ISSN 2291-8639
Volume 6, Number 1 (2014), 89-96 http://www.etamaths.com
CONVERGENCE TO COMMON FIXED POINT FOR NEARLY ASYMPTOTICALLY NONEXPANSIVE MAPPINGS IN BANACH
SPACES
G. S. SALUJA
Abstract. The purpose of this paper is to study modifiedS-iteration process to converge to common fixed point for two nearly asymptotically nonexpansive mappings in the framework of Banach spaces. Also we establish some strong convergence theorems and a weak convergence theorem for said mappings and iteration scheme under appropriate conditions.
1. Introduction
LetC be a nonempty subset of a Banach space E andT:C → C a nonlinear mapping. We denote the set of all fixed points ofT byF(T). The set of common fixed points of two mappingsS andT will be denoted byF =F(S)∩F(T). The mappingT is said to be Lipschitzian [1, 16] if for eachn∈N, there exists a constant
kn>0 such that
kTnx−Tnyk ≤ knkx−yk
for allx, y∈C.
A Lipschitzian mapping T is said to be uniformly k-Lipschitzian if kn = k for all n ∈ N and asymptotically nonexpansive [4] if kn ≥ 1 for all n ∈ N with limn→∞kn= 1.
It is easy to observe that every nonexpansive mapping T (i.e., kT x−T yk ≤ kx−yk for all x, y ∈ C) is asymptotically nonexpansive with constant sequence {1} and every asymptotically nonexpansive mapping is uniformly k-Lipschitzian withk= supn∈Nkn.
The asymptotic fixed point theory has a fundamental role in nonlinear func-tional analysis (see, [2]). The theory has been studied by many authors (see, e.g., [6], [7], [10], [12], [21]) for various classes of nonlinear mappings (e.g., Lipschitzian, uniformly k-Lipschitzian and non-Lipschitzian mappings). A branch of this theo-ry related to asymptotically nonexpansive mappings has been developed by many authors (see, e.g., [4], [5], [9], [11], [12], [14], [15], [17]-[19]) in Banach spaces with
2010Mathematics Subject Classification. 47H09, 47H10, 47J25.
Key words and phrases. Nearly asymptotically nonexpansive mapping, modifiedS-iteration process, common fixed point, strong convergence, weak convergence, Banach space.
c
2014 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.
suitable geometrical structure.
Fix a sequence {an} ⊂[0,∞) with limn→∞an = 0, then according to Agarwal et al. [1],T is said to be nearly Lipschitzian with respect to{an}if for eachn∈N, there exist constants kn ≥ 0 such that kTnx−Tnyk ≤ kn(kx−yk+an) for all
x, y ∈ C. The infimum of constants kn for which the above inequality holds is denoted byη(Tn) and is called nearly Lipschitz constant.
A nearly Lipschitzian mappingT with sequence{an, η(Tn)}is said to be nearly asymptotically nonexpansive ifη(Tn)≥1 for alln∈
Nand limn→∞η(Tn) = 1 and nearly uniformlyk-Lipschitzian ifη(Tn)≤kfor alln∈
N.
In 2007, Agarwal et al. [1] introduced the following iteration process:
x1 = x∈C,
xn+1 = (1−αn)Tnxn+αnTnyn,
yn = (1−βn)xn+βnTnxn, n≥1 (1.1)
where {αn} and{βn} are sequences in (0,1). They showed that this process verge at a rate same as that of Picard iteration and faster than Mann for con-tractions and also they established some weak convergence theorems using suitable conditions in the framework of uniformly convex Banach space.
We modify iteration scheme (1.1) for two nonlinear mappings.
Let C be a nonempty subset of a Banach space E and S, T:C → C be two nearly asymptotically nonexpansive mappings. For givenx1=x∈C, the iterative sequence{xn} defined as follows:
x1 = x∈C,
xn+1 = (1−αn)Tnxn+αnSnyn,
yn = (1−βn)xn+βnTnxn, n≥1 (1.2)
where {αn} and{βn} are sequences in (0,1). The iteration scheme (1.2) is called modifiedS-iteration scheme for two nonlinear mappings.
If we putS=T, then iteration scheme (1.2) reduces toS-iteration scheme (1.1).
The aim of this paper is to establish some strong convergence theorems and a weak convergence theorem of modified S-iteration scheme (1.2) for two nearly asymptotically nonexpansive mappings in the framework of Banach spaces.
2. Preliminaries
For the sake of convenience, we restate the following concepts.
A mappingT:C→Cis said to be semi-compact [3] if for any bounded sequence {xn} in C such thatkxn−T xnk →0 as n→ ∞, then there exists a subsequence
{xnk} ⊂ {xn} such thatxnk→x
∗∈C strongly.
We say that a Banach space E satisfies the Opial’s condition [13] if for each sequence{xn} inE weakly convergent to a pointxand for ally6=x
lim inf
n→∞ kxn−xk<lim infn→∞ kxn−yk.
The examples of Banach spaces which satisfy the Opial’s condition are Hilbert spaces and allLp[0,2π] with 1< p6= 2 fail to satisfy Opial’s condition [13].
Now, we state the following useful lemma to prove our main results.
Lemma 2.1. (See [20]) Let {αn}∞n=1, {βn}∞n=1 and {rn}∞n=1 be sequences of
nonnegative numbers satisfying the inequality
αn+1≤(1 +βn)αn+rn, ∀n≥1.
IfP∞
n=1βn<∞andP ∞
n=1rn <∞, then limn→∞αn exists.
3. Main Results
In this section, we prove some strong convergence theorems and a weak conver-gence theorem for two nearly asymptotically nonexpansive mappings in the frame-work of Banach spaces.
Theorem 3.1.Let E be a Banach space and C be a nonempty closed convex subset of E. Let S, T:C →C be two nearly asymptotically nonexpansive map-pings with sequences{a0n, η(Sn)}, {a00n, η(Tn)} andF =F(S)∩F(T)6=∅ is closed
such thatP∞
n=1an<∞and P∞
n=1
η(Sn)η(Tn)−1<∞. Let{x
n}be the
mod-ifiedS-iteration scheme defined by (1.2). Then{xn}converges to a common fixed point of the mappingsS andT if and only if lim infn→∞d(xn, F) = 0.
Proof.The necessity is obvious. Thus we only prove the sufficiency. Letq∈F. For the sake of convenience, set
Anx = (1−βn)x+βnTnx
and
Gnx = (1−αn)Tnx+αnSnAnx.
Thenyn =Anxn andxn+1=Gnxn. Moreover, it is clear thatqis a fixed point of
Gn for all n. Let η = supn∈Nη(S
n)∨sup n∈Nη(T
n) and a
n = max{a0n, a00n} for all
Consider
kAnx−Anyk = k((1−βn)x+βnTnx)−((1−βn)y+βnTny)k = k(1−βn)(x−y) +βn(Tnx−Tny)k
≤ (1−βn)kx−yk+βnη(Tn)(kx−yk+a00n)
≤ (1−βn)kx−yk+βnη(Tn)kx−yk+βnanη(Tn)
≤ (1−βn)η(Tn)kx−yk+βnη(Tn)kx−yk +βnanη(Tn)
≤ η(Tn)kx−yk+anη(Tn). (3.1)
Choosingx=xn andy=q, we get
kyn−qk ≤ η(Tn)kxn−qk+anη(Tn). (3.2)
Now, consider
kGnx−Gnyk = k((1−αn)Tnx+αnSnAnx)−((1−αn)Tny+αnSnAny)k = k(1−αn)(Tnx−Tny) +αn(SnAnx−SnAny)k
≤ (1−αn)η(Tn)(kx−yk+a00n) +αnη(Sn)(kAnx−Anyk+a0n)
≤ (1−αn)η(Tn)(kx−yk+an) +αnη(Sn)(kAnx−Anyk+an)
≤ (1−αn)η(Tn)kx−yk+αnη(Sn)kAnx−Anyk +(1−αn)anη(Tn) +αnanη(Sn).
(3.3)
Now using (3.1) in (3.3), we get
kGnx−Gnyk ≤ (1−αn)η(Tn)kx−yk+αnη(Sn)[η(Tn)kx−yk +anη(Tn)] + (1−αn)anη(Tn) +αnanη(Sn)
≤ (1−αn)η(Tn)η(Sn)kx−yk+αnη(Tn)η(Sn)kx−yk +(1−αn+ 2αn)anη(Tn)η(Sn)
≤ η(Tn)η(Sn)kx−yk+ 2anη(Tn)η(Sn)
≤ h1 +η(Tn)η(Sn)−1ikx−yk+ 2anη2
= (1 +Pn)kx−yk+Qn, (3.4)
wherePn=
η(Tn)η(Sn)−1andQ
n= 2anη2. Since by hypothesisP∞n=1
η(Sn)η(Tn)−
1<∞andP∞
n=1an<∞. It follows thatP ∞
n=1Pn <∞andP ∞
n=1Qn<∞.
Choosingx=xn andy=qin (3.4), we get
kxn+1−qk = kGnxn−qk ≤(1 +Pn)kxn−qk+Qn. (3.5)
Next, we shall prove that{xn}is a Cauchy sequence. Since 1 +x≤exforx≥0, therefore, for anym, n≥1 and for givenq∈F, from (3.5), we have
kxn+m−qk ≤ (1 +Pn+m−1)kxn+m−1−qk+Qn+m−1
≤ ePn+m−1kx
n+m−1−qk+Qn+m−1
≤ ePn+m−1[ePn+m−2kx
n+m−2−qk+Qn+m−2] +Qn+m−1
≤ e(Pn+m−1+Pn+m−2)kx
n+m−2−qk
+e(Pn+m−1+Pn+m−2)[Q
n+m−2+Qn+m−1]
≤ . . .
≤ e
Pn+m−1
k=n Pk
kxn−qk+e
Pn+m−1
k=n Pk
n+m−1 X
k=n
Qk
≤ e
P∞
n=1Pn
kxn−qk+e
P∞
n=1Pn
n+m−1 X
k=n
Qk
= Kkxn−qk+K n+m−1
X
k=n
Qk (3.6)
whereK=e
P∞
n=1Pn
<∞. Since
lim
n→∞d(xn, F) = 0,
∞
X
n=1
Qn <∞ (3.7)
for any givenε >0, there exists a positive integern1 such that
d(xn, F)<
ε
4(K+ 1),
n+m−1 X
k=n
Qk <
ε
2K ∀n≥n1.
(3.8)
Hence, there existsq1∈F such that
kxn−q1k< ε
2(K+ 1) ∀n≥n1. (3.9)
Consequently, for anyn≥n1andm≥1, from (3.6), we have
kxn+m−xnk ≤ kxn+m−q1k+kxn−q1k
≤ Kkxn−q1k+K n+m−1
X
k=n
Qk+kxn−q1k
≤ (K+ 1)kxn−q1k+K n+m−1
X
k=n
Qk
< (K+ 1) ε
2(K+ 1)+K
ε
2K =ε.
(3.10)
Theorem 3.2.Let E be a Banach space and C be a nonempty closed convex subset of E. Let S, T:C →C be two nearly asymptotically nonexpansive map-pings with sequences{a0n, η(Sn)}, {a00n, η(Tn)} andF =F(S)∩F(T)6=∅ is closed
such that P∞
n=1an <∞and P∞
n=1
η(Sn)η(Tn)−1 <∞. Let{α
n} and{βn} be sequences in [δ,1−δ] for someδ∈(0,1). Let{xn} be the modifiedS-iteration scheme defined by (1.2). If eitherSis semi-compact and limn→∞kxn−Sxnk= 0 or
T is semi-compact and limn→∞kxn−T xnk= 0, then the sequence{xn} converge strongly to a point ofF.
proof. Suppose that T is semi-compact and limn→∞kxn −T xnk = 0. Then there exists a subsequence {xnj} of{xn} such that xnj → q∈ C. Also, we have
limj→∞kxnj−T xnjk= 0 and we make use of the fact that every nearly
asymptot-ically nonexpansive mapping is nearlyk-Lipschitzian. Hence, we have
kq−T qk ≤ kq−xnjk+kxnj −T xnjk+kT xnj −T qk ≤ (1 +k)kq−xnjk+kxnj −T xnjk →0.
Thusq∈F. By (3.5),
kxn+1−qk ≤(1 +Pn)kxn−qk+Qn.
Since by hypothesisP∞
n=1Pn<∞andP ∞
n=1Qn<∞, by Lemma 2.2, limn→∞kxn−
qkexists andxnj →q∈F gives thatxn→q∈F. This shows that{xn}converges
strongly to a point ofF. This completes the proof.
As an application of Theorem 3.1, we establish another strong convergence result as follows.
Theorem 3.3.Let E be a Banach space and C be a nonempty closed convex subset of E. Let S, T:C →C be two nearly asymptotically nonexpansive map-pings with sequences{a0n, η(Sn)}, {a00n, η(Tn)} andF =F(S)∩F(T)6=∅ is closed
such that P∞
n=1an <∞and P∞
n=1
η(Sn)η(Tn)−1 <∞. Let{α
n} and{βn} be sequences in [δ,1−δ] for someδ∈(0,1). Let{xn} be the modifiedS-iteration scheme defined by (1.2). IfS andT satisfy the following conditions:
(i) limn→∞kxn−Sxnk= 0 and limn→∞kxn−T xnk= 0.
(ii) There exists a constantA >0 such that
h
a1kxn−Sxnk+a2kxn−T xnk
i
≥A d(xn, F)
wherea1 anda2are two non-negative real numbers such thata1+a2= 1.
Then the sequence{xn}converge strongly to a point ofF.
Theorem 3.4.LetEbe a Banach space satisfying Opial’s condition andCbe a nonempty closed convex subset ofE. LetS, T:C→Cbe two nearly asymptotical-ly nonexpansive mappings with sequences{a0n, η(Sn)},{an00, η(Tn)}andF =F(S)∩
F(T)6=∅is closed such thatP∞
n=1an<∞and P∞
n=1
η(Sn)η(Tn)−1<∞. Let
{αn}and{βn} be sequences in [δ,1−δ] for someδ∈(0,1). Let{xn} be the mod-ified S-iteration scheme defined by (1.2). Suppose that S and T have a common fixed point,I−S andI−T are demiclosed at zero and{xn}is an approximating common fixed point sequence for S and T, that is, limn→∞kxn−Sxnk = 0 and limn→∞kxn−T xnk= 0. Then{xn}converges weakly to a common fixed point of
S andT.
Proof: Letqbe a common fixed point ofSandT. Then limn→∞kxn−qkexists as proved in Theorem 3.1. We prove that {xn} has a unique weak subsequential limit in F = F(S)∩F(T). For, let u and v be weak limits of the subsequences {xni} and {xnj} of {xn}, respectively. By hypothesis of the theorem, we know
that limn→∞kxn−Sxnk= 0 and I−S is demiclosed at zero, therefore we obtain
Su=u. Similarly,T u=u. Thusu∈F =F(S)∩F(T). Again in the same fashion, we can prove thatv ∈F =F(S)∩F(T). Next, we prove the uniqueness. To this end, ifuandvare distinct then by Opial’s condition,
lim
n→∞kxn−uk = nlimi→∞
kxni−uk
< lim ni→∞
kxni−vk
= lim
n→∞kxn−vk = lim
nj→∞
kxnj −vk
< lim nj→∞
kxni−uk
= lim
n→∞kxn−uk.
This is a contradiction. Henceu=v∈F. Thus {xn}converges weakly to a com-mon fixed point of the mappingsS and T. This completes the proof.
Remark 3.1. Our results extend and generalize the corresponding results of [8], [14], [15], [17], [18], [20] and many others from the existing literature to the case of modified S-iteration scheme and more general class of nonexpansive and asymptotically nonexpansive mappings considered in this paper.
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