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Volume 2009, Article ID 483497, 25 pagesdoi:10.1155/2009/483497 Research Article Strong Convergence Theorems of Modified Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Ma

Volume 2009, Article ID 483497, 25 pages

doi:10.1155/2009/483497

Research Article

Strong Convergence Theorems of Modified

Ishikawa Iterations for Countable Hemi-Relatively Nonexpansive Mappings in a Banach Space

Narin Petrot,1, 2 Kriengsak Wattanawitoon,3, 4

and Poom Kumam2, 3

1 Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

2 Centre of Excellence in Mathematics, CHE, Si Ayuthaya Road, Bangkok 10400, Thailand

3 Department of Mathematics, Faculty of Science, King Mongkut’s University of Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand

4 Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology,

Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand

Correspondence should be addressed to Poom Kumam,poom.kum@kmutt.ac.th

Received 17 March 2009; Accepted 12 September 2009

Recommended by Lech G ´orniewicz

We prove some strong convergence theorems for fixed points of modified Ishikawa and Halperniterative processes for a countable family of hemi-relatively nonexpansive mappings in a uniformlyconvex and uniformly smooth Banach space by using the hybrid projection methods Moreover, wealso apply our results to a class of relatively nonexpansive mappings, and hence, we immediatelyobtain the results announced by Qin and Su’s result2007, Nilsrakoo and Saejung’s result 2008,

Su et al.’s result2008, and some known corresponding results in the literatures

Copyrightq 2009 Narin Petrot et al This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited

1 Introduction

Let C be a nonempty closed convex subset of a real Banach space E A mapping T : C → C is said to be nonexpansive if Tx−Ty ≤ x−y for all x, y ∈ C We denote by FT the set of fixed points of T, that is FT  {x ∈ C : x  Tx} A mapping T is said to be quasi-nonexpansive

if FT / ∅ and Tx − y ≤ x − y for all x ∈ C and y ∈ FT It is easy to see that if T

is nonexpansive with F T / ∅, then it is quasi-nonexpansive Some iterative processes are

often used to approximate a fixed point of a nonexpansive mapping The Mann’s iterativealgorithm was introduced by Mann 1 in 1953 This iterative process is now known asMann’s iterative process, which is defined as

2 Fixed Point Theory and Applications

where the initial guess x0 is taken in C arbitrarily and the sequence {α n}∞n0is in the interval

conditions must be satisfied

In 1974, Ishikawa4 introduced a new iterative scheme, which is defined recursivelyby

5 The problem of finding an optimal point that minimizes a given cost function overcommon set of fixed points of a family of nonexpansive mappings is of wide interdisciplinaryinterest and practical importancesee 6

Zhang and Su7 introduced the following implicit hybrid method for a finite family

of nonexpansive mappings{T i}N i1in a real Hilbert space:

where T n ≡ T n mod N,{α n } and {β n } are sequences in 0, 1 and {α n } ⊂ 0, a for some a ∈ 0, 1

and{β n } ⊂ b, 1 for some b ∈ 0, 1.

In 2008, Nakprasit et al.8 established weak and strong convergence theorems forfinding common fixed points of a countable family of nonexpansive mappings in a realHilbert space In the same year, Cho et al.9 introduced the normal Mann’s iterative processand proved some strong convergence theorems for a finite family nonexpansive mapping inthe framework Banach spaces

To find a common fixed point of a family of nonexpansive mappings, Aoyama et al.

10 introduced the following iterative sequence Let x1  x ∈ C and

for all n ∈ N, where C is a nonempty closed convex subset of a Banach space, {α n} is asequence of0, 1, and {T n} is a sequence of nonexpansive mappings Then they proved that,under some suitable conditions, the sequence{x n} defined by 1.5 converges strongly to acommon fixed point of{T n}

In 2008, by using anew hybrid method, Takahashi et al 11 proved the followingtheorem

Theorem 1.1 Takahashi et al 11 Let H be a Hilbert space and let C be a nonempty closed

convex subset of H Let {T n } and T be families of nonexpansive mappings of C into itself such that

∩∞

n1 FT n  : FT / ∅ and let x0 ∈ H Suppose that {T n } satisfies the NST-condition I with T.

For C1 C and x1  P C1x0, define a sequence {x n } of C as follows:

y n  α n x n  1 − α n T n x n ,

C n1z ∈ C n:y n − z ≤ x n − z,

x n1  P C n1 x0, n ∈ N,

1.6

where 0 ≤ α n < 1 for all n ∈ N and {T n } is said to satisfy the NST-condition I with T if for each

bounded sequence {z n } ⊂ C, lim n → ∞ z n − T n z n   0 implies that lim n → ∞ z n − Tz n   0 for all

T ∈ T Then, {x n } converges strongly to P FT x0.

Note that, recently, many authors try to extend the above result from Hilbert spaces to

a Banach space setting

Let E be a real Banach space with dual E∗ Denote by

is defined by Jx  {f ∈ E∗: 2  f2}, for all x ∈ E The function φ : E × E → R is defined by

4 Fixed Point Theory and Applications

On the other hand, Matsushita and Takahashi17 introduced the following iteration

A sequence{x n }, defined by

x n1 ΠC J−1α n Jx n  1 − α n JTx n , n  0, 1, 2, , 1.9

where the initial guess element x0 ∈ C is arbitrary, {α n } is a real sequence in 0, 1, T is a

relatively nonexpansive mapping, andΠC denotes the generalized projection from E onto a closed convex subset C of E Under some suitable conditions, they proved that the sequence {x n } converges weakly to a fixed point of T.

Recently, Kohsaka and Takahashi 18 extended iteration 1.9 to obtain a weakconvergence theorem for common fixed points of a finite family of relatively nonexpansivemappings{T i}m i1by the following iteration:

x n1 ΠC J−1

i1

w n,i α n,i Jx n  1 − α n,i JT i x n , n  0, 1, 2, , 1.10

where α n,i ⊂ 0, 1 and w n,i ⊂ 0, 1 withm i1 w n,i  1, for all n ∈ N Moreover, Matsushita and

Takahashi14 proposed the following modification of iteration 1.9 in a Banach space E:

and proved that the sequence{x n} converges strongly to ΠFT x.

Qin and Su 15 showed that the sequence {x n}, which is generated by relatively

nonexpansive mappings T in a Banach space E, as follows:

Moreover, they also showed that the sequence{x n}, which is generated by

In this paper, motivated by Qin and Su15, Nilsrakoo and Saejung 19, we considerthe modified Ishikawa iterative 1.12 and Halpern iterative processes 1.13, which isdifferent from those of 1.12–1.14, for countable hemi-relatively nonexpansive mappings

By using the shrinking projection method, some strong convergence theorems in a uniformlyconvex and uniformly smooth Banach space are provided Our results extend and improvethe recent results by Nilsrakoo and Saejung’s result 19, Qin and Su 15, Su et al 12,Takahashi et al.’s theorem11, and many others

6 Fixed Point Theory and Applications

2 Preliminaries

In this section, we will recall some basic concepts and useful well-known results

A Banach space E is said to be strictly convex if



for all x, y ∈ E with x  y  1 and x / y It is said to be uniformly convex if for any two

sequences{x n } and {y n } in E such that x n   y n  1 and

exists for each x, y ∈ U It is said to be uniformly smooth if the limit is attained uniformly for

x, y ∈ E In this case, the norm of E is said to be Gˆateaux differentiable The space E is said to

have uniformly Gˆateaux di fferentiable if for each y ∈ U, the limit 2.3 is attained uniformly for

y ∈ U The norm of E is said to be uniformly Fr´echet differentiable and E is said to be uniformly

smooth if the limit 2.3 is attained uniformly for x, y ∈ U.

In our work, the concept duality mapping is very important Here, we list some known

facts, related to the duality mapping J, as follows.

a E E∗, resp. is uniformly convex if and only if E∗E, resp. is uniformly smooth.

b Jx / ∅ for each x ∈ E.

c If E is reflexive, then J is a mapping of E onto E∗

d If E is strictly convex, then Jx ∩ Jy / ∅ for all x / y.

e If E is smooth, then J is single valued.

f If E has a Fr´echet differentiable norm, then J is norm to norm continuous.

g If E is uniformly smooth, then J is uniformly norm to norm continuous on each bounded subset of E.

h If E is a Hilbert space, then J is the identity operator.

For more information, the readers may consult20,21

If C is a nonempty closed convex subset of a real Hilbert space H and P C : H → C is the metric projection, then P Cis nonexpansive Alber22 has recently introduced a generalized

projection in Hilbert spaces

The generalized projectionΠC : E → C is a map that assigns to an arbitrary point

x ∈ E the minimum point of the functional φy, x, that is, Π C x  x∗, where x∗is the solution

to the minimization problem

φx∗, x  min

y∈C φ

y, x

Notice that the existence and uniqueness of the operatorΠCis followed from the properties of

the functional φy, x and strict monotonicity of the mapping J, and moreover, in the Hilbert

spaces setting we haveΠC  P C It is obvious from the definition of the function φ that

y − x2≤ φy, x

0 if and only if x  y, see Matsushita and Takahashi 14

To obtain our results, following lemmas are important

Lemma 2.2 Kamimura and Takahashi 23 Let E be a uniformly convex and smooth Banach

0, 2r → 0, ∞ such that g0  0 and

for all x, y ∈ B r  {z ∈ E : z ≤ r}.

Lemma 2.3 Kamimura and Takahashi 23 Let E be a uniformly convex and smooth real Banach

space and let {x n }, {y n } be two sequences of E If φx n , y n  → 0 and either {x n } or {y n } is bounded,

then x n − y n  → 0.

Lemma 2.4 Alber 22 Let C be a nonempty closed convex subset of a smooth real Banach space E

and x ∈ E Then, x0 ΠC x if and only if



Lemma 2.5 Alber 22 Let E be a reflexive strict convex and smooth real Banach space, let C be a

φ

y, Π C x

 φΠ C x, x ≤ φy, x

Lemma 2.6 Matsushita and Takahashi 14 Let E be a strictly convex and smooth real Banach

space, let C be a closed convex subset of E, and let T be a hemi-relatively nonexpansive mapping from

C into itself Then F(T) is closed and convex.

8 Fixed Point Theory and Applications

Let C be a subset of a Banach space E and let {T n } be a family of mappings from C into E For

a subset B of C, one says that

a {T n }, B satisfies condition AKTT if

∞

n1

b {T n }, B satisfies condition∗AKTT if

∞

n1

For more information, see Aoyama et al [ 10 ].

Lemma 2.7 Aoyama et al 10 Let C be a nonempty subset of a Banach space E and let {T n } be a

sequence of mappings from C into E Let B be a subset of C with {T n }, B satisfying condition AKTT,

then there exists a mapping  T : B → E such that

Tx  lim

n → ∞ T n x, ∀x ∈ B 2.11and lim sup n → ∞ { Tz − T n z : z ∈ B}  0.

Inspired byLemma 2.7, Nilsrakoo and Saejung19 prove the following results

Lemma 2.8 Nilsrakoo and Saejung 19 Let E be a reflexive and strictly convex Banach space

whose norm is Fr´echet differentiable, let C be a nonempty subset of a Banach space E, and let {T n } be a

sequence of mappings from C into E Let B be a subset of C with {T n }, B satisfies condition∗AKTT,

T : B → E such that

n → ∞ T n x, ∀x ∈ B 2.12

Lemma 2.9 Nilsrakoo and Saejung 19 Let E be a reflexive and strictly convex Banach space

whose norm is Fr´echet differentiable, let C be a nonempty subset of a Banach space E, and let {T n } be

a sequence of mappings from C into E Suppose that for each bounded subset B of C, the ordered pair

{T n }, B satisfies either condition AKTT or condition∗AKTT Then there exists a mapping T : B →

E such that

Tx  lim

n → ∞ T n x, ∀ x ∈ C. 2.13

3 Modified Ishikawa Iterative Scheme

In this section, we establish the strong convergence theorems for finding common fixedpoints of a countable family of hemi-relatively nonexpansive mappings in a uniformlyconvex and uniformly smooth Banach space It is worth mentioning that our main theoremgeneralizes recent theorems by Su et al.12 from relatively nonexpansive mappings to amore general concept Moreover, our results also improve and extend the correspondingresults of Nilsrakoo and Saejung19 In order to prove the main result, we recall a concept

as follows An operator T in a Banach space is closed if x n → x and Tx n → y, then

Tx  y.

Theorem 3.1 Let E be a uniformly convex and uniformly smooth Banach space and let C be a

nonempty bounded closed convex subset of E Let {T n } be a sequence of hemi-relatively nonexpansive

mappings from C into itself such that∞

n0 FT n  is nonempty Assume that {a n}∞n0 and {β n}∞n0 are sequences in 0, 1 such that lim sup n → ∞ α n < 1 and lim n → ∞ β n  1 and let a sequence {x n } in C by

the following algorithm be:

then {x n } converges strongly to Π FT x0, where ΠFT is the generalized projection from C onto FT.

definition of C n1 , we see that C n1 is closed for each n ≥ 0 Now we show that C n1is convex

for any n≥ 0 Since

this implies that C n1is a convex set Next, we show that∞

n0 FT n  ⊂ C n for all n≥ 0 Indeed,

10 Fixed Point Theory and Applications

This means that, p ∈ C n1 for all n ≥ 0 Consequently, the sequence {x n} is well defined

Moreover, since x n ΠC n x0and x n1 ∈ C n1 ⊂ C n, we get

for all n ≥ 0 Therefore, {φx n , x0} is nondecreasing.

By the definition of x nandLemma 2.5, we have

for all p∈∞n0 FT n  ⊂ C n Thus,{φx n , x0} is a bounded sequence Moreover, by 2.5, weknow that{x n} is bounded So, limn → ∞ φx n , x0 exists Again, byLemma 2.5, we have

φx n1 , x n   φx n1 , Π C n x0

≤ φx n1 , x0 − φΠC n x0, x0

 φx n1 , x0 − φxn , x0,

3.8

for all n ≥ 0 Thus, φx n1 , x n  → 0 as n → ∞.

Next, we show that{x n} is a Cauchy sequence UsingLemma 2.2, for m, n such that

m > n, we have

gx m − x n  ≤ φx m , x n  ≤ φx m , x0 − φxn , x0, 3.9

where g : 0, ∞ → 0, ∞ is a continuous stricly increasing and convex function with g0 

0 Then the properties of the function g yield that {x n} is a Cauchy sequence Thus, we cansay that{x n } converges strongly to p for some point p in C However, since lim n → ∞ β n  1and{x n} is bounded, we obtain

lim

n → ∞ Jx n1 − Jy n lim

n → ∞ Jx n1 − Jx n  lim

n → ∞ Jx n1 − Jz n   0. 3.14

12 Fixed Point Theory and Applications

For each n ∈ N ∪ {0}, we observe that

Jx n − JTx n  ≤ Jx n − JT n x n   JT n x n − JTx n

≤ Jx n − JT n x n   sup{JT n z − JTz : z ∈ {x n }} −→ 0. 3.22

Case 2 {T n }, {x n} satisfies condition AKTT ApplyLemma 2.7to get

lim

n → ∞ x n − Tx n   0. 3.25Since T is closed and x n → p, we have p ∈ FT Moreover, by 3.7, we obtain

for all p ∈ FT Therefore, p  Π FT x0 This completes the proof.

Since every relatively nonexpansive mapping is a hemi-relatively nonexpansivemapping, we obtain the following result for a countable family of relatively nonexpansivemappings of modified Ishikawa iterative process

Corollary 3.2 Let E be a uniformly convex and uniformly smooth Banach space and let C be a

n0 FT n  is nonempty Assume that {α n}∞n0 and {β n}∞n0

are sequences in 0, 1 such that lim sup n → ∞ α n < 1 and lim n → ∞ β n  1 and let a sequence {x n } in

C be defined by the following algorithm:

subset B of C, the ordered pair {T n }, B satisfies either condition AKTT or condition∗AKTT Let T be the mapping from C into itself defined by Tv  lim n → ∞ T n v for all v ∈ C and suppose that T is closed and FT ∞n0 FT n  If T n is uniformly continuous for all n ∈ N, then {x n } converges strongly to

ΠFT x0, whereΠFT is the generalized projection from C onto FT.

3 Modified Ishikawa Iterative Scheme

In this section, we establish the strong convergence. .. class="page_container" data-page ="8 ">

8 Fixed Point Theory and Applications

Let C be a subset of a Banach space E and let {T n } be a family of mappings from C into E For< /i>... establish the strong convergence theorems for finding common fixedpoints of a countable family of hemi-relatively nonexpansive mappings in a uniformlyconvex and uniformly smooth Banach space It