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Volume 2008, Article ID 350483, 10 pagesdoi:10.1155/2008/350483 Research Article An Implicit Iterative Scheme for an Infinite Countable Family of Asymptotically Nonexpansive Mappings in

Volume 2008, Article ID 350483, 10 pages

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,sheng-huawang@hotmail.com

Received 6 May 2008; Accepted 24 August 2008

Recommended by William Kirk

LetK be a nonempty closed convex subset of a reflexive Banach space E with a weakly continuous

dual mapping, and let{T i}∞

i1 be an infinite countable family of asymptotically nonexpansive mappings with the sequence{k in } satisfying k in ≥ 1 for each i  1, 2, , n  1, 2, , and

limn→∞k in  1 for each i  1, 2, In this paper, we introduce a new implicit iterative scheme

generated by{T i}∞

i1 and prove that the scheme converges strongly to a common fixed point of

{T i}∞i1, which solves some certain variational inequality

Copyrightq 2008 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

a mapping ThenT is called nonexpansive if

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

1, ∞ that converges to 1 as n→∞ such that

for all x, y ∈ K and all n ≥ 1 Obviously, a nonexpansive mapping is asymptotically

nonexpansive In 1, Goebel and Kirk originally introduced the concept of asymptotically nonexpansive mappings and proved that ifE is a uniformly convex Banach space and K is

a nonempty closed convex bounded subset of E, then every asymptotically nonexpansive

self-mapping onK has 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:

x n  α n x n−11− α n

T l n1

where {α n } is a sequence in 0, 1, {T i}N

mappings, where K is a nonempty closed convex subset of a uniformly convex Banach

space satisfying Opial’s condition13, and where n  l n N  r n for some integers l n ≥ 0 and 1 ≤ r n ≤ N Then the authors proved that if ∩ N

i1 FT i  / φ, then {x n} generated by 1.3 strongly converges to a common fixed point of{T i}N

i1

S : K→K be a nonexpansive mapping and let T : K→K be an asymptotically nonexpansive

mapping In10, the authors introduced the following modified Ishikawa iteration sequence with errors with respect toS and T:

y n  a n Sx n  b n T n x n  c n v n ,

where{a n }, {b n }, {c n } are three real numbers sequences in 0, 1 satisfying a n  b n  c n 

1, {a n }, {b n }, {c n } are also three real numbers sequences in 0, 1 satisfying a n  b n  c n  1, and {u n } and {v n } are given bounded sequences in K Then the authors proved that the

sequence{x n} generated by 1.4 strongly converges to a common fixed point of S and T if

some certain conditions are satisfied

contraction with efficient λ 0 < λ < 1 such that

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 {k n}

in a uniformly smooth Banach space:

x n



1−k t n

n



fx n

k t n

n T n x n ,

x n1



1−k t n

n



fx n

k t n

n T n x n ,

1.6

where{t n } is a sequence in 0, 1 They proved that the sequence {x n} converges strongly to the unique solution of some variational inequality if the sequence{t n} satisfies some certain conditions and the mappingT satisfies Tx n − x n →0 as n→∞.

Quite recently, Ceng et al 12 introduced the following two implicit and explicit iterative schemes generated by a finite family of asymptotically nonexpansive mappings

{T i}N i1 with the same sequence{k n} in a reflexive Banach space with a weakly continuous duality map:

x n



1−k1

n



x n 1− t k n

n fx nk t n

n T n

r n x n ,

x n1



1−k1

n



x n 1− t k n

n fx nk t n

n T n

r n x n ,

1.7

wherer n  n mod N and {t n } is a sequence in 0, 1 Then they proved that if the control

sequence{t n } satisfies some certain condition and T i x n −x n →0 as n→∞ for each i  1, 2, , N,

then both schemes1.7 strongly converge a common fixed point x∗of{T i}N i1which solves the variational inequality



I − fx∗, Jp − x∗

≥ 0, p ∈ N

i1

whereFT i  denotes the set of fixed points of the mapping T ifor eachi  1, 2, , N.

increasing function ϕ : R→R such that ϕ0  0 and lim t→∞ ϕt  ∞, we associate a

possibly multivalued generalized duality map J ϕ :E→2 E∗

, defined as

J ϕ x  x∗∈ E∗:x∗x  xϕx, x∗  ϕx 1.9

for everyx ∈ E We call the function ϕ a gauge If ϕt  t for all t ≥ 0, then we call J ϕ a normalized duality mapping and write it asJ.

A Banach spaceE is said to have a weakly continuous generalized duality map if there

exists a continuous strictly increasing functionϕ : R→Rsuch thatϕ0  0, lim t→∞ ϕt  ∞,

and J ϕ is single valued and sequentially continuous from E with the weak topology to E∗

with the weak∗topology For instance, everyl p-space1 < p < ∞ has a weakly continuous

generalized duality map forϕt  t p−1

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

literatures

gaugeϕ Then for all x, y ∈ E and jx  y ∈ J ϕ x  y one holds

Φx  y≤ Φxy, jx  y. 1.10 One also holds

x  y2≤ x2 2y, jx  y 1.11 for allx, y ∈ E and jx  y ∈ Jx  y.

Lemma 1.2 see 14 Let E be a Banach space satisfying a weakly continuous duality map and let K

with fixed point Then I − T is demiclosed at zero.

2 Strong convergence results

In this section, letE be a reflexive Banach space with a weakly continuous duality map J ϕ, whereϕ is a gauge and let K be a nonempty closed convex subset of E Let {T i}∞i1:K→K be

an infinite countable family of asymptotically nonexpansive mappings such that

T n

i x − T n

for allx, y ∈ K, where the sequence {k in } ⊂ 1, ∞ and lim n→∞ k in  1 for each i  1, 2,

For eachn  1, 2, , let b n  sup{k in | i  1, 2, } and assume

sup b n | n  1, 2, < ∞,

lim

Takingb n  max{b n , b} for each n  1, 2, , obviously, we have

lim

n→∞ b n  b ≥ 1,

b  sup b n | n  1, 2, < ∞. 2.3

Moreover, the following inequality

T n

i x − T n

holds for allx, y ∈ K and each i  1, 2

Take an integerr > 1 arbitrarily For each n ≥ 1, define the mapping S ni :K→K by

for eachi  1, 2, , r, that is,

S11  T1, , S1r  T r , S21  T r1 , , S2 r  T2r , 2.6

For eachi  1, 2, , r, let {α ni } ⊂ 0, 1 be a sequence real numbers For each n ≥ 1,

define the mappingW nofK into itself by

W n  U nr  α nr S n

nr U nr−11− α nrI, 2.7

where

U n1  α n1 S n

n11− α n1I,

U n2  α n2 S n

I,

U nr−1  α nr−1 S n

I.

2.8

We callW naW-mapping generated by S n1 , S n2 , , S nrandα n1 , α n2 , , α nr.

numbers{tn } ⊂ 0, b such that

lim

n→∞ t n  0, t n < b1 − b r n λ

1 − λb r

Note that sinceλ < 1/b , one has 0< b1 − b r

n λ/1 − λb r

n ≤ b Therefore, the sequence {t n} can be taken easily to satisfy the condition2.9, for example, t n  1/nb1−b r

n λ/1−λb r

n Then, we introduce an implicit iterative scheme

x n



1− b

b r1 n



x nb − t n

b r1

 t n

b r1

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

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

1, 2, If ∩∞

i1 FT i  / φ, then ∩∞

i1 FT i  ⊂ FW n  for each n  1, 2,

Lemma 2.2 Let {T i}∞

by2.7 for each n  1, 2, Then one holds

W n x − W n y  ≤ b r

for all n ≥ 1 and all x, y ∈ K.

Proof For any x, y ∈ K all n ≥ 1, we first see noting that b n≥ 1

U n1 x − U n1 y   α n1 S n

n11− α n1

Ix −α n1 S n

n11− α n1

Iy

≤ α n1 S n

n1 x − S n n1 y   1 − α n1

x − y

 α n1 T n

n−1r1 x − T n

n−1r1 y   1 − α n1

x − y

≤ α n1 k n−1r1n x − y 1− α n1x − y

≤ α n1 b n x − y 1− α n1

x − y

≤ α n1 b n x − y 1− α n1

b n x − y

 b n x − y,

U n2 x − U n2 y   α n2 S n

n2 U n11− α n2Ix −α n2 S n

n2 U n11− α n2Iy

≤ α n2 S n

n2 U n1 x − S n

n2 U n1 y   1 − α n2

x − y

 α n2 T n

n−1r2 U n1 x − T n

n−1r2 U n1 y   1 − α n2

x − y

≤ α n2 kn−1r2n U n1 x − U n1 y   1 − α n2

x − y

≤ α n2 b n U n1 x − U n1 y   1 − α n2

x − y

≤ α n2 b2

n x − y 1− α n1

b2

n x − y

 b2

n x − y.

2.12

Similarly, for eachi  3, , r − 1, we have

U ni x − U ni y ≤ b i

Hence,

W n x − W n y   α nr S n

nr U n r−11− α nrIx −α nr S n

nr U n r−11− α nrIy

≤ α nr S n

nr U n r−1 x − S n

nr U n r−1 y   1 − α nr

x − y

≤ b r

n x − y.

2.14

This completes the proof

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

b r

n λ/1 − λb r

n, we obtain

0< 1 − b

b r1

n b − t b n

Hence, the mapping



1− b

b r1 n



x  b − t n

b r1

b r1

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

Tx − Ty 

1− b

b r1 n



x − y  b − t n

b r1 n



fW n x − fW n y t n

b r1 n



W n x − W n y

≤



1− b

b r1 n



x − y  b − t n λb r n

b r1

n x − y  t n

b r1

n x − y





1− b

b r1

n b − t b n

n



x − y

≤ x − y,

2.17

which implies that the implicit scheme2.10 is well defined

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

Theorem 2.3 Assume 2.9, FT  ∩∞

i1 FT i  / φ and lim n→∞ x n −T i x n   0 for each i  1, 2,

Then {x n } converges strongly to a common fixed point x ∈ FT, where x solves the variational

inequality



I − fx, Jp − x≥ 0, p ∈ FT. 2.18

z ∈ FT, we have noting 0 < 1 − b/b r1

n  b − t n /b n λ  t n /b n < 1

x n − z2

1− b

b r1 n



x n − zb − t n

b r1 n



fW n x n

− fz t n

b r1 n



W n x n − z

 b − t n

b r1 n



fz − z2

≤

1− b

b r1 n



x n − zb − t n

b r1 n



fW n x n

− fz t n

b r1 n



W n x n − z2

2b − tn

b r1

n fz − z, jx n − z

≤ 1− b

b r1 n



x n − z  b − t n

b r1

n fW n x n− fW n z   t n

b r1

2b − tn

b r1 n



fz − z, jx n − z

≤



1− b

b r1

n b − t b n λ

n

2

x n − z2 2b − tn

b r1 n



fz − z, jx n − z

≤



1− b

b r1

n b − t b n λ

n

x

n − z2 2b − tn

b r1 n



fz − z, jx n − z

1− η n x n − z22b − tn

b r1 n



fz − z, jx n − z,

2.19

b r1 n

−b − t n

b n λ − t n

It follows from2.19 that

x n − z2≤ 2b − tn

η n b r1 n



Since limn→∞ b n  b, lim n→∞ t n 0, we have

lim

n→∞

b − t n

η n b r1

Hence,{x n} is bounded

Now we prove that{x n } strongly converges to a common fixed point x ∈ FT To see

this, we assume thatx is a weak limit point of {x n } and a subsequence {x n j } of {x n} converges weakly tox Then by the assumption of the theorem andLemma 1.2, we havex ∈ FT i for everyi  1, 2, In 2.21, replacing x nwithx n j andz with x, respectively, and then taking

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

lim

Therefore,x n j →x We further show that x solves 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

b r1 n



x n − pb − t n

b r1 n



x n − p t n

b r1 n



W n x n − pb − t n

b r1 n



fW n x n

− x n

≤ Φ

1− t n

b r1 n



x n − p t n

b r1 n



W n x n − pb − t n

b r1 n



fW n x n

− x n , J ϕ

x n − p

≤



1− t n

b r1

 Φxn − p  b − t n

b r1 n



fW n x n

− x n , J ϕ

x n − p,

2.25

which implies that



x n − fW n x n

, J ϕ

x n − p≤ b n r1 b − t − 1t n

n Φx n − p. 2.26 Now in2.26, replacing x nwithx n jand noting limn→∞ b n  b and lim n→∞ t n 0, we obtain



x − fx, J ϕ x − p lim

j→∞



x n j − fW n j x n j, J ϕx n j − p

≤ lim sup

j→∞

b r1

n j − 1t n j

b − t n j

Φxn j − p  0, 2.27

which implies thatx is a solution to 2.24

Finally, we prove that the sequence{x n } strongly converges to x It suffices to prove

that the variational inequality2.24 can have only one solution To see this, assuming that



I − fu, Ju − v≤ 0,



Adding them yields



I − fu − I − fv, Ju − v≤ 0. 2.29 However, sincef is a λ-contraction, we have that

1 − λu − v2 ≤I − fu − I − fv, Ju − v, 2.30 which implies thatu  v This completes the proof.

Remark 2.4 InTheorem 2.3, the condition that limn→∞ T i x n − x n   0 for each i  1, 2,

is necessary see 9, 12 This theorem shows that if for each n  1, 2, , the supremum

of the sequence{k in }, that is, sup{k in | i  1, 2, }, is finite and the limit of the sequence

sup{k in | i  1, 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 {T i}N i1 K→K be a finite family of asymptotically nonexpansive mappings with the sequences {k in } and let W n be a W-mapping generated by T1, T2, , T N and α n1 , α n2 , , α nN for each n  1, 2, Let the sequence {t n } ⊂ 0, 1 and satisfy t n < 1−k N

n and t n →0, where

k n  max{k1n , k2 n , , k Nn } for each n  1, 2, Assume that k  sup{k n | n  1, 2, } < ∞ Let

f be a contraction with λ0 < λ < 1/k N  Consider the implicit iterative scheme

x n



1− 1

k N1 n



x n1− t n

k N1

k N1

If {T i}N i1 satisfy the condition∩N

i1 FT i  / φ and T i x n − x n →0 as n→∞ for each i  1, 2, , N,

then {x n } converges strongly to a common fixed point x ∈ ∩ N

i1 FT i , where x solves the variational

inequality



I − fx, Jp − x≥ 0, p ∈ N

i1

FT i

Proof InTheorem 2.3, takeb n  k n , b  lim n→∞ k n  1, b  k, and r  N Then, this corollary

can obtained directly fromTheorem 2.3

Acknowledgment

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

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for all n ≥ and all x, y ∈ K.

Proof For any x, y ∈ K all n ≥ 1, we first see noting... S S Chang, K K Tan, H W J Lee, and C K Chan, “On the convergence of implicit iteration process

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