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Abstract and Applied Analysis
Volume 2012, Article ID 626547, 10 pages
doi:10.1155/2012/626547
Research Article
Strong Convergence of Parallel Iterative
Algorithm with Mean Errors for Two Finite
Families of Ćirić Quasi-Contractive Operators
Feng Gu
Institute of Applied Mathematics and Department of Mathematics, Hangzhou Normal University,
Zhejiang, Hangzhou 310036, China
Correspondence should be addressed to Feng Gu, gufeng99@sohu.com
Received 23 June 2012; Revised 19 August 2012; Accepted 21 August 2012
Academic Editor: Yonghong Yao
Copyright q 2012 Feng Gu. 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.
The purpose of this paper is to establish a strong convergence of a new parallel iterative algorithm
with mean errors to a common fixed point for two finite families of Ćirić quasi-contractive
operators in normed spaces. The results presented in this paper generalize and improve the
corresponding results of Berinde, Gu, Rafiq, Rhoades, and Zamfirescu.
1. Introduction and Preliminaries
Let X, d be a metric space. A mapping T : X → X is said to be a-contraction, if dT x, T y ≤
adx, y for all x, y ∈ X, where a ∈ 0, 1.
The mapping T : X → X is said to be Kannan mapping 1, if there exists b ∈ 0, 1/2
such that dT x, T y ≤ bdx, T x dy, T y for all x, y ∈ X.
A mapping T : X → X is said to be Chatterjea mapping 2, if there exists c ∈ 0, 1/2
such that dT x, T y ≤ cdx, T y dy, T x for all x, y ∈ X.
Combining these three definitions, Zamfirescu 3 proved the following important
result.
Theorem Z see 3. Let X, d be a complete metric space and T : X → X a mapping for which
there exist the real numbers a, b, and c satisfying a ∈ 0, 1, b, c ∈ 0, 1/2 such that for each pair
x, y ∈ X, at least one of the following conditions holds:
z1 dT x, T y ≤ adx, y,
2
Abstract and Applied Analysis
z2 dT x, T y ≤ bdx, T x dy, T y,
z3 dT x, T y ≤ cdx, T y dy, T x.
Then T has a unique fixed point p and the Picard iteration {xn } defined by
xn1 T xn ,
n∈N
1.1
converges to p for any arbitrary but fixed x1 ∈ X.
Remark 1.1. An operator T satisfying the contractive conditions z1 −z3 in the above theorem
is called Z-operator.
Remark 1.2. The conditions z1 −z3 can be written in the following equivalent form:
dx, T x d y, T y d x, T y d y, T x
,
,
d T x, T y ≤ h max d x, y ,
2
2
1.2
for all x, y ∈ X, 0 < h < 1. Thus, a class of mappings satisfying the contractive conditions
z1 −z3 is a subclass of mappings satisfying the following condition:
d x, T y d y, T x
,
d T x, T y ≤ h max d x, y , dx, T x, d y, T y ,
2
CG
0 < h < 1. The class of mappings satisfying CG is introduced and investigated by Ćirić 4
in 1971.
Remark 1.3. A mapping satisfying CG is commonly called Ćirić generalized contraction.
In 2000, Berinde 5 introduced a new class of operators on a normed space E satisfying
T x − T y ≤ ρx − y LT x − x,
1.3
for any x, y ∈ E, 0 ≤ δ < 1 and L ≥ 0.
Note that 1.3 is equivalent to
T x − T y ≤ ρx − y L min T x − x, T y − y ,
1.4
for any x, y ∈ E, 0 ≤ ρ < 1 and L ≥ 0.
Berinde 5 proved that this class is wider than the class of Zamfiresu operators and
used the Mann 6 iteration process to approximate fixed points of this class of operators in a
normed space given in the form of following theorem.
Theorem B see 5. Let C be a nonempty closed convex subset of a normed space E. Let T : C → C
be an operator satisfying 1.3 and FT / ∅. For given x0 ∈ C, let {xn } be generated by the algorithm
xn1 1 − αn xn αn T xn ,
n ≥ 0,
1.5
Abstract and Applied Analysis
where {αn } be a real sequence in [0, 1]. If
fixed point of T .
3
∞
n1
αn ∞, then {xn }∞
n0 converges strongly to the unique
In 2006, Rafiq 7 considered a class of mappings satisfying the following condition:
x − T x y − T y T x − T y ≤ h max x − y,
, x − T y, y − T x ,
2
CR
0 < h < 1. This class of mappings is a subclass of mappings satisfying the following condition:
T x − T y ≤ h max x − y, x − T x, y − T y, x − T y, y − T x ,
CQ
0 < h < 1. The class of mappings satisfying CQ was introduced and investigated by Ćirić
8 in 1974 and a mapping satisfying is commonly called Ćirić quasi-contraction.
Rafiq 7 proved the following result.
Theorem R see 7. Let C be a nonempty closed convex subset of a normed space E. Let T : C → C
be an operator satisfying the condition CR. For given x0 ∈ C, let {xn } be generated by the algorithm
xn1 αn xn βn T xn γn un ,
n ≥ 0,
1.6
where {αn }, {βn }, and {γn } be three real sequences in [0, 1] satisfying αn βn γn 1 for all n ≥ 1,
∞
{un } is a bounded sequences in C. If ∞
n1 βn ∞ and γn oαn , then {xn }n0 converges strongly
to the unique fixed point of T .
In 2007, Gu 9 proved the following theorem.
Theorem G see 9. Let C be a nonempty closed convex subset of a normed space E. Let {Ti }N
i1 :
N
∅ (the set of common fixed
C → C be N operators satisfying the condition CR with F ∩i1 FTi /
points of {Ti }N
).
Let
{α
},
{β
},
and
{γ
}
be
three
real
sequences
in
[0,
1]
satisfying αn βn γn 1
n
n
n
i1
for all n ≥ 1, {un } a bounded sequences in C satisfying the following conditions:
i
ii
∞
n1
∞
βn ∞;
n1 γn
< ∞ or γn oβn .
Suppose further that x0 ∈ C is any given point and {xn } is generated by the algorithm
xn1 αn xn βn Tn xn γn un ,
n ≥ 0,
1.7
where Tn Tn mod N . Then {xn } converges strongly to a common fixed point of {Ti }N
i1 .
Remark 1.4. It should be pointed out that Theorem G extends Theorem R from a Ćirić quasicontractive operator to a finite family of Ćirić quasi-contractive operators.
4
Abstract and Applied Analysis
Inspired and motivated by the facts said above, we introduced a new two-step parallel
k
iterative algorithm with mean errors for two finite family of operators {Si }m
i1 and {Tj }j1 as
follows:
m
xn1 1 − αn − γn xn αn
λi Si yn γn un ,
n ≥ 1,
i1
yn 1 − βn − δn xn βn
k
1.8
μi Tj xn δn vn ,
n ≥ 1,
j1
m
k
where {λi }m
i1 , {μj }j1 are two finite sequences of positive number such that
i1 λi 1 and
k
μ
1,
{α
},
{β
},
{γ
}
and
{δ
}
are
four
real
sequences
in
0,
1
satisfying
αn γn ≤ 1
n
n
n
n
j1 j
and βn δn ≤ 1 for all n ≥ 1, {un } and {vn } are two bounded sequences in C and x0 is a given
point.
Especially, if {αn }, {γn } are two sequences in 0, 1 satisfying αn γn ≤ 1 for all n ≥ 1,
{λi }m
i1 ⊂ 0, 1 satisfying λ1 λ2 · · · λm 1, {un } is a bounded sequence in C and x0 is a
given point in C, then the sequence {xn } defined by
m
xn1 1 − αn − γn xn αn
λi Si xn γn un ,
n≥1
1.9
i1
is called the one-step parallel iterative algorithm with mean errors for a finite family of
operators {Si }m
i1 .
The purpose of this paper is to study the convergence of two-steps parallel iterative
algorithm with mean errors defined by 1.8 to a common fixed point for two finite family
of Ćirić quasi-contractive operators in normed spaces. The results presented in this paper
generalized and extend the corresponding results of Berinde 5, Gu 9, Rafiq 7, Rhoades
10, and Zamfirescu 3. Even in the case of βn δn 0 or γn δn 0 for all n ≥ 1 or
m k 1 are also new.
In order to prove the main results of this paper, we need the following Lemma.
Lemma 1.5 see 11. Suppose that {an }, {bn }, and {cn } are three nonnegative real sequences
satisfying the following condition:
an1 ≤ 1 − tn an bn cn ,
where n0 is some nonnegative integer, tn ∈ 0, 1,
limn → ∞ an 0.
∞
n0 tn
∀n ≥ n0 ,
∞, bn otn and
1.10
∞
n0 cn
< ∞. Then
2. Main Results
We are now in a position to prove our main results in this paper.
Theorem 2.1. Let C be a nonempty closed convex subset of a normed space E. Let {Si }m
i1 : C → C
be m operators satisfying the condition CR and {Tj }kj1 : C → C be k operators satisfying the
Abstract and Applied Analysis
5
k
∅, where FSi and FTj are the set of fixed
condition CR with F m
i1 FSi ∩ j1 FTj /
points of Si and Tj in C, respectively. Let {αn }, {βn }, {γn }, and {δn } be four real sequences in [0, 1]
k
satisfying αn γn ≤ 1 and βn δn ≤ 1 for all n ≥ 1, {λi }m
i1 , {μj }j1 two finite sequences of positive
k
number such that m
i1 λi 1 and
j1 μj 1, {un } and {vn } two bounded sequences in C satisfying
the following conditions:
i ∞
n1 αn ∞;
ii limn → ∞ δn 0;
iii ∞
n1 γn < ∞ or γn oαn .
Suppose further that x0 ∈ C is any given point and {xn } is an iteration sequence with mane errors
k
defined by 1.8, then {xn } converges strongly to a common fixed point of {Si }m
i1 and {Tj }j1 .
Proof. Since {Si }m
i1 : C → C is m Ćirić operator satisfying the condition CR, hence there
exists 0 < hi < 1 i ∈ I {1, 2, . . . , m} such that
x − Si x y − Si y Si x − Si y ≤ hi max x − y,
, x − Si y , y − Si x .
2
2.1
For each fixed i ∈ I {1, 2, . . . , m}. Denote h max{h1 , h2 , . . . , hm }, then 0 < h < 1 and
x − Si x y − Si y Si x − Si y ≤ h max x − y,
, x − Si y, y − Si x
2
2.2
hold for each fixed i ∈ I {1, 2, . . . , m}. If from 2.2 we have
Si x − Si y ≤ h x − Si x y − Si y ,
2
2.3
Si x − Si y ≤ h x − Si x y − Si y
2
h
≤
x − Si x y − x x − Si x Si x − Si y .
2
2.4
then
Hence
1−
h Si x − Si y ≤ h x − y hx − Si x,
2
2
2.5
which yields using the fact that 0 < h < 1
Si x − Si y ≤
h
h/2 x − y x − Si x.
1 − h/2
1 − h/2
2.6
6
Abstract and Applied Analysis
Also, from 2.2, if
Si x − Si y ≤ h max x − Si y, y − Si x
2.7
holds, then
a Si x − Si y ≤ hx − Si y, which implies Si x − Si y ≤ hx − Si x hSi x − Si y
and hence, as h < 1,
Si x − Si y ≤
h
x − Si x,
1−h
2.8
or
b Si x − Si y ≤ hy − Si x, which implies
Si x − Si y ≤ hy − x hx − Si x.
2.9
Thus, if 2.7 holds, then from 2.8 and 2.9 we have
Si x − Si y ≤ hy − x h
x − Si x.
1−h
2.10
Denote
ρ1 max h,
h/2
1 − h/2
h,
L1 max h,
h
h
,
1 − h/2 1 − h
h
.
1−h
2.11
Then we have 0 < ρ1 < 1 and L1 ≥ 0. Combining 2.2,2.6, and 2.10 we get
Si x − Si y ≤ ρ1 x − y L1 x − Si x
2.12
holds for all x, y ∈ C and i ∈ I.
On the other hand, since {Tj }kj1 : C → C is k Ćirić operator satisfying the condition
CR, similarly, we can prove
Tj x − Tj y ≤ ρ2 x − y L2 x − Si x,
for all x, y ∈ C and j ∈ J {1, 2, . . . , k}, where 0 < ρ2 < 1 and L2 ≥ 0.
2.13
Abstract and Applied Analysis
Let p ∈ F m
i1
FSi ∩ 7
k
j1
FTj ; using 1.8 we have
m
xn1 − p λi Si yn − p γn un − p 1 − αn − γn xn − p αn
i1
m
≤ 1 − αn − γn xn − p αn
λi Si yn − p γn un − p
2.14
i1
m
≤ 1 − αn xn − p αn
λi Si yn − p γn M,
i1
where M supn≥1 {||un − p||, ||vn − p||}. Now for y yn and x p, 2.12 gives
Si yn − p Si yn − Si p ≤ ρ1 yn − p.
2.15
Substituting 2.15 into 2.14, we obtain that
xn1 − p ≤ 1 − αn xn − p αn ρ1 yn − p γn M.
2.16
Again it follows from 1.8 that
k
yn − p 1 − βn − δn xn − p βn
μj Tj xn − p δn vn − p j1
k
≤ 1 − βn − δn xn − p βn
μj Tj xn − p δn vn − p
2.17
j1
k
≤ 1 − βn xn − p βn
μj Tj xn − p δn M.
j1
Now for y xn and x p, 2.13 gives
Tj xn − p Tj xn − Tj p ≤ ρ2 xn − p.
2.18
Combining 2.17 and 2.18 we get
yn − p ≤ 1 − βn 1 − ρ2 xn − p δn M ≤ xn − p δn M.
2.19
Substituting 2.19 into 2.16, we obtain that
xn1 − p ≤ 1 − αn xn − p αn ρ1 xn − p δn M γn M
1 − αn 1 − ρ1 xn − p αn δn ρ1 M γn M
1 − tn xn − p bn cn ,
2.20
8
Abstract and Applied Analysis
where
tn αn 1 − ρ1 ,
bn αn δn ρ1 M,
cn γn M
2.21
or
tn αn 1 − ρ1 ,
bn αn δn ρ1 M γn M,
cn 0.
2.22
From the conditions i–iii it is easy to see that tn ∈ 0, 1, ∞
n1 tn ∞, bn otn ,
c
<
∞.
Thus
using
2.20
and
Lemma
1.5
we
have
lim
and ∞
n → ∞ ||xn − p|| 0, and so
n1 n
limn → ∞ xn p. This completes the proof of Theorem 2.1.
Theorem 2.2. Let C be a nonempty closed convex subset of a normed space E. Let {Si }m
i1 : C → C
be m operators satisfying the condition 2.12 and let {Tj }kj1 : C → C be k operators satisfying
k
∅, where FSi and FTj are the set of
the condition 2.13 with F m
i1 FSi ∩ j1 FTj /
fixed points of Si and Tj in C, respectively. Let {αn }, {βn }, {γn }, and {δn } be four real sequences in
k
[0, 1] satisfying αn γn ≤ 1 and βn δn ≤ 1 for all n ≥ 1, {λi }m
i1 , {μj }j1 two finite sequences of
k
positive number such that m
i1 λi 1, and
j1 μj 1, {un } and {vn } two bounded sequences in C
satisfying the following conditions:
i ∞
n1 αn ∞;
ii limn → ∞ δn 0;
iii ∞
n1 γn < ∞ or γn oαn .
Suppose further that x0 ∈ C is any given point and {xn } is an iteration sequence defined by 1.8,
k
then {xn } converges strongly to a common fixed point of {Si }m
i1 and {Tj }j1 .
Theorem 2.3. Let C be a nonempty closed convex subset of a normed space E. Let {Si }m
i1 : C → C
FS
∅
(the
set
of
common
fixed points
be m operators satisfying the condition CR with F m
i /
i1
m
of {Si }i1 ). Let {αn } and {γn } be two real sequences in [0, 1] satisfying αn γn ≤ 1 for all n ≥ 1,
m
{λi }m
i1 a finite sequence of positive number such that
i1 λi 1, and {un } a bounded sequence in C
satisfying the following conditions:
i ∞
n1 αn ∞;
∞
ii n1 γn < ∞ or γn oαn .
Suppose further that x0 ∈ C is any given point and {xn } is an iteration sequence with mane errors
defined by 1.9, then {xn } converges strongly to a common fixed point of {Si }m
i1 .
Theorem 2.4. Let C be a nonempty closed convex subset of a normed space E. Let {Si }m
i1 : C → C
FS
∅
(the
set
of
common
fixed points
be m operators satisfying the condition 2.12 with F m
/
i
i1
).
Let
{α
}
and
{γ
}
be
two
real
sequences
in
[0,
1]
satisfying
α
γ
≤
1
for
all n ≥ 1,
of {Si }m
n
n
n
n
i1
m
a
finite
sequence
of
positive
number
such
that
λ
1,
and
{u
}
a
bounded
sequence
in C
{λi }m
n
i1
i1 i
satisfying the following conditions:
i ∞
n1 αn ∞;
∞
ii n1 γn < ∞ or γn oαn .
Abstract and Applied Analysis
9
Suppose further that x0 ∈ C is any given point and {xn } is an iteration sequence defined by 1.9,
then {xn } converges strongly to a common fixed point of {Si }m
i1 .
Corollary 2.5 see 7. Let C be a nonempty closed convex subset of a normed space E. Let T : C →
C be an operators satisfying the condition CR. Let {αn }, {βn }, and {γn } be three real sequences in
[0, 1] satisfying αn βn γn 1 for all n ≥ 1 and {un } a bounded sequences in C satisfying the
following conditions:
i ∞
n1 βn ∞;
∞
ii n1 γn < ∞ or γn oβn .
Suppose further that x0 ∈ C is any given point and {xn } is an explicit iteration sequence as follows:
xn1 αn xn βn T xn γn un ,
n ≥ 1,
2.23
then {xn } converges strongly to the unique fixed point of T .
Proof. By Ćirić 8, we know that T has a unique fixed point in C. Taking m 1 in Theorem 2.3,
then the conclusion of Corollary 2.5 can be obtained from Theorem 2.3 immediately. This
completes the proof of Corollary 2.5.
Remark 2.6. Theorems 2.2–2.4 and Corollary 2.5 improve and extend the corresponding
results of Berinde 5, Gu 9, Rafiq 7, Rhoades 10, and Zamfirescu 3.
Acknowledgments
The present study was supported by the National Natural Science Foundation of China
11071169, 11271105 and the Natural Science Foundation of Zhejiang Province Y6110287,
Y12A010095.
References
1 R. Kannan, “Some results on fixed points,” Bulletin of the Calcutta Mathematical Society, vol. 60, pp.
71–76, 1968.
2 S. K. Chatterjea, “Fixed-point theorems,” Comptes Rendus de l’Académie Bulgare des Sciences, vol. 25, no.
6, pp. 727–730, 1972.
3 T. Zamfirescu, “Fix point theorems in metric spaces,” Archiv der Mathematik, vol. 23, no. 1, pp. 292–298,
1972.
4 L. B. Ćirić, “Generalized contractions and fixed point theorems,” Publications de l’Institut Mathématique, vol. 12, no. 26, pp. 19–26, 1971.
5 V. Berinde, Iterative Approximation of Fixed Points, Efemeride, Baia Mare, Romania, 2000.
6 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol.
4, pp. 506–510, 1953.
7 A. Rafiq, “Fixed points of Ćirić quasi-contractive operators in normed spaces,” Mathematical Communications, vol. 11, no. 2, pp. 115–120, 2006.
8 L. B. Ćirić, “A generalization of Banach’s contraction principle,” Proceedings of the American
Mathematical Society, vol. 45, no. 2, pp. 267–273, 1974.
9 F. Gu, “Strong convergence of an explicit iterative process with mean errors for a finite family of Ćirić
quasi-contractive operators in normed spaces,” Mathematical Communications, vol. 12, no. 1, pp. 75–82,
2007.
10
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10 B. E. Rhoades, “Fixed point iterations using infinite matrices,” Transactions of the American Mathematical Society, vol. 196, pp. 161–176, 1974.
11 S. S. Chang, Y. J. Cho, and H. Y. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach
Spaces, Nova Science, New York, NY, USA, 2002.
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