Approximations for fixed points of φ-hemicontractive mappings by the Ishikawa iterative process with mixed errors

MATHEMATICAL COMPUTER MODELLING Mathematrcal and Computer Modelhng 34 (2001) %18

PERGAMON

www elsevier nl/locate/mcm

Approximations for Fixed Points of $-Hemicontractive Mappings by the Ishikawa Iterative Process with Mixed Errors Department

YEOL JE CHO* of Mathematics, Gyeongsang National Umverslty GhinJu 660-701, Korea

HAIYUN ZHOU Department of Mathematics, Shrlmzhuang Mechamcal Engmeermg College Shrlrazhuang 050003, P R Chma SHIN MIN KANG* Department of Mathematics, Gyeongsang National Unrversrty Chinlu 660-701, Korea SEONG SIK KIM Department of Mathematms, Dongeur Umversrty Pusan 614-714, Korea (Recezved and accepted July 2000)

Abstract-Let X be a real uniformly smooth Banach space, K be a nonempty closed convex subset of X and T K + K be a generalized Lrpschrtzranand hemrcontractrvemapping It ISshown that the Ishlkawa iterative process with mrxed errors converges strongly to the unique fixed pomt of the mapping T As consequences, several new strong convergence results are deduced and some known results are improved @ 2001 Elsevrer Science Ltd All rrghts reserved Keywords-Bemrcontractron,

Generalized Lrpschrtzrancondltron, Ishrkawaiterative process with

mlxed errors

1. INTRODUCTION Let X be a real Banach space with norm )I dualaty mappzng J X 4 2x’ IS defined by

J(x) =

{xc* E x*

AND

PRELIMINARIES

11and X* be the dual space of X (2,x*)

The normalzzed

= lbl12= b*ll”> 9

for all z E X, where ( , ) denotes the generahzed duality palrmg It 1s well known that, If X 1s uniformly smooth, then J 1s single-valued and J(h) = tJz for all t > 0 and z E X and J normalized 1s uniformly contmuous on bounded subsets of X [1,2] We denote the single-valued duality mapping by 3 *The first and third authors wish to acknowledge the financial support of the Korea Research Foundation (KRF-99-005-D00003) 0895-7177/01/t - see front matter @ 2001 Elsevler Science Ltd PI1 SO895-7177(01)00044-9

All rights reserved

Typeset

by &&Tj$

Grant

10

Y

J CHO et al

An operator T with domain D(T) and range R(T) m X 1s said to be generalazed Lzpschztzzan If there exists a constant c > 0 such that lITa:- Tyll 5 for every 2, y E D(T)

~(1+ 112 - Ylt),

(1 1)

Without loss of generahty, we may assume that c > 1

We remark lmmedlately that, if T either 1s Llpschltzlan or has bounded range, then It 1s generahzed Llpschltzlan

On the other hand, m general, every generalized Llpschltzlan operator

neither 1s Llpschltzlan nor has the bounded range

For example, let X = (--00, t-co) and T

X + X be defined by lf 2 E (--00,-l), TX =

x - d_,

lf x E (-co,O),

x + d_,

lf x E [O,11, lf J: E (l,oo)

Clearly T 1s generalized Llpschltzlan, but T IS not Llpschltzlan and Its range 1s not bounded An operator T IS said to be accretzwe If, for every x, y E D(T), there exists 3(x - y) E J(x - y) such that (Ta: - TY,&

- Y)) 2 0

The operator T 1s said to be strongly accretzve if, for every x,y E D(T),

(1 2) there exist 3(x - y) E

J(a: - y) and a posltlve constant Ic such that, (TX - Ty,&

- y)) 2 ktb - Yii2

(13)

Without loss of generality, we may assume that k E (0,l) The operator T ISsaid to be &strongly accretzwe if, for every x, y E D(T), there exist 3(x-y) J(x - y) and a strictly increasing function #J [0, m) + [0, co) with r$(O) = 0 such that (TX - Ty,~(x - y)) > 4(11X- Y~~)~b- YII

E

(1 4)

Let N(T) = {x E X TX = 0) If N(T) # 0 and inequalities (1 2)-(1 4) hold for any x E D(T) and y E N(T), then the correspondmg operator T 1s called quasz-accretzve, strongly quaszaccretzve, and &strongly quasz-accretzve, respectively TX = x} A mappmg T D(T) c X -+ X 1s said to be a &hemLet F(T) = {x E D(T) contractwe If (I - T) 1s $-strongly quasi-accretlve, where I is the ldentlty mapping on X It IS very clear that, if T 1s &hemlcontractlve, then F(T) # 0 and, for all x E D(T) and y E F(T), there exist 3(x-y) E J(x - y) and a strictly increasing function 4 [0, m) + [0, co) with 4(O) = 0 such that (TX - Q/,3(x

- Y)) I

lb - ~11~- 4(11x - ~ll>llx - YII

(1 5)

These operators have been extensively studled and used by several authors (see [3-111) In [12,13], Lm and Xu introduced the Ishlkawa iterative schemes with errors, respectively, but note that Xu’s schemes with errors 1s a special case of Lm’s scheme with errors These operators have been also extensively studled and used by several authors (see [5,12-151) Recently, Zhou and Jla [ll] proved that the Ishlkawa iterative process converges strongly to the unique fixed point of a strong pseudo-contractlve mappmg without Llpschltz assumption m a real umformly smooth Banach space More recently, Huang [5] attempted to extend the mam results of [11,16] to the more general classes of &hemlcontractlve mappmgs and $-strongly qusslaccretlve operators, and the more general iterative process with errors Unfortunately, the proof of Theorem 1 m [5] contams a gap, 1 e , there 1s something wrong m the estlmatlon (12) of [5] Moreover, the contmulty assumption on T imposed m [5] will not be necessary

Approxrmationsfor FrxedPomts

11

It 1s our purpose m thus paper to correct, umfy, and improve the results above mentroned by using our new Ishrkawa rteratrve scheme (IS) 1 with mrxed errors defined m our mam Theorem 2 1 For this purpose, we need to introduce the followmg lemmas LEMMA 1 1 (See [12] ) Let {a,}~&,, satisfymg the followmg condition

with

o5

t,

LEMMA 1 2 satisfying

2 1,

{bn}~&,

and {~}~=c

be three nonnegative real sequences

Cr=, t, = m, b, = o(t,), and Cr=e c, < 00 Then a, + 0 as n -+ ~0

Let {pn}Eo,

{a,}Ze,

pn+1

with An E [O,l), C:=,

I

(1

-

{rn)Ec, A,

+

%)Pn

and {o(X~)%I +

4hJ

+

X, = oo, Cr’_‘=, cm -C 00, and Cr=,

rn>

b e nonnegative real sequences n

2

0,

r,, < 00 Then pn -+ 0 as n + co

{pn} = p Then p = 0 If rt 1s not the case, assume that p > 0 Then PROOF Let hmmf,,, we have pn 2 (l/2) p > 0 for sufficiently large positive integer n Hence, by usmg the fact that 0(X,) 5 X,p,, rt follows that

for sufficrently large n, whrch rmphes that llm,,W formula pn 5 (1 + a,)~,

pn exists, m fact, by virtue of the recursive

+ r,, rt follows that

for all posrtrve integers m,n

Therefore, for any fixed n 2 0, we have

and so

Thus, l~m,+oo pn exists, where we used the facts that

12

J CHOet al

Y

and

( )I M

_hm exp 71+03 [ because exp( C,“=,, Us) --) 1 as n -+ 00 {p,} It follows that

Pn = b Pn ?l-+*

C u3 j=7&

Thus, the sequence {P,},“,~

IS bounded

Set M =

sup+,,-

Pnfl

for very large n, where T, = Ma,

I

(1

-

+ T,

hL)Pn

+

7,

+4&t),

(1

6)

It follows from (1 6) and Lemma 1 1 that pn -+ 0 as

n + 00, which 1s a contradlctlon Hence, there exists a subsequence {pn,}yCO of {P~}?=~ such that pn, ---f 0 as 3 + 00 Since pn, -+ 0, Cp=‘=, a, < 00, and C,“=, T, < 00, it follows that for all c > 0, there exists nJ 2 0 such that

for all n 2 nJ By mductlon, we obtam

for all m 2 0, which shows that prL-+ 0 as n -+ 0;) This completes the proof

2. THE

MAIN

Now we prove the mam results of this paper uniformly smooth Banach space

RESULTS

In the sequel, we always assume that X 1s a real

THEOREM 2 1 Let K be a nonempty convex subset of X and T Llpschltzlan and &hemlcontractlve mapping with the property

hmmf i@ t-+m t Let q be a fixed pomt of T m K

Let {a,}~=,

K 4

K be a generalized

> 0

(2 1) be two real sequences m [0, I]

and {&}rZo

satlsfymg the followmg condltlons (1) an, A 4Oasn+co, (11) c,“=, WI = 00 Assume that {u~},“,~ (111)

u,

=

uk

+

U: = ~(a,),

ug

and {u~},“,~

are two sequences m K such that

for any sequences {u~}~=~ and {u~}~=~ m K satlsfymg C,“=, and ((TJ,[~ --f 0 as n --+ 00

lluill < 00,

Suppose that a sequence {x~}~=~ m K can be defined by xor~o,~o E K, x,+1 = (I-

a&n

+ a,Ty,

yn = (I-

P&n

+ PnTxn + 2m,

Then the sequence {x~}~!~

+ un,

(191

n20

converges strongly to the unique fixed point of T

PROOF By the definition of T, we know that, if F(T) Let q E K denote the unique fixed point Set

II(

Yn 4 l+~~x,-q~~

An=3

x,+1

%=

12 L 0,

II3

(

-

>

-3

>

-3

# 0, then F(T)

(

4

1 + 11%- QllIll ’ Yn - 4

Q

1+~~x,-q~~

xn -

(

1+

11%- QllIll

must be a smgleton

Approxlmatlons

13

for Flxed Points

Without loss of generahty, we assume that IJunI(,11~~~11 I 1 for all n 2 0 Observe that II~G

- qll

1f 11%-

IlTyn

< c

qll -

411<

-

c(2 + c),

1 + II% - qll -

’

and 11zn+1- qi’ < c(2 + c) + 1 IlGl - qll -

1+

It IS easily

seen

that, m view of the uniform contmmty of J on any bounded subset of X, A,,

B, + 0 as n -+ 00 It follows from (IS)1 and lIyn I(yn - q1125 (1 - Pn)ll~

- qllllyn

411 I (l/2)

- qll + PnPn

(1 + llyn - 411’) that

- q,dyn

- q)) + ll~nllll~n

-

411

I (1 - Pn)llGz- qllllyn- 41+ PnCk - q>.dzn - 4)

-

xn -

Yn Q 1 + IJxn- qll

Q

1+ II% - qll )>

(2 2)

x (1 + ((XT& - qllj2 + f ll%ll (1 + IlYn - ql12) I

f (1 - P,)’

1

+ Pn + ‘h&P,

[

2Pd#J(lI%

- qll)llGz

which lmphes that

cdl2 I

IlYn +4cAnPn

llx,

- ql12 + ; (I+

- qll + 2&A

+ f 11~11,

1+ (A + 4&&3n

+ lbnll

- 2Pnd(ll~z

- qli2

cdl2

IIGL -

1 - Ibnll

ll~ll)li~n

- qll)lh

(2 3)

- qll

1 - ll~nll It follows from (IS)1 that

lbn+l -

qlj2 < (1 - cy,)IIx,

I

(1 - %>ll%

5 (1 +%I

%-Jllxn

- qIIIIx:n+l

- qll + an(Tyn

- q,hCn+l

- qllllx,+1

- qll + an(Tyn

- q,.dyn

- QIIII~n+l

- qll + WI (IIYn - ql12 - dllvn

Tyn-q 1 + llxn - qll 73

- q)) + ll~nllllx~+1 - 4)) + f

G&+1 - Q

llwzll (I+ lb - ql12>

- QII)IIYYn - 411)

Yn - Q

1 + llxn - qll

- qll

1 + II% -

Qll)>

x (1+ II&l - ql12> + ; Ibnll (1 + lb%+1 - !7112)

I ; ((1- QrJ211&l - 4112+ + Qn(IlYn

- ql12 - HIIYn

+ ~(2 + +Gn <

(

(I+

; (1 - Q2

llGL+1 - qll>llvn

11~ - ql12) + ; lbnll

+ 242 + c)B,a,)

II%+1

+ f IMI)

Thus, transferrmg the term (l/2 11xn

+ lbnll

+ 2%

+1

-

-

4112 +

(I+

llxn+l

- ql12)

)1x, - c/II2 - qll) 242

+ (1/2)ll~~~~)~l2~+~ ql12 <

(2 4)

- 411)

+ % (IlYn - ql12 - @(llYn - qll)llvn + (;

ql12>

+

m&z

+

;

ll’11nll

- q)12, we have

l - I1- 4c(2 + c)BQlan (lxn

_

ql12

1 - 0~~11

(llvn - ql12 - d(llvn

- qll>llyn

1 - Ibnll

- qll) + 4c(2 + c)anB,

(2 5)

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J CHO et al

Now we consider the followmg possible cases +coasn--+oo (1) IlYn - 411

In this case, we must have that (12, - 411+ 0;) as n ---t 00 By property (2 l), we have hrn mf MlYn n+w [(&-q/1

and

all) =7>o

hmmf WI% - qll) n-+03 IIxcn-qll =w>o

Thus, we can choose no so large that

44IG - 411)> Au II% - Qll

- 2

’

4(IIYn - 41) > 1 ~ IlYn - 411 - 2 ’

P, + 4cA, I w, 4c(2 + c)& I r/2, and 11~~11/1- ll~~ll 5 (~/G)c11, for all n 2 no Substltutmg these estlmatlons mto (2 3) and (2 4) yields that

Q, Il.& - ql(2+ 1$& II%+1 - 4112 5 l -1 _(7/2) ((u,(( 2 (1 - 5 an+ ~m4I)

+ 4%)

II&I - 4112+ Mll4J

(2 6) + 44,

for all 1x1> no, where l/(1 - ll~nll) 5 M, which implies that z, -+ q as n + w by virtue of Lemma 1 2 This contradicts the assumption that llyn - 411-+ co as n + co

(11)There

exists a subsequence {yn,}~=o of {Y,},“,~ such that {y,,}r?_o 1s bounded In this case, by mductlon, we can prove that {Y~}?=~, and hence, {x~}:.~ 1s a bounded Smce T 1s generalized Llpschltzlan and T 1s bounded, it follows that both sequence {TxCn}FCo and {TY,}~_~ are bounded Usmg a slmllar argument to (I), we have

II%+1 - 4112I Let lim mfn_oo

Ih -

4112- 2%dllYn

- c7ll)llYn -

411 + WI411+ 4%)

(27)

((yn - q(( = 6 Then 6 = 0 If It 1s not the case, we assume that 6 > 0

Then there exists a posltlve constant n1 such that 11~~- 4112 b/2 for all n 2 nl and so dllvn - 4) 2 4(~/2) Th us, we can choose n2 so large that o(%J 5 4 4

0

;

Qn,

for all 722 722 Let ns = max {nl, 722) It follows from (2 7) that

II%+1 - 41125

II% - 4112 - f 4) f 0

an+ Wldll,

(2 8)

for all n 2 ns, which lmphes that $$)

2

% I II%, - 4112+ A!I 5

71=7X3

11411 < a, 71=7&s

(2 9)

which 1s a contradlctlon

and so 6 = 0 Consequently, there exists an mfimte subsequence { yn, },oO=,such that yn, + q as J --f co and so there exists an mfimte subsequence {x~,}~& such that xn, --) q as J --+ 03 By mductlon, we can prove that 5, -+ q as n -+ co This completes the proof Note that the sequence {x,}F=,, defined by (IS)l, Ishzkawa ateratave process wath maxed errors

satlsfymg Condltlon

(111) 1s called the

REMARK 2 1 Theorem 2 1 extends Theorem 3 2 of Chang et al [3] and Theorem 2 of Zhou [IG] to the more general class of operators and lteratlon processes with mixed errors

As a direct consequence of Theorem 2 1, we have the followmg

Approxlmatlonsfor FutedPoints

15

COROLLARY 2 1 Let X, K, and T be as m Theorem 2 1 Let {?J~}$&, sequences m K and {a,}$?&,, {b,}r& {c,}~&, m (0,l) satlsfymg the following condltlons

{ai}?=,,,

{b~}~&,,

{w~}~=~ be bounded

{c~}~=~

be real sequences

(1) a, + b, + c, = 1 = a:, + b:, + CL, n 2 0, (4 b,, b;, c:,-+Oasn+CQ, (111) c:lJ b, = CQ,

bv) cz,

% < 00 lteratlvely by

Define a sequence {x~}:!~

xor uo, 00 E K, x12+1 = anxn + LTy, yn = ahxn + b;Tx, Then the sequence {z~}:.~

+ Gun,

n L 0,

+ c’7L v 7%)

n>O

(IS)2

IS well defined and converges strongly to the umque fixed pomt of T

PROOF

It 1s clear that the sequence {x~}:=~

and T

K +

K 1s a self-mapping

defined by (IS) 2 1s well defined since I< 1s convex

Set (Y, = b, + c, and & = bk + CL Then the recursive

formula (IS)2 becomes %+I

in

(1 - %)Xn + anTYn - ~n(Tyn - un), = (l- Pn)xn + PnTxn - &(Txn - ~1,

n 2 0,

=

n>O

Under the assumptions of Corollary 2 1, we can prove that the sequence {x~}:=~ defined by (IS)2 1s bounded, and hence, {Tx~}~~~ and {Tyn}~Co are all bounded Thus, the conclusion of Corollary 2 1 follows from Theorem 2 1 This completes the proof THEOREM 2 2 Let T X + X be a generahzed Llpschltnan and &strongly quasi-accretlve operator with property (2 1) Let x* be a solution of the equation TX = 0 Define a mappmg S X + X by Ss = z - TX for all x E X satlsfymg the followmg condltlons

Let {a,}?=,

and {Pn}zZo

be two reaJ sequences m [0, l]

(1) %7 & ~Oa.sn-+oa, (11) cz, Q, = 00 Assume that {u~}F.~ and {D~}:=~ are two sequences In X such that u, = uk + u: for any sequences {uk}F!o and {uC}FZo in X satlsfymg cr=o Ilukll < 00, ZJ~= o(cr,), and llwnll + 0 as n -+ co For arbitrary 20 E X, define a sequence {xCn}rEo m X by %+1

=

(I-

Yn = (IThen the sequence {x,}~!~

%I)%

+

ansyn

+

f&L,

Pn)GL + PnSxn + %I,

n

2

0,

n>O

(1%

converges strongly to the umque solution x* of the equation TX = 0

PROOF We observe first that, if the equation TX = 0 has a solution x* E X, then the solution x* 1s unique since T X + X 1s &strongly quasi-accretlve By the definition of S, we have

(Sx - SY,.T(X - Y/)) I lb - Yl12 - $(IIx - Yll)llX - Yll,

(2 10)

for every x E X and y E N(T)

We observe also that T is generalized Llpschltzlan lmphes that S

1s also generahzed Llpschltzlan

Consequently, there exists a fixed positive constant c such that IlSx - SYII 5 41 + 112- Yll>l

(2 11)

for all x, y E X Therefore, the conclusion of Theorem 2 2 follows exactly from Theorem 2 1 This completes the proof From Theorem 2 2, we have the followmg

Y J CHOet al

16

COROLLARY 2 2

Let X, T, and S be as m Theorem 2 2

sequences m X and {a,}?&, {bn}r=O, {c,}F=,, m (0,l) satlsfymg the followmg condltlons

{a~}~&,,

Let {~~}n”,,,,

{wn}rZO

{bL),“==,, {c~}~!~

be bounded

be real sequences

(1) a, + b, + c, = 1 = u:, + b:, + CL, n 2 0, (11) b,, b6, CL -+ 0 as n 4 03, (4 C:, (1v) cz”=,

b, = 00, c?l < 03

Define a sequence

{~c,}~~“=on.?eratlvely by 20,~0,~0 x,+1 = anx,

+ My,

yn = abxcn + b;Sx, Then the sequence {x~},“,~ PROOF

E X7 + cn~n,

11 2 0,

+ c:v,,

n>O

(1%

converges strongly to the unique solution of the equation Ts = 0

Set cy, = b, + c, and &, = bk + CL Then the recursive formula (IS)4 becomes Xn+l

=

(1 -

yin = (1 -

+

%SYn

- Cn(SYn - WI),

n 2 0,

Pn)Gl +

PnSxn

- c;(s%

n>O

%-Jxn

- VJ,

Under the assumptions of Corollary 2 2, we can prove that the sequence {xCn}rYo defined by (Is)4 1s bounded, and hence, {SX,}~=~ and {SY,}~=~ lary 2 2 follows from Theorem 2 2

are all bounded

Thus, the conclusion of Corol-

REMARK 2 2 Theorem 2 2 extends Theorem 5 2 of [3] and Theorem 1 of [16] to the more general class of operators and iteration process with mixed errors An accretlve operator T with domain D(T) and range R(T) m X 1s said to be m-accretzwe if the range of the operator (I + XT) 1s whole space X for some posltlve constant X Recently, m [17], Zhou gave some results on the approxlmatlon methods for nonhnear operator equations of the m-accretlve type Applymg Theorem 2 2 to an m-accretlve operator, we have the followmg important result THEOREM 2 3 Let T X + X be a generalized Llpschltzlan and m-accretlve operator For any given f E X, let x* be a solution of the equation x + TX = f Let {cxn}rZo and {/?n}~=o be two real sequences m [0, l] satlsfymg the followmg condltlons (1) ok, @?I-+Oasn-+c~, (11) c,“=, a, = 00 Assume

that

{~,}p=,

and {un}rCo

are two sequences

m X such that u,

= ZL~+ ZL~ for any

sequences {z&}~=~ and {u~}~=~ m X satrsfymg C,“=, 111&11< co, U: = ~(a,), as n + w For arbitrary 50 E X, define a sequence {x~}~=~ m X by x,+1 = (Iin = Cl-

Then the sequence

{x,}~=~

GJ~,

+ a,(f

- TY~) + u,,

n > 0,

Pnh

+ PnLf

- %d

n20

converges strongly

+ G,

to the unique solutlon

and [l~~ll -+ 0

of the equation

(IS):,

x + Ts = f

PROOF Define a mapping A X + X by Ax = x + TX for each x E X Then A IS a genelahzed Llpschltzlan and strongly accretlve operator, and hence, A satisfies all assumptions of Theorem 2 2 Thus, the conclusion of Theorem 2 3 follows from Theorem 2 2 COROLLARY 2 3 Let X, T, and S be as m Theorem 2 3 Let {zL,}~=~, {‘u,}:.~ be bounded sequences m X and let {a,}rTo, {bn}rZo, {c~}?!~, {a~}~Co, {b~}~Co, {cL}~?~ be real sequences m (0,l) satlsfymg the followmg condltlons (1) a, + b, + c, = 1 = u; + 6; + CL, n > 0, (11) b,, b;, c~+Oasn+co, (111)rEzo (14 CZ,

b, = a> c,
Approximations

Define the sequence {z,}~=~

lteratlvely

17

for Fixed Points

by 50, ‘1L0,vo E x,

%+1

= &l%

y,, = a;x, Then the sequence {z,}~=~ REMARK

general

2 3 Theorem class of operators

THEOREM

{~,}Eo

m terms 2 4

and {A)%

+ cn%%,

12 L 0,

+ c;v,,

n20

+ b;Sx,

converges strongly to the urnque solution

If and only if -T

of dlsslpatlve Let T

MY,

2 3 extends Theorem and iteration process

Since T 1s m-accretlve restated

+

operators

(1%

of the equation

5 of [18] and Corollary with mixed errors

1s m-dlsslpatlve,

the result

z + Tz = f

3 1 of [14] to the more of Theorem

2 3 can be

x - XTx = f, where X > 0

for the equation

X + X be a generalized Llpschltzlan e b t wo real sequences m [0, l] sat@mg

and m-dlsslpatlve operator the followmg condltlons

Let

(1) ok, Pn -+Oasn-+oo, (11) Ix,“=, % = cfJ Assume that {un}rZo

and {v~}~=~ are two sequences

III

X such that un = uil + U: for any

sequences {u;},“=~ and {u~}$!!~ m X satlsfymg Cr=, /1z&j[ < co, U: = ~(a,), and l/v,)1 + 0 as n -+ co For any given f E X and X > 0, define a mappmg S X + X by Sx = f + XTx for each z E X For arbitrary x0 E X, define a sequence {x,}~~, m X by X n+1 = 0

- @%I

Yn = (1 - P&n Then the sequence

{x,}~=,

converges

strongly

+

%ZSY, + %,

+ P&,

+ %,

n > 0, (IS)7

n20

to the unique solutlon

of the equation

x-XTx

= f

PROOF The existence of a solution follows from the m-dlsapatlvlty of T Observe that (-XT) 1s a generalized Llpschltzlan and m-accretlve operator Thus, the conclusion of Theorem 2 4 now follows from Theorem 2 3

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4 5 6 7 8 9 10 11 12 13

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