Weak Convergence of Ishikawa Iterates for Nonexpansive Maps

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Weak Convergence of Ishikawa Iterates for

Nonexpansive Maps

Abdul Rahim Khan

1

_{, Hafiz Fukhar-ud-din}

2 ∗

Abstract_{—We establish weak convergence of the}

Ishikawa iterates of nonexpansive maps under a vari-ety of new control conditions and without employing any of the properties: (i) Opial’s property (ii) Fr´echet differentiable norm (iii) Kadec-Klee property.

Keywords: uniformly convex Banach space, nonexpan-sive map, weak convergence, Ishikawa iterates, Kadec-Klee property

1 Introduction

LetE be a real Banach space and let C be a nonempty closed convex subset of E. A map T : C → C is non-expansive if kT x−T yk ≤ kx−yk for all x, y ∈ C. We denote byF(T) the set of fixed points ofT.

Numerous problems in mathematics and physical sciences can be formulated in a fixed point problem for noexpan-sive maps.In view of practical importance of these prob-lems, methods of finding fixed points of nonexpansive maps continue to be a flourishing topic in fixed point the-ory. Iterative construction of fixed points of these maps is a fascinating field of research (see, [1, 4, 7, 9, 10]). In 1967, Browder [1] studied the iterative construction of fixed points of nonexpansive maps on closed and convex subsets of a Hilbert space (see also [3]).

For a map T of C into itself, we consider the Ishikawa iteration scheme: x1∈C,and

(

xn+1=αnT yn+ (1−αn)xn,

yn =βnT xn+ (1−βn)xn, n≥1.

(1.1)

where{αn}and{βn} are sequences in [0,1].

Set

δ(r) = inf

½

1−1

2kx+yk:kxk ≤1,kyk ≤1,kx−yk ≥r

¾

.

A Banach space E is uniformly convex if for each r ∈

(0,2],the numberδ(r)>0.

∗_{The author A. R. Khan is grateful to King Fahd}

Univer-sity of Petroleum & Minerals and SABIC for supporting RE-SEARCH PROJECT SB10012. Address1_{: Department of}

Mathe-matics and Statistics, King Fahd University of Petroleum and Min-erals, Dahran, 31261, Saudia Arabia Email: arahim@kfupm.edu.sa. Address2_: _{Department of Mathematics, The Islamia}

Univer-sity of Bahawalpur, Bahawalpur, 63100, Pakistan. Email: hfdin@yahoo.com

For a sequence, the symbol → (resp.⇀) denotes norm (resp. weak) convergence. The space E is said to sat-isfy : (i) Opial’s property [8] if for any sequence {xn}

in E, xn ⇀ x implies that lim supn→∞kxn−xk < lim supn→∞kxn−yk for all y ∈ E with y 6= x; (ii) Kadec-Klee property [6] if for every sequence{xn} inE,

xn ⇀ x and kxnk → kxk together imply xn → x as

n→ ∞.

LetS ={x∈E :kxk= 1}and letE∗

be the dual ofE, that is, the space of all continuous linear functionalsf on E.The norm ofE is : (iii)Gˆateaux differentiable [10] if

lim

t→0

kx+tyk − kxk

t

exists for each x and y in S and (iv) Fr´echet differen-tiable [10] if for each xin S, the above limit is attained uniformly fory∈S.

A mapping T : C → E is demiclosed at y ∈ E if for each sequence {xn} in C and each x∈E, xn ⇀ x and

T xn→y imply thatx∈C andT x=y.

One of the fundamental and celebrated results in the theory of nonexpansive maps is Browder’s demiclosed principle[1] which states that if C is a nonempty closed convex subset of a uniformly convex Banach space E, then for every nonexpansive map T : C → E, I−T is demiclosed at zero, i.e., for any {xn} ⊂ C, xn ⇀ x and

(I−T)xn→0 imply thatT x=x.

The above stated demiclosed principle has played an im-portant role in the study of approximation of fixed points of nonexpansive maps through weak(strong) convergence of certain iterates.

A suitable varient of Lemma 3.1 due to G´ornicki [5] for nonexpansive maps in uniformly convex Banach space is as follows:

Lemma 1.1. Let Cbe a nonempty bounded closed convex

subset of a uniformly convex Banach space Eand let T be a nonexpansive map of C into itself. If xn⇀ x({xn} ⊂

C, x∈C),then there exists strictly increasing convex map g: [0,∞)→[0,∞)with g(0) = 0 such that

g(kx−T xk)≤lim inf

n→∞ kxn−T xnk.

Note that Browder’s demiclosed principle is a simple con-sequence of Lemma 1.1.

Proceedings of the World Congress on Engineering and Computer Science 2010 Vol II WCECS 2010, October 20-22, 2010, San Francisco, USA

ISBN: 978-988-18210-0-3

ISSN: 2078-0958 (Print); ISSN: 2078-0966 (Online)

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Tan and Xu[10] and Takahashi and Tamura[9], respec-tively, proved the following interesting results.

Theorem A. Let C be a nonempty bounded closed

con-vex subset of a uniformly concon-vex Banach space E which satisfies Opial’s condition or whose norm is Fr´echet dif-ferentiable and let T be a nonexpansive map of C into itself. Then the sequence {xn} given by (1.1) converges

weakly to a fixed point of T provided the following con-dition is satisfied:

(C1) P∞

n=1αn(1−αn) =∞,P

∞

n=1βn(1−αn)<∞ and

lim supn→∞βn<1.

Theorem B. Let C be a nonempty closed convex

sub-set of a uniformly convex Banach space E which satisfies Opial’s condition or whose norm is Fr´echet differentiable and let T be a nonexpansive map of C into itself. Sup-pose that {xn} in (1.1) satisfies the condition:

(C2) αn ∈ [a,1] and βn ∈ [a, b] or αn ∈ [a, b] and

βn∈[0, b]for some a, b∈[0,1].

Then{xn} converges weakly to a fixed point ofT.

Note that Tan and Xu’s result is applicable to the case: αn= 1−1/nandβn= 1/nfor alln≥1,while Takahashi

and Tamura’s result is applicable to the case: αn=βn =

1/2 for all n ≥ 1. Moreover, in both the results, the demiclosed principle based on strong convegence of the approximate sequence{xn−T xn}to 0 has been utilized.

Using Lemma 1.1, we establish weak convergence of the Ishikawa iterates of nonexpansive maps under a variety of new parametric control conditions and without using any of the properties: (i) Opial’s property (ii) Fr´echet differentiable norm (iii) Kadec-Klee property.

In the sequel, we need the following lemmas.

Lemma 1.2 [11, Theorem 2]. Let r > 0 be a fixed real

number. Then a Banach space E is uniformly convex if and only if there is a continuous strictly increasing convex map g : [0,∞)→[0,∞)with g(0) = 0 such that for all x, y∈Br[0] ={x∈E:kxk ≤r},

kλx+ (1−λ)yk2≤λkxk2+(1−λ)kyk2−λ(1−λ)g(kx−yk)

for all λ∈[0,1].

Lemma 1.3 [12, Lemma 2.2]. Let g : [0,∞) → [0,∞)

with g(0) = 0 be a strictly increasing map. If a se-quence {xn} in [0,∞) satisfies limn→∞g(xn) = 0, then limn→∞xn= 0.

2 Weak Convergence

We establish a pair of lemmas for the development of our convergence result.

Lemma 2.1. LetCbe a nonempty closed convex subset

of a uniformly convex Banach spaceE.LetT :C→Cbe nonexpansive map with at least one fixed point.Suppose

{xn} is given by (1.1). Then limn→∞kxn−pkexists for eachp∈F(T).

Proof. For anyp∈F(T),utilizing (1.1), we have

kxn+1−pk ≤ kαn(T yn−p) + (1−αn)(xn−p)k

≤ αnkT yn−pk+ (1−αn)kxn−pk

≤ αnkβn(T xn−p) + (1−βn)(xn−p)k

+(1−αn)kxn−pk

≤ (αnβn+αn(1−βn) + 1−αn)kxn−pk

= kxn−pk.

This proves that {kxn−pk} is a non-increasing and

bounded sequence and hence limn→∞kxn−pkexists.

Lemma 2.2. LetCbe a nonempty closed convex subset

of a uniformly convex Banach space E and let T be a nonexpansive map ofCinto itself with at least one fixed point. Let{αn} and{βn}be sequences in [0,1] and

sat-isfy one of the following three sets of conditions: (C3) :P∞

n=1αn(1−αn) =∞, lim supn→∞βn <1; (C4) :P∞

n=1βn(1−βn) =∞, lim infn→∞αn>0;

(C5) : 0≤αn ≤b <1,P

∞

n=1αn=∞, βn→0 asn→ ∞.

Then lim infn→∞kxn−T xnk= 0.

Proof. Letp∈F(T). With the help of Lemma 1.2 and

the scheme (1.1), we have:

kxn+1−pk 2

≤ kαn(T yn−p) + (1−αn)(xn−p)k

2

≤ αnkT yn−pk

2

+ (1−αn)kxn−pk

2

−αn(1−αn)g(kxn−T ynk)

≤ αnkyn−pk

2

+ (1−αn)kxn−pk

2

≤ αn[βnkxn−pk

2

+ (1−βn)kxn−pk

2

−βn(1−βn)g(kxn−T xnk)]

+(1−αn)kxn−pk

2

≤ kxn−pk

2

−αnβn(1−βn)g(kxn−T xnk)

From the above estimate, we have the following two im-portant inequalities:

αn(1−αn)g(kxn−T ynk)≤ kxn−pk

2

− kxn+1−pk 2

(2.1) and

αnβn(1−βn)g(kxn−T xnk)≤ kxn−pk

2

− kxn+1−pk 2

. (2.2) Case I:αn andβn satisfy (C3).

Letm≥1. Then from the inequality (2.1), we have

Xm

n=1αn(1−αn)g(kxn−T ynk)≤

kx1−pk 2

− kxm+1−pk 2

<∞ Proceedings of the World Congress on Engineering and Computer Science 2010 Vol II

WCECS 2010, October 20-22, 2010, San Francisco, USA

ISBN: 978-988-18210-0-3

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.

Whenm→ ∞in the above inequality, we have

P∞

n=1αn(1−αn)g(kxn−T ynk)<∞.Since

P∞

n=1αn(1−

αn) =∞,therefore we have lim infn→∞g(kxn−T ynk) =

0.

From Lemma 1.3, we get that lim infn→∞kxn−T ynk=

0.

Since

kxn−T xnk ≤ kxn−T ynk+kT xn−T ynk

≤ kxn−T ynk+kxn−ynk

= kxn−T ynk+kxn−ynk

= kxn−T ynk+βnkxn−T xnk,

so we have

(1−βn)kxn−T xnk ≤ kxn−T ynk.

Therefore, from lim infn→∞kxn−T ynk = 0 and lim supn→∞βn<1, we deduce that

lim inf

n→∞

kxn−T xnk= 0.

Case II:αn andβn satisfy (C4).

From the inequality (2.2), we have

Xm

n=1αnβn(1−βn)g(kxn−T xnk) ≤ kx1−pk

2

− kxm+1−pk 2

<∞.

Letting m → ∞, we get that P∞

n=1αnβn(1 −

βn)g(kxn−T xnk)<∞.

Since P∞

n=1βn(1 − βn) = ∞, therefore

lim infn→∞αng(kxn−T xnk) = 0.

That is, (lim infn→∞αn) (lim infn→∞g(kxn−T xnk)) = 0.

As lim infn→∞αn > 0, therefore

lim infn→∞g(kxn−T xnk) = 0.

By Lemma 1.3, we get that

lim inf

n→∞ kxn−T xnk= 0. Case III:αn andβn satisfy (C5).

Using the condition ”0 ≤ αn ≤ b < 1” in the

inequal-ity(2.1), we have

αn(1−b)g(kxn−T ynk)≤ kxn−pk

2

− kxn+1−pk 2

.

Summing the first m terms of the above inequality, we have that

(1−b)Xm

n=1αng(kxn−T ynk) ≤ kx1−pk

2

− kxm+1−pk 2

<∞.

Whenm→ ∞,we get (1−b)P∞

n=1αng(kxn−T ynk)< ∞. Since P∞

n=1αn = ∞, therefore

lim infn→∞g(kxn−T ynk) = 0. Again from Lemma

1.3, we get that lim infn→∞kxn−T ynk= 0.

Since kxn−T xnk ≤ kxn−T ynk +βnkxn−T xnk ≤

kxn−T ynk+βnM for someM >0 andβn→0,therefore

we get

lim inf

n→∞

kxn−T xnk= 0. (2.3)

Now we prove our convergence result.

Theorem 2.3: LetCbe a nonempty closed convex

sub-set of a uniformly convex Banach space E and let T be a nonexpansive map of C into itself with at least one fixed point. Let {αn} and {βn} be sequences in [0,1]

and satisfy one of the three sets of conditions of Lemma 2.2. Then the sequence {xn} defined by (1.1), converges

weakly to a fixed point ofT.

Proof. Let ωw(xn), the weak ω-limit set of {xn}, be

given by:

ωw(xn) ={y∈E:xnk⇀ y for{xnk} ⊆ {xn}}.

Since limn→∞kxn−pk exists for each p∈F(T),

there-fore the sequence {xn} is bounded.Without any loss of

generality, we can suppose that C is bounded. This gives that there exists a subsequence {xni} of {xn}

such that xni ⇀ p ∈ ωw(xn) as i → ∞ and vice

versa. This shows that ωw(xn) 6= φ and so by

Lemma 1.1, g(kp−T pk) ≤ lim infk→∞kxnk−T xnkk.

But lim infk→∞kxnk−T xnkk = 0 by Lemma 2.2. That

is, g(kp−T pk) = 0. By the properties ofg,we get that

kp−T pk = 0. That is p ∈ F(T) and hence ωw(xn) ⊂

F(T). Next, we follow Chang et. al[2]to prove the weak convergence of the sequence. For any p∈ωw(xn), there

exists a subsequence{xnj} of{xn}such that

xnj ⇀ pasj→ ∞. (2.4)

Hence from (2.4) and continuity ofT, it follows that

T xni ⇀ p. (2.5)

Now from (1.1), (2.4) and (2.5), we get that

yni = (1−βni)xni+βniT xni⇀ p. (2.6)

From (2.4), (2.6) and the continuity ofT, we have that

T yni = (T yni−xni) +xni ⇀ p. (2.7)

Again from (1.1) and (2.7), we conclude that

xni+1= (1−αni)xni+αniT yni⇀ p

Continuing in this way, by induction, we can prove that, for anym>0,

xni+m⇀ p. Proceedings of the World Congress on Engineering and Computer Science 2010 Vol II

WCECS 2010, October 20-22, 2010, San Francisco, USA

ISBN: 978-988-18210-0-3

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By induction, we get that S∞

m=0{xnj+m}

con-verges weakly to p as j → ∞; in fact {xn}∞n=n1 =

S∞

m=0{xnj+m}

∞

j=1 gives thatxn⇀ pasn→ ∞.

Remark 2.4. A comparison of Theorem 2.3 with

Theo-rem A reveals that the assumptionP∞

n=1βn(1−αn)<∞

in Theorem A is superflous. Also, it is obvious that the assumption (C2) in Theorem B implies (C3) −(C4). Also Theorem A and Theorem B are established under the Opial’s property or Fr´echet differentiable norm. Moreover, Kadec-Klee property is required in Theorem 4.1[4] to establish the weak convergence of the Ishikawa iterates. The weak convergence theorem presented in this paper is valid in any uniformly convex Banach space.

References

[1] F. E. Browder,Convergence of approximates to fixed points of nonexpansive nonlinear mappings in Ba-nach spaces, Arch. Rational Mech. Anal.24 (1967), 82-90.

[2] S.S. Chang, Y.J. Cho, H. Zhou, Demi-closed prin-cipal and weak convergence problems for asymptoti-cally nonexpansive mappings,J. Korean Math. Soc. 38 (2001), No. 6, 1245–1260.

[3] J. B. Diaz, F. T. Metcalf,On the structure of the set of subsequential limit points of successive approxima-tions, Bull. Amer. Math. Soc.73(1967), 516-519.

[4] H. Fukhar-ud-din, A. R. Khan,Approximating com-mon fixed points of asymptotically nonexpansive maps in uniformly convex Banach spaces, Comput. Math. Appl. 53 (2007), 1349-1360.

[5] J. Gornicki,Nonlinear ergodic theorems for asymp-totically non-expansive mappings in Banach spaces satisfying opial’s condition, J. Math. Anal. Appl. 161(1991), 440–446.

[6] W. Kaczor,Weak convergence of almost orbits of as-ymptotically nonexpansive commutative semigroups, J. Math. Anal. Appl. 272(2002), 565-574.

[7] A. R. Khan, N. Hussain,Iterative approximation of fixed points of nonexpansive maps, Sci. Math. Jpn. 54(2001), 503-511.

[8] Z. Opial, Weak convergence of the sequence of suc-cessive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591–597.

[9] W. Takahashi, T. Tamura,Convergence theorems for a pair of nonexpansive mappings, J. Convex Anal. 5 (1995), 45–58.

[10] K.K.Tan, H.K. Xu, Approximatig fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178 (1993), 301–308.

[11] H. K. Xu, Inequalities in Banach spaces with appli-cations, Nonlinear Anal. TMA 16(12) (1991), 1127– 1138.

[12] H.Y. Zhou, G. T. Guo, H. J. Hwang, Y. J. Cho, On the iterative methods for nonlinear operator equations in Banach spaces, PanAmer Math. J., 14(4)(2004), 61-68.

Proceedings of the World Congress on Engineering and Computer Science 2010 Vol II WCECS 2010, October 20-22, 2010, San Francisco, USA

ISBN: 978-988-18210-0-3