The equivalence between the convergence of modified picard, modified mann, and modified ishikawa iterations

MATHEMATICAL COMPUTER MODELLING Mathematical and Computer Modelling 37 (2003) 985-991

PERGAMON

www.elsevier.nl/locate/mcm

The Equivalence between the Convergence of Modified Picard, Modified Mann, and Modified Ishikawa Iterations S.

S.

CHANG

Department of Mathematics, Sichuan University Chengdu, Sichuan 610064, P.R. China sszhang(Dmail.yahoo.com.cn Y. J. CHO Department of Mathematics, Gyeongsang National University College of Education Chinju, 660-730, Korea [email protected] J. K. KIM* Department of Mathematics, Kyungnan University Masan, Kyungnan 631-701, Korea jongkyukBkyungnam.ac.kr

(Received and accepted November

2002)

Abstract-some equivalence conditions between the convergence of modified Picard, modified Mann, and modified Ishikawa iterations for some kinds of nonlinear mappings in Banach spaces are obtained. @ 2003 Elsevier Science Ltd. All rights reserved. Keywords-Uniformly L-Lipschitzean mapping, Asymptotically expansive mapping, Banach contraction mapping, Modified Picard, Ishikawa iteration sequences.

1.

Throughout

INTRODUCTION

this paper,

we assume

AND

that

nonexpansive mapping, Nonmodified Mann, and modified

PRELIMINARIES

E is a real Banach

space, E* is the dual space of E, B is

a nonempty closed convex subset of E, D(T), R(T), F(T), and N are the domain, range, the set of fixed points of mapping T, and the set of all positive integers, respectively. We also assume duality mapping defined by that J : E -+ 2 E’ is the normalized

J(z) = U E E*, b.0 = IMIIlfll, llfll = 1141~, x E E. DEFINITION

1.1. Let T : D(T) c E --+ E be a mapping.

(i) T is said to be uniformly L-Lipschitzian,

IIT”a:- Tnyll I 41~ - YII,

where L > 0 is a positive constant, if for all x,y E D(T),

n E N.

(1.1)

*Author to whom all correspondence should be addressed. This work was supported by the Kyungnam University Research Fund 2002 0895-7177/03/$ - see front matter @ 2003 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(03)00113-4

Typeset

by &&W

986

S. S. CHANGet al.

(ii) T is said to be asymptotically nonexpansive [l], if there exists a sequence {kn} c (0, m) with limn+m k, = 1 such that

IIT"x - T"yll I knllx -

for all 2, y E D(T),

~11,

n E N;

(1.2)

(iii) T is said to be nonexpansive, if

IP

for all Z, y E D(T);

- TYII I 112- YIL

(14

(iv) T is said to be Banach contraction, if there exists a constant c E (0,l)

IlTx - TYII I+ REMARK 1.1. It is clear that (iv) +

-

~11,

for all z,y E D(T).

such that

(1.4

(iii) =+ (ii) +- (i), and that all the inclusions are proper.

That is, if T : D(T) + E is a Banach contraction mapping with a constant c E (0, l), then T is a nonexpansive mapping; if T : D(T) t

E is a nonexpansive mapping, then T is an asymptotically

nonexpansive mapping with a constant sequence (1); if T : D(T) --f E is an asymptotically nonexpansive mapping with a sequence {k,} L-Lipschitzian

with L = SUP,,~ {k,}.

pansive mapping and asymptotically

(0, oo) such that k, --$ 1, then it must be uniformly

c

In other words, Banach contraction mapping, nonexnonexpansive mappings are the special cases of uniformly

L-Lipschitzian mappings with L = c, L = 1, and L = SUP,>~ k,, respectively. DEFINITION 1.2. Let T : B -+ B be a mapping and {an} Then the following sequences {w,,},

{un},

and {xn}

and {fi,}

be two sequences in [0,1j.

are called Picard, Mann, and Ishikawa

iteration sequences, respectively (see, [2,3]),

wo E B, wn+l

U,+I

=

=

Twn,

uo ~6 (I- Q,&, + wJ’un, 20

(1.5)

Vn>O,

Vn>O,

E B,

(I- ~,)~n + d"~n, Vn>O, = (I- P&n + PnTxn, x,x+1 =

yn

(14

(1.7)

where the sequence {cu,} appeared in (1.6) and (1.7) is the same. REMARK 1.2. It is easy to see that if uo = ~0 and pn = 0, Vn 2 0, then the Ishikawa iterative sequence (1.7) is equivalent to the Mann iterative sequence (1.6). If (Y,, = 1, V n 2 0 and ua = wo, then the Mann iterative sequence (1.6) is equivalent to the Picard iterative sequence (1.5). Concerning the relations between the convergence of Picard (1.5), Mann (1.6), and Ishikawa iteration sequence (1.7) have been studied by several authors. In 1955, Krasnoselskii [4] showed that the Picard iterative sequence (1.5) for a nonexpansive mapping T even with a unique fixed point may fail to converge to the fixed point. But the following special Mann iterative sequence (2~~) defined by

uo E B,

un+l

=

fen +Tun),

n 2 0,

(1.8)

converges strongly to the fixed point of T. In 1974, Ishikawa [2] introduced the iterative sequence (1.7) and proved the following theorem.

The Equivalence between the Convergence

987

1.1. (See 121.) Let K be a compact convex subset of a Hilbert space H, T : K --f K be a Lipschitzian pseudocontractive mapping (i.e., for any 2, y E K, there exists j(z - y) E J(a: - y) such that (Tz - Ty, j(a: - y)) I 112- y/j2, where J : E + E* is the normalized duahty mapping) and z. be any point of K. If the real sequences {on} and {a} satisfy the following conditions:

THEOREM

(i) Olo,I&
o,A

= 00;

then the sequence {x,} defined by (1.7) converges strongly to a fixed point of T. Since its publication in 1974, it has remained an open question (see [5]) of whether or not the Mann iterative sequence defined by (1.6) converges under the setting of Theorem 1.1 to a fixed point of T if the mapping T is pseudocontractive and continuous (or even Lipschitzian with constant L > 1). In [6, Proposition 81, Borwein and Borwein gave an example of a Lipschitz map (which is not pseudocontractive) with a unique fixed point for which the Mann iterative sequence fails to converge; and in [5], Hicks and Kubicek gave an example of a discontinuous pseudocontraction with a unique fixed point for which the Mann iteration does not always converge. In [7], Liu proved that if K is a compact convex subset of a Hilbert space and T is a continuous pseudocontractive map with a finite number of fixed points, then the Ishikawa iteration sequence defined by (1.7) converges to a fixed point of T. Recently, the above open question is resolved by Chidume and Mutangadura [8] by constructing an example of a Lipschitz pseudocontraction with a unique fixed point for which every nontrivial Mann sequence fails to converge. DEFINITION

1.3. Let T : B + B be a mapping and {a,} and {p,} be two sequences in [&I].

Then the following sequences

{wn},

{u%}, and {x,}

are called modified Picard, modified Mann,

and modified Ishikawa iteration sequences, respectively,

(see, [S-11])

wo E B, wn+l

=

Tnwn,

w

Vn>O,

uo E B, &+I

= (1 -o&n

+ Q,T%,,

(1.10)

Vn>O,

B, (I- Q&, + wJ'~Y,, x0 E

xn+l

=

in = (1 -P&n

(1.11)

Vn20,

+ PJ’%,

where the sequences {on} and {/3,} are in [O,l] and the sequence and (1 .ll) is the same.

{an}

appeared

in (1 .lO)

The purpose of this paper is to study the equivalence between the convergence of the modified Picard, modified Mann, and modified Ishikawa iterative sequences for Banach contraction mappings, nonexpansive mapping, and asymptotically nonexpansive mappings in Banach spaces. In order to prove our main results, we need the following key lemma. LEMMA

1.1. (See [12, Lemma 3.11.) Let {a,},

{b,},

and {dn}

be nonnegative

real sequences

satisfying the condition a,+1

(1 - tn)an+ b + 6,

Vn L no,

integer and {tn} is a sequence d, < 00. Then a, + 0 as n + 00.

where no is some nonnegative b, = o(t,,) and CFzl

I

in [O, 11 such that C,“==, t,, = CO,

S. S. CHANGet al.

988

2. UNIFORMLY

GLIPSCHITZIAN

MAPPINGS

In this section, we shah study the equivalence between the convergence of modified Mann and modified Ishikawa iterative sequences defined by (1.9)-(1.11)

for uniformly L-Lipschitzian

mappings in Banach spaces. THEOREM 2.1. Let E be a Banacb space, B be a nonempty closed convex subset of E, T : B + B be uniformly L-Lipscbitzian, Let {u,,}

and‘{x,}

where L > 0 is a constant and x* E B be a fixed point of T.

be the modified Mann and modified Ishikawa iterative sequences defined

by (1.10) and (l.ll),

respectively.

Suppose further that sequences {on}

and {/?,}

appeared

in (1.10) and (1.11) satisfied the following conditions: (i) an, Pn E [O,11, Vn 2 0; (ii) C,“=, a, = 00; (iii) 1 - l/L < inf,>o pn and CrZo cr,&

< 00.

IfurJ = xc and R(T) is bounded, then the following statements are equivalent. (1) The modified Mann iterative sequence {un} converges strongly to Z* E F(T). (2) The modified Ishikawa iterative sequence {xn}

converges strongly to x* E F(T).

PROOF. Since the range of T is bounded, let

M={~~~llWl}+ll~oll


By induction, it is easy to prove that

ll~ll

< M,

11~~11 < M,

and

Vn>O.

llunll i M,

(2.1)

Therefore, ]]Tnznl/ 5 M

and

Vn>O.

llTnynIl I M,

Since T is uniformly L-Lipschitzian, it follows from (1.10) and (1 .ll) ]]zfi+r - ‘1~,+1]]= ]I(1 - ou,>(zc, - un) + on (T%, I (1 - on)]]z~ - %a]] +

(2.2) that

- T%J]

(2.3)

%JqlYn - unll.

Now we consider the second term on the right side of (2.3). By (1.11) and (2.2), we have

llY?a- %I( = ll(1- Pn)(G - %a) + Pd%I I (1 - Pn)llGa - unll + PnwnxnII I

(1 -

- %>ll + Ilunll~

(2.4)

Pn)]]zn - u,]] + 2MA.

Substituting (2.4) into (2.3) and simplifying, we have

112n+l

-

un+lll I [I - an(l-

L(1 - Pn>)]llxCn - 41 + 2MUM,

Vn>O.

Taking a, = ]]x, - u,]], t, = an(l - L(l - ,8,)), b, = 0, and d, = 2cu,&LM Conditions (ii) and (iii) we know that t, E (0,l) Vn 1 0 and m m t, = 00, c c d, < co. n=O n=O It follows from Lemma 1.1 that ]lz?l If u,, + x* E F(T),

%Il + O(n --$cQ).

then llxn-x*ll

Conversely, if x, + Z* E F(T),

I(l~n-~nII+ll~n--*ll

-+o(n--+~>.

then

(]u, - z*]] L (]u, - 2,ll + (Iu, - x*1( + O(n + oo). This completes the proof of Theorem 2.1.

(2.5) in (2.5), by

The Equivalence between the Convergence

989

3. BANACH CONTRACTION MAPPINGS, NONEXPANSIVE MAPPINGS, AND ASYMPTOTICALLY NONEXPANSIVE MAPPINGS In this section, we are going to study the equivalence between the convergence of modified Picard, modified Mann, and modified Ishikawa iterations for Banach contraction, nonexpansive and asymptotically

nonexpansive mappings in Banach spaces.

First, we have the following theorem for asymptotically nonexpansive mappings. THEOREM

3.1. Let E be a real Bausch space, B be a nonempty closed convex subset of E,

T : B + B be an asymptotically k, + 1 and x* E B be a tied

nonexpansive mapping with sequence {kn} c [0, oo) such that

point of T. Let {un} and {xn}

Ishikawa iterative sequences defined by (1.10) and (1 .ll),

be the modified Mann and modified

respectively, and satisfy the following

conditions: (i) on, A E [O,l], Vn L 0; (ii) C,“=c (Y, = 00; (iii) 1 - l/L < inf,>c&, and C~cc~,Jl, where L = sup,,,,

< co;

k, 2 1.

If ug = x0 and R(T)

is bounded, then the following statements are equivalent.

(1) The modified Mann iterative sequence {un} converges strongly to x* E F(T). (2) The modified Ishikawa iterative sequence {xn} converges strongly to x* E F(T). PROOF.

Since T : B +

B is an asymptotically

nonexpansive mapping with sequence {kn}

c

[0, cc) such that k, + 1 (as n + co). Let L = SUP,>~ - k, < 00, we have IIT”x - T”Yll 5 k,&

- YII 5 L * 112- Yll,

vx,y

E

B.

This implies that T is a uniformly L-Lipschitzian mapping. Hence, the conclusion of Theorem 3.1 can be obtained from Theorem 2.1 immediately.

I

Next, we have the following theorem for nonexpansive mappings. THEOREM

3.2.

Let E be a real Banach space, B be a nonempty closed convex subset of E,

T : B --f B be a nonexpansive mapping and x* E B be a fixed point of T. Let {un}

and {x~}

be the modified Mann and modified Ishilcawa iterative sequences defined by (1.10) and (l.ll), respectively, satisfying the following conditions: (i) on, A E [O,11, Vn L 0;

(ii) Cr=, a, = co; (iii) 0 < inf,zc pn and Cro a,$,,

< co.

If UO = x0 and R(T) is bounded, then the conclusion of Theorem 3.1 still holds. PROOF.

Since T : B +

B is a nonexpansive mapping, we know that T is an asymptotically

nonexpansive with a constant sequence {k, = 1). Let L=supk,=l. 7X20

Therefore, all conditions in Theorem 3.1 are satisfied. The conclusion of Theorem 3.2 can be obtained from Theorem 3.1 immediately. This completes the proof. I Finally, we have the following nice theorem for Banach contraction mappings. Let E be a real Banach space, B be a nonempty closed convex subset of E, T : B + B be a Banach contraction mapping with a constant L E (0,l) and x* E B be s fixed point and {xcn} be the modified Picard, modified Mann, and modified Isbikawa ofT. Let {G}, (4, THEOREM 3.3.

iterative sequences

defined

by (1.9)-(1 .l l), respectively,

(9 an, A E [O, 11, Vn 2 0; (ii) Cr=, cr, = 00.

and satisfy the following

conditions:

990

S.S.CHANG et 01.

If wg = uo = zo and R(T) is b ounded, then the following statements are equivalent. (1) Modified Picard iterative sequence {wn} converges strongly to z* E F(T). (2) Modified Mann iterative sequence {Us} converges strongly to z* E F(T). (3) Modified Is h ika wa iterative sequence {En} converges strongly to 2* E F(T). PROOF.

Since the range R(T) of T is bounded, denote

Again since w,, u,, x,, yn, Tnw,, T%,,

Tny,, E B, Qn 2 0, therefore, their norms are not

Px,,

greater than M. (1) M

(2). From (1.9) and (l.lO), we have

IIun+l- w,+lll 5 (I- CGJIU,- Tnw,II + a, lITnun- T’hII I (1- 4 {Ilun- wnll+ Ilwn- Tnwnll}+ ~Xl/un - wnll 5 [l - %(l - q]II% -%&II + (1- %&)1lwn - T”wnI( 5

(3-l)

[l - 41 - L)]llun - ‘wnll+ (1 - ~,)Ilw, - T”wnll.

Now we consider the second term on the right side of (3.1), we have

llwn - Tnwn/ = IjTn-‘w,+~ I Ln-lIIw,_~

- Tnw,II - Tw,II

(3.2)

5 2MLn-l. Substituting (3.2) into (3.1) and simplifying, we have IIu~+l - %+111 I [l - Qn(l - L)]ll% - %II + 2(1In (3.3), taking a, = IIu, - ~~11, b, = 0, t, = (~~(1 -L), L E (0,l) and Cr=, a, = co, we know that

c t,= 00

a,)ML”-‘.

and d, = 2(1 - cx,)ML+‘,

(3.3) since

00

cm,

n=O

c

d, <

00.

n=O

By Lemma 1.1, we know that llun - w,II ---) 0 as n -+ 00. If w, ---f x* E F(T)(n

+ co), we have

1121, - x*II I 1121, - w,II + Ilw, - x*Il --+Otn--+00). If u, --)x* E F(T)( n --f co), we have

Ilw, - x*II I Ih - u,ll + IIu, -x*11 + O(n+ m). The equivalence between Statement (i) and (ii) was proved. (1) _

(3). It follows from (1.9), (l.ll), lIXn+l - Wn+lll I (1 - %)llGl i Cl-

an>{ll~n

and (3.2) that - TnwnII + anllTnyn -wnII

- TnwnII

+ IIwn - Tnwnll}

I (1 - %z) {II% - w,II + 2ML”-l}

+ anLnJJyn - WnlJ

+ a,Lnlly,z - w,II.

(34

991

The Equivalence between the Convergence

Now we consider

IlYn -

Substituting

term on the right side of (3.4). From

the second %/I

5

(1 -

Pn)ll%

-

‘WnII + PnllTXl

=

(I-

P?a)llGl - WIII + Pn pn2n

-

(1.11)) we have

WnII

- Tn%411

I (1 - Pn)ll%

-%II

+PnJqI%

I (1 - Pn)ll%

- w,II + 2/3,ML”-l.

(3.5) into (3.4), we have II%+1 - ‘wn+lll I (1 - %[(l + 2a,.

- Jql

- Pn)]}ll%

pJt!fP-1 + 2(1-

I (1 - (Jy,(l - L)}IIzcn -z&II

- WZII

a,)ML”-l

n=O

(3.6)

+ 2ML”-l.

Take a, = 11~ - w,II, b, = 0, d, = 2MLnm1, and t, = an(l - L) in (3.6). and L E (0, l), we have M Ccl t, = Co, c d, < co. c By Lemma

(3.5)

-%-111

Since Cr_-,

cx, = co

n=O

1.1, we know that I(% - %[I --+ O(n + oo).

By the same way as given in the proof of Conclusion sion (1) M

(3) holds.

This completes

(1) M

the proof of Theorem

(2), we can prove that the Conclu3.3.

I

REFERENCES 1. K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. sot. 35 (l), 171-174, (1972). 2. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Sot. 44 (l), 147-150, (1974). 3. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Sot. 4, 506-510, (1953). 4. M.A. Krssnoselskii, Two observations about the method of successive approximations, Uspehi Math. Nauk 10 (l), 123-127, (1955). 5. T.L. Hicks and J.R. Kubicek, On the Mann iteration process in Hilbert space, J. Math. Anal. Appl. 59, 498-504, (1979). 6. D. Borwein and J.M. Borwein, Fixed point iterations for real functions, J. Math. Anal. Appl. 157, 112-126, (1991). 7. Q.H. Liu, The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings, J. Math. Anal. Appl. 148, 55-62, (1990). 8. C.E. Chidume and S.A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Ma&. Sot. 129 (8), 2359-2363, (2001). 9. S.-S. Chang, Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings, Pmt. Amer. Math. Sot. 129 (3), 845-853, (2001). 10. S.-S. Chang, The Mann and Ishikawa iterative approximation of solutions to variational inclusions with accretive type mappings, Computers Math. Applic. 37 (9), 17-24, (1999). 11. S.-S. Chang, Y.J. Cho, B.S. Lee, J.S. Jung and S.M. Kang, Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudocontractive mappings in Banach spaces, J. Math. Anal. Appl. 224, 140-165, (1998). 12. S.-S. Chang, Y.J. Cho and H. Zhou, Iterative Methods for Nonlinear Operator Equations in Banach Spaces, Nova Science Publishers, New York, (2002).