An Implicit Iterative Scheme for an Infinite Countable Family of Asymptotically Nonexpansive Mappings in Banach Spaces

(1)

Volume 2008, Article ID 350483,10pages doi:10.1155/2008/350483

Research Article

An Implicit Iterative Scheme for an Infinite

Countable Family of Asymptotically Nonexpansive

Mappings in Banach Spaces

Shenghua Wang, Lanxiang Yu, and Baohua Guo

School of Mathematics and Physics, North China Electric Power University, Baoding 071003, China

Correspondence should be addressed to Shenghua Wang,[email protected]

Received 6 May 2008; Accepted 24 August 2008 Recommended by William Kirk

LetKbe a nonempty closed convex subset of a reflexive Banach spaceEwith a weakly continuous dual mapping, and let{Ti}∞i1 be an infinite countable family of asymptotically nonexpansive mappings with the sequence{kin} satisfying kin ≥ 1 for each i 1,2, . . ., n 1,2, . . ., and

limn→∞kin 1 for eachi 1,2, . . .. In this paper, we introduce a new implicit iterative scheme

generated by{Ti}∞i1and prove that the scheme converges strongly to a common fixed point of {Ti}∞i1, which solves some certain variational inequality.

Copyrightq2008 Shenghua Wang 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.

1. Introduction and preliminaries

LetEbe a Banach space and letKbe a nonempty closed convex subset ofE. LetT :K→Kbe a mapping. ThenT is called nonexpansive if

Tx−Ty ≤ x−y 1.1

for allx, y ∈ K. T is called asymptotically nonexpansive if there exists a sequence{k_n} ⊂

1,∞that converges to 1 asn→∞such that

_Tn_x₋_Tn_y_≤_k

nx−y 1.2

(2)

self-mapping onKhas a fixed point. After that, many authors began to study the convergence of the iterative scheme generated by asymptotically nonexpansive mappings2–12.

In 8, the authors introduced an iterative scheme generated by a finite family of asymptotically nonexpansive mappings:

xnαnxn−1

1−αnTrlnn1xn, n≥1, 1.3

where {α_n} is a sequence in 0,1,{T_i}N_i₁ : K→K are N asymptotically nonexpansive mappings, where K is a nonempty closed convex subset of a uniformly convex Banach space satisfying Opial’s condition13, and where n lnN rn for some integers ln ≥ 0 and 1 ≤ r_n ≤ N. Then the authors proved that if∩N_i₁FT_i_/φ, then{x_n}generated by1.3 strongly converges to a common fixed point of{T_i}N_i₁.

LetKbe a nonempty closed convex subset of a uniformly convex Banach spaceE. Let

S:K→Kbe a nonexpansive mapping and letT :K→Kbe an asymptotically nonexpansive mapping. In10, the authors introduced the following modified Ishikawa iteration sequence with errors with respect toSandT:

ynanSxnbnTnxncnvn,

xn1anSxnbnTnyncnun, ∀n≥1,

1.4

where{a_n},{b_n},{c_n}are three real numbers sequences in0,1satisfyinga_nb_n c_n 1,{an},{bn},{cn}are also three real numbers sequences in0,1satisfyinganbncn 1, and {un} and {vn} are given bounded sequences inK. Then the authors proved that the sequence{x_n}generated by1.4strongly converges to a common fixed point ofSandT if some certain conditions are satisfied.

LetKbe a nonempty closed convex subset of a Banach spaceEand letf :K→Kbe a contraction with eﬃcientλ0< λ <1such that

_f_x₋_f_y_≤_λ_x₋_y _1.5

for all x, y ∈ K. Shahzad and Udomene 9 studied the following implicit and explicit iterative schemes for an asymptotically nonexpansive mapping T with the sequence {kn} in a uniformly smooth Banach space:

xn

1− tn

kn

fxn_ktn nT

n_x n,

xn1

1− tn

kn

fxn_ktn nT

n_x n,

1.6

where{tn}is a sequence in0,1. They proved that the sequence{xn}converges strongly to the unique solution of some variational inequality if the sequence{t_n}satisfies some certain conditions and the mappingT satisfiesTxn−xn→0 asn→∞.

(3)

{Ti}Ni1 with the same sequence{kn} in a reflexive Banach space with a weakly continuous duality map:

xn

1− 1

kn

xn 1−_ktn n f

xn_ktn nT

n rnxn,

xn1

1− 1

kn

xn 1−_ktn n f

xn_ktn nT

n rnxn,

1.7

wherern nmodN and{tn} is a sequence in0,1. Then they proved that if the control sequence{tn}satisfies some certain condition andTixn−xn→0 asn→∞for eachi1,2, . . . , N, then both schemes1.7strongly converge a common fixed pointx∗of{T_i}N_i₁which solves the variational inequality

I−fx∗_{, J}_p₋_x∗_≥₀_{, p}_∈ N

i1

FTi, 1.8

whereFT_idenotes the set of fixed points of the mappingT_ifor eachi1,2, . . . , N.

LetEbe a Banach space and letE∗be the dual space ofE. Given a continuous strictly increasing function ϕ : R→R such that ϕ0 0 and limt→∞ϕt ∞, we associate a possibly multivaluedgeneralized duality mapJ_ϕ :E→2E∗, defined as

Jϕx x∗∈E∗:x∗x xϕx,x∗ϕx 1.9

for everyx ∈ E. We call the functionϕ a gauge. Ifϕt tfor all t ≥ 0, then we call Jϕ a normalized duality mapping and write it asJ.

A Banach spaceEis said to have a weakly continuous generalized duality map if there exists a continuous strictly increasing functionϕ:R→Rsuch thatϕ0 0,limt→∞ϕt ∞, and J_ϕ is single valued and sequentially continuous from Ewith the weak topology toE∗ with the weak∗topology. For instance, everylp-space1 < p <∞has a weakly continuous generalized duality map forϕt tp−1_.

For eacht ≥ 0, let Φt t₀ϕxdx. The following property may be seen in many literatures.

Property 1.1. Let E be a real Banach space and letJϕ be the duality map associated with the gaugeϕ. Then for allx, y∈Eandjxy∈Jϕxyone holds

Φxy≤Φxy, jxy. 1.10

One also holds

xy2_≤_x2₂_{y, j}_x_y _1.11

(4)

Lemma 1.2see14. Let E be a Banach space satisfying a weakly continuous duality map and let K be a nonempty closed convex subset of E. LetT :K→Kbe an asymptotically nonexpansive mapping with fixed point. ThenI−Tis demiclosed at zero.

2. Strong convergence results

In this section, letEbe a reflexive Banach space with a weakly continuous duality mapJ_ϕ, whereϕis a gauge and letKbe a nonempty closed convex subset ofE. Let{Ti}∞i1:K→Kbe an infinite countable family of asymptotically nonexpansive mappings such that

_Tn

i x−Tiny≤kinx−y 2.1

for allx, y∈K, where the sequence{kin} ⊂1,∞and limn→∞kin 1 for eachi1,2, . . . . For eachn1,2, . . . ,letb_nsup{k_in|i1,2, . . .}and assume

supb_n|n1,2, . . .<∞, lim

n→∞bnb <∞.

2.2

Takingbnmax{bn, b}for eachn1,2, . . . ,obviously, we have

lim

n→∞bnb≥1,

b_sup_b_n_|_n₁_,₂_{, . . .}_<_∞_. 2.3

Moreover, the following inequality

_Tn

i x−Tiny≤bnx−y 2.4

holds for allx, y∈Kand eachi1,2. . . .

Take an integerr >1 arbitrarily. For eachn≥1, define the mappingSni :K→Kby

SniTn−1ri 2.5

for eachi1,2, . . . , r,that is,

(5)

For eachi 1,2, . . . , r, let{αni} ⊂ 0,1be a sequence real numbers. For eachn≥ 1, define the mappingW_nofKinto itself by

WnUnrαnrSnnrUnr−1

1−α_nrI, 2.7

where

Un1αn1Sn_n1

1−αn1

I, Un2αn2Sn_n2Un1

1−αn2

I,

.. .

Unr−1αnr−1Sn_nr₋1Unr−2

1−αnr−1

I.

2.8

We callWnaW-mapping generated bySn1, Sn2, . . . , Snrandαn1, αn2, . . . , αnr.

Let f : K→K be a λ-contraction with 0 < λ < 1/br. Take a sequence of real numbers{tn} ⊂0, bsuch that

lim

n→∞tn0, tn<

b1−br_nλ

1−λbrn , n≥1. 2.9

Note that sinceλ <1/br, one has 0< b1−br_nλ/1−λbr_n ≤b. Therefore, the sequence{tn} can be taken easily to satisfy the condition2.9, for example,tn 1/nb1−bnrλ/1−λbrn.

Then, we introduce an implicit iterative scheme

xn

1− b

br1 n

xnb−tn

br1

n f

Wnxn tn

br1

n Wnxn, n≥

1. 2.10

By using the following lemmas, we will prove that the implicit scheme2.10is well defined.

Lemma 2.1. Let {Ti}∞i1 : K→K be an infinite countable family of asymptotically nonexpansive

mappings with the sequences {kin} and let Wn be a W-mapping generated by2.7for each n 1,2, . . . .If∩∞_i₁FT_i_/φ, then∩_i∞₁FT_i⊂FW_nfor eachn1,2, . . . .

Proof. The conclusion is obtained directly from the definition ofWn.

Lemma 2.2. Let{T_i}∞_i₁ :K→Kwith the sequences{k_in}and letW_nbe theW-mapping generated by2.7for eachn1,2, . . . .Then one holds

_W_n_x₋_W_n_y_≤_br

nx−y 2.11

(6)

Proof. For anyx, y∈Kalln≥1, we first seenoting thatbn≥1

_U_n₁_x₋_U_n₁_y_α_n₁_Sn n1

1−αn1

Ix−αn1Sn_n1

1−αn1

Iy

≤αn1Sn_n1x−Snn1y

1−αn1

x−y

αn1T_nn₋₁_r₁x−T_nn₋₁_r₁y

1−α_n1

x−y ≤αn1kn−1r1nx−y1−αn1

x−y ≤αn1bnx−y1−αn1

bnx−y

bnx−y,

_U_n₂_x₋_U_n₂_y_α_n₂_Sn n2Un1

1−α_n2

Ix−αn2Sn_n2Un1

1−α_n2

Iy

≤αn2Sn_n2Un1x−Sn_n2Un1y

1−α_n2

x−y

αn2T_nn₋₁_r₂Un1x−T_nn₋₁_r₂Un1y

1−αn2

x−y ≤αn2kn−1r2nUn1x−Un1y

1−αn2

x−y ≤αn2bnUn1x−Un1y

1−αn2

x−y ≤αn2b2_nx−y

1−αn1

b2

nx−y

b2

nx−y.

2.12

Similarly, for eachi3, . . . , r−1, we have

Unix−Uniy ≤binx−y. 2.13

Hence,

_W_n_x₋_W_n_y_α_nr_Sn

nrUn r−1

1−αnrIx−αnrSnnrUn r−1

1−αnrIy

≤αnrSnnrUn r−1x−Sn_nrUn r−1y

1−αnrx−y

≤br

nx−y.

2.14

This completes the proof.

Now we prove that the implicit scheme2.10is well defined. Since 0 < t_n < b1−

br

nλ/1−λbrn, we obtain

0<1− b

br1

n

b−tn

bn λ

tn

bn <1. 2.15

Hence, the mapping

x→Tx:

1− b

br1 n

xb−tn

br1

n f

Wnx tn

br1

n Wnx

(7)

is a contraction onK. In fact, to see this, taking anyx, y∈K, byLemma 2.2we have

Tx−Ty

1− b

br1 n

x−y b−tn

br1 n

fWnx−fWny tn

br1 n

Wnx−Wny

≤

1− b

br1 n

x−yb−tnλbrn br1

n x−y

tn

br1 n b

r

nx−y

1− b

br1

n

b−tn

bn λ

tn

bn

x−y

≤ x−y,

2.17

which implies that the implicit scheme2.10is well defined.

For the implicit scheme2.10, we have strong convergence as follows.

Theorem 2.3. Assume2.9,FT ∩∞_i₁FTi/φandlimn→∞xn−Tixn0for eachi1,2, . . . .

Then{x_n} converges strongly to a common fixed point x ∈ FT, wherex solves the variational inequality

I−fx, Jp−x≥0, p∈FT. 2.18

Proof. First, we prove that{xn}is bounded. By usingProperty 1.1, Lemmas2.1,2.2, for any

z∈FT, we havenoting 0<1−b/b_nr1 b−tn/bnλtn/bn<1

xn−z2

1− b

br1 n

xn−zb−tn

br1 n

fWnxn−fz tn

br1 n

Wnxn−z

b−tn

br1 n

fz−z2

≤

1− b

br1 n

xn−zb−tn

br1 n

fWnxn−fz tn

br1 n

Wnxn−z

2

2b−tn

br1

n fz−z, jxn−z

≤

1− b

br1 n

xn−z b−tn

br1 n

_f

Wnxn−fWnz tn

br1 n

_W_n_x_n₋_W_n_z2

2b−tn

br1 n

fz−z, jxn−z

≤

1− b

br1

n

b−tnλ

bn

tn

bn

2

_x_n₋_z2 2b−tn

br1 n

fz−z, jxn−z

≤

1− b

br1

n

b−tnλ

bn

tn

bn

xn−z2 2b−tn

br1 n

fz−z, jxn−z

1−η_nx_n−z22b−tn

br1 n

fz−z, jxn−z,

(8)

where

ηn b

br1 n

−b−tn

bn λ−

tn

bn >0. 2.20

It follows from2.19that

_x_n₋_z2_≤ 2b−tn

ηnbrn1

fz−z, jxn−z. 2.21

Since limn→∞bn b,limn→∞tn0, we have

lim n→∞

b−tn

ηnbrn1 1

1−λbr. 2.22

Hence,{x_n}is bounded.

Now we prove that{xn}strongly converges to a common fixed pointx∈FT. To see this, we assume thatxis a weak limit point of{xn}and a subsequence{xnj}of{xn}converges

weakly tox. Then by the assumption of the theorem andLemma 1.2, we havex∈FT_ifor everyi1,2, . . . .In2.21, replacingxnwithxnj andzwithx, respectively, and then taking

the limit asj→∞, we obtain by the weak continuity of the duality mapJ

lim

j→∞xnj−x0. 2.23

Therefore,x_n_j→x. We further show thatxsolves the variational inequality

I−fx, Jp−x≥0, p∈FT. 2.24

To see this result, taking anyp ∈ FT, then by usingProperty 1.1, Lemmas2.1and2.2we compute

Φxn−p

Φ

1− b

br1 n

xn−pb−tn

br1 n

xn−p tn

br1 n

Wnxn−pb−tn

br1 n

fWnxn−xn

≤Φ

1− tn

br1 n

xn−p tn

br1 n

Wnxn−p

b−tn

br1 n

fWnxn−xn, Jϕxn−p

≤

1− tn

br1 n tn

Φxn−p b−tn

br1 n

fWnxn−xn, Jϕxn−p,

(9)

which implies that

xn−fWnxn, Jϕxn−p≤ b

r1 n −1tn

b−tn Φ

xn−p. 2.26

Now in2.26, replacingxnwithxnjand noting limn→∞bnband limn→∞tn0, we obtain

x−fx, Jϕx−plim

j→∞

xnj−f

Wnjxnj

, Jϕxnj−p

≤lim sup j→∞

br1 nj −1tnj

b−tnj

Φxnj−p0,

2.27

which implies thatxis a solution to2.24.

Finally, we prove that the sequence{xn}strongly converges tox. It suﬃces to prove that the variational inequality2.24can have only one solution. To see this, assuming that bothu∈FTandv∈FTare solutions to2.24, we have

I−fu, Ju−v≤0,

I−fv, Jv−u≤0. 2.28

Adding them yields

I−fu−I−fv, Ju−v≤0. 2.29

However, sincefis aλ-contraction, we have that

1−λu−v2 ≤I−fu−I−fv, Ju−v, 2.30

which implies thatuv.This completes the proof.

Remark 2.4. InTheorem 2.3, the condition that limn→∞Tixn −xn 0 for each i 1,2, . . . is necessary see9, 12. This theorem shows that if for eachn 1,2, . . ., the supremum of the sequence{kin}, that is, sup{kin | i 1,2, . . .}, is finite and the limit of the sequence

sup{kin|i1,2, . . .}∞n1 exists, then by choosing the contraction constantλ and the control

sequence{t_n}we can obtain the common fixed point of{T_i}∞_i₁.

Corollary 2.5. Let{Ti}Ni1K→Kbe a finite family of asymptotically nonexpansive mappings with the

sequences{kin}and letWn be aW-mapping generated byT1, T2, . . . , TN andαn1, αn2, . . . , αnN for

eachn1,2, . . . .Let the sequence{t_n} ⊂0,1and satisfyt_n<1−kN_n λ/1−λk_nNandt_n→0, where knmax{k1n, k2n, . . . , kNn}for eachn1,2, . . . .Assume thatksup{kn|n1,2, . . .}<∞. Let

fbe a contraction withλ0< λ <1/kN. Consider the implicit iterative scheme

xn

1− 1

kN1

n

xn1−tn

kN1

n f

Wnxn tn

kN1

n Wnxn.

(10)

If {Ti}Ni1 satisfy the condition∩Ni1FTi/φ and Tixn−xn→0 as n→∞ for each i 1,2, . . . , N,

then{x_n}converges strongly to a common fixed pointx∈ ∩_iN₁FT_i, wherexsolves the variational inequality

I−fx, Jp−x≥0, p∈

N

i1

FTi. 2.32

Proof. InTheorem 2.3, takebnkn, blimn→∞kn1, bk,andr N. Then, this corollary can obtained directly fromTheorem 2.3.

Acknowledgment

The work was supported by Youth Foundation of North China Electric Power University.

References

1 K. Goebel and W. A. Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 35, pp. 171–174, 1972.

2 J. Schu, “Iterative construction of fixed points of asymptotically nonexpansive mappings,”Journal of Mathematical Analysis and Applications, vol. 158, no. 2, pp. 407–413, 1991.

3 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991.

4 S.-S. Chang, “On the approximation problem of fixed points for asymptotically nonexpansive mappings,”Indian Journal of Pure and Applied Mathematics, vol. 32, no. 9, pp. 1297–1307, 2001. 5 M. O. Osilike and S. C. Aniagbosor, “Weak and strong convergence theorems for fixed points of

asymptotically nonexpansive mappings,”Mathematical and Computer Modelling, vol. 32, no. 10, pp. 1181–1191, 2000.

6 Z. Liu, J. K. Kim, and K. H. Kim, “Convergence theorems and stability problems of the modified Ishikawa iterative sequences for strictly successively hemicontractive mappings,” Bulletin of the Korean Mathematical Society, vol. 39, no. 3, pp. 455–469, 2002.

7 Z. Liu, J. K. Kim, and S.-A. Chun, “Iterative approximation of fixed points for generalized asymptotically contractive and generalized hemicontractive mappings,”Pan-American Mathematical Journal, vol. 12, no. 4, pp. 67–74, 2002.

8 S. S. Chang, K. K. Tan, H. W. J. Lee, and C. K. Chan, “On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 313, no. 1, pp. 273–283, 2006.

9 N. Shahzad and A. Udomene, “Fixed point solutions of variational inequalities for asymptotically nonexpansive mappings in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 3, pp. 558–567, 2006.

10 Z. Liu, C. Feng, J. S. Ume, and S. M. Kang, “Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings,”Taiwanese Journal of Mathematics, vol. 11, no. 1, pp. 27–42, 2007.

11 Y. Yao and Y.-C. Liou, “Strong convergence to common fixed points of a finite family of asymptotically nonexpansive map,”Taiwanese Journal of Mathematics, vol. 11, no. 3, pp. 849–865, 2007.

12 L.-C. Ceng, H.-K. Xu, and J.-C. Yao, “The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces,”Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 4, pp. 1402–1412, 2008.

13 Z. Opial, “Weak convergence of the sequence of successive approximations for nonexpansive mappings,”Bulletin of the American Mathematical Society, vol. 73, pp. 591–597, 1967.