Volume 2008, Article ID 324575,12pages doi:10.1155/2008/324575
Research Article
Implicit Iteration Process for Common Fixed
Points of Strictly Asymptotically Pseudocontractive
Mappings in Banach Spaces
You Xian Tian,1Shih-sen Chang,2Jialin Huang,2
Xiongrui Wang,2 and J. K. Kim3
1College of Mathematics and Physics, Chongqing University of Post Telecommunications, Chongqing 400065, China
2Department of Mathematics, Yibin University, Yibin, Sichuan 644007, China 3Department of Mathematics, Kyungnam University, Masan 631-701, South Korea
Correspondence should be addressed to You Xian Tian,[email protected]
Received 25 May 2008; Accepted 3 September 2008
Recommended by Nanjing Huang
In this paper, a new implicit iteration process with errors for finite families of strictly asymptot-ically pseudocontractive mappings and nonexpansive mappings is introduced. By using the iterative process, some strong convergence theorems to approximating a common fixed point of strictly asymptotically pseudocontractive mappings and nonexpansive mappings are proved. The results presented in the paper are new which extend and improve some recent results of Osilike et al.2007, Liu1996, Osilike2004, Su and Li2006, Gu2007, Xu and Ori2001.
Copyrightq2008 You Xian Tian et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction and preliminaries
Throughout this paper, we assume that Eis a real Banach space,C is a nonempty closed convex subset ofE,E∗is the dual space ofE, andJ:E→E∗is the normalized duality mapping defined by
Jx f∈E∗:x, f||x||2||f||2, x∈E. 1.1
Recall that a setC ⊂ Eis said to beclosed, convex, and pointed coneif it is a closed set and satisfies the following conditions:1CC⊂C;2λC⊂C for eachλ≥0;3ifx∈C
Definition 1.1. LetT:C→Cbe a mapping:
1T is said to be λ,{kn}-strictly asymptotically pseudocontractive if there exist a constant λ ∈ 0,1 and a sequence {kn} ⊂ 1,∞ with kn→1 such that for all
x, y∈C, and for alljx−y∈Jx−y,
Tnx−Tny, jx−y≤kn||x−y||2−λx−Tnx−y−Tny2 ∀n≥1
, 1.2
2T is said to beλ-strictly pseudocontractive in the terminology of Browder-Petryshyn1
if there exist a constant λ∈0,1such that for allx, y∈C,
Tx−Ty, jx−y≤ ||x−y||2−λx−Tx−y−Ty2 ∀jx−y∈Jx−y, 1.3
3T is said to beuniformlyL-Lipschitzianif there exists a constantL >0 such that Tnx−Tny≤L||x−y|| ∀n≥1. 1.4
The class of λ,{kn}-strictly asymptotically pseudocontractive mappings was first introduced in Hilbert spaces by Liu2. In the case of Hilbert spaces, it is shown by2that
1.2is equivalent to the inequality
Tnx−Tny2≤kn||x−y||2
λI−Tnx−I−Tny2. 1.5
Concerning the convergence problem of iterative sequences for strictly pseudocontrac-tive mappings has been studied by several authorssee, e.g.,1,3–7. Concerning the class of strictly asymptotically pseudocontractive mappings, Liu2and Osilike et al.8proved the following results.
Theorem 1.2Liu2. LetHbe a real Hilbert space, letCbe a nonempty closed convex and bounded subset ofH, and letT :C→Cbe a completely continuous uniformlyL-Lipschitzianλ,{kn}-strictly asymptotically pseudocontractive mapping such that ∞n1k2n −1 < ∞. Let {αn} ⊂ 0,1 be a
sequence satisfying the following condition:
0< ≤αn≤1−λ− ∀n≥1and some >0. 1.6
Then, the sequence{xn}generated from an arbitraryx1 ∈Cby
xn1
1−αnxnαnTnxn ∀n≥1 1.7
converges strongly to a fixed point ofT.
Theorem 1.3 Oslike et al.8. LetEbe a real q-uniformly smooth Banach space which is also uniformly convex, letC be a nonempty closed convex subset of E, let T : C→C be a λ,{kn} -strictly asymptotically pseudocontractive mapping such that ∞n1kn−1 < ∞, and letFT/∅. Let{αn} ⊂0,1be a real sequence satisfying the following condition:
0< a≤αq−1n ≤b <
q1−k
2cq 1L
−q−2 ∀n≥1. 1.8
Let{xn}be the sequence defined by1.7. Then,
1limn→ ∞||xn−p||exists∀p∈FT, 2limn→ ∞||xn−Txn||0,
3{xn}converges weakly to a fixed point ofT.
It is our purpose in this paper to introduce the following new implicit iterative process with errors for a finite family of strictly asymptotically pseudocontractive mappings{Ti}and a finite family of nonexpansive mappings{Si}:
x1∈C, xnαnSnxn−11−αnTn
nxnun ∀n≥1,
1.9
whereCis a closed convex cone ofE,Sn SnmodN,Tnn TnnmodN, and{un}is a bounded
sequence inC. Also, we aim to prove some strong convergence theorems to approximating a common fixed point of{Si}and {Ti}. The results presented in the paper are new which extend and improve some recent results of2–8.
In order to prove our main results, we need the following lemmas.
Lemma 1.4see9. LetEbe a real Banach space, letCbe a nonempty subset ofE, and letT :C→C
be aλ,{kn}-strictly asymptotically pseudocontractive mapping, thenTis uniformlyL-Lipschitzian.
Lemma 1.5. LetEbe a real Banach space, letCbe a nonempty closed convex subset ofE, and let
Ti :C→Cbe aλi,{kni}-strictly asymptotically pseudocontractive mapping,i1,2, . . . , N, then
there exist a constantλ∈0,1, a constantL >0, and a sequence{kn} ⊂1,∞withlimn→ ∞kn1
such that for anyx, y∈Cand for eachi1,2, . . . , Nand eachn≥1, the following hold:
Tinx−Tiny, jx−y≤kn||x−y||2−λx−Tinx−y−Tiny2 1.10
for eachjx−y∈Jx−yand
Tn
ix−Tiny≤L||x−y||. 1.11
Proof. Since for eachi1,2, . . . , N,Tiisλi,{kni}-strictly asymptotically pseudocontractive,
Takingkn max{kni, i 1,2, . . . , N}and λ min{λi, i 1,2, . . . , N}, hence, for eachi
1,2, . . . , N, we have
Tn
ix−Tiny, jx−y
≤kni||x−y||2−λix−Tinx−
y−Tn iy
2
≤kn||x−y||2−λx−Tn ix−
y−Tn i y
2
. 1.12
The conclusion1.10is proved. Again, takingL max{Li :i1,2, . . . N}for anyx, y∈C, we have
Tn
ix−Tiny≤Li||x−y|| ≤L||x−y|| ∀n≥1. 1.13
This completes the proof ofLemma 1.5.
Lemma 1.6see9. Let{an},{bn}, and{cn}be three nonnegative real sequences satisfying the following condition:
an1≤
1bnancn ∀n≥n0, 1.14
wheren0 is some nonnegative integer such that ∞n1bn < ∞and ∞n1cn < ∞, thenlimn→ ∞an
exists.
In addition, if there exists a subsequence {ani} ⊂ {an} such that ani→0, then an→
0n→ ∞.
2. Main results
We are now in a position to prove our main results in this paper.
Theorem 2.1. LetEbe a real Banach space, letCbe a nonempty closed pointed convex cone ofE, let
Ti :C→C, i1,2, . . . , N, be a finite family ofλi,{kni}-strictly asymptotically pseudocontractive
mappings, and letSi:C→C, i1,2, . . . , N,be a finite family of nonexpansive mappings with
F N
i1 FSi
N
i1
FTi/∅ 2.1
(the set of common fixed points of{Si}and {Ti}). Let {αn}be a sequence in 0,1, let {un} be a bounded sequence inC, letλ min{λi :i1,2, . . . , N},kn max{kni, i 1,2, . . . , N}, and let Lmax{Li :i1,2, . . . , N}>0be positive numbers defined by1.10and1.11, respectively. If the following conditions are satisfied:
i0<max{λ, 1−1/L}<lim infn→ ∞αn≤αn <1, ii ∞n11−αn ∞,
then the iterative sequence{xn}with errors defined by1.9has the following properties:
1limn→ ∞||xn−p||exists for eachp∈F, 2limn→ ∞dxn, Fexists,
3lim infn→ ∞||xn−Tnnxn||0,
4the sequence{xn}converges strongly to a common fixed pointp∈Fif and only if
lim inf
n→ ∞ d
xn, F0. 2.2
Proof. We divide the proof ofTheorem 2.1into four steps.
IFirst, we prove that the mappingGn:C→C, n1,2, . . . ,defined by
Gnx αnSnxn−11−αnTnnxun, x∈C 2.3
is a Banach contractive mapping.
Indeed, it follows from conditionithat 1−1/L < αn, that is,1−αnL < 1. Hence,
fromLemma 1.5, for anyx, y∈C, we have
Gnx−GnyαnSnxn−1
1−αnTnnxun−
αnSnxn−11−αnTnnyun
1−αnTnnx−Tnny
≤1−αnL||x−y||, n1,2, . . . ,
2.4
that is, for eachn 1,2, . . . , Gn : C→Cis a Banach contraction mapping. Therefore, there exists a unique fixed pointxn ∈Csuch thatxn Gxn. This shows that the sequence{xn} defined by1.9is well defined.
IIThe proof of conclusions1and2.
For any givenp∈Fand for anyjxn−p∈Jxn−yfromLemma 1.5, we have xn−p2
αnSnxn−1−p1−αnTnnxn−p
un2 αnSnxn−1−p, jxn−p1−αnTnnxn−p, j
xn−pun, jxn−p ≤αnxn−1−pxn−p1−αnknxn−p2−λxn−Tnnxn
2
unxn−p. 2.5
Simplifying it, we have
xn−p≤ αn
1−1−αnknxn−1−p
un 1−1−αnkn −
1−αnλ
1−1−αnkn·
xn−Tn nxn
2
xn−p . 2.6
By virtue of conditionsiandiii, we have
kn≤ 1−λ
1−lim infn→ ∞αn ≤
1−λ
and so
0< λ≤1−1−αnkn<1. 2.8
It follows from2.6and2.8that
xn−p≤ αn
1−1−αnknxn−1−p
un
λ −
1−αnλ·xn−T n nxn
2
xn−p
1
1−αnkn−1 1−1−αnkn
xn−1−pun λ −
1−αnλ·xn−T n nxn
2
xn−p .
2.9
Lettingbn 1−αnkn−1/1−1−αnknandcnun/λ, then we have xn−p≤
1bnxn−1−pcn ∀n≥1. 2.10
By using2.8,
bn≤
1−αnkn−1
λ < kn−1
λ . 2.11
By conditions iii and iv, ∞n1bn < ∞ and ∞n1cn < ∞. By virtue of Lemma 1.6, limn→ ∞||xn−p||exists; and so{xn}is a bounded sequence inC. Denote
Msup
n≥1
xn−p. 2.12
From2.10, we have
dxn, F≤1bndxn−1, Fcn ∀n≥1. 2.13
By usingLemma 1.6again, we know that limn→ ∞dxn, Fexists.
The conclusions1and2are proved.
IIIThe proof of conclusion3. It follows from2.9that
xn−p≤
1bnxn−1−pcn−1−αnλ·xn−T n nxn
2
M
≤xn−1−pbnMcn−1−αnλ·xn−T n nxn
2
M ,
2.14
that is,
1−αnλ·xn−T n nxn
2
For any positive numbern1, we have
λ M
n1
n1
1−αnxn−Tn
nxn2≤x0−p−xn1−p n1
n1
bnMcn
≤x0−p n1
n1
bnMcn.
2.16
Lettingn1→ ∞, we have
λ M
∞
n1
1−αnxn−Tnnxn 2≤
x0−p ∞
n1
bnMcn<∞. 2.17
By conditionii, we have
lim inf
n→ ∞ xn−T n
nxn0. 2.18
IVNext, we prove the conclusion4.
Necessity
If{xn}converges strongly to some pointp ∈ F, then from 0 ≤ dxn, F ≤ xn−p →0, we have
lim inf
n→ ∞ d
xn, F0. 2.19
Sufficiency
If lim infn→ ∞dxn, F 0, it follows from the conclusion2that limn→ ∞dxn, F 0.
Next, we prove that{xn}is a Cauchy sequence inC. In fact, since for anyt >0, 1t≤
expt, therefore, for anym, n≥1 and for givenp∈F, from2.10, we have xnm−p≤
1bnmxnm−1−pcnm ≤expbnmxnm−1−pcnm ≤expbnm
expbnm−1xnm−2−pcnm−1cnm expbnmbnm−1}xnm−2−pexpbnm
cnm−1cnm ≤ · · ·
≤exp nm
in1 bi
xn−p nm in1
exp
nm
ji1 bj
ci
≤K
xn−p nm in1
ci
<∞,
whereKexp{ ∞j1bj}<∞. Since
lim
n→ ∞d
xn, F0, ∞
n1
cn<∞ 2.21
for any given >0, there exists a positive integern1such that
dxn, F<
4K1,
∞
in1 ci<
2K ∀n≥n1. 2.22
Hence, there existsp1 ∈Fsuch that
xn−p1<
2K1 ∀n≥n1. 2.23
Consequently, for anyn≥n1andm≥1, from2.20, we have xnm−xn≤xnm−p1xn−p1
≤K
xn−p1 nm in1
ci
xn−p1
≤K1xn−p1K
nm
in1 ci
≤K1
2K1K
2K .
2.24
This implies that{xn}is a Cauchy sequence inC. Letxn→x∗∈C. Since limn→ ∞dxn, F 0,
and sodx∗, F 0. Again, since{Sn}is a finite family of nonexpansive mappings and{Tn} is a finite family of strictly asymptotically pseudocontractive mappings, byLemma 1.5, it is a finite family of uniformly Lipschitzian mappings. Hence, the setFof common fixed points of{Sn}and{Tn}is closed and sox∗∈F.
This completes the proof ofTheorem 2.1.
Remark 2.2. Theorem 2.1is a generalization and improvement of the corresponding results in Osilike et al.8and Liu2which is also an improvement of the corresponding results in
3,5–7.
The following theorem can be obtained fromTheorem 2.1immediately.
Theorem 2.3. LetEbe a real Banach space, letCbe a nonempty closed pointed convex cone ofE, let
T :C→Cbe aλ,{kn}-strictly asymptotically pseudocontractive mappings, and let{Si:C→C, i
1,2, . . . , N}be a finite family of nonexpansive mappings with
F N
i1
(the set of common fixed points of{Si}andT). Let{αn}be a sequence in0,1, let{un}be a bounded sequence inC. If the following conditions are satisfied:
i0<max{λ, 1−1/L}<lim infn→ ∞αn≤αn<1,whereL >0is a constant appeared in
Lemma 1.4,
ii ∞n11−αn ∞,
iii ∞n1kn−1<∞ and 1≤kn<1−λ/1−lim infn→ ∞αn, iv ∞n1||un||<∞,
then the conclusions inTheorem 2.1still hold.
Theorem 2.4. Let Ebe a real Banach space, let C be a nonempty closed convex subset ofE, and
{Ti:C→C, i1,2, . . . , N}be a finite family ofλi,{kni}-strictly asymptotically pseudocontractive
mappings, and let{Si:C→C, i1,2, . . . , N}be a finite family of nonexpansive mappings with
F N
i1 FSi
N
i1
FTi/∅ 2.26
(the set of common fixed points of{Si}and{Ti}). Let{xn}be the sequence defined by the following: for any givenx1∈C,
xnαnSnxn−1βnTnnxnγnun ∀n≥1, 2.27
whereSn SnmodN,Tnn TnnmodN,{αn},{βn}, and{γn}are sequences in0,1withαnβn
γn 1,{un} is a bounded sequence inC,λ min{λi : i 1,2, . . . , N},kn max{kni, i
1,2, . . . , N}, and L max{Li : i 1,2, . . . , N} > 0are positive numbers defined by1.10and
1.11, respectively. If the following conditions are satisfied:
i0< λ <lim infn→ ∞αn≤αn<1, ii ∞n11−αn ∞,
iii0< βn≤lim supn→ ∞βn≤min{1−λ, 1/L}<1,
iv ∞n1kn−1<∞ and 1≤kn<1−λ/1−lim infn→ ∞αn, v ∞n1γn<∞,
then the conclusions ofTheorem 2.1for sequence{xn}defined by2.27still hold.
Proof. By the same method as given in the proof of Theorem 2.1, we can prove that the mappingWn:C→Cdefined by
Wnx αnSnxn−1βnTnnxγnun, x∈C, n≥1, 2.28
For eachp∈F, we have xn−p2
αnSnxn−1−p, jxn−pβnTnnxn−p, j
xn−pγnun−p, jxn−p ≤αnxn−1−pxn−pβnknxn−p2−λxn−Tnnxn
2
γnun−pxn−p. 2.29
Simplifying it, we have
xn−p2≤ αnxn−1−pxn−p
1−βnkn − βnλ
1−βnknxn−T n nxn
2 γn
1−βnknun−pxn−p. 2.30
Since
lim sup
n→ ∞ βnlim supn→ ∞
1−αn−γn≤lim sup
n→ ∞
1−αn1−lim inf
n→ ∞ αn, 2.31
by conditionsi,iii, andiv, we have
kn≤ 1−λ
1−lim infn→ ∞αn ≤
1−λ
lim supn→ ∞βn ≤
1−λ
βn , 2.32
that is, 1−βnkn≥λ >0.Hence, we have
xn−p≤ αnxn−1−p
1−βnkn γn
λun−p
1βnkn−βn−γn 1−βnkn
xn−1−pγn
λun−p
≤
1βnkn−βn 1−βnkn
xn−1−p γn
λun−p.
2.33
By conditioniv,
∞
n1
βnkn−βn
1−βnkn ≤
1
λ ∞
n1
kn−1<∞. 2.34
Again, since{||un−p||}is bounded, by conditionv, we have
∞
n1
γnun−p
λ <∞. 2.35
It follows from2.33andLemma 1.6that limn→ ∞||xn −p|| exists, and so{xn}is bounded.
Since{Ti}is uniformly Lipschitzian,{Tn
Now, we rewrite2.27as follows:
xnαnSnxn−11−αnTnnxnvn ∀n≥1, 2.36
wherevnγnun−Tn
nxn. By conditionv, ∞
n1
vn<∞. 2.37
These imply that all conditions in Theorem 2.1are satisfied. Therefore, the conclusion of
Theorem 2.4can be obtained fromTheorem 2.1immediately.
This completes the proof ofTheorem 2.4.
Theorem 2.5. LetEbe a real Banach space, letCbe a nonempty closed convex subset ofE, and let
{Ti:C→C, i1,2, . . . , N}be a finite family ofλi,{kni}-strictly asymptotically pseudocontractive
mappings with
F N
i1
FTi/∅ 2.38
(the set of common fixed points of{Ti}. Let{xn}be the sequence defined by the following: for any givenx1∈C,
xnαnxn−1βnTnnxnγnun ∀n≥1, 2.39
where Tnn TnnmodN,{αn},{βn}, and {γn} are sequences in [0, 1] withαnβnγn 1,{un} is a bounded sequence inC,λ min{λi : i 1,2, . . . , N},kn max{kni, i 1,2, . . . , N}, and Lmax{Li :i1,2, . . . , N}>0are positive numbers defined by1.10and1.11, respectively. If the following conditions are satisfied:
i0< λ <lim infn→ ∞αn≤αn<1, ii ∞n11−αn ∞,
iii0< βn≤lim supn→ ∞βn≤min{1−λ, 1/L}<1,
iv ∞n1kn−1<∞ and 1≤kn<1−λ/1−lim infn→ ∞αn, v ∞n1γn<∞,
then the conclusions ofTheorem 2.1for sequence{xn}defined by2.39still hold.
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