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MATHEMATICAL AND COMPUTER MODELLING
PERGAMON Mathematical and Computer Modelling 34 (2001) 319-330 www.elsevier.nl /locate/ mcm
Stability of Ishikawa Iteration Methods with Errors for Strong Pseudocontract ions and Nonlinear Equations Involving Accretive
Operators in Arbitrary Real Banach Spaces
Z E Q I N G L I U D e p a r t m e n t of M a t h e m a t i c s Liaoning Norma l Univers i ty
Dal ian, Liaoning 116029, P.R. C h i n a
S H I N M I N K A N G * D e p a r t m e n t of M a t h e m a t i c s
Gyeongsang Na t iona l Univers i ty Ch in ju 660-701, Korea
smkang©nongae, g snu . ac . k r
(Received September 2000; accepted November 2000)
A b s t r a c t - - L e t X be an arbitrary real Banach space and T : X -* X be a Lipschitz strongly pseudocontraction. It is proved that certain Ishikawa iteration procedures with errors are bo th convergent and T-stable. A few related results deal with the convergence and stability of the iteration procedures for the iterative approximation of solutions of nonlinear equations involving accretive operators. Our results are the improvements and extension of the results obtained previously by Chidume [1,2], Liu [3], and Osilike [4,5]. (~) 2001 Elsevier Science Ltd. All rights reserved.
K e y w o r d s - - S t a b i l i t y , Ishikawa iteration methods with errors, Strongly accretive operators, Strongly pseudocontractive operators, Accretive operators, Fixed points, Real Banach spaces.
1. I N T R O D U C T I O N
Let X be an arbitrary real Banach space with norm I1" II and dual X*, and J denote the normalized duality map form X into 2 X* given by
J x = ( f ~ X * : LIfll 2 = IIx[I 2 = ( x , f ) } ,
where (., .) is the generalized duality pairing. In the sequel, I denotes the identity operator on X. An operator T with domain D(T) and range R(T) in X is called accretive if for all x, y E D(T)
and r > 0, there holds the inequality
II x - Yll <- II x - Y + r ( T x - T y ) H . (i .1)
The first author was supported in part by the National Natural Science Foundation of China (69973019) and the second author was supported by Korea Research Foundation Grant (KRF-99-005-D00003). *Author to whom all correspondence should be addressed.
0895-7177/01/$ - see front mat ter (D 2001 Elsevier Science Ltd. All rights reserved. Typeset by AAzIS-TFjX PII: S0895-7177(01)00064-4
3 2 0 Z. LIU AND S. M. KANG
I f T is accretive and ( I + r T ) ( D ( T ) ) = X for all r > 0, then T is called m-accretive. The accretive operators were introduced independently in 1967 by Browder [6] and Kato [7]. An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem
du d---[ + T u = O, u(O) = uo,
is solvable if T is locally Lipschitzian and accretive on X.
Recall that an operator T is said to be strongly accretive if for all x, y E D ( T ) , there exist j ( x - y) E J ( x - y) and a constant k > 0 such that
( T x - T y , j ( x - y)) >_ ki[x - ylI 2. (1.2)
Without loss of generality we may assume k E (0, 1). I t is well known that T is accretive if and only if (1.2) holds with k = 0. Closely related to the class of strongly accretive operators is the class of strongly pseudocontractive operators where an operator T is called strongly pseudocontractive if for all x, y E D ( T ) there exist j ( x - y) E J ( x - y) and t > 1 satisfying
1 ( T x - Ty , j ( x - y)) < ~ IIx - yl[ 2. (1.3)
I t follows from inequalities (1.2) and (1.3) that T is strongly pseudocontractive if and only if ( I - T) is strongly accretive. The classes of operators introduced above have been studied by several researchers (see, for example [1-12]).
The following iterative processes were introduced by Ishikawa [9], Mann [10], and Liu [3], respectively.
(a) Let K be a nonempty convex subset of X and let T : K --~ K be an operator. For any given Xo E K, the s e q u e n c e {Xn}n°°=o defined by
x~+l = (1 - an)Xn + anTyn , n >>_ O,
yn = (1 - bn)xn + bnTxn , n >_ 0,
(b)
is called the Ishikawa iteration sequence, where {an}n~=0 and {bn}n~=o are real sequences in [0, 1] satisfying appropriate conditions.
- - OO In particular, if bn = 0 for all n > 0, then the sequence {xn},~=0 defined by
x o E K , xn+l = ( 1 - a n ) X n + a n T x n , n > O ,
(c) is called the M a n n iteration sequence.
Let K be a nonempty convex subset of X and let T : K --* X be an operator. For any given x0 E K the sequence {x,~}n~=0 C_/( defined iteratively by
Xn+l = (1 - an)X n + a n T y n + Un, 7t > O,
Yn = (1 -- bn)xn + bnTxn + vn, n > O,
- - V o o o o where {Un}n~=o and { ~}n=o are two summable sequences in X. (That is, ~ n = o [lun]l < oc a cx~ b oo and )--~n°°=o IIvnil < oc), { n}n=O and { n}n=O are real sequences in [0, 1] satisfying suitable
conditions, is called the Ishikawa iteration sequence with errors. X (x~ (d) If, with K, T and x0 as in (c), the sequence { n}n=O C K defined iteratively by
Xn+l = (1 -- an)xn + a n T x n + Un, n > O,
where (Un}~=o is summable sequence in X and {an}n~=o is real sequence in [0, 1] satisfying suitable conditions, is called the Mann iteration sequence with errors.
Stability of Ishikawa Iteration Methods 321
It is easy to see that the Ishikawa and Mann iterative processes are all special cases of the
Ishikawa and Mann iterative processes with errors, respectively.
Suppose that T is an operator on X. Let x0 be a point in X and let Xn+l = f (T , xn) denote an iteration procedure which yields a sequence of points {xn}n=o in X. Assume that F ( T ) = {x : T x = x E X } ~ 0 and that {xn}~=o converges strongly to q E F(T) . Let {Yn}~=o be an arbitrary sequence in X, and set en = [lYn+l - f (T , yn)[I. If limn--.oo en = 0 implies that l i m n - ~ Yn = q, then the iteration procedure defined by Xn+l = f (T , Xn) is said to be T-stable
or stable with reslSect to T. In [1-5,12], serveral authors applied the Mann and Ishikawa iteration methods to approxi-
mate fixed points of Lipschitz strongly pseudocontractions and to approximate solutions of equa- tions T x = f and x + T x = f . In [4,5,12-18], the authors obtained the stability of some iteration methods for certain nonlinear mappings. Harder [13] studied applications of stability results to first-order differential equations. Harder and Hicks [15] showed how such sequences {yn}n°°__0 could arise in practice and demonstrated the importance of investigating the stability of various iteration procedures for various classes of nonlinear mappings. Recently, Osilike [4,5] established the stability and convergence of certain Mann and Ishikawa iteration procedures for fixed points of Lipschitz strongly pseudocontractions, and solutions of nonlinear accretive operator equations.
It is our purpose in this paper to prove that certain Ishikawa iteration procedures with errors as both convergent and stable with respect to Lipschitz strongly pseudocontractive operators in arbi trary real Banach spaces. A few related results deal with the convergence and stability of the iteration procedures for the iterative approximation of solutions of nonlinear equations involving accretive operators. Our results extend, improve, and unify important known results in [1-5].
2. P R E L I M I N A R I E S
The following results will be needed in the sequel.
LEMMA 2.1. (See [19].) Suppose that {an}~n=o and {j3n}n~=o are nonnegative sequences satisfying
the following inequality:
an-kl <~-- "~OLn "~ ~n, n >_ O,
where 7 C [0, 1) and limn-~oo ~n = O. Then l i m n - ~ an = 0.
LEMMA 2.2. (See [8].) Suppose that X is an arbitrary real Banach space.
(i) I f T : X - * point.
(ii) I f T : X --~ solution for
LEMMA 2.3. (See is continuous and
solution for any f
X is continuous and strongly pseudocontractive, then T has a unique fixed
X is continuous and strongly accretive, then equation T x = f has a unique
any f E X .
[11].) Suppose that X is an arbitrary real Banach space and that T : X --* X accretive, then T is m-accretive. Moreover, the equation x + T x = f has a
c X .
3. M A I N R E S U L T S
In the sequel, L denotes the Lipschitz constant of T, L, -- 1 ÷ L, and k and t be the constants appearing in (1.2) and (1.3), respectively. Now, we prove the following results.
THEOREM 3. i . Suppose that X is an arbitrary real Banach space and T : X --* X is a Lip- schitz strongly pseudocontractive operator. Assume that {un}~=o, {vn}~-o are sequenes in X and {an}~=o, {bn}n~=o are sequences in [0, 1] satisfying the following conditions:
0 < a < an, n > 0, (3.1)
3 2 2
where a is cons tan t ;
where r E (0, k ) ;
Z. L I u AND S. M . KANG
L(1 + L ) ( a n + b~) + (L - 1)L2a=bn
+ ( 2 - k)(1 + L + ( L - 1)b=)an < k - r,
l im U n = l im v n = 0 . n ~ o o n---*oo
For arb i t ra ry xo E X def ine the sequence {xn}n°°=o by
zn = (1 - bn)xn + b n T x n + Vn, n > O,
Xn+l = (1 -- an)Xn + a n T z n + Un, n > O.
L e t {Yn}n~__o be a n y sequence in X and def ine { n )n=0 by
n _ > 0 ,
T h e n
(i)
W n ~-- (1 - b n ) y n + b ~ T y n + Vn, n > O,
( n = [ [Yn+l - - ( 1 - - an)yn - a n T w n - unl], n > O.
(3.2)
(3.3)
(3.4)
(3.5)
t he s e q u e n c e {Xn }n~_--0 converges s t r o n g l y to the un ique f i xed-po in t q o f T ; moreover ,
1 ~'* llx,~ - qll < 1 - ~ a r ) Ilx0 - qlt + 2 (1 - (1 - ( 1 / 2 ) a r ) n) C, n > O, (3.6)
a r
w h e r e C = s u p { ( 3 L + L2)llv~[I + (3 + L)llunll : n > 0},
(ii) I lyn+l - qll < (1 - (1 /2 ) ar)l lyn - ql[ + en + I l l - If + (3L + n2)Hv~]l for a11 n >_ O;
(iii) l i m n - o o Yn = q i f and on ly i f l imn__,~ en = 0.
PROOF. L e m m a 2.2 y i e lds t h a t T has a u n i q u e f ixed p o i n t q E X . S ince T is s t r o n g l y p s e u d o -
c o n t r a c t i v e , so t h a t for a n y x, y E X t h e r e ex i s t j ( x - y) E J ( x - y) a n d k E (0, 1) s a t i s f y i n g
( ( I - T ) x - ( I - T ) y , j ( x - y)) > kl]x - yll 2,
which imp l i e s t h a t ( ( I - T - k I ) x - ( I - T - k I ) y , j ( x - y)) _> 0, a n d i t fol lows f rom L e m m a 1.1 of K a t o [7] t h a t
IIx - yN < II x - y + r [ ( / - T - k Z ) x - ( I - T - k l ) y ] l l , ( 3 . 7 )
for all x , y E X a n d r > 0. Us ing (3.4), we eas i ly c o n c l u d e t h a t for al l n >_ 0,
X.n = X n + l ~- a n x n - - a n T Z n - - u n
= (1 + an)X~+l + an ( (1 - k ) I - T ) x n + l - (1 - k ) a n x n
+ a n ( T x n + l - T z n ) + (2 - k)a2n(Xn - T z n ) - (1 + an(2 - k ) ) u n ,
(3.s)
and
q = (1 + an)q + a~( (1 - k ) I - T ) q - (1 ~- k)anq .
F r o m (3 .7 ) - (3 .9 ) , we have
(3.9)
Xn+l an T)q] IIz,~ - q]] >~ (1 + an) - q + ~ [((1 - k ) I - T ) x n + l - ((1 - k ) I -
- (1 - k)anl[xn - q[[ - a n l [ r x n + l - - Tznl l - (2 - k)a2n[lxn - Tznl[ - 3[[un]]
> (1 + a,~)llXn+ t - q[] - (1 - k)aT~llx,~ - q[[ - anllTx,~+, - r z~ l l
- (2 - k)a~l lXn -- Tznl l - 311u,~tl,
Stability of Ishikawa Iteration Methods 323
which implies tha t
IlXn+~ - qll <
+
an IITXn+l - Tznll 1 + l+an(1 - k)an ilxn _ qll + ~ (3.10)
3 (2 - k)a~ IIXn -- T'nil + ~ II~nll,
1 + an
for all n > 0. It follows from (3.2) and (3.4) tha t
IIx,, - Tznll <_ IlXn -- qH + IITzn - qll < Ilxn - qll + LIIzn - qll
<_ Ilxn - qN + El(1 - bn)llxn - qN + bnllTxn - qll + II'Vnll] (3.11)
<_ (1 + L + L(L - 1 ) b n ) l l z n - qll + Lllvnll,
and [ [ T x n + l - TznlL <_ L l l x n + l - Znll
< L(anllXn - Tznl[ + bnllxn - Txnll + IlUnll + ][Vnl[)
< L[an(1 + L + n ( n - 1)bn)l[Xn - ql] + nanl]Vnt[ (3.12)
+ (1 + L)bn]]Xn - qll + IlUnN + [IVnll]
< [L(1 + L)(an + b~) + L 2 ( L - 1)a~bn] IlXn -- qll
4- (L 4- an n2) IlVnll 4- nllunll,
for all n > 0. Subst i tut ing (3.11) and (3.12) into (3.10), we infer tha t
1 [1 + (1 - k)an + L(1 + L)an(an + bn) + L 2 ( L - 1)a~bn IlXn+l -- qll <- 1 + an
+ ( 2 - k)a~(1 + L + L ( L - 1)bn)] ]lzn - qll
(2 - k)a,~, ( an (L + L2an) + L ) II~nll
+ \ 1 + a n 1 + a n
( an L + Ilunll + 1 + a n
an [k - L((an + b)(1 + L) + L (L - 1)anbn) (3.13) < 1 1 + a n
--(2-- k)an(1 + L + L ( L - 1)bn)]} IlXn qll
+ Jan (L + L2an) 4- (2 - - k)a~L] ]lvn]l + (3 + L)l[Unll
anr ~ (3L + L 2) IlVn]] + (3 + L) llUnl[ _< 1 1Tan/ l l~n- -q l l+
<_ ( 1 - ~ a r ) N X n - q I [ + ( 3 L + L 2) l'vnN + (3 + L)llunll,
for all n > O. Set
1 = 1 - 5 at, ~n = IIXn --qll,
~n = (3L + L 2) I[vn[[ + (3 + L)llUn[I, n > O.
In view of Lemma 2.1, (3.3), and (3.13), we deduce tha t an --~ 0 as n --~ oc. Tha t is, Xn --~ q as n ~ oc. Furthermore, using (3.13), we have
[ Ixn-q l l <- 1 - - ~ a r I l X n - l - q l l + C
( 1 ~n 2 ( 1 - ( 1 - ( 1 / 2 ) a ~ ) n) c , < ~, l--~ar] [Ixo-qll+ ar
324 Z. Liu AND S. M. KANG
for all n _> 0, c o m p l e t i n g t h e p roo f of (i). P u t Pn = (1 - a n ) y n + a n T w n for all n > 0. No te t h a t
f ly .+1 - qll -< I ]Yn+l - - P ~ -- Unll + IlUnll ~ IlPn -- qll _< en + [lUnll + IIp,~ - ql], (3.14)
for all n > O. Us ing (3.5) a n d (3.2), we conc lude t h a t
I lyn - T w n l l <_ NYn -- qll + L l l w n - qJ[
< ]IY,~ - qJf + L [ (1 - bn)rlYn - ql] + bnL l lYn - qlf + IlVnlI]
_< (1 + L + L ( L - 1)bn)llyn - ql] + LIIvnll, (3.15)
a n d
IITp,~ - T w n l l < L l l p n - Wnll <_ L(a,~ l lyn - T w n l l + bnllYn - Tyn l l + ]Fv,~II)
< L[a,~(1 + L + L ( L - 1)bn)l lyn - qll
+ (1 + anL)[ Iv,~l l + bn(1 + L)l]y,~ - q l l ]
< L[(an + bn)(1 + L) + anbnL(L - 1)]llyn - qll + (L + L2an) Ilvnl[,
(3.16)
for all n _> O. Obse rv e t h a t
Yn = Pn -t- a n Y n -- a n T w n
= (1 + a n ) P n + an( (1 - k ) I - T ) p n - (1 - k ) a n Y n
+ (2 - k)a2n(Yn - T w n ) + a n ( T p n - T w n ) ,
(3.17)
a n d
q = (1 + a n ) q + an( (1 -- k ) I - T ) q - (1 - k ) a n q ,
for all n > 0. B y v i r t u e of (3 .15)- (3 .18) a n d (3.7), we o b t a i n t h a t
(3.18)
Pr~ an T)q] Ilyn - qll -> (1 + a n ) - q + ~ [((1 - k ) I - T ) p n - ((1 - k ) I -
- (1 - k )an l lYn - q[[ - (2 - k)a2n]]Yn - Twn] l - a n l l T p n - T W n H
> (1 -t- an) l lPn -- qll -- (1 -- k )an l lYn - q[I
- (2 - k)a2[Jyn - T W n H - a n ] l T p n - Twn]] ,
which impl ies t h a t
Ilpn - qll -< 1 + (1 - k ) a n 2 - k 1 + an Ifun - qll + ~ a~llYn - Tmnl l
an + l---+~a~ [[Tpn - T w n l [ ,
(3.19)
for a n y n > 0. S imi la r to (3.11) a n d (3.12). we have also the fol lowing
Ityn - r w n l l <_ (1 + L + L ( L - 1)bn) l lyn - qll + LIIvnlf, (3.20)
a n d
IITpn - Twnll <_ [L(1 + L)(an + bn) + L2(L - 1)anb~] IlYn -- qll
+ (L + L2an) Ilvnll, (3.21)
Stability of Ishikawa Iteration Methods 325
for any n > O. Subs t i tu t ing (3.20) and (3.21) into (3.19), we obta in t ha t
I l p n - ql l - - - 1 [1 + (1 - k)an + (2 - k)a2(1 + L + L ( L - 1)bn) l + a n + n ( 1 + n)an(an + bn) + n 2 ( n - 1)a2nbn] Ilyn - q]l
1 + ~ [(2 - k)La~ + (L + L 2) an)an] Ilvnll
1 <- --1+an (1 + an + ( - k + k - r)an)l]y~ - qll + (3L + L 2) IIvnll
( < 1 - r ~ IlYn - qH + (3L + L 2) Ilvnll
(1) <_ 1 - -~ a r Ilyn - qll + (3L + L 2) IlVnll,
(3.22)
for all n > 0. Therefore , (ii) follows immedia te ly form (3.14) and (3.22).
Suppose t h a t l imn-- .~ en = 0. Note t h a t ra c (0, 1). I t follows from (3.3), (ii), and L e m m a 2.1
t h a t l imn-- .~ Yn = q. Suppose t h a t limn--.c¢ Yn = q. Using (3.22) and (3.3), we immedia te ly conclude t h a t
~n = IiYn+l -- (1 -- a n ) y n - a n T w n - UnH
_< ][Yn+l - qH + IlPn - qll Jr llu~ll (1) < ]lyn+l - q]l + 1 - ~ a r IlYn - Nit + (3L + L 2) IIv~ll + Ilunll
---* O,
n --~ 0. T h a t is, limn--oo en = 0. Hence, (iii) holds. This completes the proof.
In case Un = Vn = 0 for all n _> 0, Theo rem 3.1 reduces to the following.
a 0 0 COROLLARY 3.1. Suppose that X , T, r, a, { n}n=o,
arbitrary xo C X define the sequences {Xn}nC¢= o by
zn = (1 - bn)x~ + bnTxn, n >_ O,
xn+l = (1 - an)xn + anTzn , n > O.
{bn}n=o are as in Theorem 3.1. For
Let {Yn}nC¢=o be any sequence in X and define {Cn},~__0 by
w~ = (1 - b~)y~ + b=Ty~,
~n = ]]Yn+l -- (1 -- an)Yn -- anTwni],
n > 0 ,
n > _ 0 .
Then
(iv)
(v) (vi)
REMARK 3. i.
the sequence {xn}n~=o converges strongly to the unique f ixed-point q o f T . Moreover,
( 1 ) ~ Ilxn - qll < 1 - -~ ar Nxo - qll <- anllxo - qH, n > O,
where a = e-(1/2) ar;
[lYn+l -- qH < (1 - (1/2) ar)iiy n - qH A- en for all rt > O; limn--.oo Yn = q i f and only i f limn--oo (~ = 0.
Suppose t ha t the sequences {an}n~=o, {bn}nC%_o are in [0, 1] and sat isfy the condi t ion
k ( 1 - k ) bn <a, , , n>_O, (3.23) a <_ an <_ L3 Jr 5L 2 Jr 4L + 3 '
326 Z. LIu AND S. N'I. KANG
where a c (0, 1). I t follows from (3.23) t ha t for any n _> 0,
L(1 + L)(a~ + bn) + ( L - 1)L2anbn + ( 2 - k)(1 + L + L ( L - 1)bn)an
3 2 (2 - k) (1 L < 2L(1 + L)a~ + L a n + + + L 2) an
_< [ L ( I + L ) ( 2 + L ) + ( 2 - k ) ( I + L + L 2 ) ] a n
L 3 + (5 - k )L 2 + (4 - k )L + (2 - k) k(1 - k) L 3 + 5L 2 + 4L + 3
< k(1 - k),
which implies t h a t (3.2) is satisfied with r = k s. Thus, Theorem 3 of Osilike [5] is a par t icular
case of our Corol lary 3.1.
THEOREM 3.2. Suppose that X is an arbitrary real Banach space and T : X -+ X is a Lipschitz
s trongly accretive operator. A s s u m e that {un}n~=o, {Vn}n~_0 are sequences in X and {an}~=o, {bn}~-o are sequences in [0, 1] satisfying (3.1), (3.3), and the following condition:
L. (1 + L . ) (an + bn) + (L . - 1)L2.anbn
+(2 - k)(1 + L . + L . ( L . - 1)bn)an < k - r, n > _ 0 , (3.24)
where r E (0, k). For arbitrary xo, f E X define the sequence {Xn}n°°__ 0 b y
zn = ( 1 - b,~)xn + bn ( f + xn - T z n ) + vn, n >_ O,
X n + 1 = (1 - a~)xn + a n ( f + zn - Tzn ) + Un, n > O.
Le t {yn}n~_-0 be any sequence in X and define {en}n~__0 by
W n = ( 1 - - bn)y,~ + b n ( f + Yn -- Tyn ) ~- Vn, Tt ~ O,
e n = I l Y n + x - - ( 1 - - an)yn -- a n ( f + w~ - T w n ) - u n l l , n > 0.
' /~ l lc l l
(i) X (3o the sequence { n}n=O converges strongly to the unique solution q o f the equation T x = f ;
moreover,
( 1 ) ° I l xn - qll ~ 1 - ~ a~ IIx0 - qll +
2 (1 - (1 - (1/2) a ~ y ) c , n > o, a r
where C = sup{(3L. + f2 , )Llvnl l + (3 + L . ) l l u n l [ :n ~ 0}; (ii) ]]Yn+l - q[[ -< (1 - (1/2) ar)]]yn - q]] + cn + []Unll + (3L, + L2.)]]vn[[ for all n >_ O;
(iii) l i m n ~ y~ = q i f and only i f l i m n ~ (n = 0.
PROOF. Define S : X --~ X by S x = f + ( I - T ) x for all x E X . I t follows f rom L e m m a 2.2, t h a t the equat ion T x = f has a unique solution q E X. Hence, S has a unique fixed point q. Obviously, S is Lipschitz with constant L , = 1 + L. Observe t ha t for all x, y E X , there exist j ( x - y) E J ( x - y) and k c (0, 1) such t ha t
( ( ± - s ) x - ( I - s ) u , j ( x - u ) ) = ( T x - T u , j ( x - y ) ) > k l l x - Ull ~.
T h a t is, S is a s t rongly pseudocont rac t ive opera tor . Thus, Theo rem 3.2 follows immedia te ly f rom
T h e o r e m 3.1. This completes the proof.
S t a b i l i t y of I s h i k a w a I t e r a t i o n M e t h o d s
COROLLARY 3.2. Suppose that X , T , r, a, { a n } n ~ = 0 , {bn}n°°__ 0 are as in Theorem X oo arbitrary Xo, f ~ X define the sequence { ~}n=0 by
z , = (1 - b , )x~ + b , ( f + x n - - Tx~) , n >_ O,
Xn+l = (1 - an)Xn + a n ( f + zn - Tzn) , n > O.
Le t {yn},~__o be any sequence in X and define {e~},~°°__ o by
w , = (1 - bn)y , + b , ( f + y~ - Ty~) , n > 0,
= Ily.+ - (1 - a . ) y . - + - n > 0 .
T h e n
327
3.2. For
(iv) the sequence {Xn}n~__o converges strongly to the unique solution q o f the equation T x = f ; moreover,
( 1 ) ° Ilx~ - qU <- 1 - ~ ar I l x o - q U - < c y ~ U x o - q [ t , n_>O,
w h e r e gr = e - ( 1 / 2 ) ar;
(v) [[Yn+l - q[[ _< (1 - (1/2) ar)[[yn - q[[ + e~ for all n >_ O;
(vi) l i ln~_,~ y~ = q i f and only iflim,~--,oo en = O.
REMARK 3.2. T h e o r e m 3.2 reveals tha t the Ishikawa i terat ion me thod with errors is bo th S-s tab le and converges s t rongly to the fixed point of S which is tile solution of the equat ion T x = f .
R E M A R K 3 . 3 . Let {an}~__0, {b,~}~°~= 0 be sequences in [0, 1] and satisfy the condit ion
k(1 - k) a < _ a n < _ L 3 + 4 L l , + L , + 3 , bn<_a,,, n > 0 , (3.25)
where a E (0, 1). Thus, (3.25) yields t ha t
L , (1 + n,)(a~, + bn) + L2,(L, - 1)a~b,~ + (2 - k)(1 + L , + L , ( L , - 1)b,~)a~, 2 < [2(1 + L , ) L , + L , ( L , - 1) + (2 - k) (1 + n.2)] an
L 3 + ( 3 - k)Ll, + 2L, + ( 2 - k) < * k ( 1 - k )
- n 3 + 4L, 2 + L , + 3
< k(1 - k),
for all n _> 0. T h a t is, (3.24) holds wi th r = k 2. Thus, Corol lary 3.2 generalizes T h e o r e m 4 of Osilike [5].
In the rest of this paper , let k s tand for any fixed cons tant in (0, 1).
THEOREM 3.3. Suppose that X is all arbitrary real Banach space and T : X ---* X is a Lipschi tz , O 0 O0 b O0 accretive operator. A s s u m e that {un}~=0 , {v,,}n°°=o are sequences in X and {a,~}~=o , { ~}~=o
are sequences ill [0, 1] satisfying (3.1), (3.3), and the following condition:
L(1 + n)(an + b,~) + (L - 1)L2anbn (3.26)
+(1 + L + L ( L - 1)b,,)an < k, n >_ O.
For arbitrary xo, f C X define the sequence {xn}n°C=o by
Zn = (1 - bn)xn -+- b n ( f - T X n ) q- Vn, r~ > O,
Xn+ 1 = (1 - an)X n n t- a n ( f - Tzn ) + u,,, n > O. (3.27)
328 Z. LIu AND S. ~i. KANG
Let {Yn}n=o be any sequence in X and define {en}n=o by
Wn = (1 -- bn)Yn + bn( f - Tyn) + vn, n > O,
~n ~-~ [ [ Y n + l - - (1 - an)Yn - a n ( f - Twn) - Unl[, n >_ O. (3.28)
Then
(i) the sequence {xn}~=o converges strongly to the unique solution q of the equation x + T x = f ; moreover,
[IXn - qll -< 1 - ~ (1 - k)a [Ixo - qlJ + (1 - k)a C, n >_ O, ( 3 . 2 9 )
where C = sup{(2L + L2)llv.l[ + (1 + L)llun[[ : n _> 0}; (ii) IlYn+l - qI[ -< (1 - (1/2) (1 - k)a)llyn - ql[ + en + IlUnl[ + (2L + L2)llvni[ for 311 n >_ O;
(iii) limn--.oo Yn = q i f and only if l i m n - ~ en = 0.
PROOF. Define S : X ~ X b y S x = f - T x for a l l x E X. It follows from Lemma 2.3 tha t the equat ion x + T x = f has a unique solution q E X. Tha t is, S has a unique fixed-point q. Equat ion (1.1) ensures tha t
II x - Yl[ <- l[ x - Y + r ( r x - Ty) l I = t l x - y - r ( S x - Sy)ll, (3.30)
for all x, y c X, and r > 0. Observe tha t
X n = X n + l + a n X n -- a n S Z n -- t tn
= (1 + a n ) X n + l -- a n S X n + l + a n ( S X n + l - - S Z n )
+ a~(xn - Szn) - (1 + an)tt n
and
q = (1 + an)q - anSq,
for any n > O. Using (3.30), we have
X n + l a n Ilxn - qll >- (1 + an) - q 1 + a------:~ (Sxn+~ - Sq)
-- a n l l S X n + l -- S Z n l l - a2nllXn -- S Z n l l - - (1 + a n ) l l u n l l
> (1 + an)llxn+l - qH - anllSXn+l -- SznN
- a~llx~ - Sz~ l l - (1 + anDIlu.ll,
an 1 Ilxn - qll + l t s x n + , - Sznl l 1 +
2 a n + ~ ILzn - ¢oznLL + Llunll
1 <- - - 1 + a n [1 + L ( 1 + L)a,~(an + bn) + L 2 (L - 1)a~bn
+(1 + L + L ( L - 1)bn)a~] [[xn - qH
( an L + l ) llunl, 1 [Lan(1 + anL) + La2n] ][vnl[ + 1 + a,~ + l + a n
<_ 11 + a - - - ' ~ + k a n ][xn -q[[ + (2L + L 2) [IVn[I + (1 + n)[[unH
[ 1 ( 1 - k)a] I lXn - q[I + (2L + L 2) [IVnll + (1 + L)llunl[, < 1 - ~
for all n > 0, which implies tha t (3.29) holds. The rest of the argument is now essentially the same as in the proof of Theorem 3.1 and is, therefore, omitted. This completes the proof.
which means that
Ilmn+l -- qll -----
Stability of Ishikawa Iteration Methods 329
COROLLARY 3.3. Suppose that X , T , a, {a~}~=o, (b~)n~=o are as in Theorem 3.3. For arbi- x oo t r a r y x0, f E X define the sequence ( n}n=o by
z~ = (1 - bn)xn + b n ( / - T z n ) , 'n >_ O,
Xn+l = (1 - an)Xn 4" a n ( f -- Tzn) , n > O.
Le t {y,}n°%_o be a n y sequence ill X and define {en}n°~__ 0 by
W n ---- ( 1 - bn)Yn 4- b n ( f - Tyn) , n > O,
c n = I l Y n + l - (1 - an)yn - a n ( f - Twn)[[, n _> 0.
Then
(iv) the sequence {Xn }n°°_ 0 converges s trongly to the unique solution q o f the equation x + T x = f ; moreover,
( 1 )° I l x n - q[I - 1 - ~ ( 1 - k)a Hxo - all ~ ~ n l l x 0 - a ir , n _ > 0 ,
where a : e - (1 /2) (1-k)a;
(v) t[Yn+l -- qll --~ (1 -- (1 /2) (1 -- k)a)Ilyn - q]l 4- en for all n >_ O;
(vi) l imn-.oo yn = q f f a n d only f f l imn- .o~ cn -- 0.
R E M A R K 3 . 4 . Assume t h a t (an}~=0, {bn}~=o are sequences in [0, 1] and sa t i s fy t he cond i t ion
k a < an <_ L 3 + 4 L 2 + 3 L + 2 , bn <_ an, n > O, (3.31)
where a E (0, 1). As in the proof of R e m a r k 3.2, we deduce from (3.31) t h a t for all n >_ 0,
L(1 + L)(an 4- bn) 4- (L - 1)L2anbn 4- (1 4- L 4- L ( L - 1)bn)an
L 3 4- 2L 2 4- 2L 4- 1 < k < k . - L 3 4- 4L 2 4- 3L 4- 2
T h a t is, (3.26) holds. Thus , Coro l l a ry 3.3 general izes T he o re m 5 of Osil ike [5].
REMARK 3.5. T h e o r e m s 3.1-3.3 ex tend the cor respond ing resul ts of [1-5] to the I sh ikawa i t-
e r a t i on m e t h o d wi th er rors and to a r b i t r a r y real Banach spaces which do not d e p e n d on any
geomet r i c s t r uc tu r e of the under l ing Banach spaces, respectively.
REMARK 3.6. P r o t o t y p e s for {an}~-0 and {bn}~=0 in Theo rems 3.1-3.3 as follows, respect ively.
n + l k - r n + l k - r bn = - - n > 0;
an = 5(n + 2) " 5(1 + L)2(1 + L2) ' 3(n + 2) " 5 ( 1 + L) 2 '
n + l k - r n + l k - r bn = - - n __ 0;
an -- 6(n + 2) " 5 . ( 1 + 5 . ) : ( 1 + 5.2) ' 8(n + 2) " 5 . ( 1 + 5 . ) 2 '
n + l k n + l k b~ = - - n > 0 .
a s -- 8(n + 2) " L(1 + L)~(1 + 52) ' 6(n + 2) " L(1 + L) 2 '
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