Volume 2008, Article ID 469357,8pages doi:10.1155/2008/469357
Research Article
Common Fixed Point Theorems on Weakly
Contractive and Nonexpansive Mappings
Jian-Zhong Xiao and Xing-Hua Zhu
Department of Mathematics, Nanjing University of Information Science and Technology, Nanjing 210044, China
Correspondence should be addressed to Jian-Zhong Xiao,[email protected]
Received 8 August 2007; Accepted 26 November 2007
Recommended by Jerzy Jezierski
A family of commuting nonexpansive self-mappings, one of which is weakly contractive, are stud-ied. Some convergence theorems are established for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point. The error estimates of these iterations are also given.
Copyrightq2008 J.-Z. Xiao and X.-H. Zhu. 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, dbe a metric space andD⊂X. A mappingT :D→Xis said to be nonexpansive if
dTx, Ty≤dx, y, ∀x, y∈D, 1.1
and it is said to be weakly contractive if
dTx, Ty≤dx, y−ψdx, y, ∀x, y∈D, 1.2
whereψ : 0,∞ → 0,∞is continuous and nondecreasing such thatψ is positive on0,∞, ψ0 0,and limt→∞ψt ∞.
It is evident thatT is contractive if it is weakly contractive withψt 1−αt, where α∈0,1, and it is nonexpansive if it is weakly contractive.
Inspired by2,5, 6, the purpose of this paper is to study a family of commuting non-expansive mappings, one of which is weakly contractive, in arbitrary complete metric spaces and Banach spaces.
We will establish some convergence theorems for the iterations of types Krasnoselski-Mann, Kirk, and Ishikawa to approximate a common fixed point and to give their error esti-mates.
Throughout this paper, we assume thatFTis the set of fixed points of a mapping T, that is,FT {x: Tx x};Φis defined by the antiderivativeindefinite integralof 1/ψt on0,∞, that is,Φt dt/ψt, andΦ−1is the inverse function ofΦ.
We define iterations which will be needed in the sequel.
Suppose that X is a metric space and D ⊂ X, {Tr}kr0 is a family of commuting
self-mappings ofDandx0∈D. The iteration{xn}n∞0⊂Dof type Krasnoselski-Mannsee7,8is
cyclically defined by
x1T1x0, . . . , xk Tkxk−1, xk1T0xk,
xk2T1xk1, . . . , x2k1T0x2k1, x2k11T1x2k1, . . . .
1.3
For convenience, we write
xnTnmodk1xn−1, 1.4
where the modk1 function takes values in{0,1,2, . . . , k}.
LetDbe a closed convex subset of the normed spaceX. Then the iteration{xn}∞n0 ⊂ D
of type Kirksee5,9is defined by
xnSnx0, n1,2, S
k
i0
aiTi, a0>0, ai≥0 i1,2, . . . , k, k
i0
ai1. 1.5
Again, the iteration{xn}∞n0⊂Dof type lshikawa with errorsee10–12is defined by
xn11−an1−bn1
xnan1T1yn1bn1un1,
yn1
1−an2−bn2
xnan2T2yn2bn2un2,
.. . ynk−11−ank−bnk
xnankTkynkbnkunk,
ynk
1−an0−bn0
xnan0T0xnbn0un0,
1.6
where{uni}∞n0⊂Di0,1, . . . , k,{ani}∞n0⊂0,1,{bni}∞n0⊂0,1 i0,1, . . . , k, and
max
0≤i≤k
anibni
≤1 n0,1,2, . . .. 1.7
Lemma 1.1see12. Suppose that{ρn},{σn}are two sequences of nonnegative numbers such that
ρn1≤ρnσn, for alln≥n0. If∞n0σn<∞, thenlimn→∞ρnexists.
2. Main result
Theorem 2.1. Let X, d be a complete metric space and let {Tr}kr0 be a family of commuting self-mappings, whereTii 1,2, . . . , kare all nonexpansive andT0 is weakly contractive, then there is a
unique common fixed pointp∈kr0FTrand the iteration{xn}of type Krasnoselski-Mann generated by1.4converges in metric top, with the following error estimate:
dxn, p
≤Φ−1
Φdx0, p
− n
k1
n0,1,2, . . ., 2.1
wheren/k1is the Gauss integer ofn/k1.
Proof. The uniqueness of fixed point ofT0is clear from1.2. Hence, the common fixed point
of {Tr}kr0 is unique. Let X0 be an arbitrary point in X and let {xn} be an iteration of type
Krasnoselski-Mann generated by1.4. Since{Tr}kr0 is commutative, then we have k
r0Tr k
r1TrT0. Suppose thatnimodk1andn/k1 j. Then,
xnxjk1i k
r0
Tr
xj−1k1i i0,1,2, . . . , k, j1,2, . . .. 2.2
Writeyj xjk1i for fixed i. Then{yj}∞j0 is a subsequence of{xn}. Since k
r1Tr is
nonex-pansive andT0is weakly contractive, then we obtain
dyj1, yj
d
k
r1
Tr
T0yj, k
r1
Tr
T0yj−1
≤dT0yj, T0yj−1
≤dyj, yj−1
−ψdyj, yj−1
,
2.3
which shows dyj1, yj ≤ dyj, yj−1, that is, {dyj1, yj}∞j0 is a nonincreasing sequence of
nonnegative real numbers. Therefore, it tends to a limitd≥0. Ifd >0, then, by nondecreasity ofψ,ψdyj1, yj≥ψd, for allj≥0.Thus, from2.3it follows that
dyjk1, yjk
≤dyj1, yj
−kψd, 2.4
a contradiction forklarge enough. Therefore,
lim
j→∞d
yj1, yj
0. 2.5
By2.5, for any givenε >0, there existsNsuch that
dyj1, yj
<min
ε 2, ψ
ε 2
, ∀j≥N. 2.6
We claim that
dyjm, yj
In fact, from 2.6 we see that 2.7 holds when m 1. Suppose that dyjm−1, yj < ε. If
dyjm−1, yj< ε/2, then from2.6we get
dyjm, yj≤dyjm, yjm−1 dyjm−1, yj< 2ε ε2 ε. 2.8
Ifdyjm−1, yj≥ε/2, thenψdyjm−1, yj≥ψε/2,we also get
dyjm, yj
≤dyjm, yj1dyj1, yj
d
k
r0
Tr
yjm−1,
k
r0
Tr
yj
dyj1, yj
≤dyjm−1, yj
−ψdyjm−1, yj
dyj1, yj
< ε−ψ
ε 2
ψ
ε 2
ε.
2.9
Therefore, by induction we derive that 2.7 holds. Since εis arbitrary, {yj} is a Cauchy
se-quence. AsXis complete, we have
lim
j→∞xjk1ipi∈X i0,1,2, . . . , k. 2.10
Observe thatTi i0,1,2, . . . , kare all continuous, so iskr0Tr. From2.10, it follows that k
r0
Tr
pilim j→∞
k
r0
Tr
xjk1ilim
j→∞xj1k1ipi i0,1,2, . . . , k, 2.11
Ti1pilim
j→∞Ti1xjk1ijlim→∞xjk1i1pi1
i0,1,2, . . . , k; Tk1T0; pk1p0
2.12
By1.1,1.2, and2.11, we deduce
dps, pt
d
k
r0
Tr
ps, k
r0
Tr
pt
≤dps, pt
−ψdps, pt
, ∀t /s∈ {0,1,2, . . . , k},
2.13
which shows
pspt, that is, pip i0,1,2, . . . , k. 2.14
From2.12, it implies that pis a common fixed point of{Tr}kr0,that is,p∈kr0FTr. Hence, k
r0FTr {p}. By2.10and2.14, we conclude limn→∞xn p. Setαj dxjk1, p.From
2.3, we have
αj ≤αj−1−ψ
αj−1
, ∀j ∈Z. 2.15
Sinceψ is continuous and nondecreasing, using2.15, it yields
Φαj−1
−Φαj
αj−1
αj dt ψt ≥
αj −αj−1
ψαj−1
≥1,
αj ≤Φ−1
Φα0
−j.
Observe that
dxn, p
dxjk1i, p
dTi· · ·T2T1xjk1, Ti· · ·T2T1p
≤dxjk1, p
, 1≤i≤k. 2.17
From2.16and2.17, we obtain the error estimate2.1. This completes the proof.
Remark 2.2. If Ti I i 1,2, . . . , kin Theorem 2.1, where I is the identity mapping of X,
then we conclude that the sequence {xn}converges to the unique common fixed point p of
weakly contractive mappingT0, with the error estimatedxn, p≤Φ−1Φdx0, p−n, where
xnT0nx0. Thus, ourTheorem 2.1is a generalization of the corresponding theorem of Rhoades
5.
Theorem 2.3. LetXbe a Banach space and let D ⊂ Xbe a nonempty closed convex set. Let {Tr}kr0 be a family of commuting self-mappings, whereTi :D→ D, i1,2, . . . , kare all nonexpansive and
T0 : D → Dis weakly contractive. Then, for any x0 ∈ X, the iteration{xn}of type Kirk generated by1.5converges strongly to a unique common fixed pointp ∈ ki0FTr, with the following error estimate:
xn1−p≤a0Φ−1
1 a0Φ
k
i0
aiTix0−p
−n
n1,2, . . .. 2.18
Proof. ApplyingTheorem 2.1, we can suppose thatpis a unique common fixed point of{Tr}kr0.
Since
Sp
k
i0
aiTi
p
k
i0
ai
Tip
k i0
aipp, 2.19
we derive thatpis a fixed point ofS. SinceTi i1,2, . . . , kare all nonexpansive,T0is weakly
contractive, anda0 / 0, then we have
Sx−Sy
k
i0
ai
Tix−Tiy
≤ k
i0
aiTix−Tiy
≤a0 x−y −a0ψ
x−y
k
i1
ai x−y
x−y −a0ψ
x−y .
2.20
The inequality2.20shows thatSis weakly contractive. Thus,pis a unique fixed point ofS. Setψ1a0ψ.Then,
Φ1
1
a0Φ, Φ
−1
1 a0Φ−1, 2.21
and{xn}converges topwith the following error estimateseeRemark 2.2: xn1−p≤Φ−1
1
Φ1x1−p−n
. 2.22
Observe that
x1−pSx0−p≤k i0
aiTix0−p. 2.23
Theorem 2.4. LetXbe a Banach space and let D ⊂ Xbe a nonempty closed convex set. Let {Tr}kr0
be a family of commuting self-mappings, whereTi: D→ Di1,2, . . . , kare all nonexpansive and
T0:D→Dis weakly contractive. For anyx0∈D, let{xn}be the iteration of type Ishikawa generated by1.6, where
∞ n0 k i0
ani ∞,
∞
n0
max
0≤i≤kbni <∞, 2.24
and{uni}∞n0⊂Xi0,1, . . . , kare all bounded. Then,{xn}converges strongly to a unique common fixed pointp∈sr0FTrwith the following estimate:
xn1−p≤Φ−1
Φx0−p−
n j0 k i0 aji M n j0 k i0
bji, 2.25
whereMmax0≤i≤ksupn≥1 uni−p .
Proof. Applying Theorem 2.1, we can suppose that p is a unique common fixed point of {Tr}kr0. Since {uni}i 0,1, . . . , k are all bounded, we have M max0≤i≤k
supn≥1 uni−p <∞. SinceTii1,2, . . . , kare all nonexpansive andT0is weakly contractive,
we obtain in proper order that
ynk−p≤
1−an0−bn0xn−pan0T0xn−pbn0un0−p
≤1−an0xn−pan0xn−p−ψxn−pbn0M
≤xn−p−an0ψxn−pbn0M,
ynk−1−p≤
1−ank−bnkxn−pankTkynk−pbnkunk−p
≤1−ankxn−pankynk−pbnkM
≤1−ankxn−pankxn−p−an0ψxn−pbn0M
bnkM
≤xn−p−an0ankψxn−p
bn0bnk
M, ..
.
yn1−p≤xn−p−an0k i2
ani
ψxn−p
bn0
k i2 bni M,
xn1−p≤
1−an1−bn1xn−pan1T1yn1−pbn1un1−p
≤1−an1xn−pan1yn1−pbn1M
≤1−an1xn−pbn1M
an1
xn−p−an0k i2
ani
ψxn−p
bn0
k i2 bni M
≤xn−p− k
i0
ani
ψxn−pM k
i0
bni.
2.26
Writeβn xn−p , θnM k
i0bni.Then ∞
n0θn<∞,and2.26yields
βn1≤βnθn, 2.27
n j0 k i0
ajiψ
βj≤
n
j0
βj−βj1
n
j0
θj ≤β0
n
j0
From 2.27 and Lemma 1.1, it implies that limn→∞βn exists, and so does limn→∞ψβn by the continuity of ψ. From 2.28, it implies that ∞n0ki0aniψβn < ∞. Since
∞ n0
k
i0ani ∞, we conclude that limn→∞ψβn 0. Therefore, limn→∞βn 0, that is,
xn converges strongly top. To establish the error estimate, we set n
j0θj Γn andΓ−1 0.
Then,2.26yields
βn1≤βn−
k
i0
ani
ψβn Γn−Γn−1. 2.29
Setλnβn−Γn−1.From2.29we have
λn1≤λn−
k
i0
ani
ψλn Γn−1
. 2.30
Sinceψ is nondecreasing, from2.30we deduce
Φλn
−Φλn1
λn
λn1 dt ψt ≥
λn−λn1
ψλn
≥ λn−λn1
ψλn Γn−1
≥k
i0
ani. 2.31
Thus,
Φλ0
−Φλn1
n
j0
Φλj
−Φλj1≥
n
j0
k
i0
aji,
λn1≤Φ−1
Φλ0
−n
j0
k
i0
aji
.
2.32
Hence, the estimate2.25holds. This completes the proof.
Acknowledgment
This work is supported by the National Natural Science Foundation of China10671094.
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