Iterative Methods with Mixed Errors for Perturbed m ... - Science Direct

MATHEMATICAL COMPUTEfR MODELLING Mathematical and Computer Modelling 35 (2002) 55-62

PERGAMON

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Iterative Methods with Mixed Errors for Perturbed m-Accretive Operator Equations in Arbitrary Banach Spaces JONG Soo JUNG Department of Mathematics, Dong-A University Pusan 604-714, Korea [email protected] YEOL JE CHO* Department of Mathematics, Gyeongsang National University Chinju 660-701, Korea yjchocDnongae.gsnu.ac.kr HAIYUN ZHOU Department of Mathematics, Shijiazhuang Mechanical Engineering College Shijiazhuang 050003, P.R. China [email protected],cn (Received April 2001; accepted May 2001)

Abstract-Let X be a real Banach space, A : X + 2x a uniformly continuous m-accretive operator with nonempty closed values and bounded range R(A), and S : X + X a uniformly continuous strongly accretive operator with bounded range R(I - S). It is proved that the Ishikawa and Mann iterative processes with mixed errors converge strongly to unique solution of the equation z E Sz + XAr for given z E X and X > 0. As an immediate consequence, in case that X = 0 and S : X -+ 2x is uniformly continuous strongly accretive, some convergence theorems of Ishikawa and Mann type iterative processes with mixed errors for approximating unique solution of the equation .z E Sr are also obtained. @ 2001 Elsevier Science Ltd. All rights reserved. Keywords-m-accretive cesses, Nonexpansive

operator, retraction.

Strongly accretive operator,

Ishikawa and Mann iterative pro-

1. INTRODUCTION Let X be a real Banach duality

mapping

space with norm 11. 11and X* be the dual space of X.

The normalized

J from X into the family of nonempty subset of X* is defined by

J(z) = {j E X* : b,d = 11412, lbll = Ibll} > for all z E X, where (e, -) denotes the generalized of the Hahn-Banach a bounded

mapping,

theorem

duality pairing.

It is an immediate

consequence

that

J(z) is nonempty for each x E X. It is well known that J is i.e., for any bounded subset B c X, J(B) is a bounded subset in X*.

*Author to whom all correspondence should be addressed. This work was supported by Korea Research Foundation Grant (KRF-2000-DPO013). 0895-7177/01/$ - see front matter @ 2001 Elsevier Science Ltd. All rights reserved. PII: SO895-7177(01)00148-O

Typc=t hy d&@-W

56

J. S. JUNG et al.

An operator A : D(A) c X --+ 2x with domain D(A) and range R(A) is said to be k-accretiwe (k E W) if, for each z,y E D(A) there exists j(~ - y) E J(z - y) such that

for all u E Ax and v E Ay. For k > 0 in inequality (l), we say that A is strongly accretive,

while

for k = 0, A is simply called accretive. In addition, if the range of I + XA is precisely X for all X > 0, then A is said to be m-accretive. Along with the family of k-accretive mappings, we find a family of operators intimately related to it which is known as k-pseudo-contractive

(see [l]). This latter family is formed by mappings

written as I - A, where I is the identity mapping and A is k-accretive. case, an operator T is said to be k-pseudo-contractive j(~ - y) E J(x - y) such that

(TX -

TY,~(x

-

Y))

if, for each x,y

In the single-valued E D(T),

there exists

57

Perturbed m-Accretive Operator Equations with mixed errors to the unique solution of equation z E Sx in case that S : X + multivalued uniformly continuous strongly accretive operator.

2x is a

Our results improve, generalize,

and unify most of the recent, results given by many authors.

LEMMAS

2.

In the sequel, we need the following lemmas for the proof of our main results. The first lemma is actually Lemma 1 of [27]. Also Asplund [28] proved a general result for single-valued duality mappings, which can be used to derive this lemma. LEMMA 1. Let X be a real Banach space and J be the normalized duality mapping.

Then, for

any given x, y E X, we have (Ix + Y112I lbl12 + 2(Y,j(X + Y)), for all j(x + Y) E J(x + y). PROOF. Let x, y E X and j(x + y) E J(x + y). Then we have

112+ Yl12 = (x + Y,j(X + Y))

= (Xlj) +

(Y>j(X

+ Y))

I f (llxl12+ lljl12)+ (Ylj(a:

+ Y))*

Therefore, it follows that

I/x + Yl125 11412 + 2(Y?j(X + Y)). This completes the proof. {!I~}, {cn}, and {w,}

LEMMA 2. (See 1241.) Let {a,}, fying an+1

where no is some positive CF_o w, < co. Then lim,,,

I(1

-h&n

+w&n

+b,

be four nonnegative real sequences satis-

+cn,

72 2 no,

integer, 0 I t, < 1, C,“=. t, = co, b, = o(tn),

C,“=. c, < 00, and

a, = 0.

We need also the following lemmas for our main result. LEMMA 3. Let X be a Banach space and A : D(A)

c X -+ 2x be a multivalued m-accretive operator. Assume that S : X + X is a continuous strongly accretive operator. Then for each z E X and X > 0, the equation z E Sx + XAx has the unique solution x* in D(A). PROOF. Let k > 0 be the strongly accretive constant. written as k(z/k) E Sx + k(X/k)Ax, that is,

The inclusion z E Sx + XAx may be

+A

x >

for each z E X and X > 0. Since ((l/k)S - 1)/(X/k) is also continuous, everywhere defined, accretive operator from X into itself and A is an m-accretive operator, ((l/k)S - 1)/(X/k) +-A is m-accretive (cf. [29, p. 158, Theorem 3.11). Thus, the inclusion (z/k) E (l/k)Sx+ (A/k)Ax has a solution in D(A) ( since k > 0 is fixed), and hence, the inclusion z E Sx + XAx has a solution x*. The uniqueness of x* is a consequence of the strong accretiveness of S + XA. REMARK 1. The inclusion z E Sx + XAx was also studied in [30]. Finally, we recall the following definition.

J . S . JUNG et al.

58

DEFINITION 1. Let X be a Banach space and C B ( X )

be the f a m i l y of a / / b o u n d e d closed subsets o f X . A m u l t i v a l u e d m a p p i n g T : X --* C B ( X ) is said to be u n i f o r m l y continuous if, for a n y ¢ > O, t h e r e e x i s t s a 5 > O s u c h that, f o r a n y x , y E X , w h e n Hx-yi[ < 5, w e h a v e H ( T x , T y ) < e,

where

f

H(A,B)-=

m a x ~ s u p inf I ] x - y [ i , s u p inf H x - y [ [ ~ ( xEA yEB

yEB xEA

J

for a n y A , B E C B ( X ) . LEMMA 4. (See [31].) L e t A , B E C B ( X ) t h a t d(a, b) O, t h e n there exists b E B such

3. M A I N 3.1. C o n v e r g e n c e T h e o r e m s

RESULTS

for P e r t u r b e d

m-Accretive Operators

Now we give the first main result. THEOREM 1. L e t X be a B a n a c h space and A : X -* 2 x be a m u l t i v a l u e d u n i f o r m l y c o n t i n u o u s m-accretive operator w i t h n o n e m p t y closed values and bounded range R ( A ) . A s s u m e t h a t S : X --* X is u n i f o r m l y continuous s t r o n g l y accretive and the range o f ( I - S ) , R ( I - S ) , is bounded. L e t {un},{vn} be two sequences in X and ( a n } , ( ~ n } be real sequences in [0, 1] s a t i s f y i n g t h e following conditions: I It (i) u~ = un' + u~ for a n y sequences {un},{un} in X and n > -- 0 w i t h Ilu"ll = o ( ~ ) ,

(ii) limn-~oo

O0

~n:0

IlU/nll < OO a n d

II~nll -- 0,

(iii) l i m n - ~ an = limn-~oo 13n = 0 and )-~n=0°°an = c~. T h e n , for a n y xo E X and z E X , the Ishikawa t y p e iterative sequence w i t h errors { x n } g e n e r a t e d f r o m zo b y x n + l E (1 - an)x,~ + a n ( z + ( I - S - AA)yn) + un, n _> 0, (2) y~ e (1 -- Z ~ ) x n + Z n ( z + ( I -- S -- ~ A ) ~ n ) + vn, n > O, converges s t r o n g l y to the unique solution o f the equation z E S x + A A x for A > O.

PROOF. It follows from L e m m a 3 t h a t the equation z E S x + )~Ax has a unique solution x* E X. Set G x = z + x - S x - A A x and observe t h a t x* is a fixed point of G, i.e., x* E Gx* and the range of G is also bounded. Furthermore, for all x, y E X , u E G x , and v E G y , there exist E A x and ~ E A y such t h a t u = z + x - S x - ~fi and v = z + y - S y - ~ , and hence, since S is strongly accretive, (u - v , j ( x - y)) = (z + x - S x - ) ~ - (z + y - S y - ) ~ ) , j ( x - Y)I < (1 - k ) l l x - yll 2

(3) //

for some j ( x - y ) E J(x-y) and some k E (0,1). By Assumption (i), we have u n = e n a n , where en --~ 0 as n --~ oo. So, due to the choice of zn E G x n and wn E Gyn, equation (2) can be rewritten as xn+l = (1 - c~n)Xn + a n ( w n + en) + u ' , Yn = (1 - 13n)Xn + t3nzn + vn,

n >_ O.

Since the the range of G is bounded, we may denote by d = sup Iiwn + en -- x*lI + sup I[zn - x*ll + Hxo - x*II n_>0

and

n>0 OQ

M = d+ ~ n=0

Ilu'll.

(4)

Perturbed m-Accretive

This implies

Operator Equations

59

that

11x1 -x*11 = /I(1 - CYO)(XO- z*) + QO(WO + e0 -x*) + z&II 5

QO)II~O -2*ll

5 d + ll4,ll By induction,

+ ~0lJw0 + e0 - 4

+ 11411

5 M.

we obtain II% - X*/I I M

and

~115 M + ll~nll,

IIP -&&n

llYn for all n 2 Now, by

1,

have

112,+i

z*) + CX,(W,

e, - z*)

&+1,j(%+1 +

2%L(%+1

-

2*,.&&+1

-x*))

+

for all j(zc,+i - z*) E J(z,+i - x*). Since .z,+i j(zn+i - z*) E J(x,+i - x*) such that (%+1

Also, observe

-x*))

E Gzn+i

-

xc*)),

and z* E Gx*, by (3), there

5 (1 - ~)llx,+1

- 2*l12.

exists

(6)

that 2(4,.+n+l

Substituting

- x*,.&s+1

2(4,j(Gz+1

(5)

-z*))

-

x:*))

I2ll4llll~n+l

-x*11

(6) and (7) into (5) and simplifying,

I

-

(1+

11x,+1 -x*112).

- x*/l2 + 2a,&

x*l12+ ll~;ll (1 + lIxn+1

(8) -

x*I12)

,

where d, = (w, + e, - %+1,j(Ga+1 -z*)). We prove that d, + 0 as n -+ 00. In fact, since {x,+1 - x*} is a bounded that {J(x,+i -z*)} On the other hand,

is bounded since Yn -

Xn+l

and so {j(xn+i

-xc*)} is a bounded

=

+ A&

(%

-

P&n

(7)

we have

lIxn+i - x*l12 I (I - o,J211x, +2oY,(1 - k)lbn+l

II&II

-

%Lwt

sequence

+ %

-

sequence, in {J(x,+~

we know -x*)}.

%,

where {x,), {.dl {wd, {%I, (4

a 11are bounded sequences, by Conditions (i)-(iii), we have yn - x,+1 -+ 0 as n + co. Again, since, for each n 2 0, w, E Gy,, Gy, E CB(X), and Gx n+1 E CB(X), by Lemma 4, there exists zn+i E Gx,+i such that

IlwnBy the uniform

continuity

H(Gyn,

%x+1).

of G, we have H(GynI,, Gxn+l)

-+ 0

as n --+ 00, and so

II% as n --t DC).

WG~mGxn+l)

Since e, + 0 as n -+ 00, it follows from (9) that

+

0

d, -+ 0 as n -+ co.

(9)

60

J.

S.

JUNG

et al.

Now, since (II, +

0 and IlukII -+ 0 as n + co, there exists a positive integer no such that k) - 1/z&11 > 0, for all n 1 n 0. Thus, it follows from (8) that, for all n 2 no,

1 - 2a,(l-

II%+1

-

z*I12

- k) - IWI - 2%~ + Ibill 1 - 2a,(l - k) - IlU;ll

= 1 - 2M1

+ =

+ Ilu;ll]

1 k) - IlU&ll[(1 - %)211% - x*1i2 +2&d,

1 - 2a,(l-

6

1 - 2a,(l

+ 4 ,,& _ 2*,,2

2cwL

ll4zll

- k) - l/u;11 + 1 - 2&(1 - k) 2k - cy, ’ - 1 _ 2a,(l _ k) _ IIu;IILy, “% - 2*112

ll%II

(10)

II4 II

+ 1-

2a,(1 - k) - lluhlj JJz, - 2*112

+I -

2a,(1 - k) - Ilu;ll + 1 - 2a,(l

ll4tll

2dn

where d, = (wn + e, - ~,+~,j(x,+~ a positive integer n1 such that

-k)

- llu;ll’

Since cy, ---f 0 and ~~u~~~ -+ 0 as n + 00, there exists

-x*)).

2k - cr, 2a,(l

I-

(11)

- k) - llu;ll ’ ”

for all n > nl. Let s E (0,l) be a real number. Again, since ~11,+ 0 and ~~u~~~ -+ 0 as n + oo, there exists a positive n2 such that

k) + lludllI s,

2a,(lfor all n 2 n2 and so we have

1

1 1 - 2a,(l

- k) - IIu:,II ’

for all n > n2. Thus, it follows from (lo)-(12) l(xn+l

-

x*l)2 < (1 - ka,)llz,

for all n 2 max{n0,n~,n2}.

an = 11%-

(12)

1 - s’

that

- x*l12 + Eilzn

-x*112 + s

+ $+$

(13)

In (13), let

t, =

x*l12,

2w,dn

b, = -

kcxn,

1-s

_ II4 II wn 1-s

for all n 2 max{no, nl, n2). Then inequality (13) reduces to an+1

i

(1 -t&n

+ ~,a, i-b, + w,,

for all n 2 max{no, nl, n2). Also, we have x7=0 W, < 00 and b, = o(tn). Hence, by Lemma 2, we have a, -+ 00 as n -+ co, that is, {zcn} converges strongly to x*. This completes the proof. Ram Theorem 1, we have the following. COROLLARY 1. Let X, A, R(A), S, and R(I - S) be as in Theorem 1. Let {un} be a bounded sequence in X and {a,} a real sequences in [0, l] satisfying the following conditions: (i) 21, = uk + UK for any sequences

{z&},{u~}

ll4ill = 4wz>, (ii) lim,,,

a, = 0 and Cr__, a, = co.

in X and n 2 0 with c,“=,

ljuLl1 < cm and

Perturbedm-AccretiveOperatorEquations

61

Then, for any x0 E X and z E X, the Mann type iterative sequence with errors {x,}

generated

from x0 by x,+1 copverges

E (1 - Q,)x,

+ a,(~ + (I - S - XA)x,)

+ IL,,

n 2 0,

strongly to the unique solution of the equation z E Sx + XAx for X > 0.

2. Theorem 1 and Corollary 2 also extend and improve the corresponding results of Chidume and Osilike [15], Zhu [17], Ding [18], and Osilike [21] and in several aspects.

REMARK

REMARK 3. It is clear that the hypothesis that the ranges of A and (I-S)

are bounded imposed in

Theorem 1 and Corollary 1 can be replaced by the assumption that the sequences {(I-S-XA)x,} and {(I - S - XA)y,} 3.2.

are bounded.

Convergence Theorems for Strongly Accretive Operators

In case that X = 0 and S is a multivalued strongly accretive operator, we obtain the following theorem, which can be easily proved by virtue of the technique of Theorem 1 and so its proof is omitted. THEOREM 2. Let X be a Banach space and S : X --+ 2x be a multivalued uniformly continuous strongly accretive operator with nonempty closed values. Suppose that z E Sx has a solution. Let {un},{un} be two sequences in X and {cryn},{Pn} real sequences in [0, l] satisfying the following conditions: (i) uL, = u; + ui for any sequences {u~},{u~}

in X and n > 0 with C,“=.

11uk11 < 00 and

11~~11 = 0(&d, (ii) lim,,, (iii) lim,,,

Ilv,II = 0, a, = limn_+m 0, = 0 and c,“=,

an = 00.

Then, for any zo E X and x E X, the Ishikawa type iterative sequence with errors {x,}

generated

from ICOby

(1 - %Jxn + &2(z + (IYn E converges

S)Yn)

+

Unr

(1 - Pn)xn+ Pn(z+ (I- Sh) + %z,

n 2 0, n 2 0,

strongly to the unique solution z* of the equation z E Sx provided that {(I - S)x,}

and {(I - S)y,}

are bounded.

COROLLARY 2. Let X be a Banach space and S : X -+ 2x be a multivalued uniformly continuous strpngly accretive operator with nonempty closed values. Suppose that z E Sx has a solution. Suppose that {un} and {an} be as in Theorem 2. Then, for any x0 E X and z E X, the Mann type iterative sequence {xn} generated from x0 by x,+1 E (l-a,)x,+a,(t+(I-S)x,)+u,, converges strongly bounded.

to the unique solution

of the equation

n > 0,

z = Sx, provided

that {(I - S)xn}

is

REMARK 4. Theorem 2 improves results of Osilike [21, Theorem 31 (and Chidume and Osilike [16, Theorem 2] with u, = w, = 0) to the case of multivalued operator S. Theorem 4.1 of Chang [4] is a special case of Theorem 2 for which u, = 0 and V, = 0. Theorem 2 also generalizes the corresponding results of Chidume [12,13]. REMARK 5. Corollary 2 is a multivalued version of Chidume [14, Corollary 21 (and Chidume and Osilike [16, Corollary 21 with un = 0). In turn, Theorem 2 and Corollary 2 generalize many previous results, in particular, of Chidume [13, Theorem 2 and Theorem 41, Zeng [23, Theorem 1 and Theorem 31, and Zhu [17, Theorem 1 and Corollary]. REMARK 6. By putting ug = 0, n 2 0, in iterative processes with mixed errors defined in our main results, we also obtain Liu’s processes with errors.

62

J. S. JUNG et al.

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