Approximation Methods for Common Fixed Points of Mean Nonexpansive Mapping in Banach Spaces

Volume 2008, Article ID 471532,7pages doi:10.1155/2008/471532

Research Article

Approximation Methods for Common Fixed Points

of Mean Nonexpansive Mapping in Banach Spaces

Zhaohui Gu1 _{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,[email protected]

Received 17 October 2007; Accepted 2 January 2008

Recommended by Tomonari Suzuki

LetXbe a uniformly convex Banach space, and letS, Tbe a pair of mean nonexpansive mappings. In this paper, it is proved that the sequence of Ishikawa iterations associated withSandTconverges to the common fixed point ofSandT.

Copyrightq2008 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

LetXbe a Banach space and letS,Tbe mappings fromXtoX. The pair of mean nonexpansive mappings was introduced by Bose in1:

Sx−Ty ≤ax−ybx−Sxy−Tycx−Tyy−Sx, 1.1

for allx, y ∈X,a, b, c∈0,1,a2b2c≤1.

The Ishikawa iteration sequence{xn}ofSandT was defined by

yn1−β_nxnβ_nSxn,

xn11−αnxnαnTyn, 1.2

wherex0 ∈X,αn, β_n ∈0,1. The recursion formulas1.2were first introduced in 1994

by Rashwan and Saddeek2in the framework of Hilbert spaces.

to a unique common fixed point in Hilbert spacessee2. Recently, ´Ciri´c has proved that if the sequence of Ishikawa iterations sequence{xn}associated withSandT converges top, thenp

is the common fixed point ofSandT see7. In4,6, 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 1.1see1. LetXbe a uniformly convex Banach space andKa nonempty closed convex subset ofX,S:K→KandT :K→Kare a pair of mean nonexpansive mappings, andc /0. Then,

iSandThave a common fixed pointu;

iifurther, ifb /0, then

auis the unique common fixed point and unique as a fixed point of eachSandT,

bthe sequence{xn}defined byx1Sx0, x2Tx1, x3Sx2. . ., for anyx0 ∈K, converges

strongly tou.

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. LetXbe a uniformly convex Banach space,S :X→XandT :X→Xare a pair of mean nonexpansive with a nonempty common fixed points set; ifb >0, 0 < α≤αn ≤1/2, 0≤β_n ≤β <1,

then the Ishikawa sequence{xn}converges to the common fixed point ofSandT.

Proof. First, we show that the sequence{xn}is bounded. For a common fixed pointpofSand

T, we have

Tx−pTx−Sp

≤ax−pbx−Txp−Spcx−Spp−Tx ≤ax−pbx−pp−Txcx−Spp−Tx.

2.1

LetL abc/1−b−c, bya2b2c≤1, it is easy to see thatabc≤1−b−c, thus 0≤L≤1 andTx−p ≤Lx−p ≤ x−p.

Similarly, we haveSx−p ≤Lx−p ≤ x−p,

_{xn1−p1−αn}

xnαnTyn−p

1−αnxn−pαnTyn−p

≤1−αnxn−pαnTyn−p

≤1−αnxn−pαnLyn−p

≤1−αnxn−pαn1−βn

xnβnSxn−p

1−αnxn−pαn1−β_nxn−pβ_nSxn−p

≤1−αnxn−pαn1−β_nxn−pαnβ_nSxn−p

≤1−αnxn−pαn1−β_nxn−pαnβ_nxn−p 1−αnαn1−β_nαnβ_nxn−pxn−p.

So

_{xn1−p≤xn−p≤xn−1−p≤ · · · ≤x0−p. 2.3}

Hence,{xn}is bounded.

Second, we show that

lim

n→∞xn−Tyn0. 2.4

We recall that Banach spaceXis called uniformly convex ifδε>0 for everyε >0, where the modulusδεof convexity ofXis defined by

δε inf

1−xy 2

:x ≤1, y ≤1, x−y ≥ε

, 2.5

for everyεwith 0≤ε≤2. It is easy to see that Banach spaceXis uniformly convex if and only if for anyxn, yn∈BX {x| x ≤1}, xnyn →2 impliesxn−yn →0.

Assume that limn→∞xn−Tyn/0, then there exist a subsequence {xnk}of {xn}and a

real numberε0>0, such that

_{xnk−Tynk≥ε0, k 1,2,3, . . . .} 2.6

On the other hand, for a common fixed pointpofTandS, we have

_{xnk−Tynk≤xnk−pTynk−p}

≤xnk−pLynk−p

xnk−pL1−β_nkxnkβnkSxnk−p

xnk−pL1−β_nkxnk−pβ_nkSxnk−p

≤xnk−p1−βnk

Lxnk−pβnkLSxnk−p

≤11−β_nkLβ_nkL2xnk−p

≤1Lxnk−p≤2xnk−p.

2.7

Thus,

_xnk−p≥ 1

2xnk−Tynk≥ ε0

2 ε1>0. 2.8

Because

_{Tyn−p≤yn−p≤1−βn}

xnβnSxn−p

1−β_nxn−pβ_nSxn−p≤1−β_nxn−pβ_nSxn−p

≤1−β_nxn−pβ_nxn−p≤xn−p,

2.9

we know{xn}is bounded, then there existsM >0, such thatxn−p ≤M. Thus,Tyn−p ≤

Furthermore, we have

xnk−p

_xnk−p −Tynk−p

xnk−p

_{xnk−Tynk xnk−p} ≥ ε1

M >0. 2.10 From

xnk−p

_xnk−p1,

Tynk−p

_xnk−p≤L≤1, 2.11

and the fact thatXis uniformly convex Banach space, there existsδ >0, such that

xnk−p

_xnk−p Tynk−p

xnk−p

≤2−δ. 2.12

Thus,

_{xnk1−p1−αnk}

xnkαnkTynk−p

≤1−2αnkxnk−pαnkxnk−pαnkTynk−p

≤1−2αnkxnk−pαnkxnk−p·

xnk−p

_xnk−p Tynk−p

xnk−p

≤1−2αnkxnk−p 2−δαnkxnk−p≤1−δαnkxnk−p xnk−p−δαnkxnk−p≤xnk−p−δαε1.

2.13

Using2.3, we obtain that

_{xnk1−p≤xnk−p−δαε1≤xnk−1−p−δαε1}

≤xnk−2−p−δαε1≤ · · · ≤xnk−11−p−δαε1

≤xnk−1−p−2δαε1.

2.14

So

_{xnk−p≤xnk−1 −p−δαε1≤xnk−2−p−2δαε1≤ · · · ≤xn1 −p−k−1δαε1. 2.15}

Letk→∞, then we havexnk−p<0. It is a contradiction. Hence, limn→∞xn−Tyn0.

Third, we show that

lim

n→∞xn−Sxn0. 2.16

Since

_{xn−Sxn≤xn−TynTyn−Sxn}

≤xn−Tynaxn−ynbxn−Sxnyn−Tyn cxn−Tynyn−Sxn

1cxn−Tynaxn−ynbxn−Sxn byn−Tyncyn−Sxn

1cxn−Tyna1−β_nxnβ_nSxn−xn bxn−Sxnb1−βn

xnβnSxn−Tyn c1−β_nxnβ_nSxn−Sxn

≤1cxn−Tynaβ_nxn−Sxn

bxn−Sxnbβ_nxn−Sxnbxn−Tync1−β_nxn−Sxn

1bcxn−Tynaβ_nbbβ_nc1−β_nxn−Sxn,

we have

1−aβ_n−b−bβ_n−c1−β_nxn−Sxn≤1bcxn−Tyn. 2.18

LetM11−aβ_n−b−bβ_n−c1−β_n, then

M11−aβ_n−b−bβ_n−ccβ_n1−b−c−ab−cβ_n

≥abc−ab−cβ_n ab1−β_nc1β_n

≥ab1−β c >0.

2.19

So

_xn−Sxn≤ 1bc M1

_{xn−Tyn. 2.20}

Using2.4, we get that

lim

n→∞xn−Sxn0. 2.21

Forth, we show that if the Ishikawa sequence{xn}converges to some pointp ∈X, then

pis the common fixed point ofSandT. By

yn1−β_nxnβ_nSxn,

xn11−αnxnαnTyn, 2.22

we havexn−Tyn 1/αnxn1−xn. Since{xn}is a convergent sequence, we get limn→∞xn−

Tyn 0. It is easy to see thatxn−yn β_nxn−SxnandSxn−yn 1−β_nxn−Sxn.

On the other hand,

_{yn−Tyn1−βn}

xnβ_nSxn−Tyn≤1−β_nxn−Tynβ_nSxn−Tyn. 2.23

By1.1, we obtain

_{Tyn−Sxn≤axn−ynbxn−Sxnyn−Tyncxn−Tynyn−Sxn}

≤aβ_nxn−Sxnbxn−Sxnb1−βnxn−Tyn bβ_nSxn−Tyncxn−Tync1−β_nxn−Sxn aβ_nbc1−β_nxn−Sxn

b1−β_ncxn−Tynbβ_nSxn−Tyn.

2.24

Since

_{xn−Sxn≤Sxn−Tynxn−Tyn, 2.25}

we get

_Tyn−Sxn≤

b1−β_ncaβ_nbc1−β_nxn−Tyn

So

1−b−c−ab−cβ_nTyn−Sxn≤b1−βn

caβ_nbc1−β_nxn−Tyn. 2.27

LetM21−b−c−ab−cβ_n, Since 0≤β_n≤β <1, we have

M2≥abc−ab−cβ_n≥ab

1−β_nc1β_n≥ab1−β c >0. 2.28

It is easy to see that

b1−β_ncaβ_nbc1−β_n>0. 2.29

Note that limn→∞xn−Tyn0, then we get

lim

n→∞Sxn−Tyn0, nlim→∞yn−Tyn0. 2.30 So limn→∞xn−ynlimn→∞β_nSxn−xn0.

Letp limn→∞xn, then limn→∞yn p, limn→∞Sxn p, limn→∞Tyn p. By1.1, we

have

_{Sxn−Tp≤axn−pbxn−Sxnp−Tp}

cxn−Tpp−Sxn. 2.31

Letn→∞, then we get

p−Tp ≤bcp−Tp. 2.32

Sincebc <1, it follows that

p−Tp0, that isTpp. 2.33

Similarly, we can prove thatSpp. Sopis the common fixed point ofSandT. Finally, we show that{Sxn}is a Cauchy sequence. For anym, n∈N,

_{Sxn−Sxnm≤Sxn−TynmSxnm−Tynm}

≤axn−ynmbxn−Sxnynm−Tynm cxn−Tynmynm−SxnSxnm−Tynm

≤axn−SxnSxn−SxnmSxnm−ynm bxn−Sxnynm−Tynm

cxn−SxnSxn−Sxnm

Sxnm−Tynmynm−Sxnm Sxnm−SxnSxnm−Tynm.

2.34

Sinceb >0, thus we get 1−a−2c >0.Simplify, then we have

_{Sxn−Sxnm≤Axn−SxnBynm−Tynm}

Cynm−SxnmDSxnm−Tynm, 2.35

whereA abc/1−a−2c≥0, Bb/1−a−2c≥0, C ac/1−a−2c≥0, and D 1c/1−a−2c≥0.By2.16and2.30, we know that

So it is easy to see thatynm−Sxnm→0. Thus,Sxn−Sxnm→0, that is{Sxn}is a Cauchy

sequence. Hence, there existsp, such thatp limn→∞Sxn. We know thatp limn→∞xnandp

is the common fixed point ofSandT. 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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