Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 457024, 11 pages
doi:10.1155/2012/457024
Research Article
Strong Convergence of the Iterative Methods
for Hierarchical Fixed Point Problems of an Infinite
Family of Strictly Nonself Pseudocontractions
Wei Xu1 and Yuanheng Wang2
1
2
Tongji Zhejiang College, Zhejiang 314000, China
Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China
Correspondence should be addressed to Yuanheng Wang, wangyuanheng@yahoo.com.cn
Received 20 August 2012; Accepted 11 September 2012
Academic Editor: Yonghong Yao
Copyright q 2012 W. Xu and Y. Wang. 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.
This paper deals with a new iterative algorithm for solving hierarchical fixed point problems of
an infinite family of pseudocontractions in Hilbert spaces by yn βn Sxn 1 − βn xn , xn1 n
PC αn fxn 1 − αn ∞
i1 μi Ti yn , and ∀n ≥ 0, where Ti : C → H is a nonself ki -strictly
pseudocontraction.
Under certain approximate conditions, the sequence {xn } converges strongly
to x∗ ∈ ∞
i1 FTi , which solves some variational inequality. The results here improve and extend
some recent results.
1. Introduction
Let H be a real Hilbert space with inner product ·
and norm · . Let C be a nonempty
closed convex subset of H. A mapping f : C → H is called a contraction with coefficient γ if
there exits a constant γ ∈ 0, 1 such that
fx − f y ≤ γ x − y, ∀x, y ∈ C.
1.1
A mapping T : C → C is called nonexpansive if
T x − T y ≤ x − y, ∀x, y ∈ C.
1.2
A mapping T : C → H is called k-strictly pseudocontraction if there exits a constant
k ∈ 0, 1 such that
T x − T y2 ≤ x − y2 kI − T x − I − T y2 , ∀x, y ∈ C.
1.3
2
Abstract and Applied Analysis
Write FT as the set of fixed points of T , that is, FT {x ∈ C, T x x}. In 2000,
Moudafi 1 introduced an iterative scheme for nonexpansive mappings
xn1 αn fxn 1 − αn T xn ,
∀n ≥ 0,
1.4
where f be a contraction on H and the sequence {xn } started with arbitrary initial x0 ∈ H.
In 2004, Xu 2 proved that the sequence {xn } generated by 1.4 converges strongly to a
fixed point of T under certain conditions on the parameters, which also solves the variational
inequality
I − f x∗ , x − x∗ ≥ 0,
∀x ∈ FT .
1.5
Recently, some authors studied the problems of fixed points of nonexpansive
mappings with strongly positive operators, Lipschitizian, strongly monotone operators, and
extragradient methods, and many convergence results were obtained such as, see 3–9.
In 2008, Yao et al. 10 introduced the following iterative scheme:
x0 x ∈ C,
yn βn xn 1 − βn T xn ,
xn1 αn fxn 1 − αn yn ,
1.6
∀n ≥ 0,
where f is a contraction on C and T : C → C is nonexpansive mapping. In 2012, Song et al.
11 analyzed the following iterative algorithm:
x0 x ∈ C,
yn PC βn xn 1 − βn
xn1 αn fxn γn xn ∞
i1
n
μi Ti xn
1 − γn I − αn F yn ,
,
1.7
∀n ≥ 0,
where Ti is a ki -strictly pseudocontraction, F : C → C is a lipschitzian and strongly monotone
operator, f : C → C is a contraction, and PC is the metric projection from H onto C.
Under certain conditions on the parameters, the sequence {xn } generated by 1.7 converges
strongly to a fixed point of a countable family of ki -strictly pseudocontraction, which is the
solution of some variational inequality.
On the other hand, in 2010, Yao et al. 12 introduced the iterative algorithm for solving
hierarchical fixed point of nonexpansive mappings and gave the following theorem.
Theorem YCL
Let C be a nonempty closed convex subset of a real Hilbert space H. Let f : C → H be a
contraction with coefficient γ ∈ 0, 1. Suppose the following conditions are satisfied:
i limn → ∞ αn 0 and Σ∞
n0 αn ∞;
ii limn → ∞ βn /αn 0;
iii limn → ∞ |αn1 − αn |/αn 0 and limn → ∞ |βn1 − βn |/βn 0.
Abstract and Applied Analysis
3
Then the sequence {xn } generated by
xn1
x0 x ∈ C,
yn βn Sxn 1 − βn xn ,
PC αn fxn 1 − αn T yn ,
1.8
∀n ≥ 0,
converges strongly to a point of x∗ ∈ H, which is the unique solution of the variational
inequality
x∗ ∈ FT , I − f x∗ , x − x∗ ≥ 0, ∀x ∈ FT .
1.9
Motivated and inspired by the iterative schemes 1.7 and 1.8, we introduce and
study the hybrid iterative algorithm for solving some hierarchical fixed point problem of
infinite family of strictly nonself pseudocontractions:
xn1 PC
x0 x ∈ C,
yn βn Sxn 1 − βn xn ,
∞
n
αn fxn 1 − αn μi Ti yn ,
1.10
∀n ≥ 0,
i1
where S, f, and PC are the same in 1.8, Ti : C → H is a nonself ki -strictly pseudocontraction.
Under certain conditions on the parameters, the sequence {xn } generated by 1.10 converges
strongly to a common fixed point of infinite family of ki -strictly pseudocontractions, which
solves the variational inequality
x∗ ∈
∞
FTi ,
I − f x∗ , x − x∗ ≥ 0,
∀x ∈
i1
∞
FTi .
i1
1.11
So, our results extend and improve some results of other authors such as 10–12 from
self-mappings to nonself-mappings, from nonexpansive mappings to ki -strictly pseudocontraction, and from one mapping to a infinite family mappings.
2. Preliminaries
In this section, we recall some basic facts that will be needed in the proof of the main results.
Lemma 2.1 see 13 demiclosedness principle. Let C be a nonempty closed convex subset of a
real Hilbert space H and let T : C → C be a nonexpansive mapping with FT / ∅. If {xn } is a
sequence in C weakly converging to x and if {I − T xn } converges strongly to y, then I − T x y;
in particular if y 0, then x ∈ FT .
Lemma 2.2 see 9. Let x ∈ H and z ∈ C be any points. The following results hold:
1 that z PC x if and only if there holds the relation:
x − y, y − z ≤ 0, ∀y ∈ C;
2.1
4
Abstract and Applied Analysis
2 that z PC x if and only if there holds the relation:
2 2
x − z2 ≤ x − y − y − z ,
∀y ∈ C;
2.2
3 there holds the ration:
2
PC x − PC y, x − y ≥ PC x − PC y ,
∀x, y ∈ H.
2.3
Lemma 2.3 see 14. For all x, y ∈ H, the following inequality holds:
x y2 ≤ x2 2 y, x y .
2.4
Lemma 2.4 see 3. Let f : C → H be a contraction with coefficient γ ∈ 0, 1 and let T : C → C
be a nonexpansive mapping. Then for all x, y ∈ C:
1 the mapping I − f is strongly monotone with coefficient 1 − γ, that is,
2
x − y, I − f x − I − f y ≥ 1 − γ x − y ,
2.5
2 the mapping I − T is monotone:
x − y, I − T x − I − T y ≥ 0.
2.6
Lemma 2.5 see 15. Let H be a Hilbert space and let C be a nonempty convex subset of H. Let
T : C → H be a k-strictly pseudocontractive mapping with FT /
∅. Then FPC T FT .
Lemma 2.6 see 16. Let H be a Hilbert space and let C be a nonempty convex subset of H. Let
T : C → H be a k-strictly pseudocontractive mapping. Define a mapping Jx δx 1 − δT x for all
x ∈ C. Then as δ ∈ k, 1, J is a nonexpansive mapping such that FJ FT .
Lemma 2.7 see 11. Let H be a Hilbert space and let C be a nonempty convex subset of H.
Assume that Ti : C → H is a countable family of ki -strictly pseudocontraction for some 0 ≤ ki < 1
and sup{ki : i ∈ N} < 1 such that ∞
/ ∅. Assume that {μi } is a positive sequence such that
i1 FTi ∞
∞
: C → H is a k-strictly pseudocontraction with coefficient k sup{ki :
i1 μi 1. Then
i1 μi Ti
∞
i ∈ N} and F ∞
i1 FTi .
i1 μi Ti Lemma 2.8 see 17. Let {αn } be a sequence of nonnegative real numbers satisfying the following
relation: αn1 ≤ 1 − γn αn δn , where 1 {γn } ⊂ 0, 1, Σ∞
n1 γn ∞; 2 lim supn → ∞ δn /γn ≤ 0
∞
or Σn1 |δn | < ∞, then limn → ∞ αn 0.
3. Main Results
In this section, we prove some strong convergence results on the iterative algorithm for
solving hierarchical fixed point problem.
Abstract and Applied Analysis
5
Theorem 3.1. Let H be a real Hilbert space and let C be a nonempty closed convex subset of H.
Let f : C → H be a (possibly nonself) contraction with coefficient γ ∈ 0, 1, and let S : C → C
be a nonexpansive mapping. Let Ti : C → H be a countable family of ki -strictly (possibly nonself)
pseudocontraction with 0 ≤ ki ≤ k < 1 such that ∞
/ ∅. Let the sequence {xn } be generated
i1 FTi ∞ n
n
by 1.10 with {αn }, {βn } in 0, 1. Suppose for each n, i1 μi 1, for all n and μi ≥ 0, for
all i ∈ N. Assume that the parameters satisfy the following conditions:
i limn → ∞ αn 0 and Σ∞
n0 αn ∞;
ii limn → ∞ βn /αn 0;
iii limn → ∞ |αn1 − αn |/αn 0, limn → ∞ |βn1 − βn |/αn 0, and limn → ∞ n−1
μi
|/αn 0;
iv limn → ∞
∞
i1
n
|μi
− μi | 0 and
∞
i1
∞
i1
n
|μi
−
μi 1 μi > 0.
Then the sequence {xn } converges strongly to x∗ ∈
inequality
I − f x∗ , x − x∗ ≥ 0,
∞
∀x ∈
i1
FTi , which solves the variational
∞
FTi .
i1
3.1
Proof. The proof is divided into four steps.
Step 1. We show that the sequences {xn } and {yn } are bounded.
n
For each n ≥ 0, write Bn ∞
i1 μ
i Ti and by Lemma 2.7, we have Bn is a k-strictly
pseudocontraction on C and FBn ∞
i1 FTi , for all n ∈ N. Therefore, the iterative
algorithm 1.10 can be written as
xn1
x0 x ∈ C,
yn βn Sxn 1 − βn xn ,
PC αn fxn 1 − αn Bn yn ,
3.2
∀n ≥ 0.
By condition ii, without loss of generality, we may assume βn ≤ αn , for all n ≥ 0. Take
p∈ ∞
i1 FTi and we estimate Bn yn − p. For fixed approximate δ ∈ k, 1, define a mapping
Jx δx 1 − δBn x and by Lemma 2.6, J is a nonexpansive mapping and FixJ FixBn .
So
Bn yn − p 1 Jyn − δyn − p
1 − δ
≤
1 Jyn − Jp δyn − p
1−δ
≤
1 δ
yn − p.
1−δ
3.3
6
Abstract and Applied Analysis
Together with 3.2 and 3.3, we get
xn1 − p PC αn fxn 1 − αn Bn yn − PC p
≤ αn fxn 1 − αn Bn yn − p
≤ αn fxn − f p αn f p − p 1 − αn Bn yn − p
1 δ
yn − p
≤ αn γ xn − p αn f p − p 1 − αn 1−δ
1δ ≤ αn γ xn − p αn f p − p 1 − αn βn Sxn − p
1−δ
1 − αn 1 δ
1 − βn xn − p
1−δ
1δ ≤ αn γ xn − p αn f p − p 1 − αn βn Sxn − Sp
1−δ
1 − αn 3.4
1δ 1 δ
βn Sp − p 1 − αn 1 − βn xn − p
1−δ
1−δ
1 δ
xn − p 1 δ αn Sp − p αn f p − p
≤ αn γ xn − p 1 − αn 1−δ
1−δ
1δ
1 − αn
− γ xn − p
1−δ
1
1δ
1 δ
Sp − p f p − p .
αn
−γ
1−δ
1 δ/1 − δ − γ 1 − δ
Therefore, we obtain
xn − p ≤ max x0 − p,
1
1 δ/1 − δ − γ
1 δ
Sp − p f p − p ,
1−δ
3.5
which gives the results that the sequence {xn } is bounded and so are {fxn }, {yn }, {Bn xn },
{Bn yn }.
Step 2. Now we show that xn1 − xn → 0 as n → ∞. Let
vn αn fxn 1 − αn Bn yn .
Next we estimate xn1 − xn . From 3.2,we have
xn1 − xn PC vn − PC vn−1 ≤ vn − vn−1 αn fxn 1 − αn Bn yn − αn−1 fxn−1 − 1 − αn−1 Bn−1 yn−1 αn fxn − fxn−1 αn − αn−1 fxn−1 1 − αn Bn yn − Bn yn−1
1 − αn Bn yn−1 1 − αn−1 Bn yn−1 − Bn−1 yn−1 3.6
Abstract and Applied Analysis
7
≤ αn fxn − fxn−1 |αn − αn−1 |fxn−1 − Bn yn−1 1 − αn ∞ 1 δ
yn − yn−1 1 − αn−1 μn − μn−1 Ti yn−1 i
i
1−δ
i1
1 δ
yn − yn−1 1−δ
∞ n
n−1 M |αn − αn−1 | μi − μi
,
≤ αn γxn − xn−1 1 − αn i1
3.7
where M is a constant such that
sup fxn−1 − Bn yn−1 Sxn−1 − xn−1 Ti yn−1 ≤ M.
n∈N
3.8
From 3.2, we also obtain
yn − yn−1 βn Sxn 1 − βn xn − βn−1 Sxn−1 − 1 − βn−1 xn−1 βn Sxn − Sxn−1 βn − βn−1 Sxn−1
1 − βn xn − xn−1 βn−1 − βn xn−1 3.9
βn Sxn − Sxn−1 1 − βn xn − xn−1 βn − βn−1 Sxn−1 − xn−1 ≤ xn − xn−1 βn − βn−1 M.
Together with 3.7 and 3.9, we have
xn1 − xn ≤ vn − vn−1 ≤ αn γxn − xn−1 1 − αn 1 δ
1δ
M
xn − xn−1 1 − αn βn − βn−1 1−δ
1−δ
∞ n
n−1 |αn − αn−1 |M 1 − αn−1 μi − μi
M
i1
≤ 1 − αn
1δ
−γ
1−δ
xn − xn−1 ∞ 1 δ n
n−1 M |αn − αn−1 | βn − βn−1 μi − μi
1 − δ i1
8
Abstract and Applied Analysis
1δ
1δ
− γ xn − xn−1 M αn
−γ
1 − αn
1−δ
1−δ
1
×
αn 1 δ/1 − δ − γ
∞ 1 δ n
n−1 μ − μi
× |αn − αn−1 | βn − βn−1
.
1 − δ i1 i
3.10
By Lemma 2.8 and conditions i–iii, we immediately get xn1 − xn → 0 as n → ∞.
Step 3. Next we prove that xn − PC ∞
i1 μi Ti xn → 0 as n → ∞.
∞
Let B i1 μi Ti . By Lemma 2.7 and condition iv, we get the results that B : C → H
is a k-strictly pseudocontraction with FB ∞
i1 FTi and Bn x → Bx as n → ∞, for any
x ∈ C,
xn − PC Bxn ≤ xn − xn1 xn1 − PC Bxn xn − xn1 PC vn − PC Bxn ≤ xn − xn1 αn fxn 1 − αn Bn yn − Bn xn Bn xn − Bxn xn − xn1 αn fxn − Bn xn 1 − αn Bn yn − Bn xn Bn xn − Bxn 3.11
≤ xn − xn1 αn fxn − Bn xn 1 δ
yn − xn Bn xn − Bxn 1−δ
xn − xn1 αn fxn − Bn xn 1 − αn 1 − αn 1δ
βn Sxn − xn Bn xn − Bxn .
1−δ
Because αn → 0, βn → 0, xn1 −xn → 0, and Bn x → Bx, so we obtain xn − ∞
i1 μi Ti xn →
0 as n → ∞.
Step 4. Now we show that lim supn → ∞ fx∗ − x∗ , xn − x∗ ≤ 0, where x∗ PFB fx∗ .
Since the sequence {xn } is bounded, we take s subsequence {xnk } of {xn } such that
lim supn → ∞ fx∗ − x∗ , xn − x∗ limk → ∞ fx∗ − x∗ , xnk − x∗ and xnk x . Notice that
xn − PC Bxn → 0 and by Lemmas 2.1 and 2.5, we have x ∈ FixPC B FB ∞
i1 FTi .
Then
lim fx∗ − x∗ , xnk − x∗ fx∗ − x∗ , x − x∗ ≤ 0,
k→∞
x ∈ FB.
Now, by Lemma 2.2, we get PC vn − vn , PC vn − x∗ ≤ 0. Therefore, we have
xn1 − x∗ 2 PC vn − x∗ , xn1 − x∗ PC vn − vn , xn1 − x∗ vn − x∗ , xn1 − x∗ 3.12
Abstract and Applied Analysis
9
PC vn − vn , PC vn − x∗ vn − x∗ , xn1 − x∗ ≤ vn − x∗ , xn1 − x∗ αn fxn 1 − αn Bn yn − x∗ , xn1 − x∗
αn
fxn − fx∗ , xn1 − x∗ αn fx∗ − x∗ , xn1 − x∗
1 − αn Bn yn − x∗ , xn1 − x∗
≤ αn γxn − x∗ · xn1 − x∗ αn fx∗ − x∗ , xn1 − x∗
1 − αn Bn βn Sxn 1 − βn xn − Bn x∗ , xn1 − x∗
≤ αn γxn − x∗ · xn1 − x∗ αn fx∗ − x∗ , xn1 − x∗
1 − αn βn Sxn − Sx∗ βn Sx∗ − x∗ 1 − βn xn − x∗ · xn1 − x∗ ≤ αn γxn − x∗ · xn1 − x∗ αn fx∗ − x∗ , xn1 − x∗
1 − αn xn − x∗ · xn1 − x∗ 1 − αn βn Sx∗ − x∗ · xn1 − x∗ 1 − 1 − γ αn xn − x∗ · xn1 − x∗ αn fx∗ − x∗ , xn1 − x∗
1 − αn βn Sx∗ − x∗ · xn1 − x∗ 1 − 1 − γ αn ≤
xn − x∗ 2 xn1 − x∗ 2 αn fx∗ − x∗ , xn1 − x∗
2
1 − αn βn Sx∗ − x∗ · xn1 − x∗ .
3.13
Hence it follows that
2 1 − γ αn
2αn
fx∗ − x∗ , xn1 − x∗ xn1 − x ≤ 1 −
xn − x∗ 2 1 1 − γ αn
1 1 − γ αn
∗ 2
21 − αn βn
Sx∗ − x∗ · xn1 − x∗ 1 1 − γ αn
2 1 − γ αn
1−
xn − x∗ 2
1 1 − γ αn
2 1 − γ αn
1 fx∗ − x∗ , xn1 − x∗
1 1 − γ αn 1 − γ
1 − αn βn
∗
∗
∗
Sx − x · xn1 − x .
1 − γ αn
3.14
10
Abstract and Applied Analysis
Now, by Lemma 2.8, conditions i–iii, and
1 − αn βn
1 ∗
∗
∗
∗
∗
∗
lim sup fx − x , xn1 − x Sx − x · xn1 − x ≤ 0,
1−γ
1 − γ αn
n→∞
we have xn → x∗ ∈
∞
i1
3.15
FTi as n → ∞ and x∗ also solves the variational inequality
I − f x∗ , x − x∗ ≥ 0,
∀x ∈
∞
FTi .
i1
3.16
This completes the proof.
From Theorem 3.1, if we take S I or βn 0, for all n ∈ N, we get the following
corollary.
Corollary 3.2. Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Let
f : C → H be a (possibly nonself) contraction with coefficient γ ∈ 0, 1 and let Ti : C → H be a
countable family of ki -strictly (possibly nonself) pseudocontraction with 0 ≤ ki < 1 and sup{ki : i ∈
∅. Let the sequence {xn } be generated by
N} < 1 such that ∞
i1 FTi /
x0 x ∈ C,
xn1 PC
∞
n
αn fxn 1 − αn μi Ti xn ,
∀n ≥ 0,
3.17
i1
n
n
with {αn } in 0, 1. Suppose for each n, ∞
1, for all n and μi ≥ 0, for all i ∈ N. Assume
i1 μi
that the parameters satisfied the following conditions:
i limn → ∞ αn 0 and Σ∞
n0 αn ∞;
n
n−1
ii limn → ∞ |αn1 − αn |/αn 0, and limn → ∞ ∞
|/αn 0;
i1 |μi − μi
∞
n
iii limn → ∞ ∞
i1 |μi − μi | 0 and
i1 μi 1 μi > 0.
Then the the sequence {xn } converges strongly to x∗ ∈
inequality
I − f x∗ , x − x∗ ≥ 0,
∞
∀x ∈
i1
FTi , which solves the variational
∞
FTi .
i1
3.18
Remark 3.3. Theorem 3.1 extends and improves Theorem YCL in the following way. The
nonexpasnsive self-mapping T : C → C is extended to a infinite family of nonself ki strictly pseudocontraction Ti : C → H. If we take ki 0, i ∈ N in Theorem 3.1, then
∞
B i1 μi Ti reduces to a nonexpasnsive possibly nonself mapping, thus Theorem 3.1
reduces to Theorem YCL.
Abstract and Applied Analysis
11
Acknowledgments
The authors would like to thank editors and referees for many useful comments and
suggestions for the improvement of the paper. This paper was partially supported by the
Natural Science Foundation of Zhejiang Province Y6100696 and NSFC 11071169.
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International Journal of
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International Journal of
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Optimization
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