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Volume 2008, Article ID 167535, 14 pagesdoi:10.1155/2008/167535 Research Article Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces Jong Soo Jung Depar

Volume 2008, Article ID 167535, 14 pages

doi:10.1155/2008/167535

Research Article

Convergence on Composite Iterative Schemes for Nonexpansive Mappings in Banach Spaces

Jong Soo Jung

Department of Mathematics, Dong-A University, Busan 604-714, South Korea

Correspondence should be addressed to Jong Soo Jung, jungjs@mail.donga.ac.kr

Received 13 January 2008; Revised 5 April 2008; Accepted 3 May 2008

Recommended by Mohammed Khamsi

Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable norm Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings Let C

be a nonempty closed convex subset of E, f : C → C a contractive mapping or a weakly contractive

mapping, and T : C → C nonexpansive mapping with the fixed point set FT / ∅ Let {x n} be

generated by a new composite iterative scheme: y n  λ n fx n 1−λ n Tx n , x n1  1−β n y n β n Ty n,

n ≥ 0 It is proved that {x n } converges strongly to a point in FT, which is a solution of

certain variational inequality provided that the sequence{λ n } ⊂ 0, 1 satisfies lim n→∞ λ n  0 and

∞

n1 λ n  ∞, {β n } ⊂ 0, a for some 0 < a < 1 and the sequence {x n} is asymptotically regular Copyright q 2008 Jong Soo Jung 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 E be a real Banach space and let C be a nonempty closed convex subset of E Recall that a mapping f : C → C is a contraction on C if there exists a constant k ∈ 0, 1 such that fx −

above inequality That is,ΣC  {f : C → C | f is a contraction with constant k} Note that each

Now let T : C → C be a nonexpansive mapping recall that a mapping T : C → C is

is, FT  {x ∈ C : x  Tx}.

We consider the iterative scheme: for T nonexpansive mapping, f ∈ Σ C and λ n ∈ 0, 1,



As a special case of1.1, the following iterative scheme:

where u, z0 ∈ C are arbitrary but fixed, has been investigated by many authors; see, for

example, Cho et al.1 , Halpern 2 , Lions 3 , Reich 4,5 , Shioji and Takahashi 6 , Wittmann

7 , and Xu 8 The authors above showed that the sequence {z n} generated by 1.2 converges

strongly to a point in the fixed point set FT under appropriate conditions on {λ n} in either Hilbert spaces or certain Banach spaces Recently, many authors also considered the iterative scheme 1.2 for finite or countable families of nonexpansive mappings {T i}i∈{1,2, ,r or ∞}; see, for instance,9 14

The viscosity approximation method of selecting a particular fixed point of a given nonexpansive mapping in a Hilbert space was proposed by Moudafi15 see 16 for finding hierarchically a fixed point In 2004, Xu 17 extended Theorem 2.2 of Moudafi 15 for the iterative scheme1.1 to a Banach space setting using the following conditions on {λ n}:

H1 limn→∞ λ n 0;∞n0 λ n  ∞ or, equivalently,∞n0 1 − λ n  0;

H2∞n0 |λ n1 − λ n | < ∞ or lim n→∞ λ n /λ n1  1

We also refer to 18–23 for the iterative scheme 1.1 for finite of countable families of nonexpansive mappings {T i}i∈{1,2, ,r or ∞} For the iterative scheme 1.1 with generalized contractive mappings instead of contractions, see22,24 We can refer to 25 for the general iteration method for finding a zero of accretive operator

Recently, Kim and Xu 26 provided a simpler modification of Mann iterative scheme

1.3 in a uniformly smooth Banach space as follows:

x0 x ∈ C,

1− α n



y n ,

1.3

where u ∈ C is an arbitrary but fixed element, and {α n } and {β n } are two sequences in 0, 1.

They proved that {x n} generated by 1.3 converges to a fixed point of T under the control

conditions:

i limn→∞ α n  0, lim n→∞ β n 0;

ii∞n0 α n  ∞, or equivalently, ∞n0 1 − α n   0,∞n0 β n ∞;

iii∞n0 |α n1 − α n | < ∞,∞n0 |β n1 − β n | < ∞.

In this paper, motivated by the above-mentioned results, as the viscosity approximation

method, we consider a new composite iterative scheme for nonexpansive mapping T:

x0 x ∈ C,



1− λ nTx n ,

IS

where {β n }, {λ n } ⊂ 0, 1 First, we prove the strong convergence of the sequence {x n} generated by IS under the suitable conditions on the control parameters {β n } and {λ n} and the asymptotic regularity on {x n} in reflexive Banach space with a uniformly Gˆateaux differentiable norm together with the assumption that every weakly compact convex subset of

E has the fixed point property for nonexpansive mappings Moreover, we show that the strong

limit is a solution of certain variational inequality Next, we study the viscosity approximation with the weakly contractive mapping to a fixed point of nonexpansive mapping in the same Banach space The main results improve and complement the corresponding results of 1

8,15,17 In particular, if β n  0, for all n ≥ 0, then IS reduces to 1.1 We point out that the iterative schemeIS is a new one for finding a fixed point of T.

2 Preliminaries and lemmas

Let E be a real Banach space with norm · and let E∗be its dual The value of f ∈ E∗at x ∈ E

will be denoted by n } is a sequence in E, then x n → x resp., x n  x will

denote strongresp., weak convergence of the sequence {x n } to x.

Thenormalized duality mapping J from E into the family of nonempty by Hahn-Banach

theorem weak-star compact subsets of its dual E∗is defined by

2.1

for each x ∈ E 27

The norm of E is said to be Gˆateaux differentiable and E is said to be smooth if

lim

t→0

x  ty − x

exists for each x, y in its unit sphere U  {x ∈ E : x  1} The norm is said to be uniformly

Gˆateaux differentiable if for y ∈ U, the limit is attained uniformly for x ∈ U The space E is said

to have a uniformly Fr´echet di fferentiable norm and E is said to be uniformly smooth if the limit

in2.2 is attained uniformly for x, y ∈ U × U It is known that E is smooth if and only if each duality mapping J is single-valued It is also well known that if E has a uniformly Gˆateaux

differentiable norm, J is uniformly norm to weak continuous on each bounded subset of E

27

Let C be a nonempty closed convex subset of E C is said to have the fixed point property

for nonexpansive mappings if every nonexpansive mapping of a bounded closed convex

subset D of C has a fixed point in D.

Let D be a subset of C Then, a mapping Q : C → D is said to be a retraction from C onto

D if Qx  x for all x ∈ D A retraction Q : C → D is said to be sunny if QQx  tx − Qx  Qx

for all x ∈ C and t ≥ 0 with Qxtx−Qx ∈ C A subset D of C is said to be a sunny nonexpansive

retract of C if there exists a sunny nonexpansive retraction of C onto D In a smooth Banach

space E, it is well known 28, page 48 that Q is a sunny nonexpansive retraction from C onto

D if and only if the following condition holds:



We need the following lemmas for the proof of our main results.Lemma 2.1was also given by Jung and Morales 29 andLemma 2.2is essentially Lemma 2 of Liu30 also see

8 .

Lemma 2.1 Let X be a real Banach space and let J be the duality mapping Then, for any given x, y ∈

X, one has

x  y2≤ x2 2y, jx  y

2.4

for all jx  y ∈ Jx  y.

Lemma 2.2 Let {s n } be a sequence of nonnegative real numbers satisfying

s n1 ≤ 1 − α n s n  α n γ n  δ n , n ≥ 0, 2.5

i {α n } ⊂ 0, 1 and∞

n0 α n  ∞ or, equivalently,∞

n0 1 − α n   0,

ii lim supn→∞ γ n ≤ 0 or∞n1 α n γ n < ∞,

iii δ n ≥ 0 n ≥ 0,∞n0 δ n < ∞.

Recall that a mapping A : C → C is said to be weakly contractive if

Ax − Ay ≤ x − y − ψx − y, ∀x, y ∈ C, 2.6

where ψ : 0, ∞ → 0, ∞ is a continuous and strictly increasing function such that ψ is

positive on 0, ∞ and ψ0  0 As a special case, if ψt  1 − kt for t ∈ 0, ∞, where

k ∈ 0, 1, then the weakly contractive mapping A is a contraction with constant k Rhoades

31 obtained the following result for weakly contractive mapping

Lemma 2.3 see 31, Theorem 2  Let X, d be a complete metric space, and A a weakly contractive

to p.

The following lemma was given in32,33

Lemma 2.4 Let {s n } and {γ n } be two sequences of nonnegative real numbers and {λ n } a sequence of

positive numbers satisfying the conditions

i∞n0 λ n  ∞ or, equivalently,∞n0 1 − λ n   0,

ii limn→∞ γ n /λ n   0.

Let the recursive inequality

s n

be given where ψt is a continuous and strict increasing function on 0, ∞ with ψ0  0 Then,

limn→∞ s n  0.

Finally, the sequence{x n } in E is said to be asymptotically regular if

lim

3 Main results

First, using the asymptotic regularity, we study a strong convergence theorem for a composite iterative scheme for the nonexpansive mapping with the contractive mapping

For T : C → C nonexpansive and so for any t ∈ 0, 1 and f ∈ Σ C , tf  1 − tT : C → C

defines a strict contraction mapping Thus, by the Banach contraction mapping principle, there

exists a unique fixed point x f t satisfying

x t f  tfx t f

 1 − tTx f

For simplicity, we will write x t for x f t provided no confusion occurs

In 2006, the following result was given by Jung18 see also Xu 17 for the result in uniformly smooth Banach spaces

Theorem J see Jung 18  Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable

norm Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings Let C be a nonempty closed convex subset of E and T nonexpansive mapping

Qf : lim

then Qf solves a variational inequality



I − fQf

≤ 0, f ∈ Σ C , p ∈ FT. 3.2



Hence by2.3, Q reduces to the sunny nonexpansive retraction from C to FT Namely, FT

is a sunny nonexpansive retraction of C.

Using Theorem J and the asymptotic regularity on the sequence {x n}, we have the following result

Theorem 3.2 Let E be a reflexive Banach space with a uniformly Gˆateaux differentiable norm Suppose

that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings Let C be a nonempty closed convex subset of E and T nonexpansive mappings from C into itself with

B1 β n ∈ 0, a for some 0 < a < 1 for all n ≥ 0,

C1 limn→∞ λ n 0;∞

n0 λ n  ∞.

x0 x ∈ C,



1− λ nTx n ,

IS

If {x n } is asymptotically regular, then {x n } converges strongly to Qf ∈ FT, where Qf is the

unique solution of the variational inequality



I − fQf

≤ 0, f ∈ Σ C , p ∈ FT. 3.4

Proof We notice that by Theorem J, there exists a solution Qf of a variational inequality



I − fQf

≤ 0, f ∈ Σ C , p ∈ FT. 3.5

Namely, Qf  lim t→0x t , where x tis defined byR We will show that x n → Qf.

We proceed with the following steps

and so{x n }, {y n }, {fx n }, {Tx n }, and {Ty n} are bounded

Indeed, let z ∈ FT Then, we have



− z1− λ nTx n − z

≤ λ n f

≤ λ n f



1−1− kλ n x n − z n fz − z

≤ max n − z , 1

,

≤1− β n y n − z n y n − z

.

3.6

Using an induction, we obtain

3.7

for all n ≥ 0 Hence, {x n } is bounded, and so are {y n }, {Tx n }, {Ty n }, and {fx n} Moreover, it follows from conditionC1 that



B1

≤ β n Ty n − Tx n n − y n

≤ a y n − x n n − y n

≤ a y n − x n1 n1 − x n n − y n

3.9

which implies that

x n1 − y n 1− a x n1 − x n n − y n 3.10

So, by asymptotic regularity of{x n} and 3.8, we have x n1 − y n → 0, and also

subsequence{y n j } of {y n} be such that

lim sup

n→∞



 lim

j→∞



3.13

and y n j  p for some p ∈ E From Step3, it follows that limj→∞ y n j − Ty n j  0

Now let Qf  lim t→0x t , where x t  tfx t   1 − tTx t Then, we can write

x t



− y n j 1 − tTx t − y n j. 3.14 Putting

a j t  1 − t2

Ty n j − y n j 2 x t − y n j n j − y n j 3.15

by Step3and usingLemma 2.1, we obtain

x t − y n j2≤ 1 − t2Tx t − y n j2 2tf

x t



− y n j , J

≤ 1 − t2

Tx t − Ty n j   Ty n j − y n j2 2tf

x t



− x t , J

x t − y n j  2tx t − y n j2

≤ 1 − t2x t − y n j2 a j t  2tf

x t



− x t , J



 2tx t − y n j2.

3.16 The last inequality implies





2x t − y n j2 1

It follows that

lim sup

j→∞







≤ t

where M > 0 is a constant such that M ≥ x t − y n2 for all n ≥ 0 and t ∈ 0, 1 Taking the lim sup as t → 0 in 3.18 and noticing the fact that the two limits are interchangeable due to

the fact that J is uniformly continuous on bounded subsets of E from the strong topology of E

to the weak∗topology of E∗, we have

lim sup

j→∞





Indeed, letting t → 0, from 3.18 we have

lim sup

t→0

lim sup

j→∞





x t − y n j ≤ 0. 3.20

So, for any ε > 0, there exists a positive number δ1such that for any t ∈ 0, δ1,

lim sup

j→∞



x t − fx t

Moreover, since x t → Qf as t → 0, the set {x t − y n j } is bounded and the duality mapping J

is norm-to-weak∗uniformly continuous on bounded subset of E, there exists δ2 > 0 such that,

for any t ∈ 0, δ2,



−x t − fx t





− Jx t − y n j



−x t − fx t



−x t − fx t x t − y n j < ε

2.

3.22

Choose δ  min{δ1, δ2}, we have for all t ∈ 0, δ and j ∈ N,





<





ε

which implies that

lim sup

j→∞





≤ lim sup

j→∞







 ε

2. 3.24 Since lim supj→∞ t − fx t , Jx t − y n j  ≤ ε/2, we have

lim sup

j→∞





Since ε is arbitrary, we obtain that

lim sup

j→∞





x n1 − Qf n − Qf nf



− Qf1− λ nTx n − Qf 3.27

ApplyingLemma 2.1, we obtain

x n1 − Qf 2≤ y n − Qf 2

≤1− λ n2

Tx n − Qf 2 2λ nf



− Qf, Jy n − Qf

≤1− λ n

2

x n − Qf2 2λ n





− fQf

y n − Qf

 2λ nf

− Qf, Jy n − Qf

≤1− λ n2

x n − Qf 2 2kλ n x n − Qf y n − Qf

 2λ nf

− Qf, Jy n − Qf

≤1− λ n

2

x n − Qf 2 2kλ n x n − Qf 2

 2λ nf

− Qf, Jy n − Qf .

3.28

It then follows that

x n1 − Qf 2≤1− 21 − kλ n  λ2

n x n − Qf 2 2λ nQf − f



≤1− 2 − kλ n x n − Qf 2 λ2

n M2 2λ n





,

3.29

where M  sup n≥0 x n − Qf Put

α n  21 − kλ n ,

21 − kM2

1

1− k





From the conditionC1 and Step4, it follows that α n → 0,∞n0 α n ∞, and lim supn→∞ γ n≤ 0 Since3.29 reduces to

x n1 − Qf2≤ 1 − α n x n − Qf2 α n γ n , 3.31

from Lemma 2.2with δ n  0, we conclude that limn→∞ x n − Qf  0 This completes the

proof

Corollary 3.3 Let E be a uniformly smooth Banach space Let C, T, f, {β n }, {λ n }, f, x0, and {x n } be

Proof Since E is a uniformly smooth Banach space, E is reflexive and the norm is uniformly

Gˆateaux differentiable norm and its every nonempty weakly compact convex subset of E has the fixed point property for nonexpansive mappings Thus, the conclusion of Corollary 3.3

follows fromTheorem 3.2immediately

Remark 3.4 1 If {β n } and {λ n} inTheorem 3.2satisfy the conditions

B2∞n0 |β n1 − β n | < ∞,

C1 limn→∞ λ n  0,∞n0 λ n  ∞,

C2∞n0 |λ n1 − λ n | < ∞, or

C3 limn→∞ λ n /λ n1  1, or

C4 |λ n1 − λ n | ≤ ◦λ n1   σ n , ∞

then the sequence {x n} generated by IS is asymptotically regular Now, we only give the proof in case when{β n } and {λ n} satisfy the conditions B2, C1, and C4 Indeed, from IS,

we have for every n ≥ 1,



1− λ n





1− λ n−1Tx n−1 , 3.32

and so, for every n ≥ 1, we have





 λ n





− fx n−1



λ n − λ n−1





− Tx n−1

≤1− λ n x n − x n−1 n − λ n−1 n x n − x n−1

1− 1 − kλ n x n − x n−1 n − λ n−1 ,

3.33

where L  sup{fx n  − Tx n  : n ≥ 0}.

On the other hand, byIS, we also have for every n ≥ 1,

x n1− β n−1y n−1  β n−1 Ty n−1 3.34 Simple calculations show that





 β n





β n − β n−1





, 3.35 then it follows that

x n1 − x n n y n − y n−1 n y n − y n−1 n − β n−1 Ty n−1 − y n−1 3.36 Substituting3.33 into 3.36 and using the condition C4, we derive

≤1− 1 − kλ n x n − x n−1 n



 σ n−1



 M β n − β n−1 , 3.37

where M  sup{Ty n − y n  : n ≥ 0} By taking s n1  x n1 − x n , α n  1 − kλ n , α n γ n  L ◦ λ n , and δ n  Lσ n−1  M|β n − β n−1|, we have

s n1≤1− α ns n  α n γ n  δ n 3.38

follows fromTheorem 3.2immediately