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Volume 2009, Article ID 819036, 12 pagesdoi:10.1155/2009/819036 Research Article Strong Convergence Theorems for Common Fixed Points of Multistep Iterations with Errors in Banach Spaces

Volume 2009, Article ID 819036, 12 pages

doi:10.1155/2009/819036

Research Article

Strong Convergence Theorems for

Common Fixed Points of Multistep Iterations

with Errors in Banach Spaces

1 Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University,

Hangzhou, Zhejiang 310036, China

2 Mathematics group, West Lake High Middle School, Hangzhou, Zhejiang 310012, China

Correspondence should be addressed to Feng Gu,gufeng99@sohu.com

Received 19 November 2008; Revised 11 January 2009; Accepted 9 April 2009

Recommended by Yeol Je Cho

We establish strong convergence theorem for multi-step iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in Banach spaces Our results extend and improve the recent ones announced by Plubtieng and Wangkeeree2006, and many others

Copyrightq 2009 F Gu and Q Fu 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

Let C be a subset of real normal linear space X A mapping T : C → C is said to be asymptotically nonexpansive on C if there exists a sequence {r n } in 0, ∞ with lim n → ∞ r n 0

such that for each x, y ∈ C,

T n x − T n y  ≤ 1  r nx − y, ∀n ≥ 1. 1.1

If r n ≡ 0, then T is known as a nonexpansive mapping T is called asymptotically

nonexpansive in the intermediate sense1 provided T is uniformly continuous and

lim sup

n → ∞

sup

x,y∈C

T n x − T n y  − x − y ≤ 0. 1.2

From the above definitions, it follows that asymptotically nonexpansive mapping must be asymptotically nonexpansive in the intermediate sense

Let C be a nonempty subset of normed space X, and Let T i : C → C be m mappings For a given x1∈ C and a fixed m ∈ N N denotes the set of all positive integers, compute the iterative sequences x1n , , x m n defined by

x1n  α1n T i k x n  β1n x n  γ n1u1n ,

x2n  α2n T i k x1n  β2n x n  γ n2u2n ,

x3n  α3n T i k x2n  β3n x n  γ n3u3n ,

x m−1 n  α m−1 n T k

i x n m−2  β m−1 n x n  γ n m−1 u m−1 n ,

x n1  x n m  α m n T i k x m−1 n  β n m x n  γ n m u m n , ∀n ≥ 1,

1.3

where n  k − 1m  i, {u1n }, {u2n }, , {u m n } are bounded sequences in C and {α i n },

{β i n }, {γ n i }, are appropriate real sequences in 0, 1 such that α i n  β i n  γ n i  1 for each

i ∈ {1, 2, , m}.

The purpose of this paper is to establish a strong convergence theorem for common fixed points of the multistep iterative scheme with errors for asymptotically nonexpansive mappings in the intermediate sense in a uniformly convex Banach space The results presented in this paper extend and improve the corresponding ones announced by Plubtieng and Wangkeeree2, and many others

2 Preliminaries

Definition 2.1see 1 A Banach space X is said to be a uniformly convex if the modulus of convexity of X is

δ X   inf



1− x  y

2 : x y   1, x − y  > 0, ∀ ∈ 0, 2. 2.1

Lemma 2.2 see 3 Let {a n }, {b n }, and {γ n } be three nonnegative real sequences satisfying the

following condition:

a n1≤1 γ n



a n  b n , ∀n ≥ 1, 2.2

where∞

n1 γ n < ∞ and∞

n1 b n < ∞ Then

1 limn → ∞ a n exists;

2 If lim inf n → ∞ a n  0, then lim n → ∞ a n  0.

Lemma 2.3 see 4 Let X be a uniformly convex Banach space and 0 < α ≤ t n ≤ β < 1 for all

n ≥ 1 Suppose that {x n } and {y n } are two sequences of X such that

lim sup

n → ∞

x n ≤ a,

lim sup

n → ∞

y n  ≤ a,

lim

n → ∞t n x n  1 − t n y n   a,

2.3

for some a ≥ 0 Then

lim

3 Main Results

Lemma 3.1 Let X be a uniformly convex Banach space, {x n }, {y n } are two sequences of X, α, β ∈

0, 1 and {α n } be a real sequence If there exists n0∈ N such that

i 0 < α ≤ α n ≤ β < 1 for all n ≥ n0;

ii lim supn → ∞ x n ≤ a;

iii lim supn → ∞ y n ≤ a;

iv limn → ∞ α n x n  1 − α n y n  a,

then lim n → ∞ x n − y n  0.

Proof The proof is clear byLemma 2.3

Lemma 3.2 Let X be a uniformly convex Banach space, let C be a nonempty closed bounded convex

subset of X, and let T i : C → C be m asymptotically nonexpansive mappings in the intermediate

sense such that F m

i1 FT i  / ∅ Put

G ik sup

x,y∈C

T k

i x − T k

so that∞

k1 G ik < ∞ Let {α i n }, {β n i }, and {γ n i } be real sequences in 0, 1 satisfying the following

condition:

i α i n  β i n  γ n i  1 for all i ∈ {1, 2, , m} and n ≥ 1;

ii∞

n1 γ n i < ∞ for all i ∈ {1, 2, , m}.

If {x n } is the iterative sequence defined by 1.3, then, for each p ∈ F  m

i1 FT i , the limit

limn → ∞ x n − p exists.

Proof For each q ∈ F, we note that



x n1− q α1

n T i k x n  β n1x n  γ n1u1n − q

≤ α1n T k

i x n − q  β1

n x n − q  γ1

n u1

n − q

≤ α1n x n − q  α1

n G ik  β1n x n − q  γ1

n u1

n − q

 α1n  β n1 x n − q  α1

n G ik  γ n1u1

n − q

≤x n − q  d1

n ,

3.2

where d1n  α1n G ik  γ n1 u1n − q Since

∞

n1

G ik

i∈I

∞

k1

we see that

∞

n1

It follows from3.2 that



x2n − q ≤ α2

n x1

n − q  α2

n G ik  β2n x n − q  γ2

n u2

n − q

≤ α2n

x

n − q  d1

n  α2n G ik  β2n x n − q  γ2

n u2

n − q

 α2n  β2n x n − q  α2

n d1n  α2n G ik  γ n2u2

n − q

≤x n − q  d2

n ,

3.5

where d2n  α2n d n1 α2n G ik  γ n2 u2n − q Since

∞

n1

G ik < ∞,

∞

n1

we see that

∞

n1

It follows from3.5 that



x3n − q ≤ α3

n x2

n − q  α3

n G ik  β3n x n − q  γ3

n u3

n − q

≤ α3n

x

n − q  d1

n  α3n G ik  β3n x n − q  γ3

n u3

n − q

 α3n  β3n x n − q  α3

n d2n  α3n G ik  γ n3u3

n − q

≤x n − q  d3

n ,

3.8

where d3n  α3n d n2 α3n G ik  γ n3 u3n − q , and so

∞

n1

By continuing the above method, there are nonnegative real sequences{d k n } such that

∞

n1

d k n < ∞,



x k n − q ≤x

n − q  d k

n , ∀k ∈ {1, 2, , m}.

3.10

This together withLemma 2.2gives that limn → ∞ x n − q exists This completes the proof.

Lemma 3.3 Let X be a uniformly convex Banach space, let C be a nonempty closed bounded convex

subset of X, and let T i : C → C be m asymptotically nonexpansive mappings in the intermediate

sense such that F m

i1 FT i  / ∅ Put

G ik sup

x,y∈C

T k

i x − T i k y −x − y 3.11

so that∞

k1 G ik < ∞ Let the sequence {x n } be defined by 1.3 whenever {α i n }, {β n i }, {γ n i } satisfy

the same assumptions as in Lemma 3.2 for each i ∈ {1, 2, , m} and the additional assumption that there exists n0∈ N such that 0 < α ≤ α m−1 n , α m n ≤ β < 1 for all n ≥ n0 Then we have the following:

1 limn → ∞ T k

i x n m−1 − x n  0;

2 limn → ∞ T k

i x n m−2 − x n  0.

Proof 1 Taking each q ∈ F, it follows fromLemma 3.2that limn → ∞ x n − q exists Let

lim

for some a ≥ 0 We note that



x m−1 n − q ≤x

n − q  d m−1

where{dm−1 n } is a nonnegative real sequence such that

∞

n1

d m−1 n < ∞. 3.14

It follows that

lim sup

n → ∞



x n m−1 − q ≤ lim sup

n → ∞

x n − q

 lim

n → ∞x n − q

 a,

3.15

which implies that

lim sup

n → ∞



T k

i x m−1 n − q ≤ lim sup

n → ∞

x m−1

n − q  G

ik

 lim

n → ∞



x n m−1 − q

≤ a.

3.16

Next, we observe that



T k

i x m−1 n − q  γ n m

u m n − x n  ≤T k

i x m−1 n − q  γ m

n 

u m n − x n . 3.17 Thus we have

lim sup

n → ∞



T k

i x n m−1 − q  γ n m

u m n − x n  ≤ a. 3.18

Also,



x n − q  γ n m

u m n − x n  ≤x n − q  γ m

n u m

n − x n 3.19

gives that

lim sup

n → ∞



x n − q  γ n m u m n − x n  ≤ a. 3.20

Note that

a  lim

n → ∞



x n m − q

 lim

n → ∞



α m n T i k x n m−1  β m n x n  γ n m u m n − q

 lim

n → ∞



α m n T i k x n m−1 1− α m n x n − γ n m x n

γ n m u m n − 1− α m n q − α m n q

 lim

n → ∞



α m n T k

i x n m−1 − α m n q  α m n γ n m u m n − α m n γ n m x n

 1− α m n q − γ n m x n  γ n m u m n − α m n γ n m u m n  α m n γ n m x n

 lim

n → ∞



α m n

T i k x n m−1 − q  γ n m

u m n − x n

 1− α m n x n − q  γ n m

u m n − x n .

3.21

This together with3.18, 3.20, andLemma 3.1, gives

lim

n → ∞



T k

i x m−1 n − x n  0. 3.22

This completes the proof of1

2 For each n ≥ 1,

x n − q x n − T k

i x m−1 n  T k

i x n m−1 − q

≤x

n − T k

i x m−1 n  x m−1

n − q  G

ik

3.23

Since

lim

n → ∞



x n − T k

i x n m−1  0  lim

we obtain

a  lim

n → ∞x n − q ≤ liminf

n → ∞



x m−1 n − q. 3.25

It follows that

a ≤ lim inf

n → ∞



x n m−1 − q

≤ lim sup

n → ∞



x m−1 n − q

≤ a,

3.26

which implies that

lim

n → ∞



x n m−1 − q  a. 3.27

On the other hand, we note that



x m−2 n − q ≤x

n − q  d m−2

where{d m−2 n } is a nonnegative real sequence such that

∞

n1

d m−2 n < ∞. 3.29

Thus we have

lim sup

n → ∞



x n m−2 − q ≤ lim sup

n → ∞

x n − q

 a,

3.30

and hence

lim sup

n → ∞



T k

i x m−2 n − q ≤ lim sup

n → ∞

x m−2

n − q  G

ik

≤ a.

3.31

Next, we observe that



T k

i x m−2 n − q  γ n m−1

u m−1 n − x n  ≤T k

i x n m−2 − q  γ m−1

n u m−1

n − x n. 3.32

Thus we have

lim sup

n → ∞



T k

i x m−2 n − q  γ n m−1 u m−1 n − x n  ≤ a. 3.33



x n − q  γ n m−1

um−1n − x n  ≤x n − q  γ m−1

n u m−1

n − x n 3.34

gives that

lim sup

n → ∞



x n − q  γ n m−1

u m−1 n − x n  ≤ a. 3.35

Note that

a  lim

n → ∞



x m−1 n − q

 lim

n → ∞



α m−1 n T i k x n  β n m−1 x n  γ n m−1 u m−1 n − q

 lim

n → ∞



α m−1 n

T i k x m−2 n − q  γ n m−1

u m−1 n − x n

 1− α m−1 n x n − q  γ n m−1

u m−1 n − x n .

3.36

Therefore, it follows from3.33, 3.35, andLemma 3.1that

lim

n → ∞



T k

i x m−2 n − x n  0. 3.37 This completes the proof

Theorem 3.4 Let X be a uniformly convex Banach space and let C be a nonempty closed bounded

convex subset of X Let T i : C → C be m asymptotically nonexpansive mappings in the intermediate

sense such that F  m

i1 FT i  / ∅ and there exists one member T in {T i}m

i1 which is completely continuous Put

G ik sup

x,y∈C

T k

i x − T i k y −x − y 3.38

so that∞

k1 G ik < ∞ Let the sequence {x n } be defined by 1.3 whenever {α i n }, {β n i }, {γ n i } satisfy

the same assumptions as in Lemma 3.2 for each i ∈ {1, 2, , m} and the additional assumption that there exists n0 ∈ N such that 0 < α ≤ α m−1 n , α m n ≤ β < 1 for all n ≥ n0 Then {x n k } converges

strongly to a common fixed point of the mappings {T i}m

i1 Proof FromLemma 3.3, it follows that

lim

n → ∞



T k

i x m−1 n − x n  0  lim

n → ∞



T k

i x n m−2 − x n, 3.39

which implies that

x n1 − x n x m

n − x n

≤ α m n T k

i x m−1 n − x n  γ m−1

n u m−1

n − x n −→ 0, n −→ ∞, 3.40

and so

x nl − x n −→ 0, n −→ ∞. 3.41

It follows from3.22, 3.37 that



T k

n x n − x n ≤T k

i x n − T k

i x n m−1 T k

i x m−1 n − x n

≤x

n − x m−1 n   G

ikT k

i x n m−1 − x n

≤ α m−1 n T k

i x m−2 n − x n  G

ik  γ n m−1u m−1

n − x n

T k

i x m−1 n − x n −→ 0, n −→ ∞.

3.42

Let σ n  T k

i x n − x n for all n > n0 Then we have

x n − T n x n ≤x

n − T k

n x n T k

n x n − T n x n

≤x

n − T k

i x n  LT k−1

n x n − x n

≤ σ n  LT k−1

n x n − T k−1

n−m x n−m T k−1

n−m x n−m − x n−m  x

n−m − x n

3.43

Notice that n ≡ n − mmodm Thus T n  T n−mand the above inequality becomes

x n − T n x n ≤ σ n  L2 x n − x n−m  Lσ n−m  x n−m − x n , 3.44 and so

lim

Since

x n − T nl x n ≤ x n − x nl  x nl − T nl x nl  T nl x nl − T nl x n

≤ 1  L x n − x nl  x nl − T nl x nl , ∀l ∈ {1, 2, , m}, 3.46

we have

lim

n → ∞ x n − T nl x n  0, ∀l ∈ {1, 2, , m}, 3.47

and so

lim

n → ∞ x n − T l x n  0, ∀l ∈ {1, 2, , m}. 3.48

Since{x n } is bounded and one of T i is completely continuous, we may assume that T1 is completely continuous, without loss of generality Then there exists a subsequence{T1x n k} of

{T1x n } such that T1x n k → q ∈ C as k → ∞ Moreover, by 3.48, we have

lim

n → ∞ x n k − T1x n k  0, 3.49

which implies that x n k → q as k → ∞ By 3.48 again, we have

q − T l q  lim

n → ∞ x n k − T l x n k  0, ∀l ∈ {1, 2, , m}. 3.50

It follows that q ∈ F Since lim n → ∞ x n − q exists, we have

lim

that is,

lim

n → ∞ x m n  lim

Moreover, we observe that



x k n − q ≤x

n − q  d k

for all k  1, 2, , m − 1 and

lim

Therefore,

lim

for all k  1, 2, , m − 1 This completes the proof.

Remark 3.5. Theorem 3.4improves and extends the corresponding results of Plubtieng and Wangkeeree2 in the following ways

1 The iterative process {x n} defined by 1.3 in 2 is replaced by the new iterative process{x n} defined by 1.3 in this paper

a asymptotically nonexpansive mappings in the intermediate sense to a finite family of asymptotically nonexpansive mappings in the intermediate sense

Remark 3.6 If m  3 and T1  T2  T3  T inTheorem 3.4, we obtain strong convergence

theorem for Noor iteration scheme with error for asymptotically nonexpansive mapping T in

the intermediate sense in Banach space, we omit it here

References

1 K Goebel and W A Kirk, “A fixed point theorem for asymptotically nonexpansive mappings,”

Proceedings of the American Mathematical Society, vol 35, no 1, pp 171–174, 1972.

2 S Plubtieng and R Wangkeeree, “Strong convergence theorems for multi-step Noor iterations with

errors in Banach spaces,” Journal of Mathematical Analysis and Applications, vol 321, no 1, pp 10–23,

2006

3 Q Liu, “Iterative sequences for asymptotically quasi-nonexpansive mappings with error member,”

Journal of Mathematical Analysis and Applications, vol 259, no 1, pp 18–24, 2001.

4 J Schu, “Iterative construction of fixed points of strictly pseudocontractive mappings,” Applicable

Analysis, vol 40, no 2-3, pp 67–72, 1991.