Volume 2008, Article ID 471532, 7 pagesdoi:10.1155/2008/471532 Research Article Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces Zhaohui Gu 1 a
Trang 1Volume 2008, Article ID 471532, 7 pages
doi:10.1155/2008/471532
Research Article
Approximation Methods for Common Fixed Points
of Mean Nonexpansive Mapping in Banach Spaces
Zhaohui Gu 1 and Yongjin Li 2
1 Department of Foundation, Guangdong Finance and Economics College, Guangzhou 510420, China
2 Institute of Logic and Cognition, Department of Mathematics, Sun Yat-Sen University,
Guangzhou 510275, China
Correspondence should be addressed to Yongjin Li, stslyj@mail.sysu.edu.cn
Received 17 October 2007; Accepted 2 January 2008
Recommended by Tomonari Suzuki
LetX be a uniformly convex Banach space, and let S, T be a pair of mean nonexpansive mappings.
In this paper, it is proved that the sequence of Ishikawa iterations associated withS and T converges
to the common fixed point ofS and T.
Copyright q 2008 Z Gu and Y Li 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
LetX be a Banach space and let S, T be mappings from X to X The pair of mean nonexpansive
mappings was introduced by Bose in1:
Sx − Ty ≤ ax − y bx − Sx y − Ty cx − Ty y − Sx, 1.1 for allx, y ∈ X, a, b, c ∈ 0, 1, a 2b 2c ≤ 1.
The Ishikawa iteration sequence{xn} of S and T was defined by
y n1− β nx n β n Sx n ,
wherex0 ∈ X, αn , β n ∈ 0, 1 The recursion formulas 1.2 were first introduced in 1994
by Rashwan and Saddeek2 in the framework of Hilbert spaces
In recent years, several authorssee 2 6 have studied the convergence of iterations to
a common fixed point for a pair of mappings Rashwan has studied the convergence of Mann iterations to a common fixed pointsee 5 and proved that the Ishikawa iterations converge
Trang 2to a unique common fixed point in Hilbert spacessee 2 Recently, ´Ciri´c has proved that if the sequence of Ishikawa iterations sequence{xn} associated with S and T converges to p, then p
is the common fixed point ofS and T see 7 In 4,6, the authors studied the same problem
In1, Bose defined the pair of mean nonexpansive mappings, and proved the existence of the fixed point in Banach spaces In particular, he proved the following theorem
Theorem 1.1 see 1 Let X be a uniformly convex Banach space and K a nonempty closed convex
subset of X, S : K→K and T : K→K are a pair of mean nonexpansive mappings, and c / 0 Then,
i S and T have a common fixed point u;
ii further, if b / 0, then
a u is the unique common fixed point and unique as a fixed point of each S and T,
b the sequence {xn} defined by x1 Sx0, x2 Tx1, x3 Sx2 , for any x0 ∈ K, converges
strongly to u.
It is our purpose in this paper to consider an iterative scheme, which converges to a common fixed point of the pair of mean nonexpansive mappings Theorem 2.1 extends and improves the corresponding results in1
2 Main results
Now we prove the following theorem which is the main result of this paper
Theorem 2.1 Let X be a uniformly convex Banach space, S : X→X and T : X→X are a pair of mean
nonexpansive with a nonempty common fixed points set; if b > 0, 0 < α ≤ α n ≤ 1/2, 0 ≤ β n ≤ β < 1,
then the Ishikawa sequence {xn} converges to the common fixed point of S and T.
Proof First, we show that the sequence {xn} is bounded For a common fixed point p of S and
T, we have
Tx − p Tx − Sp
≤ ax − p bx − Tx p − Sp cx − Sp p − Tx
≤ ax − p bx − p p − Tx cx − Sp p − Tx.
2.1
LetL a b c/1 − b − c, by a 2b 2c ≤ 1, it is easy to see that a b c ≤ 1 − b − c, thus
0≤ L ≤ 1 and Tx − p ≤ Lx − p ≤ x − p.
Similarly, we haveSx − p ≤ Lx − p ≤ x − p,
xn1 − p 1 − αn
x n αn Ty n − p
1− αn
x n − p αnTy n − p
≤1− αnxn − p αnTyn − p
≤1− αnxn − p αn L yn − p
≤1− α nxn − p αn1− β n
x n β n Sx n − p
1− αnxn − p αn1− β n
x n − p β nSx n − p
≤1− αnxn − p αn
1− β n xn − p αn β n Sxn − p
≤1− αnxn − p αn
1− β n xn − p αn β n xn − p
1− αn αn1− β n αn β n xn − p xn − p.
2.2
Trang 3xn1 − p ≤ xn − p ≤ xn−1 − p ≤ ··· ≤ x0− p. 2.3 Hence,{x n} is bounded
Second, we show that
lim
We recall that Banach spaceX is called uniformly convex if δε > 0 for every ε > 0, where the
modulusδε of convexity of X is defined by
δε inf1−
x y2 : x ≤ 1, y ≤ 1, x − y ≥ ε, 2.5 for everyε with 0 ≤ ε ≤ 2 It is easy to see that Banach space X is uniformly convex if and only
if for anyx n , y n ∈ BX {x | x ≤ 1}, xn yn → 2 implies xn − yn → 0.
Assume that limn→∞ xn − Tyn / 0, then there exist a subsequence {xn k } of {xn} and a
real numberε0> 0, such that
xn k − Tyn k ≥ ε0, k 1, 2, 3, 2.6
On the other hand, for a common fixed pointp of T and S, we have
xn k − Ty n k ≤ xn k − p Tyn k − p
≤xn k − p Lyn k − p
xn k − p L1 − β n k
x n k βn k Sx n k − p
xn k − p L1 − β n k
x n k − p β n kSx n k − p
≤xn k − p 1 − β n k
L xn k − p β n k L Sxn k − p
≤11− β n kL β n k L2xn k − p
≤ 1 Lxn k − p ≤ 2xn k − p.
2.7
Thus,
xn k − p ≥ 1
2xn k − Ty n k ≥ ε0
Because
Tyn − p ≤ yn − p ≤ 1 − β n
x n β n Sx n − p
1− β n
x n − p β nSx n − p ≤ 1 − β n xn − p β n Sxn − p
≤1− β n xn − p β n xn − p ≤ xn − p, 2.9
we know{x n } is bounded, then there exists M > 0, such that x n − p ≤ M Thus, Ty n − p ≤
xn − p ≤ M.
Trang 4Furthermore, we have
x n k − p
xn k − p − Ty n k − p
xn k − p
xn k − Tyn k
xn k − p ≥ ε1
From
x n k − p
xn k − p 1,
Ty n k − p
xn k − p ≤ L ≤1, 2.11
and the fact thatX is uniformly convex Banach space, there exists δ > 0, such that
x n k − p
xn k − p Ty n k − p
Thus,
xn k1− p 1 − αn k
x n k αn k Ty n k − p
≤1− 2αn k xn k − p αn k
x n k − p αn kTy n k − p
≤1− 2α n k xn k − p αn k xn k − p· x n k − p
xn k − p Ty n k − p
xn k − p
≤1− 2αn k xn k − p 2 − δαn k xn k − p ≤ 1 − δαn k xn k − p
xn k − p − δαn k xn k − p ≤ xn k − p − δαε1.
2.13
Using2.3, we obtain that
xn k1− p ≤ xn k − p − δαε1≤xn k−1− p − δαε1
≤xn k−2− p − δαε1≤ · · · ≤xn k−11− p − δαε1
≤xn k−1 − p − 2δαε1.
2.14
So
xn k − p ≤ xn k−1 − p − δαε1≤xn k−2 − p − 2δαε1≤ · · · ≤xn1 − p − k − 1δαε1 2.15
Letk→∞, then we have x n k − p < 0 It is a contradiction Hence, lim n→∞xn − Tyn 0.
Third, we show that
lim
Since
xn − Sxn ≤ xn − Tyn Tyn − Sxn
≤xn − Ty n axn − y n bxn − Sx n yn − Ty n
cxn − Tyn yn − Sxn
1 cxn − Tyn axn − yn bxn − Sxn
byn − Tyn cyn − Sxn
1 cxn − Tyn a1 − β n
x n β n Sx n − xn
bxn − Sx n b1 − β n
x n β n Sx n − Ty n
c1− β n
x n β n Sx n − Sxn
≤ 1 cxn − Tyn aβ n xn − Sxn
bxn − Sxn bβ n xn − Sxn bxn − Tyn c1 − β n xn − Sxn
1 b cxn − Tyn aβ n b bβ n c1− β n xn − Sxn,
2.17
Trang 5we have
1− aβ n − b − bβ n − c1− β n xn − Sxn ≤ 1 b cxn − Tyn. 2.18 LetM1 1 − aβ n − b − bβ n − c1 − β n, then
M1 1 − aβ n − b − bβ n − c cβ n 1 − b − c − a b − cβ n
≥ a b c − a b − cβ n a b1− β n c1 β n
≥ a b1 − β c > 0.
2.19
So
xn − Sxn ≤ 1 b c M
Using2.4, we get that
lim
Forth, we show that if the Ishikawa sequence{xn} converges to some point p ∈ X, then
p is the common fixed point of S and T By
y n1− β nx n β n Sx n ,
we havex n −Tyn 1/αnxn1 −xn Since {xn} is a convergent sequence, we get lim n→∞ xn−
Ty n 0 It is easy to see that xn − yn β n xn − Sxn and Sxn − yn 1 − β n xn − Sxn.
On the other hand,
yn − Tyn 1 − β n
x n β n Sx n − Tyn ≤ 1 − β n xn − Tyn β n Sxn − Tyn 2.23
By1.1, we obtain
Tyn − Sxn ≤ axn − yn bxn − Sxn yn − Tyn cxn − Tyn yn − Sxn
≤ aβ n xn − Sxn bxn − Sxn b1 − β n xn − Tyn
bβ n Sxn − Tyn cxn − Tyn c1 − β n xn − Sxn
aβ n b c1− β n xn − Sx n
b1− β n cxn − Tyn bβ n Sxn − Tyn.
2.24 Since
we get
Tyn − Sxn ≤ b1 − β n
c aβ n b c1− β n xn − Tyn
bβ n aβ n b c1− β n Sxn − Tyn. 2.26
Trang 6
1− b − c − a b − cβ n Tyn − Sx n ≤ b1 − β n
c aβ n b c1− β n xn − Ty n.
2.27 LetM2 1 − b − c − a b − cβ n, Since 0≤ β n ≤ β < 1, we have
M2≥ a b c − a b − cβ n ≥ a b1− β n c1 β n≥ a b1 − β c > 0. 2.28
It is easy to see that
b1− β n c aβ n b c1− β n> 0. 2.29 Note that limn→∞ x n − Ty n 0, then we get
lim
n→∞ Sxn − Tyn 0, lim
So limn→∞xn − yn lim n→∞ β n Sxn − xn 0.
Letp lim n→∞ x n, then limn→∞ y n p, lim n→∞ Sx n p, lim n→∞ Ty n p By 1.1, we have
Sxn − Tp ≤ axn − p bxn − Sxn p − Tp cxn − Tp p − Sxn 2.31
Letn→∞, then we get
Sinceb c < 1, it follows that
Similarly, we can prove thatSp p So p is the common fixed point of S and T.
Finally, we show that{Sxn} is a Cauchy sequence For any m, n ∈ N,
Sxn − Sxnm ≤ Sxn − Tynm Sxnm − Tynm
≤ axn − y nm bxn − Sx n ynm − Ty nm
cxn − Tynm ynm − Sxn Sxnm − Tynm
≤ axn − Sxn Sxn − Sxnm Sxnm − ynm
bxn − Sxn ynm − Tynm
cxn − Sxn Sxn − Sxnm
Sxnm − Ty nm ynm − Sx nm
Sxnm − Sxn Sxnm − Tynm.
2.34
Sinceb > 0, thus we get 1 − a − 2c > 0 Simplify, then we have
Sxn − Sxnm ≤ Axn − Sxn Bynm − Tynm
Cynm − Sxnm DSxnm − Tynm, 2.35 whereA a b c/1 − a − 2c ≥ 0, B b/1 − a − 2c ≥ 0, C a c/1 − a − 2c ≥ 0, and
D 1 c/1 − a − 2c ≥ 0 By 2.16 and 2.30, we know that
xn − Sxn −→ 0, ynm − Tynm −→ 0, Sxnm − Tynm −→ 0. 2.36
Trang 7So it is easy to see thatynm − Sxnm→0 Thus, Sxn − Sxnm→0, that is {Sxn} is a Cauchy
sequence Hence, there existsp, such that p lim n→∞ Sx n We know thatp lim n→∞ x nandp
is the common fixed point ofS and T This completes the proof of the theorem.
Acknowledgment
The work was partially supported by the Emphases Natural Science Foundation of Guangdong Institution of Higher Learning, College and Universityno 05Z026
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