Báo cáo hóa học: " Research Article Iteration Scheme with Perturbed Mapping for Common Fixed Points of a Finite Family of Nonexpansive Mappings" doc

Hindawi Publishing CorporationFixed Point Theory and Applications Volume 2007, Article ID 29091, 10 pages doi:10.1155/2007/29091 Research Article Iteration Scheme with Perturbed Mapping

Hindawi Publishing Corporation

Fixed Point Theory and Applications

Volume 2007, Article ID 29091, 10 pages

doi:10.1155/2007/29091

Research Article

Iteration Scheme with Perturbed Mapping for Common Fixed Points of a Finite Family of Nonexpansive Mappings

Yeong-Cheng Liou, Yonghong Yao, and Rudong Chen

Received 17 December 2006; Revised 6 February 2007; Accepted 6 February 2007 Recommended by H´el`ene Frankowska

We propose an iteration scheme with perturbed mapping for approximation of common fixed points of a finite family of nonexpansive mappings{ T i } N

i =1 We show that the pro-posed iteration scheme converges to the common fixed pointx ∗ ∈N

i =1Fix(T i) which solves some variational inequality

Copyright © 2007 Yeong-Cheng Liou et al This is an open access article distributed un-der the Creative Commons Attribution License, which permits unrestricted use, distri-bution, and reproduction in any medium, provided the original work is properly cited

1 Introduction and preliminaries

LetH be a real Hilbert space with inner product ·,·and norm · , respectively A mappingT with domain D(T) and range R(T) in H is called nonexpansive if

 Tx − T y  ≤  x − y , ∀ x, y ∈ D(T). (1.1) Let{ T i } N

i =1be a finite family of nonexpansive self-maps ofH Denote the common fixed

points set of{ T i } N

i =1 byN

i =1Fix (T i) LetF : H → H be a mapping such that for some

constants k,η > 0, F is k-Lipschitzian and η-strongly monotone Let { α n } ∞

n =1⊂(0, 1),

{ λ n } ∞

n =1⊂[0, 1) and take a fixed numberμ ∈(0, 2η/k2) The iterative schemes concern-ing nonlinear operators have been studied extensively by many authors, you may refer

to [1–12] Especially, in [13], Zeng and Yao introduced the following implicit iteration process with perturbed mappingF.

For an arbitrary initial pointx0∈ H, the sequence { x n } ∞

n =1is generated as follows:

x n = α n x n −1+

1− α n



T n x n − λ n μF

T n x n



, n ≥1, (1.2) whereT n:= T nmodN

Using this iteration process, they proved the following weak and strong convergence theorems for nonexpansive mappings in Hilbert spaces

Theorem 1.1 (see [13]) Let H be a real Hilbert space and let F : H → H be a mapping such that for some constants k,η > 0, F is k-Lipschitzain vcommentand η-strongly mono-tone Let { T i } N

i =1 be N nonexpansive self-mappings of H such thatN

i =1Fix (T i) Let

μ ∈(0, 2η/k2) and x0∈ H Let { λ n } ∞

n =1⊂ [0, 1) and { α n } ∞

n =1⊂ (0, 1) satisfying the condi-tions∞

n =1λ n < ∞ and α ≤ α n ≤ β, n ≥ 1, for some α,β ∈ (0, 1) Then the sequence { x n } ∞

n =1

defined by ( 1.2 ) converges weakly to a common fixed point of the mappings { T i } N

i =1.

Theorem 1.2 (see [13]) Let H be a real Hilbert space and let F : H → H be a mapping such that for some constants k,η > 0, F is k-Lipschitzain and η-strongly monotone Let { T i } N

i =1be

N nonexpansive self-mappings of H such that N

i =1Fix (T i) Let μ ∈(0, 2η/k2) and

x0∈ H Let { λ n } ∞

n =1⊂ [0, 1) and { α n } ∞

n =1⊂ (0, 1) satisfying the conditions ∞

n =1λ n < ∞

and α ≤ α n ≤ β, n ≥ 1, for some α,β ∈ (0, 1) Then the sequence { x n } ∞

n =1 defined by ( 1.2 ) converges strongly to a common fixed point of the mappings { T i } N

i =1if and only if

lim inf

n →∞ d



x n,

N

i =1

Fix

T i



Very recently, Wang [14] considered an explicit iterative scheme with perturbed map-pingF and obtained the following result.

Theorem 1.3 Let H be a Hilbert space, let T : H → H be a nonexpansive mapping with F(T) , and let F : H → H be an η-strongly monotone and k-Lipschitzian mapping For any given x0∈ H, { x n } is defined by

x n+1 = α n x n+

1− α n

T λ n+1 x n, n ≥0, (1.4)

where T λ n+1 x n = Tx n − λ n+1 μF(Tx n ), { α n } and { λ n } ⊂ [0, 1) satisfy the following condi-tions:

(1)α ≤ α n ≤ β for some α,β ∈ (0, 1);

(2)∞

n =1λ n < ∞ ;

(3) 0< μ < 2η/k2.

Then

(1){ x n } converges weakly to a fixed point of T,

(2){ x n } converges strongly to a fixed point of T if and only if

lim inf

n →∞ d

x n,F(T)

This naturally brings us the following questions

Questions 1.4 Let T i:H → H (i =1, 2, ,N) be a finite family of nonexpansive mappings

andF is k-Lipschitzain and η-strongly monotone.

(i) Could we construct an explicit iterative algorithm to approximate the common fixed points of the mappings{ T i } N

i =1? (ii) Could we remove the assumption (2) imposed on the sequence{ x n }?

Yeong-Cheng Liou et al 3 Motivated and inspired by the above research work of Zeng and Yao [13] and Wang [14], in this paper, we will propose a new explicit iteration scheme with perturbed map-ping for approximation of common fixed points of a finite family of nonexpansive self-mappings ofH We will establish strong convergence theorem for this explicit iteration

scheme To be more specific, letα n1,α n2, ,α nN ∈(0, 1], n ∈ N Given the mappings

T1,T2, ,T N, following [15], one can define, for eachn, mappings U n1,U n2, ,U nN by

U n1 = α n1 T1+

1− α n1

I,

U n2 = α n2 T2U n1+

1− α n2

I,

U n,N −1= α n,N −1T N −1U n,N −2+

1− α n,N −1



I,

W n:= U nN = α nN T N U n,N −1+

1− α nN

I.

(1.6)

Such a mappingW nis called theW-mapping generated by T1, ,T Nandα n1, ,α nN First we introduce the following explicit iteration scheme with perturbed mappingF.

For an arbitrary initial pointx0∈ H, the sequence { x n } ∞

n =1is generated iteratively by

x n+1 = βx n+ (1− β)

W n x n − λ n μF

W n x n

, n ≥0, (1.7) where{ λ n } is a sequence in (0, 1),β is a constant in (0,1), F is k-Lipschitzian and

η-strongly monotone, andW nis theW-mapping defined by (1.6)

We have the following crucial conclusion concerningW n

Proposition 1.5 (see [15]) Let C be a nonempty closed convex subset of a Banach space

E Let T1,T2, ,T N be nonexpansive mappings of C into itself such thatN

i =1Fix (T i ) is nonempty, and let α n1,α n2, ,α nN be real numbers such that 0 < α ni ≤ b < 1 for any i ∈ N For any n ∈ N, let W n be the W-mapping of C into itself generated by T N,T N −1, ,T1

and α nN,α n,N −1, ,α n1 Then W n is nonexpansive Further, if E is strictly convex, then

Fix (W n)=N

i =1Fix (T i ).

Now we recall some basic notations Let T : H → H be nonexpansive mapping and

F : H → H be a mapping such that for some constants k,η > 0, F is k-Lipschitzian and η-strongly monotone; that is, F satisfies the following conditions:

 Fx − F y  ≤ k  x − y , ∀ x, y ∈ H,

 Fx − F y,x − y  ≥ η  x − y 2, ∀ x, y ∈ H, (1.8)

respectively We may assume, without loss of generality, thatη ∈(0, 1) and k ∈[1,∞) Under these conditions, it is well known that the variational inequality problem—find

x ∗ ∈N

i =1Fix (T i) such that

VI



F, N

i =1

Fix

T i

:

F

x ∗

,x − x ∗

≥0, ∀ x ∈ N

i =1

Fix

T i

, (1.9)

has a unique solution x ∗ ∈N

i =1Fix (T i) [Note: the unique existence of the solution

x ∗ ∈N

i =1Fix (T i) is guaranteed automatically becauseF is k-Lipschitzian and η-strongly

monotone overN

i =1Fix (T i).]

For any given numbersλ ∈[0, 1) andμ ∈(0, 2η/k2), we define the mappingT λ:H →

H by

Concerning the corresponding result ofT λ x, you can find it in [16]

Lemma 1.6 (see [16]) If 0 ≤ λ < 1 and 0 < μ < 2η/k2, then there holds for T λ:H → H,

T λ x − T λ y (1− λτ)  x − y , ∀ x, y ∈ H, (1.11)

where τ =1−1− μ(2η − μk2)∈ (0, 1).

Next, let us state four preliminary results which will be needed in the sequel.Lemma 1.7is very interesting and important, you may find it in [17], the original prove can be found in [18] Lemmas1.8and1.9well-known demiclosedness principle and subdiffer-ential inequality, respectively.Lemma 1.10is basic and important result, please consult it

in [19]

Lemma 1.7 (see [17]) Let { x n } and { y n } be bounded sequences in a Banach space X and let { β n } be a sequence in [0, 1] with

0< liminf

n →∞ β n ≤lim sup

Suppose

x n+1 =1− β n



y n+β n x n, (1.13)

for all integers n ≥ 0 and

lim sup

n →∞ y n+1 − y n x n+1 − x n 0. (1.14)

Then, lim n →∞  y n − x n  = 0.

Lemma 1.8 (see [20]) Assume that T is a nonexpansive self-mapping of a closed convex subset C of a Hilbert space H If T has a fixed point, then I − T is demiclosed That is, when-ever { x n } is a sequence in C weakly converging to some x ∈ C and the sequence {(I − T)x n }

strongly converges to some y, it follows that (I − T)x = y Here, I is the identity operator

of H.

Lemma 1.9 (see [21])  x + y 2≤  x 2+ 2 y,x + y  for all x, y ∈ H.

Lemma 1.10 (see [19]) Assume that { a n } is a sequence of nonnegative real numbers such that

a n+1 ≤1− γ n

Yeong-Cheng Liou et al 5

where { γ n } is a sequence in (0, 1) and { δ n } is a sequence such that

(1)∞

n =1γ n = ∞ ,

(2) lim supn →∞ δ n /γ n ≤ 0 or∞

n =1| δ n | < ∞ Then lim n →∞ a n = 0.

2 Main result

Now we state and prove our main result

Theorem 2.1 Let H be a real Hilbert space and let F : H → H be a k-Lipschitzian and η-strongly monotone mapping Let { T i } N

i =1be a finite family of nonexpansive self-mappings

of H such thatN

i =1Fix (T i) Let μ ∈(0, 2η/k2) Suppose the sequences { α n,i } N

i =1 sat-isfy lim n →∞(α n,i − α n −1,i)= 0, for all i =1, 2, ,N If { λ n } ∞

n =1⊂ [0, 1) satisfy the following conditions:

(i) limn →∞ λ n = 0;

(ii)∞

n =0λ n = ∞ ,

then the sequence { x n } ∞

n =1defined by ( 1.7 ) converges strongly to a common fixed point x ∗ ∈

N

i =1Fix (T i ) which solves the variational inequality ( 1.9 ).

Proof Let x ∗be an arbitrary element ofN

i =1Fix (T i) Observe that

x n+1 − x ∗ βx n+ (1− β)W λ n

n x n − x ∗

≤ β x n − x ∗ + (1− β) W λ n

n x n − x ∗ , (2.1) whereW λ n

n x : = W n x − λ n μF(W n x) Note that

W λ n

n x ∗ = x ∗ − λ n μF

x ∗

UtilizingLemma 1.6, we have

W λ n

n x n − x ∗ W λ n

n x n − W λ n

n x ∗+W λ n

n x ∗ − x ∗

≤ W λ n

n x n − W λ n

n x ∗ + W λ n

n x ∗ − x ∗

≤1− λ n τ x n − x ∗ +λ n μ F

x ∗

(2.3)

From (2.1) and (2.3), we have

x n+1 − x ∗ β + (1 − β)

1− λ n τ x n − x ∗ + (1− β)λ n μ F

x ∗

=1−(1− β)λ n τ x n − x ∗ + (1− β)λ n μ F

x ∗

≤max x0− x ∗ ,μ



(2.4)

Hence,{ x n }is bounded We also can obtain that{ W n x n },{ T i U n, j x n }(i =1, ,N; j =

1, ,N), and { F(W n x n)}are all bounded

We will useM to denote the possible different constants appearing in the following

reasoning

We note that

W λ n+1

n+1 x n+1 − W λ n

n x n

= W n+1 x n+1 − W n x n − λ n+1 μF

W n+1 x n+1



+λ n μF

W n x n

≤ W n+1 x n+1 − W n x n +λ n+1 μ F

W n+1 x n+1 +λ n μ F

W n x n

≤ W n+1 x n+1 − W n+1 x n + W n+1 x n − W n x n +

λ n+1+λ n

M

≤ x n+1 − x n + W n+1 x n − W n x n +

λ n+1+λ n

M.

(2.5)

From (1.6), sinceT NandU n,Nare nonexpansive,

W n+1 x n − W n x n

= α n+1,N T N U n+1,N −1x n+

1− α n+1,N



x n − α n,N T N U n,N −1x n −1− α n,N



x n

≤ α n+1,N T N U n+1,N −1x n − α n,N T N U n,N −1x n +α n+1,N − α n,N x n

≤ α n+1,N

T N U n+1,N −1x n − T N U n,N −1x n +α n+1,N − α n,N T N U n,N −1x n

+α

n+1,N − α n,N x n

≤ α n+1,N U n+1,N −1x n − U n,N −1x n + 2Mα n+1,N − α n,N.

(2.6) Again, from (1.6), we have

U n+1,N −1x n − U n,N −1x n

= α n+1,N −1T N −1U n+1,N −2x n+

1− α n+1,N −1



x n

− α n,N −1T N −1U n,N −2x n −1− α n,N −1



x n

≤ α n+1,N −1T N −1U n+1,N −2x n − α n,N −1T N −1U n,N −2x n

+α n+1,N −1− α n,N −1 x n

≤α n+1,N −1− α n,N −1 x n +α n+1,N −1− α n,N −1M

+α n+1,N −1 T N −1U n+1,N −2x n − T N −1U n,N −2x n

≤2Mα n+1,N −1− α n,N −1+α n+1,N −1 U n+1,N −2x n − U n,N −2x n

≤2Mα

n+1,N −1− α n,N −1 + U

n+1,N −2x n − U n,N −2x n

(2.7)

Yeong-Cheng Liou et al 7 Therefore, we have

U n+1,N −1x n − U n,N −1x n

≤2Mα n+1,N −1− α n,N −1+ 2Mα n+1,N −2− α n,N −2

+ U n+1,N −3x n − U n,N −3x n

≤2M

N−1

i =2

α n+1,i − α n,i+ U n+1,1 x n − U n,1 x n

= α n+1,1 T1x n+

1− α n+1,1

x n − α n,1 T1x n −1− α n,1

x n

+ 2M

N−1

i =2

α n+1,i − α n,i,

(2.8)

then

U n+1,N −1x n − U n,N −1x n

≤α n+1,1 − α n,1 x n + α n+1,1 T1x n − α n,1 T1x n

+ 2M

N−1

i =2

α n+1,i − α n,i  ≤2M

N−1

i =1

α n+1,i − α n,i. (2.9)

Substituting (2.9) into (2.6), we have

W n+1 x n − W n x n 2Mα n+1,N − α n,N+ 2α n+1,N M N−1

i =1

α n+1,i − α n,i

≤2M N



i =1

α

Substituting (2.10) into (2.5), we have

W λ n+1

n+1 x n+1 − W λ n

n x n x n+1 − x n + 2M

N



i =1

α n+1,i − α n,i+

λ n+1+λ n

M, (2.11) which implies that

lim sup

n →∞ W λ n+1

n+1 x n+1 − W λ n

n x n x n+1 − x n 0. (2.12)

We note thatx n+1 = βx n+ (1− β)W λ n

n x nand 0< β < 1, then fromLemma 1.7and (2.12),

we have limn →∞  W λ n

n x n − x n  =0 It follows that lim

n →∞ x n+1 − x n lim

n →∞(1− β) W λ n

n x n − x n 0. (2.13)

On the other hand,

x n − W n x n x n+1 − x n + x n+1 − W n x n

≤ x n+1 − x n +β x n − W n x n + (1− β)λ n μ F

W n x n , (2.14)

that is,

x n − W n x n 1

1− β x n+1 − x n +λ n μ F

W n x n , (2.15) this together with (i) and (2.13) imply

lim

n →∞ x n − W n x n 0. (2.16)

We next show that

lim sup

n →∞

− F

x ∗

,x n − x ∗

To prove this, we pick a subsequence{ x n i}of{ x n }such that

lim sup

n →∞

− F

x ∗

,x n − x ∗

=lim

i →∞

− F

x ∗

,x n i − x ∗

Without loss of generality, we may further assume thatx n i → z weakly for some z ∈ H.

z ∈Fix

W n



this together withProposition 1.5imply that

z ∈ N

i =1

Fix

T i

Sincex ∗solves the variational inequality (1.9), then we obtain

lim sup

n →∞

− F

x ∗

,x n − x ∗

=− F

x ∗

,z − x ∗

≤0. (2.21) Finally, we show thatx n → x ∗ Indeed, fromLemma 1.9, we have

x n+1 − x ∗ 2

= β

x n − x ∗

+ (1− β)

W λ n

n x n − W λ n

n x ∗

+ (1− β)

W λ n

n x ∗ − x ∗ 2

≤ β

x n − x ∗

+ (1− β)

W λ n

n x n − W λ n

n x ∗ 2+ 2(1− β)

W λ n

n x ∗ − x ∗,x n+1 − x ∗

≤β x n − x ∗ + (1− β) W λ n

n x n − W λ n

n x ∗ 2+ 2(1− β)λ n μ

− F

x ∗

,x n+1 − x ∗

≤β x n − x ∗ + (1− β)

1− λ n τ x n − x ∗ 2+ 2(1− β)λ n μ

− F

x ∗

,x n+1 − x ∗

≤1−(1− β)τλ n x n − x ∗ 2+ (1− β)τλ n



2μ τ

− F

x ∗

,x n+1 − x ∗

.

(2.22) Now applyingLemma 1.10and (2.21) to (2.22) concludes thatx n → x ∗(n → ∞) This

Yeong-Cheng Liou et al 9

Acknowledgments

The authors thank the referees for their suggestions and comments which led to the present version The research was partially supposed by Grant NSC 95-2221-E-230-017

References

[1] L.-C Zeng and J.-C Yao, “Stability of iterative procedures with errors for approximating com-mon fixed points of a couple ofq-contractive-like mappings in Banach spaces,” Journal of Math-ematical Analysis and Applications, vol 321, no 2, pp 661–674, 2006.

[2] Y.-C Lin, N.-C Wong, and J.-C Yao, “Strong convergence theorems of Ishikawa iteration

pro-cess with errors for fixed points of Lipschitz continuous mappings in Banach spaces,” Taiwanese Journal of Mathematics, vol 10, no 2, pp 543–552, 2006.

[3] L.-C Zeng, N.-C Wong, and J.-C Yao, “Strong convergence theorems for strictly

pseudocon-tractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol 10, no 4,

pp 837–849, 2006.

[4] Y.-C Lin, “Three-step iterative convergence theorems with errors in Banach spaces,” Taiwanese Journal of Mathematics, vol 10, no 1, pp 75–86, 2006.

[5] L.-C Zeng, G M Lee, and N.-C Wong, “Ishikawa iteration with errors for approximating fixed

points of strictly pseudocontractive mappings of Browder-Petryshyn type,” Taiwanese Journal of Mathematics, vol 10, no 1, pp 87–99, 2006.

[6] S Schaible, J.-C Yao, and L.-C Zeng, “A proximal method for pseudomonotone type

variational-like inequalities,” Taiwanese Journal of Mathematics, vol 10, no 2, pp 497–513,

2006.

[7] L.-C Zeng, L J Lin, and J.-C Yao, “Auxiliary problem method for mixed variational-like

in-equalities,” Taiwanese Journal of Mathematics, vol 10, no 2, pp 515–529, 2006.

[8] L.-C Zeng, N.-C Wong, and J.-C Yao, “On the convergence analysis of modified hybrid steepest-descent methods with variable parameters for variational inequalities,” to appear in

Journal of Optimization Theory and Applications.

[9] L C Ceng, P Cubiotti, and J.-C Yao, “Approximation of common fixed points of families of

nonexpansive mappings,” to appear in Taiwanese Journal of Mathematics.

[10] L C Ceng, P Cubiotti, and J.-C Yao, “Strong convergence theorems for finitely many

nonex-pansive mappings and applications,” to appear in Nonlinear Analysis.

[11] L.-C Zeng, S Y Wu, and J.-C Yao, “Generalized KKM theorem with applications to generalized

minimax inequalities and generalized equilibrium problems,” Taiwanese Journal of Mathematics,

vol 10, no 6, pp 1497–1514, 2006.

[12] L C Ceng, C Lee, and J.-C Yao, “Strong weak convergence theorems of implicit hybrid

steepest-descent methods for variational inequalities,” to appear in Taiwanese Journal of Mathe-matics.

[13] L.-C Zeng and J.-C Yao, “Implicit iteration scheme with perturbed mapping for common fixed

points of a finite family of nonexpansive mappings,” Nonlinear Analysis, vol 64, no 11, pp.

2507–2515, 2006.

[14] L Wang, “An iteration method for nonexpansive mappings in Hilbert spaces,” Fixed Point The-ory and Applications, vol 2007, Article ID 28619, 8 pages, 2007.

[15] W Takahashi and K Shimoji, “Convergence theorems for nonexpansive mappings and

feasibil-ity problems,” Mathematical and Computer Modelling, vol 32, no 11–13, pp 1463–1471, 2000.

[16] H K Xu and T H Kim, “Convergence of hybrid steepest-descent methods for variational

in-equalities,” Journal of Optimization Theory and Applications, vol 119, no 1, pp 185–201, 2003.

[17] T Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter

nonexpansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Ap-plications, vol 305, no 1, pp 227–239, 2005.

[18] T Suzuki, “Strong convergence theorems for infinite families of nonexpansive mappings in

gen-eral Banach spaces,” Fixed Point Theory and Applications, vol 2005, no 1, pp 103–123, 2005 [19] H.-K Xu, “Viscosity approximation methods for nonexpansive mappings,” Journal of Mathe-matical Analysis and Applications, vol 298, no 1, pp 279–291, 2004.

[20] K Geobel and W A Kirk, Topics in Metric Fixed Point Theory, vol 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990.

[21] G Marino and H.-K Xu, “Convergence of generalized proximal point algorithms,” Communi-cations on Pure and Applied Analysis, vol 3, no 4, pp 791–808, 2004.

Yeong-Cheng Liou: Department of Information Management, Cheng Shiu University,

Kaohsiung 833, Taiwan

Email address:simplex liou@hotmail.com

Yonghong Yao: Department of Mathematics, Tianjin Polytechnic University, Tianji 300160, China

Email address:yuyanrong@tjpu.edu.cn

Rudong Chen: Department of Mathematics, Tianjin Polytechnic University, Tianji 300160, China

Email address:chenrd@tjpu.edu.cn

[13] L.-C Zeng and J.-C Yao, “Implicit iteration scheme with perturbed mapping for common. .. J.-C Yao, “Approximation of common fixed points of families of< /small>

nonexpansive mappings,” to appear in Taiwanese Journal of Mathematics.

[10] L...

that is,

x n − W n x n

1−