk∑i=1k-1αizin.
Since (1/1-αk)∑i=1k-1αi=1, we get limsupn→∞∥tn∥≤d. On the other hand, one has
(3.4)d=lim⁡inf⁡n→∞‖∑i=1kαizin‖=lim⁡inf⁡n→∞‖(1-αk)tn+αkzkn‖≤(1-αk)lim⁡inf⁡n→∞‖tn‖+αklimsup⁡n→∞‖zkn‖≤(1-αk)lim⁡inf⁡n→∞‖tn‖+αkd.
We then obtain liminfn→∞∥tn∥≥d which means limn→∞∥tn∥=d. In this case, according to the assumption of induction with the sequence tn, we can conclude that limn→∞∥zin-zjn∥=0, if 1≤i,j≤k-1.
Sincelimn→∞∥(1-αk)tn+αkzkn∥=d due to Lemma 2.2, one gets
(3.5)lim⁡n→∞‖tn-zkn‖=0.
If 1≤j≤k-1, then the following inequality
(3.6)‖zjn-zkn‖≤‖zjn-tn‖+‖tn-zkn‖≤11-αk∑i=1k-1αi‖zin-zjn‖+‖tn-zkn‖
implies that limn→∞∥zjn-zkn∥=0. This completes the proof.Lemma 3.2.
LetX be a uniformly convex Banach space, and let α*,α* be two constants with 0<α*<α*<1. Suppose that {αin}n=1∞⊂[α*,α*], i=1,m¯ are any sequences with ∑i=1mαin=1 for all n∈ℕ. Suppose that {zin}n=1∞, i=1,m¯ are sequences in X such that
(3.7)lim⁡n→∞‖∑i=1mαinzin‖=d,limsup⁡n→∞‖zin‖≤d,∀i=1,m¯
hold for some d≥0, then limn→∞∥zin∥=d and limn→∞∥zin-zjn∥=0 for any i,j=1,m¯.Proof.
Analogously as in the proof of Lemma3.1, it is easy to show that limn→∞∥zin∥=d. Therefore, let us prove the statement limn→∞∥zin-zjn∥=0 for any i,j=1,m¯. Suppose to the contrary, that there exist two numbers i0,j0 such that
(3.8)limsup⁡n→∞‖zi0n-zj0n‖=βi0,j0>0
then there exists a subsequence {zi0nk-zj0nk}k=1∞ of {zi0n-zj0n}n=1∞ such that limk→∞∥zi0nk-zj0nk∥=βi0j0.
Let us consider the subsequences{αink}k=1∞ of {αin}n=1∞, here i=1,m¯. Since {αink}k=1∞⊂[α*,α*], there exists a subsequence {nkl}l=1∞ of {nk}k=1∞ such that liml→∞αinkl=αi for all i=1,m¯. Since ∑i=1mαin=1, for all n∈ℕ, one gets ∑i=1mαi=1, and αi∈[α*,α*], for all i=1,m¯. We know that
(3.9)d=lim⁡l→∞‖∑i=1mαinklzinkl‖=lim⁡inf⁡l→∞‖∑i=1m((αinkl-αi)zinkl+αizinkl)‖≤lim⁡inf⁡l→∞(∑i=1m|αinkl-αi|‖zinkl‖+‖∑i=1mαizinkl‖)≤∑i=1mlimsup⁡l→∞(|αinkl-αi|‖zinkl‖)+lim⁡inf⁡l→∞‖∑i=1mαizinkl‖.
It then follows that liminfl→∞∥∑i=1mαizinkl∥≥d. On the other hand, we have
(3.10)limsup⁡l→∞‖∑i=1mαizinkl‖≤∑i=1mαilimsup⁡l→∞‖zinkl‖≤d.
Therefore, liml→∞∥∑i=1mαizinkl∥=d. Consequently, Lemma 3.1 implies that liml→∞∥zi0nkl-zj0nkl∥=0. However, it contradicts to
(3.11)lim⁡l→∞‖zi0nkl-zj0nkl‖=lim⁡k→∞‖zi0nk-zj0nk‖=βi0j0>0.
This completes the proof.Proposition 3.3.
LetX be a real Banach space, and let K be a nonempty closed-convex subset of X. Let {Ti}i=1m: K→K be a finite family of total asymptotically Ii-nonexpansive mappings with sequences {μin}n=1∞, {λin}n=1∞, where i=1,m¯, and let {Ii}i=1m: K→K be a finite family of total asymptotically nonexpansive mappings with sequences {μ̃in}n=1∞, {λ̃in}n=1∞, where i=1,m¯. Suppose that there exist Mi,Mi*,Ni,Ni*>0, i=1,m¯ such that ϕi(ξi)≤Mi*ξi, for all ξi≥Mi and φi(ζi)≤Ni*ζi for all ζi≥Ni, where i=1,m¯, then the following holds for any x,y∈K and for any i=1,m¯:
(3.12)‖Iinx-Iiny‖≤(1+μ̃inNi*)‖x-y‖+μ̃inφi(Ni)+λ̃in,‖Tinx-Tiny‖≤(1+μinMi*)(1+μ̃inNi*)‖x-y‖+μ̃in(1+μinMi*)φi(Ni)(3.13)+λ̃in(1+μinMi*)+μinϕi(Mi)+λin.Proof.
Sinceϕi,φi: ℝ+→ℝ+ are the strictly increasing continuous functions, where i=1,m¯, it follows that ϕi(ξi)≤ϕi(Mi) and φi(ζi)≤ϕi(Ni) whenever ξi≤Mi and ζi≤Ni, where i=1,m¯. By the hypothesis of Proposition 3.3, for all ξi,ζi≥0 and i=1,m¯, we then get
(3.14)ϕi(ξi)≤ϕi(Mi)+Mi*ξi,(3.15)φi(ζi)≤φi(Ni)+Ni*ζi.
Since {Ti}i=1m:K→K, {Ii}i=1m:K→K are total asymptotically Ii-nonexpansive and total asymptotically nonexpansive mappings, respectively, from (3.14) and (3.15), one gets
(3.16)‖Iinx-Iiny‖≤‖x-y‖+μ̃inφi(‖x-y‖)+λ̃in≤‖x-y‖+μ̃in(φi(Ni)+Ni*‖x-y‖)+λ̃in=(1+μ̃inNi*)‖x-y‖+μ̃inφi(Ni)+λ̃in,‖Tinx-Tiny‖≤‖Iinx-Iiny‖+μinϕi(‖Iinx-Iiny‖)+λin≤‖Iinx-Iiny‖+μin(ϕi(Mi)+Mi*‖Iinx-Iiny‖)+λin=(1+μinMi*)‖Iinx-Iiny‖+μinϕi(Mi)+λin≤(1+μinMi*)(1+μ̃inNi*)‖x-y‖+μ̃in(1+μinMi*)φi(Ni)+λ̃in(1+μinMi*)+μinϕi(Mi)+λin.Lemma 3.4.
LetX be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let {Ti}i=1m: K→K be a finite family of total asymptotically Ii-nonexpansive mappings with sequences {μin}n=1∞, {λin}n=1∞, where i=1,m¯, and let {Ii}i=1m: K→K be a finite family of total asymptotically nonexpansive mappings with sequences {μ̃in}n=1∞, {λ̃in}n=1∞, where i=1,m¯, such that F:=⋂i=1m(F(Ti)∩F(Ii))≠∅. Suppose that ∑n=1∞μin<∞, ∑n=1∞λin<∞, ∑n=1∞μ̃in<∞, ∑n=1∞λ̃in<∞ for all i=1,m¯, and there exist Mi,Mi*,Ni,Ni*>0, i=1,m¯ such that ϕi(ξi)≤Mi*ξi, for all ξi≥Mi and φi(ζi)≤Ni*ζi for all ζi≥Ni, where i=1,m¯. If {xn} is the explicit iterative sequence defined by (1.27), then for each p∈F, the limit limn→∞∥xn-p∥ exists.Proof.
SinceF≠∅, for any given p∈F, it follows from (1.27) and (3.13) that
(3.17)‖xn+1-p‖=‖(1-∑i=1mαin)(xn-p)+∑i=1mαin(Tinyn-p)‖≤(1-∑i=1mαin)‖xn-p‖+∑i=1mαin‖Tinyn-p‖≤(1-∑i=1mαin)‖xn-p‖+∑i=1mαin(1+μinMi*)(1+μ̃inNi*)‖yn-p‖+∑i=1m(αinμ̃in(1+μinMi*)φi(Ni)+αinλ̃in(1+μinMi*))+∑i=1m(αinμinϕi(Mi)+αinλin).
Again from (1.27) and (3.12), we derive that
(3.18)∥yn-p∥=‖(1-∑i=1mβin)(xn-p)+∑i=1mβin(Iinxn-p)‖≤(1-∑i=1mβin)‖xn-p‖+∑i=1mβin‖Iinxn-p‖=(1+∑i=1mμ̃inβinNi*)‖xn-p‖+∑i=1m(μ̃inβinφi(Ni)+λ̃inβin).
Then from (3.17) and (3.18), one finds
(3.19)‖xn+1-p‖≤(1+bn)‖xn-p‖+cn.
Here
(3.20)bn=∑i=1m(μinαinMi*+μ̃inαinNi*+αin∑i=1mμ̃inβinNi*)+∑i=1mμinμ̃inαinMi*Ni*+∑i=1mμinαinMi*⋅∑i=1mμ̃inβinNi*+∑i=1mμ̃inαinNi*⋅∑i=1mμ̃inβinNi*,cn=∑i=1m(μ̃inβinφi(Ni)+λ̃inβin)⋅∑i=1mαin(1+μinMi*)(1+μ̃inNi*)+∑i=1m(μ̃inαin(1+μinMi*)φi(Ni)+λ̃inαin(1+μinMi*))+∑i=1m(μinαinϕi(Mi)+λinαin).
Denoting an=∥xn-p∥ in (3.19), one gets
(3.21)an+1≤(1+bn)an+cn.
Since ∑n=1∞bn<∞ and ∑n=1∞cn<∞, it follows from Lemma 2.1 the existence of the limit limn→∞an. This means the limit
(3.22)lim⁡n→∞‖xn-p‖=d
exists, where d≥0 is a constant. This completes the proof.Now we prove the following result.Theorem 3.5.
LetX be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let {Ti}i=1m:K→K be a finite family of total asymptotically Ii-nonexpansive continuous mappings with sequences {μin}n=1∞,{λin}n=1∞, where i=1,m¯, and let {Ii}i=1m:K→K be a finite family of total asymptotically nonexpansive continuous mappings with sequences {μ̃in}n=1∞, {λ̃in}n=1∞, where i=1,m¯, such that F:=⋂i=1m(F(Ti)∩F(Ii))≠∅. Suppose that ∑n=1∞μin<∞, ∑n=1∞λin<∞, ∑n=1∞μ̃in<∞, ∑n=1∞λ̃in<∞ for all i=1,m¯, and there exist Mi,Mi*,Ni,Ni*>0,i=1,m¯ such that ϕi(ξi)≤Mi*ξi, for all ξi≥Mi and φi(ζi)≤Ni*ζi for all ζi≥Ni, where i=1,m¯, then the explicit iterative sequence {xn} defined by (1.27) converges strongly to a common fixed point in F if and only if
(3.23)lim⁡inf⁡n→∞d(xn,F)=0.Proof.
The necessity of condition (3.23) is obvious. Let us prove the sufficiency part of the theorem.
Since{Ti}i=1m,{Ii}i=1m:K→K are continuous mappings, the sets F(Ti) and F(Ii) are closed. Hence, F=⋂i=1m(F(Ti)∩F(Ii)) is a nonempty closed set.
For any givenp∈F, we have (see (3.19))
(3.24)‖xn+1-p‖≤(1+bn)‖xn-p‖+cn.
Hence, one finds
(3.25)d(xn+1,F)≤(1+bn)d(xn,F)+cn.
From (3.25) due to Lemma 2.1, we obtain the existence of the limit limn→∞d(xn,F). By condition (3.23), one gets
(3.26)lim⁡n→∞d(xn,F)=lim⁡inf⁡n→∞d(xn,F)=0.
Let us prove that the sequence{xn} converges strongly to a common fixed point in F. We first show that {xn} is Cauchy sequence in X. In fact, due to 1+t≤exp(t) for all t>0, and from (3.24), we obtain
(3.27)‖xn+1-p‖≤exp⁡(bn)(‖xn-p‖+cn).
Thus, for any positive integers m,n, from (3.27) with ∑n=1∞bn<∞,∑n=1∞cn<∞, we find
(3.28)‖xn+m-p‖≤exp⁡(bn+m-1)(‖xn+m-1-p‖+cn+m-1)≤exp⁡(bn+m-1+bn+m-2)(‖xn+m-2-p‖+cn+m-1+cn+m-2)≤⋯≤exp⁡(∑i=nn+m-1bi)(‖xn-p‖+∑i=nn+m-1ci)≤exp⁡(∑i=n∞bi)(‖xn-p‖+∑i=n∞ci).
Therefore, we get
(3.29)‖xn+m-xn‖≤‖xn+m-p‖+‖xn-p‖≤(1+exp⁡(∑i=n∞bi))‖xn-p‖+exp⁡(∑i=n∞bi)∑i=n∞ci≤W(‖xn-p‖+∑i=n∞ci),
for all p∈F, where 0<W-1=exp(∑i=n∞bi)<∞. Taking infimum over p∈F in (3.29) gives
(3.30)‖xn+m-xn‖≤W(d(xn,F)+∑i=n∞ci).
Sincelimn→∞d(xn,F)=0 and ∑i=1∞ci<∞, given ɛ>0, there exists an integer N0>0 such that for all n>N0, we have d(xn,F)<(ɛ/2W) and ∑i=n∞ci<(ɛ/2W). Consequently, for all integers n≥N0 and m≥1 and from (3.30), we derive
(3.31)‖xn+m-xn‖≤ɛ,
which means that {xn} is Cauchy sequence in X, and since X is complete, there exists x*∈X such that the sequence {xn} converges strongly to x*.
Now we show thatx* is a common fixed point in F. Suppose for contradiction that x*∉F. Since F is closed subset of X, we have that d(x*,F)>0. However, for all p∈F, we have
(3.32)‖x*-p‖≤‖xn-x*‖+‖xn-p‖.
This implies that
(3.33)d(x*,F)≤‖xn-x*‖+d(xn,F),
so that as n→∞ we obtain d(x*,F)=0 which contradicts d(x*,F)>0. Hence, x* is a common fixed point in F. This proves the required assertion.To formulate and prove the main result, we need once more an auxiliary result.Proposition 3.6.
LetX be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let {Ti}i=1m:K→K be a finite family of total asymptotically Ii-nonexpansive continuous mappings with sequences {μin}n=1∞,{λin}n=1∞, where i=1,m¯, and let {Ii}i=1m: K→K be a finite family of total asymptotically nonexpansive continuous mappings with sequences {μ̃in}n=1∞, {λ̃in}n=1∞, where i=1,m¯, such that F:=⋂i=1m(F(Ti)∩F(Ii))≠∅. Suppose that ∑n=1∞μin<∞, ∑n=1∞λin<∞,∑n=1∞μ̃in<∞, ∑n=1∞λ̃in<∞ for all i=1,m¯, and {αjn}n=1∞,{βjn}n=1∞ are sequences with {αjn}n=1∞⊂[α*,α*] and {βjn}n=1∞⊂[β*,β*], for all j=0,m¯, here 0<α*<α*<1, 0<β*<β*<1, and there exist Mi,Mi*,Ni,Ni*>0, i=1,m¯ such that ϕi(ξi)≤Mi*ξi, for all ξi≥Mi and φi(ζi)≤Ni*ζi for all ζi≥Ni, where i=1,m¯. then the explicit iterative sequence {xn} defined by (1.27) satisfies the following:
(3.34)lim⁡n→∞‖xn-Tinxn‖=0,(3.35)lim⁡n→∞‖xn-Iinxn‖=0,
for all i=1,m¯.Proof.
According to Lemma3.4 for any p∈F, we have limn→∞∥xn-p∥=d. It follows from (1.27) that
(3.36)‖xn+1-p‖=‖α0n(xn-p)+∑i=1mαin(Tinyn-p)‖⟶d,
as n→∞. By means of ∑n=1∞μin<∞, ∑n=1∞λin<∞,∑n=1∞μ̃in<∞,∑n=1∞λ̃in<∞, for all i=1,m¯, from (3.18), one yields that
(3.37)limsup⁡n→∞‖yn-p‖≤limsup⁡n→∞[(1+∑i=1mμ̃inβinNi*)‖xn-p‖]+limsup⁡n→∞[∑i=1m(μ̃inβinφi(Ni)+λ̃inβin)]=limsup⁡n→∞‖xn-p‖=d,
and from (3.13), (3.37), we have
(3.38)limsup⁡n→∞‖Tinyn-p‖≤limsup⁡n→∞[limsup⁡n→∞μ̃in(1+μinMi*)φi(Ni)(1+μinMi*)(1+μ̃inNi*)‖yn-p‖+limsup⁡n→∞μ̃in(1+μinMi*)φi(Ni)]+limsup⁡n→∞[λ̃in(1+μinMi*)+μinϕi(Mi)+λin]≤d,
for all i=1,m¯. Now using
(3.39)limsup⁡n→∞‖xn-p‖=d,
with (3.38) and applying Lemma 3.2 to (3.36), one finds
(3.40)lim⁡n→∞‖xn-Tinyn‖=0,
for all i=1,m¯. Now from (1.27) and (3.40), we infer that
(3.41)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞‖∑i=1mαin(Tinyn-xn)‖=0.
On the other hand, from (3.13), we have
(3.42)‖xn-p‖≤‖xn-Tinyn‖+‖Tinyn-p‖≤‖xn-Tinyn‖+(1+μinMi*)(1+μ̃inNi*)‖yn-p‖+μ̃in(1+μinMi*)φi(Ni)+λ̃in(1+μinMi*)+μinϕi(Mi)+λin,
which implies
(3.43)‖xn-p‖-‖xn-Tinyn‖≤(1+μinMi*)(1+μ̃inNi*)‖yn-p‖+μ̃in(1+μinMi*)φi(Ni)+λ̃in(1+μinMi*)+μinϕi(Mi)+λin.
The last inequality with (3.22), (3.40) yields
(3.44)lim⁡inf⁡n→∞‖yn-p‖=d.
Combining (3.44) with (3.37), we get
(3.45)lim⁡n→∞‖yn-p‖=d.
Again from (1.27), we can see that
(3.46)‖yn-p‖=‖β0n(xn-p)+∑i=1mβin(Iinxn-p)‖⟶d,n⟶∞.
From (3.12) and (3.22), one finds
(3.47)limsup⁡n→∞‖Iinxn-p‖≤limsup⁡n→∞((1+μ̃inNi*)‖xn-p‖+μ̃inφi(Ni)+λ̃in)=d
for all i=1,m¯. Now applying Lemma 3.2 to (3.46), we obtain
(3.48)lim⁡n→∞‖xn-Iinxn‖=0,
for all i=1,m¯. We then have
(3.49)lim⁡n→∞‖yn-xn‖=lim⁡n→∞‖∑i=1mβin(Iinxn-xn)‖=0.
Consider
(3.50)‖xn-Tinxn‖≤‖xn-Tinyn‖+‖Tinyn-Tinxn‖≤‖xn-Tinyn‖+(1+μinMi*)(1+μ̃inNi*)‖yn-xn‖+(1+μinMi*)(μ̃inφi(Ni)+λ̃in)+μinϕi(Mi)+λin,
for all i=1,m¯. Then from (3.40) and (3.49), we get
(3.51)lim⁡n→∞‖xn-Tinxn‖=0,
for all i=1,m¯.Now we are ready to formulate a main result concerning strong convergence of the sequence{xn}.Theorem 3.7.
LetX be a uniformly convex real Banach space, and let K be a nonempty closed-convex subset of X. Let {Ti}i=1m:K→K be a finite family of total asymptotically Ii-nonexpansive continuous mappings with sequences {μin}n=1∞,{λin}n=1∞, where i=1,m¯, and let {Ii}i=1m:K→K be a finite family of total asymptotically nonexpansive continuous mappings with sequences {μ̃in}n=1∞, {λ̃in}n=1∞, where i=1,m¯, such that F:=⋂i=1m(F(Ti)∩F(Ii))≠∅. Suppose that ∑n=1∞μin<∞, ∑n=1∞λin<∞, ∑n=1∞μ̃in<∞, ∑n=1∞λ̃in<∞ for all i=1,m¯, and {αjn}n=1∞,{βjn}n=1∞ are sequences with {αjn}n=1∞⊂[α*,α*] and {βjn}n=1∞⊂[β*,β*], for all j=0,m¯, here 0<α*<α*<1, 0<β*<β*<1, and there exist Mi,Mi*,Ni,Ni*>0, i=1,m¯ such that ϕi(ξi)≤Mi*ξi, for all ξi≥Mi and φi(ζi)≤Ni*ζi for all ζi≥Ni, where i=1,m¯. If at least one mapping of the mappings {Ti}i=1m and {Ii}i=1m is compact, then the explicitly iterative sequence {xn} defined by (1.27) converges strongly to a common fixed point of {Ti}i=1m and {Ii}i=1m.Proof.
Without any loss of generality, we may assume thatT1 is compact. This means that there exists a subsequence {T1nkxnk}k=1∞ of {T1nxn}n=1∞ such that {T1nkxnk}k=1∞ converges strongly to x*∈K. Then from (3.34), we have that {xnk}k=1∞ converges strongly to x*. Also from (3.34), we obtain that {Tinkxnk}k=1∞ converges strongly to x*, for all i=2,m¯. Since {Ti}i=1m are continuous mappings, so {Tink+1xnk}k=1∞ converges strongly to Tix*, for all i=1,m¯. On the other hand, from (3.35) and continuousness of {Ii}i=1m, we obtain that {Iinkxnk}k=1∞ converges strongly to x*, and {Iink+1xnk}k=1∞ converges strongly to Iix*, for all i=1,m¯. Due to (3.41), {∥xnk+1-xnk∥} converges to 0, as k→∞. Then, {xnk+1}k=1∞ converges strongly to x* and moreover, (3.13) and (3.12) imply that {∥Tink+1xnk+1-Tink+1xnk∥} and {∥Iink+1xnk+1-Iink+1xnk∥} converge to 0, as k→∞, for all i=1,m¯. From (3.34), (3.35), it yields that ∥xnk+1-Tink+1xnk+1∥ and ∥xnk+1-Iink+1xnk+1∥ converge to 0 as k→∞, for all i=1,m¯. Observe that
(3.52)‖x*-Tix*‖≤‖x*-xnk+1‖+‖xnk+1-Tink+1xnk+1‖+‖Tink+1xnk+1-Tink+1xnk‖+‖Tink+1xnk-Tix*‖,‖x*-Iix*‖≤‖x*-xnk+1‖+‖xnk+1-Iink+1xnk+1‖+‖Iink+1xnk+1-Iink+1xnk‖+‖Iink+1xnk-Iix*‖,
for all i=1,m¯. Taking limit as k→∞, we have that x*=Tix* and x*=Iix*, for all i=1,m¯, which means x*∈F. However, by Lemma 3.4, the limit limn→∞∥xn-x*∥ exists, then
(3.53)lim⁡n→∞‖xn-x*‖=lim⁡nk→∞‖xnk-x*‖=0,
which means {xn} converges strongly to x*∈F. This completes the proof.Remark 3.8.
If one has that allIi are identity mappings, then the obtained results recover and correctly prove the main result of [24].Remark 3.9.
Suppose that we are given two families{Ti}i=1m:K→K and {Si}i=1m:K→K of total asymptotically nonexpansive continuous mappings such that ⋂i=1m(F(Ti)∩F(Si))≠∅. Define the following explicit iterative process:
(3.54)x0∈K,xn+1=α0nxn+∑i=1mαinTinyn+αm+1,nun,yn=β0nxn+∑i=1mβinSinxn+βm+1,nvn,
such that ∑j=0m+1αjn=1 and ∑j=0m+1βjn=1.Under suitable conditions, by the same argument and methods used above, one can prove, with either little mirror or no modifications, the strong convergence of the explicit iterative process defined by (3.54) to a common fixed point of the given families.Remark 3.10.
Let{Ti}i=1m: K→K be a finite family of total asymptotically nonexpansive continuous mappings with sequences {μin}n=1∞, {λin}n=1∞, where i=1,m¯. It is clear for each operator Ti that one has
(3.55)‖Tinx-Tiny‖≤‖Tinx-Tiny‖+μin‖Tinx-Tiny‖,
and this means that Ti is total asymptotically Ti-nonexpansive mappings with sequence {μin}n=1∞ and the function ϕ(λ)=λ. Hence, our iteration scheme can be written as follows:
(3.56)x0∈K,xn+1=α0nxn+∑i=1mαinTinyn,yn=β0nxn+∑i=1mβinTinxn,
where {αjn}n=1∞,{βjn}n=1∞ in (0,1), (j=0,m¯) with ∑j=0mαjn=1, ∑j=0mβjn=1.The defined scheme is a new iterative method generalizing one given in [24]. So, according to our main results for the defined sequence {xn} (see (3.56)), we obtain strong convergence theorems. On the other hand, playing with numbers {αjn}n=1∞,{βjn}n=1∞ and by means of the defined method, one may introduce lots of different schemes. All of the them strongly converge to a common fixed point of {Ti}i=1m. Moreover, the recursion formula (3.56) is much simpler than the others studied earlier for this problem [21, 23, 29, 30, 32, 35–38]. Therefore, all presented results here generalize, unify, and extend the corresponding main results of the mentioned papers. Note that one can consider the method (1.27) with errors, and all the theorems could be carried over for such iteration scheme as well with little or no modifications.We stress that all the theorems of this paper carry over to the class of total asymptotically quasi-I-nonexpansive mappings (see [24]), [39] with little or no modifications.



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*Source: 281383-2012-05-16.xml*