Strong Convergence Theorems for

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Volume 2009, Article ID 819036,12pages doi:10.1155/2009/819036

Research Article

Strong Convergence Theorems for

Common Fixed Points of Multistep Iterations with Errors in Banach Spaces

Feng Gu

¹

and Qiuping Fu

²

1Department of Mathematics, Institute of Applied Mathematics, Hangzhou Normal University, Hangzhou, Zhejiang 310036, China

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

Correspondence should be addressed to Feng Gu,[email protected] 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.

Copyrightq2009 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 onCif there exists a sequence{rn}in0,∞with lim_n_{→ ∞}rn0 such that for eachx, y∈C,

Tⁿx−Tⁿy≤1rnx−y, ∀n≥1. 1.1

If rn ≡ 0, then T is known as a nonexpansive mapping. T is called asymptotically nonexpansive in the intermediate sense1providedT is uniformly continuous and

lim sup

n→ ∞ sup

x,y∈C

Tⁿx−Tⁿy−x−y≤0. 1.2

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

LetCbe a nonempty subset of normed spaceX, and LetTi :C → Cbemmappings.

For a givenx1∈Cand a fixedm∈NNdenotes the set of all positive integers, compute the iterative sequencesx¹n , . . . , x^mn defined by

x¹n α¹n T_i^kxnβ¹n xnγn¹u¹n , x²n α²n T_i^kx¹n β²n xnγn²u²n , x³n α³n T_i^kx²n β³n xnγn³u³n ,

...

x^m−1n α^m−1n T_i^kxn^m−2β^m−1n xnγn^m−1u^m−1n , xn1 xn^mα^mn T_i^kx^m−1n βn^mxnγn^mu^mn , ∀n≥1,

1.3

where n k −1m i, {u¹n },{u²n }, . . . ,{u^mn } are bounded sequences in C and {αⁱn }, {βⁱn },{γnⁱ}, are appropriate real sequences in0,1such thatαⁱn βⁱn γnⁱ 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 2.1see1. A Banach spaceX is said to be a uniformly convex if the modulus of convexity ofXis

δX inf

1− xy

2 : x y1, x−y

>0, ∀∈0,2. 2.1

Lemma 2.2see3. Let{an},{bn}, and{γn}be three nonnegative real sequences satisfying the following condition:

an1 ≤ 1γn

anbn, ∀n≥1, 2.2

where_∞

n1γn<∞and_∞

n1bn<∞. Then 1lim_n_{→ ∞}anexists;

2If lim inf_n_{→ ∞}an0, then lim_n_{→ ∞}an0.

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Lemma 2.3see4. LetX be a uniformly convex Banach space and 0 < α ≤ tn ≤ β < 1 for all n≥1. Suppose that{xn}and{yn}are two sequences ofXsuch that

lim sup

n→ ∞ xn ≤a, lim sup

n→ ∞

yn≤a,

nlim→ ∞tnxn 1−tnyna,

2.3

for somea≥0. Then

nlim→ ∞xn−yn0. 2.4

3. Main Results

Lemma 3.1. LetXbe a uniformly convex Banach space,{xn},{yn}are two sequences ofX,α, β ∈ 0,1and{αn}be a real sequence. If there existsn0∈Nsuch that

i0< α≤αn≤β <1 for alln≥n0; iilim sup_n_{→ ∞} xn ≤a;

iiilim sup_n_{→ ∞} yn ≤a;

ivlim_n_{→ ∞} αnxn 1−αnyn a, then lim_n_{→ ∞} xn−yn 0.

Proof. The proof is clear byLemma 2.3.

Lemma 3.2. LetXbe a uniformly convex Banach space, letCbe a nonempty closed bounded convex subset ofX, and letTi : C → Cbemasymptotically nonexpansive mappings in the intermediate sense such thatF _m

i1FTi/∅. Put

Gik sup

x,y∈C T_i^kx−T_i^ky−x−y∨0, ∀k≥1, 3.1 so that_∞

k1Gik<∞. Let{αⁱn },{βnⁱ}, and{γnⁱ}be real sequences in0,1satisfying the following condition:

iαⁱn βⁱn γnⁱ1 for alli∈ {1,2, . . . , m}andn≥1;

ii_∞

n1γnⁱ <∞for alli∈ {1,2, . . . , m}.

If {xn} is the iterative sequence defined by 1.3, then, for each p ∈ F _m

i1FTi, the limit lim_n_{→ ∞} xn−p exists.

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Proof. For eachq∈F, we note that

xn¹−qα¹n T_i^kxnβn¹xnγn¹u¹n −q

≤α¹n T_i^kxn−qβ¹n xn−qγn¹u¹n −q

≤α¹n xn−qα¹n Gikβ¹n xn−qγn¹u¹n −q α¹n βn¹

xn−qα¹n Gikγn¹u¹n −q

≤xn−qdn¹,

3.2

whered¹n α¹n Gikγn¹ u¹n −q . Since ∞ n1

Gik

i∈I

∞ k1

Gik<∞, 3.3

we see that

∞ n1

d¹n <∞. 3.4

It follows from3.2that

x²n −q≤α²n x¹n −qα²n Gikβ²n xn−qγn²u²n −q

≤α²n xn−qd¹n

α²n Gikβ²n xn−qγn²u²n −q α²n β²n

xn−qα²n d¹n α²n Gikγn²u²n −q

≤xn−qd²n ,

3.5

whered²n α²n dn¹α²n Gikγn² u²_n −q . Since ∞

n1

Gik <∞, ^∞

n1

dn¹<∞, 3.6

we see that

∞ n1

d²n <∞. 3.7

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It follows from3.5that

x³n −q≤α³n x²n −qα³n Gikβ³n xn−qγn³u³n −q

≤α³n xn−qd¹n

α³n Gikβ³n xn−qγn³u³n −q α³n β³n

xn−qα³n d²n α³n Gikγn³u³n −q

≤xn−qd³n ,

3.8

whered³n α³n dn²α³n Gikγn³ u³n −q , and so ∞

n1

d³n <∞. 3.9

By continuing the above method, there are nonnegative real sequences{d^kn }such that ∞

n1

d^kn <∞,

x^kn −q≤xn−qdn^k, ∀k∈ {1,2, . . . , m}.

3.10

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

Lemma 3.3. LetXbe a uniformly convex Banach space, letCbe a nonempty closed bounded convex subset ofX, and letTi : C → Cbemasymptotically nonexpansive mappings in the intermediate sense such thatF _m

i1FTi/∅. Put

Gik sup

k1Gik<∞. Let the sequence{xn}be defined by1.3whenever{αⁱn },{βnⁱ},{γnⁱ}satisfy the same assumptions as inLemma 3.2for eachi∈ {1,2, . . . , m}and the additional assumption that there existsn0∈Nsuch that 0< α≤α^m−1n , α^mn ≤β <1 for alln≥n0. Then we have the following:

1lim_n_{→ ∞} T_i^kxn^m−1−xn 0;

2limn→ ∞ T_i^kxn^m−2−xn 0.

Proof. 1Taking eachq∈F, it follows fromLemma 3.2that lim_n_{→ ∞} xn−q exists. Let

nlim→ ∞xn−qa, 3.12

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for somea≥0. We note that

x^m−1n −q≤xn−qd^m−1n , ∀n≥1, 3.13

where{d^m−1_n }is a nonnegative real sequence such that ∞

n1

d^m−1n <∞. 3.14

It follows that

lim sup

n→ ∞

xn^m−1−q≤lim sup

n→ ∞

xn−q lim

n→ ∞xn−q a,

3.15

which implies that

lim sup

n→ ∞

T_i^kx^m−1n −q≤lim sup

n→ ∞ x^m−1n −qGik

lim

n→ ∞

xn^m−1−q

≤a.

3.16

Next, we observe that

T_i^kx^m−1n −qγn^m u^mn −xn≤T_i^kx^m−1n −qγn^m u^mn −xn. 3.17

Thus we have

lim sup

n→ ∞

T_i^kxn^m−1−qγn^m u^mn −xn≤a. 3.18

Also,

xn−qγn^m u^mn −xn≤xn−qγn^mu^mn −xn 3.19

gives that

lim sup

n→ ∞

xn−qγn^m u^mn −xn≤a. 3.20

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Note that

a lim

n→ ∞

xn^m−q lim

n→ ∞

α^mn T_i^kxn^m−1β^mn xnγn^mu^mn −q lim

n→ ∞

α^mn T_i^kxn^m−1 1−α^mn

xn−γn^mxn

γn^mu^mn − 1−α^mn

q−α^mn q lim

n→ ∞

α^mn T_i^kxn^m−1−α^mn qα^mn γn^mu^mn −α^mn γn^mxn

1−α^mn

q−γn^mxnγn^mu^mn −α^mn γn^mu^mn α^mn γn^mxn lim

n→ ∞

α^mn T_i^kxn^m−1−qγn^m u^mn −xn

1−α^mn xn−qγn^m u^mn −xn.

3.21

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

nlim→ ∞

T_i^kx^m−1n −xn0. 3.22

This completes the proof of1.

2For eachn≥1,

xn−qxn−T_i^kx^m−1n T_i^kxn^m−1−q

≤xn−T_i^kx^m−1n x^m−1n −qGik.

3.23

Since

nlim→ ∞

xn−T_i^kxn^m−10 lim

n→ ∞Gik, 3.24

we obtain

a lim

n→ ∞xn−q≤lim inf

n→ ∞

x^m−1n −q. 3.25

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It follows that

a≤lim inf

n→ ∞

xn^m−1−q

≤lim sup

n→ ∞

x^m−1n −q

≤a,

3.26

which implies that

nlim→ ∞

xn^m−1−qa. 3.27

On the other hand, we note that

x^m−2n −q≤xn−qd^m−2n , ∀n≥1, 3.28

where{d^m−2_n }is a nonnegative real sequence such that ∞

n1

d^m−2n <∞. 3.29

Thus we have

lim sup

n→ ∞

xn^m−2−q≤lim sup

n→ ∞

xn−q a,

3.30

and hence

lim sup

n→ ∞

T_i^kx^m−2n −q≤lim sup

n→ ∞ x^m−2n −qGik

≤a.

3.31

Next, we observe that

T_i^kx^m−2n −qγn^m−1 u^m−1n −xn≤T_i^kxn^m−2−qγn^m−1u^m−1n −xn. 3.32

Thus we have

lim sup

n→ ∞

T_i^kx^m−2n −qγn^m−1 u^m−1n −xn≤a. 3.33

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Also,

xn−qγn^m−1 u^m−1n −xn≤xn−qγn^m−1u^m−1n −xn 3.34

gives that

lim sup

n→ ∞

xn−qγn^m−1 u^m−1n −xn≤a. 3.35

Note that

a lim

n→ ∞

x^m−1n −q lim

n→ ∞

α^m−1n T_i^kxnβn^m−1xnγn^m−1u^m−1n −q lim

n→ ∞

α^m−1n T_i^kx^m−2n −qγn^m−1 u^m−1n −xn

1−α^m−1n xn−qγn^m−1 u^m−1n −xn.

3.36

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

nlim→ ∞

T_i^kx^m−2n −xn0. 3.37

This completes the proof.

Theorem 3.4. LetX be a uniformly convex Banach space and letCbe a nonempty closed bounded convex subset ofX. LetTi :C → Cbemasymptotically nonexpansive mappings in the intermediate sense such that F _m

i1FTi/∅ and there exists one memberT in {Ti}^m_i1which is completely continuous. Put

Gik sup

k1Gik<∞. Let the sequence{xn}be defined by1.3whenever{αⁱn },{βnⁱ},{γnⁱ}satisfy the same assumptions as inLemma 3.2for eachi∈ {1,2, . . . , m}and the additional assumption that there existsn0 ∈Nsuch that 0< α ≤ α^m−1n , α^mn ≤ β < 1 for alln ≥ n0. Then{xn^k}converges strongly to a common fixed point of the mappings{Ti}^m_i1.

Proof. FromLemma 3.3, it follows that

nlim→ ∞

T_i^kx^m−1n −xn0 lim

n→ ∞

T_i^kxn^m−2−xn, 3.39

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which implies that

xn1−xn x^mn −xn

≤α^mn T_i^kx^m−1n −xnγn^m−1u^m−1n −xn−→0, n−→ ∞, 3.40

and so

x_nl−xn −→0, n−→ ∞. 3.41

It follows from3.22,3.37that

T_n^kxn−xn≤T_i^kxn−T_i^kxn^m−1T_i^kx^m−1n −xn

≤xn−x^m−1n GikT_i^kxn^m−1−xn

≤α^m−1n T_i^kx^m−2n −xnGikγn^m−1u^m−1n −xn T_i^kx^m−1n −xn−→0, n−→ ∞.

3.42

Letσn T_i^kxn−xn for alln > n0. Then we have xn−Tnxn ≤xn−T_n^kxnT_n^kxn−Tnxn

≤xn−T_i^kxnLT_n^k−1xn−xn

≤σnLT_n^k−1xn−T_n−m^k−1xn−mT_n−m^k−1xn−m−xn−m xn−m−xn

.

3.43

Notice thatn≡n−mmodm. ThusTnTn−mand the above inequality becomes

xn−Tnxn ≤σnL² xn−xn−m Lσn−m xn−m−xn , 3.44

and so

nlim→ ∞ xn−Tnxn 0. 3.45

Since

xn−T_nlxn ≤ xn−x_nl x_nl−T_nlx_nl T_nlx_nl−T_nlxn

≤1L xn−xnl xnl−Tnlxnl , ∀l∈ {1,2, . . . , m}, 3.46

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we have

nlim→ ∞ xn−Tnlxn 0, ∀l∈ {1,2, . . . , m}, 3.47

and so

nlim→ ∞ xn−Tlxn 0, ∀l∈ {1,2, . . . , m}. 3.48

Since{xn} is bounded and one of Ti is completely continuous, we may assume that T1 is completely continuous, without loss of generality. Then there exists a subsequence{T1xnk}of {T1xn}such thatT1xnk → q∈Cask → ∞. Moreover, by3.48, we have

nlim→ ∞ xnk −T1xnk 0, 3.49

which implies thatxnk → qask → ∞. By3.48again, we have q−Tlq lim

n→ ∞ xnk−Tlxnk 0, ∀l∈ {1,2, . . . , m}. 3.50 It follows thatq∈F. Since limn→ ∞ xn−q exists, we have

nlim→ ∞xn−q0, 3.51

that is,

nlim→ ∞x^mn lim

n→ ∞xnq. 3.52

Moreover, we observe that

x^kn −q≤xn−qd^kn , 3.53

for allk1,2, . . . , m−1 and

nlim→ ∞d^kn 0. 3.54

Therefore,

nlim→ ∞x^kn q, 3.55

for allk1,2, . . . , m−1. This completes the proof.

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Remark 3.5. Theorem 3.4improves and extends the corresponding results of Plubtieng and Wangkeeree2in the following ways.

1The iterative process{xn}defined by1.3in2is replaced by the new iterative process{xn}defined by1.3in this paper.

2 Theorem 3.4 generalizes Theorem 3.4 of Plubtieng and Wangkeeree 2 from a asymptotically nonexpansive mappings in the intermediate sense to a finite family of asymptotically nonexpansive mappings in the intermediate sense.

Remark 3.6. Ifm 3 andT1 T2 T3 T in Theorem 3.4, we obtain strong convergence theorem for Noor iteration scheme with error for asymptotically nonexpansive mappingT 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.