Int. J. Nonlinear Anal. Appl. 3 (2012) No. 1, 9-16
ISSN: 2008-6822 (electronic)
http://www.ijnaa.semnan.ac.ir
Weak and Strong Convergence Theorems for a
Finite Family of Generalized Asymptotically
Quasi-Nonexpansive Nonself-Mappings
P. Yatakoat, S. Suantai∗
Department of Mathematics, Faculty of Science, Chiang Mai University, Chiang Mai 50200, Thailand.
(Communicated by M. Eshaghi Gordji)
Abstract
In this paper, we introduce and study a new iterative scheme to approximate a common fixed point for
a finite family of generalized asymptotically quasi-nonexpansive nonself-mappings in Banach spaces.
Several strong and weak convergence theorems of the proposed iteration are established. The main
results obtained in this paper generalize and refine some known results in the current literature.
Keywords: Generalized Asymptotically Quasi-Nonexpansive Nonself-Mappings, Common Fixed
Points, Weak and Strong Convergence.
2010 MSC: 47H09, 47H10.
1. Introduction
Let X be a real Banach space, C a nonempty closed convex subset of X and T a self-mapping of C.
The fixed point set of T is denote by F (T ).
Definition 1.1. The mapping T is said to be;
(i) Nonexpansive if kT x − T yk ≤ kx − yk for all x, y ∈ C;
(ii) Quasi-nonexpansive if F (T ) 6= ∅ and kT x − pk ≤ kx − pk for all x, y ∈ C and p ∈ F (T );
(iii) Asymptotically nonexpansive if there exists a sequence {rn } in [0, ∞) with limn→∞ rn = 0 and
kT n x − T n yk ≤ (1 + rn )kx − yk, for all x, y ∈ C and n ≥ 1;
∗
Corresponding author
Email addresses: [email protected] (P. Yatakoat ), [email protected] (S. Suantai )
Received: January 2011
Revised: June 2012
10
Yatakoat, Suantai
(iv) Asymptotically quasi-nonexpansive if F (T ) 6= ∅ and there exists a sequence {rn } in [0, ∞) with
limn→∞ rn = 0 and kT n x − pk ≤ (1 + rn )kx − pk, for all x ∈ C, p ∈ F (T ) and n ≥ 1;
(v) Aeneralized quasi-nonexpansive if F (T ) 6= ∅ and there exists a sequences {sn } in [0, ∞) with
sn → 0 as n → ∞ such that kT n x − pk ≤ kx − pk + sn , for all x ∈ C and p ∈ F (T ) and n ≥ 1;
(iv) Aeneralized asymptotically quasi-nonexpansive if F (T ) 6= ∅ and there exist two sequences {rn }
and {sn } in [0, ∞) with rn → 0 and sn → 0 as n → ∞ such that kT n x−pk ≤ (1+rn )kx−pk+sn ,
for all x ∈ C, p ∈ F (T ) and n ≥ 1.
(vii) Uniformly L-Lipschitzian if there exists a constant L > 0 such that
kT n x − T n yk ≤ Lkx − yk, for all x, y ∈ C and n ≥ 1.
From the above definition (1.1) it follows that
(i) a nonexpansive mapping is a generalized asymptotically quasi-nonexpansive;
(ii) a quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive;
(iii) an asymptotically nonexpansive mapping with nonempty fixed points set is a generalized
asymptotically quasi-nonexpansive;
(iv) an asymptotically quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive;
(v) a generalized quasi-nonexpansive mapping is a generalized asymptotically quasi-nonexpansive.
The concept of asymptotically nonexpansive nonself-mappings was introduced by Chidume et al. [1]
as an important generalization of asymptotically nonexpansive mappings.
Recently, Deng and Liu [2] generalized the concept of generalized asymptotically quasi-nonexpansive
self-mappings defined by Shahzad and Zegeye [3] to the case of nonself-mappings. Those mappings
are defined as follows ;
Definition 1.2. (see [1],[4]) Let X be a real Banach space and C a nonempty closed convex subset
of X and let P : X → C be the nonexpansive retraction of X onto C. A nonself-mappins T : C → X
is said to be
(i) Asymptotically nonexpansive if there exists a sequence {rn } in [0, ∞) with limn→∞ rn = 0 such
that kT (P T )n−1 x − T (P T )n−1 yk ≤ (1 + rn )kx − yk, for all x, y ∈ C and n ≥ 1;
(i) Asymptotically quasi-nonexpansive if F (T ) 6= ∅ and there exists a sequence {rn } in [0, ∞) with
limn→∞ rn = 0 such that kT (P T )n−1 x − pk ≤ (1 + rn )kx − pk, for all x ∈ C, p ∈ F (T ) and
n ≥ 1;
(iii) Generalized asymptotically quasi-nonexpansive [5] if F (T ) =
6 ∅ and there exist two sequences
{rn } and {sn } in [0, ∞) with rn → 0 and sn → 0 as n → ∞ such that kT (P T )n−1 x − pk ≤
(1 + rn )kx − pk + sn , for all x ∈ C, p ∈ F (T ) and n ≥ 1.
(iv) Uniformly L-Lipschitzian if there exists a constant L > 0 such that
kT (P T )n−1 x − T (P T )n−1 yk ≤ Lkx − yk, for all x, y ∈ C and n ≥ 1.
Weak and Strong Convergence Theorems for a Finite Family ...3 (2012) No. 1,9-16
11
If T is self-mapping, then P becomes the identity mapping, so that (i) − (iii) of Definition 1.2 reduce
to (iii), (iv) and (vi) of Definition 1.1, respectively.
In this paper, we introduced a new iteration process for a finite family {Ti : i = 1, 2, 3, ..., m} of
generalized asymptotically quasi-nonexpansive nonself-mappings as follows:
Let X be a real Banach space, C a nonempty closed convex subset of X and P : X → C a
nonexpansive retraction of X onto C, and let Ti : C → X (i = 1, 2, 3, ..., m) be nonself-mappings.
Let {xn } be a sequence defined by
x0 ∈ C,
xn+1 = Sn xn ,
∀n ≥ 1
(1.1)
where Sn = P (α0n I + α1n T1 (P T1 )n−1 + P
α2n T2 (P T2 )n−1 + α3n T3 (P T3 )n−1 + . . . + αmn Tm (P Tm )n−1 )
with αin ∈ [0, 1] for i = 1, 2, 3, ..., m and m
i=0 αin = 1.
The main purpose of this paper is to prove strong convergence theorems of the iterative scheme
(1.1) to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive
nonself-mappings in real Banach spaces.
2. Preliminaries and lemmas
In this section, we give some definitions and lemmas used in the main results.
A subset C of
X is said to be retract of X if there exists a continuous mappings P : X → C such that P (x) = x
for all x ∈ C.
A mappings T : C → X with F (T ) 6= ∅ is said to be;
(i) Demiclosed at 0 if for each sequence {xn } converging weakly to x and {T xn } converging strongly
to 0, we have T x = 0;
(ii) semi-compact if for each sequence {xn } with limn→∞ kxn −T xn k = 0, there exists a subsequence
{xnk } of {xn } such that xnk → p;
(iii) completely continuous if for every bounded sequence {xn } ⊂ C, there is a subsequence {xnk }
such that {T xnk } is convergent.
A Banach space X is said to satisfy Opial0 s property (see [6])if for each x ∈ X and each sequence
{xn } weakly converges to x, the following condition holds for all x 6= y:
lim inf kxn − xk < lim inf kxn − yk.
n→∞
n→∞
Lemma 2.1. [7] Let the sequences {an } , {δn } and {cn } of real numbers satisfy:
an+1 ≤ (1 + δn )an + cn , where an ≥ 0, δn ≥ 0, cn ≥ 0 for all n = 1, 2, 3, . . .
P
P∞
and ∞
δ
<
∞,
n
n=1
n=1 cn < ∞. Then
(i) limn→∞ an exists;
(ii) if lim inf n→∞ an = 0, then limn→∞ an = 0.
Lemma 2.2. [8] Let X be a Banach space which satisfy Opial0 s property and let {xn } be a sequence
in X. Let x, y ∈ X be such that limn→∞ kxn − xk and limn→∞ kxn − yk exists. If {xnk } and {xmk }
are subsequences of {xn } which converge weakly to x and y, then x = y.
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Yatakoat, Suantai
Lemma 2.3. [9] Let X be a uniformly convex Banach. Then there exists a continuous strictly
increasing convex function g : [0, ∞) → [0, ∞) with g(0) = 0 such that for each m ∈ N and j ∈
{1, 2, 3, ..., m},
k
m
X
α i xi k 2 ≤
i=1
m
X
αi kxk2 −
i=1
m
αi X
(
αi g(kxj − xi k)),
m − 1 i=1
for all xi ∈ Br (0) and αi ∈ [0, 1] for all i = 1, 2, 3, ..., m with
Pm
i=1
αi = 1.
3. Main Results
The aim of this section is to establish weak and strong convergence of the iterative scheme (1.1)
to a common fixed point of a finite family of generalized asymptotically quasi-nonexpansive nonselfmappings in a Banach space under some appropriate conditions.
Lemma 3.1. Let C be a nonempty closed convex subset of a real Banach space X, and Ti : C →
X, (i = 1, 2, 3, . . . , m) be family of generalized asymptotically quasi-nonexpansive nonself-mappings
Tm
with the sequence {rin }, {sin } ⊂ [0, ∞) and pi ∈ F (Ti ), i = 1, 2, . . . , m. P
Suppose that F = P
i=1 F (Ti ) 6=
∞
∅, and the iterative sequence {xn }, is defined by (1.1). Assume that n=1 rin < ∞ and ∞
n=1 sin <
∞. Then we get
P∞
P
δ
<
∞,
(i) there exists two sequences {δn }, {cn } in [0, ∞) such that ∞
n
n=1 cn < ∞ and
n=1
kxn+1 − pk ≤ (1 + δn )kxn − pk + cn for all p ∈ F (T ) and n ≥ 1;
(ii) there exist L, D > 0 such that kxn+k − pk ≤ Lkxn − pk + D, for all p ∈ F (T ) and n, k ∈ N.
ProofP
. (i) Let p ∈ F andPrn = max1≤i≤k {rin }, sn = max1≤i≤k {sin } for all n.
P∞
∞
∞
Since
n=1 rn < ∞ and
n=1 sin < ∞ for all i = 1, 2, 3, . . . , m, we obtain that
n=1 rin < ∞ and
P∞
s
<
∞.
n=1 n
For i = 1, 2, 3, . . . , m, we have
kxn+1 − pk = kSn xn − pk
= kP (α0n I + α1n T1 (P T1 )n−1 + α2n T2 (P T2 )n−1 + . . .
+αmn Tm (P Tm )n−1 )xn − P (p)k
≤ α0n kxn − pk + α1n kT1 (P T1 )n−1 xn − pk + α2n kT2 (P T2 )n−1 xn − pk + . . .
+αmn kTm (P Tm )n−1 xn − pk
≤ α0n kxn − pk + α1n ((1 + r1n )kxn − pk + s1n ) + . . .
+α2n ((1 + r2n )kxn − pk + s2n ) + αmn ((1 + rmn )kxn − pk + smn )
≤ (α0n + α1n (1 + rn ) + α2n (1 + rn ) + . . . + αmn (1 + rn ))kxn − pk
+s1n + s2n + s3n + . . . + smn
≤ (1 + mrn )kxn − pk + msn
= (1 + δn )kxn − pk + cn
where δn = mrn and cn = msn .
(3.1)
Weak and Strong Convergence Theorems for a Finite Family ...3 (2012) No. 1,9-16
13
(ii) If t ≥ 0 then 1 + t ≤ et . Thus, from part (i) and for n, k ∈ N, we have
kxn+k − pk ≤
≤
≤
≤
..
.
(1 + δn+k−1 )kxn+k−1 − pk + cn+k−1
exp{δn+k−1 }kxn+k−1 − pk + cn+k−1
exp{δn+k−1 }[(1 + δn+k−2 )kxn+k−2 − pk + cn+k−2 ] + cn+k−1
exp{δn+k−1 }exp{δn+k−2 }kxn+k−2 − pk + exp{δn+k−1 }cn+k−2 + cn+k−1
k−1
k−1
k−1
X
X
X
≤ exp{
δn+i }kxn − pk + exp{
δn+i }
δn+i
i=0
i=0
(3.2)
i=0
P
P∞
Setting L = exp{ ∞
i=1 δi } and D = L
i=1 ci , we obtain kxn+k − pk ≤ Lkxn − pk + D.
Thus (ii) is satisfied. Lemma 3.2. Let C be a nonempty closed convex subset of a uniformly convex Banach space X, and
Ti : C → X, (i = 1, 2, 3, . . . , m) be family of uniformly Li -Lipschitzian and generalized asymptotically
quasi-nonexpansive nonself-mappings
with the sequence {rin }, {sin } ⊂ [0, ∞) and pi ∈ F (Ti ), i =
Tm
1, 2, . . . , m. Suppose that F = i=1
(Ti ) 6= ∅, and lim
inf n→∞ α0n αin > 0, ∀i = 1, 2, 3, . . . , m and
P∞
PF
∞
{xn } is defined by (1.1) such that n=1 rin < ∞ and n=1 sin < ∞. Then we get
(i) limn→∞ kxn − pk exists, ∀p ∈ F (T );
(ii) limn→∞ kxn − Ti xn k = 0, for each i = 1, 2, . . . , m.
Proof .
(i) By lemmas 2.1 and 3.1(i), we obtain that limn→∞ kxn − pk exists.
(ii) From (i), we have that {xn } is bounded. For each i = 1, 2, 3, . . . , m, we have
kTi (P Ti )n−1 xn − pk ≤ (1 + rin )kxn − pk + sin
≤ (1 + rn )kxn − pk + sn
It follows that {Ti (P Ti )n−1 xn − p} is bounded ∀i = 1, 2, 3, . . . , m.
Put r = sup{kTi (P Ti )n−1 xn − pk : 1 ≤ i ≤ m, n ∈ N} + sup{kxn − pk : n ∈ N}.
Let 1 ≤ i ≤ m. By Lemma 2.3, there is a continuous strictly increasing convex function g : [0, ∞) →
[0, ∞) with g(0) = 0 such that
k
m
X
i=1
2
α i xi k ≤
m
X
i=1
m
αi X
(
αi g(kxj − xi k)),
αi kxk −
m − 1 i=1
2
for all xi ∈ Br (0) and αi ∈ [0, 1] for all i = 1, 2, 3, ..., m with
By (3.3), we have
(3.3)
Pm
i=1
αi = 1.
kxn+1 − pk2 = kα0n (xn − p) + α1n (T1 (P T1 )n−1 xn − p) + . . . + αmn (Tm (P Tm )n−1 xn − p)k2
≤ α0n kxn − pk2 + α1n ((1 + r1n )kxn − pk + s1n )2 + . . .
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Yatakoat, Suantai
m
α0 X
+αmn ((1 + rmn )kxn − pk + smn ) − (
αi g(kxn − Ti (P Ti )n−1 xn k))
m i=1
2
≤ α0n kxn − pk2 +
m
X
αin kxn − pk2 + 2
i=1
+
+
m
X
i=1
m
X
m
X
αin (1 + rn )sn kxn − pk
i=1
αin s2n
2
m
α0 X
αi g(kxn − Ti (P Ti )n−1 xn k))
− (
m i=1
≤ kxn − pk + 2
m
X
2
αin rn kxn − pk +
i=1
m
X
m
X
αin rn2 kxn − pk2
i=1
αin (1 + rn )sn kxn − pk +
i=1
−
αin rn kxn − pk2
i=1
αin rn2 kxn − pk2 + 2
i=1
+2
m
X
m
X
αin s2n
i=1
m
X
α0
(
αi g(kxn − Ti (P Ti )n−1 xn k)).
m i=1
It follows that
m
m
X
α0 X
(
αi g(kxn − Ti (P Ti )n−1 xn k)) ≤ (kxn+1 − pk2 − kxn − pk2 ) + 2
αin rn kxn − pk2
m i=1
i=1
+
+
m
X
i=1
m
X
αin rn2 kxn
2
− pk + 2
m
X
αin (1 + rn )sn kxn − pk
i=1
αin s2n .
i=1
Since limn→∞ kxn − pk exists,limn→∞ rn = 0 = limn→∞ sn and lim inf n→∞ α0n αin > 0 for each i =
1, 2, 3, . . . , m, it follows that limn→∞ g(kxn − Ti (P Ti )n−1 xn k) = 0.
Since g is continuous strictly increasing with g(0) = 0, we can conclude that
lim kxn − Ti (P Ti )n−1 xn k = 0, ∀i = 1, 2, 3, . . . , m.
n→∞
(3.4)
For each n ∈ N, we have
kxn+1 − T1 (P T1 )n−1 xn k = kα0n xn + α1n T1 (P T1 )n−1 xn + α2n T2 (P T2 )n−1 xn + . . .
+αmn Tm (P Tm )n−1 xn − T1 (P T1 )n−1 xn k
≤ α0n kxn − T1 (P T1 )n−1 xn k + α2n kT2 (P T2 )n−1 xn − T1 (P T1 )n−1 xn k +
. . . + αmn kTm (P Tm )n−1 xn − T1 (P T1 )n−1 xn k
≤ α0n kxn − T1 (P T1 )n−1 xn k + α2n kT2 (P T2 )n−1 xn − xn k
+α2n kxn − T1 (P T1 )n−1 xn k + . . .
+αmn kTm (P Tm )n−1 xn − xn k + αmn kxn − T1 (P T1 )n−1 xn k.
(3.5)
From (3.4) and (3.5), we have
kxn+1 − T1 (P T1 )n−1 xn k → o as n → ∞
(3.6)
Weak and Strong Convergence Theorems for a Finite Family ...3 (2012) No. 1,9-16
15
From (3.4) and (3.6), where n → ∞
kxn+1 − xn k ≤ kxn+1 − T1 (P T1 )n−1 xn k + kT1 (P T1 )n−1 xn − xn k → 0.
(3.7)
Since Ti is uniformly Li -Lipschitzian, we have
kxn − Ti xn k ≤ kxn+1 − xn k + kxn+1 − Ti (P Ti )n−1 xn+1 k +
kTi (P Ti )n−1 xn+1 − Ti (P Ti )n−1 xn k + kTi (P Ti )n−1 xn − Ti xn k
≤ kxn+1 − xn k + kxn+1 − Ti (P Ti )n−1 xn+1 k +
Lkxn+1 − xn k + Lk(P Ti )n−1 xn − xn k
≤ kxn+1 − xn k + kxn+1 − Ti (P Ti )n−1 xn+1 k +
Lkxn+1 − xn k + LkTi (P Ti )n−1 xn − xn k.
(3.8)
It follows from (3.4), (3.6) and (3.7) that
kxn − Ti xn k = 0
This completes the proof. Theorem 3.3. Under the hypotheses of Lemma 3.2, assume that one of Ti is completely continuous.
Then the iterative sequence {xn } defined by (1.1) converges strongly to a common fixed point of the
family {Ti : i = 1, 2, 3, . . . , m}.
Proof . Suppose that Ti0 is completely continuous for some i0 ∈ {1, 2, . . . , m}. Since {xn } is bounded,
{xn } has a subsequence {xnk } such that Ti0 xnk → p. By Lemma 3.2 (ii), we have limn→∞ kxn −
Ti xn k = 0, ∀i = 1, 2, . . . , m. It follows that
kxnk − pk ≤ kxnk − Ti0 xnk k + kTi0 xnk − pk → 0.
Thus xnk → p. By the continuity of Ti , we have
kp − Ti pk = lim kxnk − Ti xnk k = 0, ∀i = 1, 2, . . . , m.
k→∞
Hence p ∈ F . By Lemma 3.2 (i), we have that limn→∞ kxn −pk exists. This implies that limn→∞ kxn −
pk = 0. Theorem 3.4. Under the hypotheses of Lemma 3.2, assume that one of Ti is semi-compact. Then
the iterative sequence {xn } defined by (1.1) converges strongly to a common fixed point of the family
{Ti : i = 1, 2, 3, . . . , m}.
Proof . Suppose that Ti0 is semi-compact for some i0 ∈ {1, 2, . . . , m}. By Lemma 3.2 (ii), we have
limn→∞ kxn − Ti xn k = 0, ∀i = 1, 2, . . . , m. Since {xn } is bounded and Ti0 is semi-compact, {xn } has
a convergent subsequence {xnk } such that xnk → p. By the continuity of Ti , we have
kp − Ti pk = lim kxnk − Ti xnk k = 0, ∀i = 1, 2, . . . , m.
k→∞
Hence p ∈ F . By Lemma 3.2 (i), we have that limn→∞ kxn −pk exists. This implies that limn→∞ kxn −
pk = 0. Theorem 3.5. Under the hypotheses of Lemma 3.2, assume that (I − Ti ) is demiclosed at 0, for
each i = 1, 2, . . . , m. Then the iterative sequence {xn } defined by (1.1) converges weakly to a common
fixed point of the family {Ti : i = 1, 2, 3, . . . , m}.
16
Yatakoat, Suantai
Proof . Let p ∈ F. By Lemma 3.2 (i), we have limn→∞ kxn − pk exists, and hence {xn } is bounded.
Since a uniformly convex Banach space is reflexive, there exists a subsequence {xnk } of {xn } converging weakly q1 ∈ C. By Lemma 3.2 (ii), limn→∞ kxn − Ti xn k = 0. Since (I − Ti ) is demiclosed
at 0, for each i = 1, 2, . . . , m, we obtain that Ti q1 = q1 . That is, q1 ∈ F . Next, we show that {xn }
converges weakly to q1 . Take another subsequence {xnk } of {xn } converging weakly to some q2 ∈ C.
Again, as above, we can conclude that q2 ∈ F . By Lemma 3.2, we obtain that q1 = q2 . This show
that {xn } converges weakly to a common fixed point of the family {Ti : i = 1, 2, . . . , m}. Remark 3.6. If {Ti : C → C}m
i is a finite family of self-mappings, then the mapping Sn in (1.1)
is reduced to Sn = α0n I + α1n T1 + α2n T2 + α3n T3 + . . . + αmn Tm by Cholamjiak and Suantai [9]. So
results obtained in the paper generalized those in [9].
4. Acknowledgments
This research was supported by grant from under the program Strategic Scholarships for Frontier Research Network for the Ph.D. Program Thai Doctoral degree from the Office of the Higher
Education Commission, the Graduate School of Chiang Mai University and the Thailand Research
Fund.
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