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Abstract and Applied Analysis
Volume 2012, Article ID 386983, 9 pages
doi:10.1155/2012/386983
Research Article
On the Convergence of Multistep
Iteration for Uniformly Continuous
Φ-Hemicontractive Mappings
Zhiqun Xue,1 Arif Rafiq,2 and Haiyun Zhou3
1
Department of Mathematics and Physics, Shijiazhuang Tiedao University, Shijiazhuang 050043, China
Hajvery University, 43-52 Industrial Area, Gulberg III, Lahore 54660, Pakistan
3
Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, China
2
Correspondence should be addressed to Zhiqun Xue, xuezhiqun@126.com
Received 18 July 2012; Revised 25 August 2012; Accepted 28 August 2012
Academic Editor: Xiaolong Qin
Copyright q 2012 Zhiqun Xue 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.
It is shown that the convergence of the multistep iterative process with errors is obtained for
uniformly continuous Φ-hemicontractive mappings in real Banach spaces. We also revise the
problems of C. E. Chidume and C. O. Chidume 2005.
1. Introduction
Let E be a real Banach space with norm · and let E∗ be its dual space. The normalized
∗
duality mapping J : E → 2E is defined by
2 Jx f ∈ E∗ : x, f x2 f ,
∀x ∈ E,
1.1
where ·, · denotes the generalized duality pairing. The single-valued-normalized duality
mapping is denoted by j.
A mapping T with domain DT and range RT in E is said to be strongly
pseudocontractive if there is a constant k ∈ 0, 1, and for all x, y ∈ DT , ∃jx − y ∈ Jx − y
such that
2
T x − T y, j x − y ≤ kx − y .
1.2
2
Abstract and Applied Analysis
The mapping T is called Φ-pseudocontractive if there exists a strictly increasing continuous
function Φ : 0, ∞ → 0, ∞ with Φ0 0 such that
2
T x − T y, j x − y ≤ x − y − Φ x − y
1.3
holds for all x, y ∈ DT . It is well known that the strongly pseudocontractive mapping must
be the Φ-pseudocontractive mapping in the special case in which Φt 1 − kt2 , but the
converse is not true in general. That is, the class of strongly pseudocontractive mappings is
a proper subclass of the class of Φ-pseudocontractive mappings. Let FT {x ∈ DT :
T x x} / ∅. If the inequalities 1.2 and 1.3 hold for any x ∈ DT and y ∈ FT , then
the corresponding mapping T is called strongly hemicontractive and Φ-hemicontractive,
respectively.
Let NT {x ∈ E : T x 0} /
∅. An operator T : DT ⊆ E → E is called
strongly quasiaccretive, Φ-quasiaccretive if and only if I − T is strongly hemicontractive,
Φ-hemicontractive, respectively, where I denotes the identity mapping on E. That is, if T
∅ and there exists a strictly increasing continuous function
is Φ-quasi-accretive, then NT /
Φ : 0, ∞ → 0, ∞ with Φ0 0 such that
T x − T y, j x − y ≥ Φ x − y
1.4
holds for all x ∈ DT and y ∈ NT . Many authors have studied extensively the strongly
convergence problems of the iterative algorithms for the class of operators.
In 2004, Rhoades and Soltuz 1 introduced the multistep iteration as follows.
Let D be a nonempty closed convex subset of real Banach space E and let T : D → D
be a mapping. The multistep iteration {xn } is defined by
p−1
p−1
x0 ∈ D,
p−1
yn 1 − bn xn bn T xn , n ≥ 0, p ≥ 2,
ynk 1 − bnk xn bnk T ynk1 , k p − 2, p − 3, . . . , 2, 1,
xn1 1 − an xn an T yn1 ,
1.5
n ≥ 0,
where {an }, {bnk } k 1, 2, . . . , p−1 in 0, 1 satisfy certain conditions. Obviously, the iteration
defined above is generalization of Mann, Ishikawa, and Noor iterations.
Inspired and motivated by the work of Xu 2 and the iteration above, we discuss the
following multistep iteration with errors:
p−1
yn
p−1
bn
p−1
dn
x0 ∈ D,
p−1
p−1
p−1
xn bn T xn dn wn , n ≥ 0, p ≥ 2,
1−
−
ynk 1 − bnk − dnk xn bnk T ynk1 dnk wnk , k p − 2, p − 3, . . . , 2, 1,
xn1 1 − an − cn xn an T yn1 cn un ,
1.6
n ≥ 0,
where {an }, {cn }, {bnk }, {dnk } k 1, 2, . . . , p − 1 in 0, 1 with an cn ≤ 1, bnk dnk ≤ 1,
{un }, {wnk } k 1, 2, . . . , p − 1 are the bounded sequences of D.
Abstract and Applied Analysis
3
In 2005, C. E. Chidume and C. O. Chidume 3 proved the convergence theorems
for fixed points of uniformly continuous generalized Φ-hemicontractive mappings and
published in 3. However, there exists a gap in the proof course of their theorems.
The aim of this paper is to show the convergence of the multistep iteration with errors
for fixed points of uniformly continuous Φ-hemicontractive mappings and revise the results
of C. E. Chidume and C. O. Chidume 3. For this, we need the following Lemmas.
∗
Lemma 1.1 see 4. Let E be a real Banach space and let J : E → 2E be a normalized duality
mapping. Then
x y2 ≤ x2 2 y, j x y ,
1.7
for all x, y ∈ E and for all jx y ∈ Jx y.
∞
∞
Lemma 1.2 see 5. Let {δn }∞
n0 , {λn }n0 and {γn }n0 be three nonnegative real sequences and let
Φ : 0, ∞ → 0, ∞ be a strictly increasing and continuous function with Φ0 0 satisfying
the following inequality:
2
δn1
≤ δn2 − λn Φδn1 γn ,
where λn ∈ 0, 1 with
∞
n0
n ≥ 0,
1.8
λn ∞, γn oλn . Then δn → 0 as n → ∞.
2. Main Results
Theorem 2.1. Let E be an arbitrary real Banach space, D a nonempty closed convex subset of E,
and T : D → D a uniformly continuous Φ-hemicontractive mapping with q ∈ FT /
∅. Let
{an }, {bnk }, {cn }, {dnk } be real sequences in 0, 1 and satisfy the conditions:
i an cn ≤ 1, bnk dnk ≤ 1, k 1, 2, . . . , p − 1;
ii an , cn , bnk , dnk → 0 as n → ∞, k 1, 2, . . . , p − 1;
iii cn oan , ∞
n0 an ∞.
p−1
For some x0 ∈ D, let {un }, {wn1 }, {wn2 }, . . . , {wn } be any bounded sequences of D, and
let {xn } be the multistep iterative sequence with errors defined by 1.6. Then 1.6 converges
strongly to the fixed point q of T .
Proof. Since T : D → D is Φ-hemicontractive mapping, then there exists a strictly increasing
continuous function Φ : 0, ∞ → 0, ∞ with Φ0 0 such that
2
T x − T q, j x − q ≤ x − q − Φ x − q ,
2.1
for x ∈ D, q ∈ FT , that is
T x − x, j x − q ≤ −Φ x − q .
@
Choose some x0 ∈ D and x0 /
T x0 such that x0 − T x0 · x0 − q ∈ RΦ and denote that
r0 x0 − T x0 · x0 − q, RΦ is the range of Φ. Indeed, if Φr → ∞ as r → ∞, then
4
Abstract and Applied Analysis
r0 ∈ RΦ; if sup{Φr : r ∈ 0, ∞} r1 < ∞ with r1 < r0 here, we only give a example.
If r0 2, Φt t/1 t, then sup{Φr : r ∈ 0, ∞} r1 1 < 2 r0 , then for q ∈ D,
q. Furthermore,
there exists a sequence {wn } in D such that wn → q as n → ∞ with wn /
we obtain that T wn → T q as n → ∞. So {wn − T wn } is the bounded sequence. Hence, there
exists natural number n0 such that wn − T wn · wn − q < r1 /2 for n ≥ n0 , then we redefine
x0 wn0 and x0 − T x0 · x0 − q ∈ RΦ. This is to ensure that Φ−1 r0 is defined well.
Step 1. We show that {xn } is a bounded sequence.
Set R Φ−1 r0 , then from above formula @, we obtain that x0 − q ≤ R. Denote
B1 x ∈ D : x − q ≤ R ,
B2 x ∈ D : x − q ≤ 2R .
2.2
Since T is the uniformly continuous, so T is a bounded mapping. We let
M sup T x − q 1
x∈B2
p−1
max sup wn1 − q , sup wn2 − q , . . . , sup wn − q , sup un − q .
n
n
n
n
2.3
Next, we want to prove that xn ∈ B1 . If n 0, then x0 ∈ B1 . Now, assume that it holds
for some n, that is, xn ∈ B1 . We prove that xn1 ∈ B1 . Suppose that it is not the case, then
xn1 − q > R > R/2. Since T is uniformly continuous, then for 0 ΦR/2/8R, there exists
δ > 0 such that T x − T y < 0 when x − y < δ. Denote
ΦR/2
δ
R
,
,
.
τ0 min 1,
M 8RM 2R 2M 4R
2.4
Since an , bnk , cn , dnk → 0 as n → ∞ for k 1, 2, . . . , p − 1. Without loss of generality, we assume
that 0 ≤ an , bnk , cn , dnk ≤ τ0 for any n ≥ 0. Since cn oan , let cn < an τ0 . Now, estimate ynk − q
for k 1, 2, . . . , p − 1. By using 1.6, we have
p−1
p−1 p−1 p−1 p−1
p−1
yn − q ≤ 1 − bn − dn xn − q bn T xn − q dn wn − q
≤ R τ0 M
2.5
≤ 2R,
p−1
then yn
∈ B2 . Similarly, we have
p−2
p−2 p−2 p−1
p−2 p−2
p−2
yn − q ≤ 1 − bn − dn xn − q bn T yn − q dn wn − q
≤ R τ0 M
≤ 2R,
2.6
Abstract and Applied Analysis
p−2
then yn
5
∈ B2 , . . .. We have
1
yn − q ≤ 1 − bn1 − dn1 xn − q bn1 T yn2 − q dn1 wn1 − q
≤ R τ0 M
2.7
≤ 2R,
then yn1 ∈ B2 . Therefore, we get
xn1 − q ≤ 1 − an − cn xn − q an T yn1 − q cn un − q
≤ R τ0 M
2.8
≤ 2R.
And we have
xn1 − xn ≤ an T yn1 − xn cn un − xn ≤ an T yn1 − q xn − q cn un − q xn − q
≤ τ0
1
T yn − q un − q 2xn − q
≤ τ0 M 2R
ΦR/2
,
8R
xn1 − yn1 ≤ an T yn1 − xn cn un − xn bn1 T yn2 − xn dn1 wn1 − xn ≤ an T yn1 − q xn − q cn un − q xn − q
≤
bn1 T yn2 − q xn − q dn1 wn1 − q xn − q
≤ τ0 T yn1 − q un − q 2xn − q
T yn2 − q wn1 − q 2xn − q
≤ τ0 2M 4R
≤ δ.
2.9
Further, by using uniform continuity of T , we have
ΦR/2
.
T xn1 − T yn1 <
8R
2.10
6
Abstract and Applied Analysis
In view of Lemma 1.1 and the above formulas, we obtain
xn1 − q2
2
xn − q an T yn1 − xn cn un − xn 2
≤ xn − q 2an T yn1 − xn , j xn1 − q 2cn un − xn , j xn1 − q
2
≤ xn − q 2an T xn1 − xn1 xn1 − xn − T xn1 T yn1 , j xn1 − q
2cn un − xn · xn1 − q
2
≤ xn − q − 2an Φ xn1 − q 2an xn1 − xn · xn1 − q
2an T xn1 − T yn1 · xn1 − q 2cn un − q xn − q xn1 − q
2
ΦR/2
ΦR/2
R
≤ xn − q − 2an Φ
2an
· 2R 2an
· 2R 2an τ0 R M2R
2
8R
8R
2
ΦR/2
R
2an
≤ xn − q − an Φ
R M2R
2
8RM 2R
2 an
R
≤ xn − q − Φ
≤ R2 ,
2
2
2.11
which is a contradiction. Hence, xn1 ∈ B1 , that is, {xn } is a bounded sequence; it leads to that
p−1
{yn1 }, {yn2 }, . . . , {yn } are all bounded sequences as well.
Step 2. We want to prove xn − q → 0 as n → ∞.
Since an , bnk , cn , dnk → 0 as n → ∞ and {xn }, {yn1 } are bounded. From 2.9, we obtain
lim xn1 − xn 0,
n→∞
lim xn1 − yn1 0,
n→∞
lim T xn1 − T yn1 0.
n→∞
2.12
By 2.11, we have
xn1 − q2 2
xn − q an T yn1 − xn cn un − xn 2
≤ xn − q 2an T yn1 − xn , j xn1 − q 2cn un − xn , j xn1 − q
2
≤ xn − q 2an T xn1 − xn1 xn1 − xn − T xn1 T yn1 , j xn1 − q
2cn un − xn · xn1 − q
Abstract and Applied Analysis
7
2
≤ xn − q − 2an Φ xn1 − q 2an xn1 − xn · xn1 − q
2an T xn1 − T yn1 · xn1 − q 2cn un − xn · xn1 − q
2
xn − q − 2an Φ xn1 − q oan ,
2.13
where 2an xn1 −xn ·xn1 −q2an T xn1 −T yn1 ·xn1 −q2cn un −xn ·xn1 −q oan .
By Lemma 1.2, we obtain that limn → ∞ xn − q 0.
Theorem 2.2. Let E be an arbitrary real Banach space and let T : E → E be a uniformly continuous
Φ-quasi-accretive operator with q ∈ NT /
∅. Let {an }, {bnk }, {cn }, {dnk } be real sequences in 0, 1
and satisfy the conditions:
i an cn ≤ 1, bnk dnk ≤ 1, k 1, 2, . . . , p − 1;
ii an , cn , bnk , dnk → 0 as n → ∞, k 1, 2, . . . , p − 1;
iii cn oan , ∞
n0 an ∞.
p−1
For some x0 ∈ E, let {un }, {wn1 }, {wn2 }, . . . , {wn } be any bounded sequences of E, and let {xn } be
the multistep iterative sequence with errors defined by
x0 ∈ D,
p−1
yn
p−1
p−1
p−1
p−1 p−1
1 − bn − dn xn bn Sxn dn wn ,
ynk 1 − bnk − dnk xn bnk Synk1 dnk wnk ,
n ≥ 0, p ≥ 2,
k p − 2, p − 3, . . . , 2, 1,
xn1 1 − an − cn xn an Syn1 cn un ,
2.14
n ≥ 0,
where S : E → E is defined by Sx x − T x for all x ∈ E. Then 2.14 converges strongly to the fixed
point q of S.
Proof. We find easily that S is a uniformly continuous Φ-hemicontractive. Then the
conclusion of Theorem 2.2 is obtained directly by Theorem 2.1.
Remark 2.3. In Theorems 2.1 and 2.2, if bnk dnk 0 k p − 1, p − 2, . . . , 2, 1, then, the
conclusions are as follows.
Corollary 2.4. Let E be an arbitrary real Banach space, D a nonempty closed convex subset of E, and
T : D → D a uniformly continuous Φ-hemicontractive mapping with q ∈ FT /
∅. Let {an }, {cn }
be real sequences in 0, 1 and satisfy the conditions i an cn ≤ 1; ii an , cn → 0 as n → ∞;
iii cn oan and ∞
n0 an ∞. For some x0 ∈ D, let {un } be any bounded sequence of D, and let
{xn } be Mann iterative sequence with errors defined by xn1 1 − an − cn xn an T xn cn un , n ≥ 0.
Then {xn } converges strongly to the fixed point q of T .
Corollary 2.5. Let E be an arbitrary real Banach space and let T : E → E be a uniformly continuous
Φ-quasi-accretive operator with q ∈ NT /
∅. Let {an }, {cn } be real sequences in 0, 1 and satisfy
the conditions i an cn ≤ 1; ii an , cn → 0 as n → ∞; iii cn oan and ∞
n0 an ∞. For
8
Abstract and Applied Analysis
some x0 ∈ E, let {un } be any bounded sequence of E, and let {xn } be Mann iterative sequence with
errors defined by xn1 1 − an − cn xn an Sxn cn un , n ≥ 0. where S : E → E is defined by
Sx x − T x for all x ∈ E. Then {xn } converges strongly to the fixed point q of S.
Remark 2.6. It is mentioned to notice that there exists a serious shortcoming in the proof
process of Theorem 2.3 of 3. That is, M1 cn ≤ Φ/4αn does not hold in line 15 of Claim
∞
2 of page 552. The reason is that the conditions ∞
n0 cn < ∞ and
n0 bn ∞, bn → 0 as
n → ∞ can not obtain cn obn .
Counterexample, let the iteration parameters be an 1 − bn − cn , bn , cn in the following:
{bn } : b0 b1 0, bn 1
,
n
n ≥ 2,
1
1
1
1
1 1
1 1 1 1 1 1 1 1 1 1
{cn } : 0, , 2 , 2 , , 2 , 2 , 2 2 , , 2 , 2 , 2 , 2 , 2 , 2 , ,
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
2.15
1
1
1
1 1 1
1 1 1
,
,..., 2, 2, , 2,..., 2, , 2,...
172 182
23 24 25 26
35 36 37
Then,
∞
n0
bn ∞,
∞
n0 cn
<2
∞
2
n1 1/n < ∞, but cn /
obn .
Application 1. Let E R be a real number space with the usual norm and D 0, ∞. Define
T : D → D by
x3
1 x2
Tx 2.16
for all x ∈ D. Then T is uniformly continuous with FT {0}. Define Φ : 0, ∞ → 0, ∞
by
Φt t2
,
1 t2
2.17
then Φ is a strictly increasing function with Φ0 0. For all x ∈ D, q ∈ FT , we obtain that
T x − T q, j x − q x3
− 0, jx − 0
1 x2
x3
,x
1 x2
x4
1 x2
2
x − q −
2.18
x − q2
2
1 x − q
2
x − q − Φ x − q .
Abstract and Applied Analysis
9
Therefore, T is a Φ-hemicontractive mapping. Set
an 1
,
n2
cn 1
n 2
2
,
bnk dnk 1
,
n2
k 1, 2, . . . , p − 1
2.19
for all n ≥ 0.
Acknowledgment
This work is supported by Hebei Natural Science Foundation under Grant no. A2011210033.
References
1 B. E. Rhoades and S. M. Soltuz, “The equivalence between Mann-Ishikawa iterations and multistep
iteration,” Nonlinear Analysis: Theory, Methods & Applications, vol. 58, no. 1-2, pp. 219–228, 2004.
2 Y. Xu, “Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator
equations,” Journal of Mathematical Analysis and Applications, vol. 224, no. 1, pp. 91–101, 1998.
3 C. E. Chidume and C. O. Chidume, “Convergence theorems for fixed points of uniformly continuous
generalized Φ-hemi-contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 303,
no. 2, pp. 545–554, 2005.
4 K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985.
5 C. Moore and B. V. C. Nnoli, “Iterative solution of nonlinear equations involving set-valued uniformly
accretive operators,” Computers & Mathematics with Applications, vol. 42, no. 1-2, pp. 131–140, 2001.
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