Applied Mathematics and Computation 133 (2002) 389–406 www.elsevier.com/locate/amc
Iterative processes with mixed errors for nonlinear equations with perturbed m-accretive operators in Banach spaces q Jong Soo Jung a, Yeol Je Cho
b,*
, Haiyun Zhou
c
a
b
Department of Mathematics, Dong-A University, Pusan 604-714, South Korea Department of Mathematics, Gyeongsang National University, Chinju 660-701, South Korea c Department of Mathematics, Shijiazhuang Mechanical Engineering College, Shijiazhuang 050003, People’s Republic of China
Abstract Let X be a real uniformly smooth Banach space of which the dual X has a Frechet differentiable norm. Let A : DðAÞ X ! 2X be an m-accretive operator with closed domain DðAÞ and bounded range RðAÞ and S : X ! X a continuous and a-strongly accretive operator with bounded range RðI SÞ. It is proved that the Ishikawa and Mann iterative processes with mixed errors converge strongly to the unique solution of the equation z 2 Sx þ kAx for given z 2 X and k > 0. Ó 2002 Elsevier Science Inc. All rights reserved. Keywords: m-accretive operator; a-strongly accretive operator; Ishikawa iterative process; Mann iterative process; Nonexpansive retraction
1. Introduction Let X be a real Banach space with norm k k whose dual space is denoted by X . The normalized duality mapping J from X into the family of nonempty subsets of X is defined by 2
J ðxÞ ¼ fj 2 X : hx; ji ¼ kxk ; kjk ¼ kxkg; q
Supported by Korea Research Foundation Grant (KRF-2000-DP0013). Corresponding author. E-mail address:
[email protected] (Y.J. Cho).
*
0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 2 3 9 - 9
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where h ; i denotes the generalized duality pairing. It is an immediate consequence of the Hahn–Banach theorem that J ðxÞ is nonempty for each x 2 X . Moreover, it is known that J is single valued if and only if X is smooth, while if X is uniformly smooth, then the mapping J is uniformly continuous on bounded sets. An operator A : DðAÞ X ! 2X with domain DðAÞ and range RðAÞ is said to be k-accretive (k 2 R) if for each x; y 2 DðAÞ there exists jðx yÞ 2 J ðx yÞ such that hu v; jðx yÞi P kkx yk2
ð1Þ
for all u 2 Ax and v 2 Ay. For k > 0 in the inequality (1), we say that A is strongly accretive, while for k ¼ 0, A is simply called accretive. In addition, if the range of I þ kA is precisely X for all k > 0, then A is said to be m-accretive (cf. [9]). Along with the family of k-accretive mappings, we find a family of operators intimately related to this one, which is known as k-pseudo-contractive (see [23]). This latter family is formed by mappings written as I A, where I is the identity and A is k-accretive. In the single-valued case, an operator T is said to be k-pseudo-contractive if for each x; y 2 DðT Þ there exists jðx yÞ 2 J ðx yÞ such that 2
hTx Ty; jðx yÞi 6 kkx yk : Once again if k < 1, T is called strongly pseudo-contractive, while if k ¼ 1, T is called pseudo-contractive. Incidentally, these operators were introduced by Browder [2], while the notion of accretive operators was independently introduced by Browder [2] and Kato [16]. In the case X ¼ H is a Hilbert space, one of the earliest problems in the theory of accretive operators was the solvability of the equation z ¼ x þ Ax for a given z 2 H with A accretive (see for instance [3,11,22]). In [2], Browder actually proved that if A is locally Lipschitzian and accretive with DðAÞ ¼ X , then A is m-accretive. In particular, for any z 2 X , the equation z ¼ x þ Ax has a unique solution. This result was later generalized by Martin [21] to continuous accretive operators, and most recently Morales [24] extended it to the multi-valued case. Nevertheless, in this paper we are mainly interested in a class of operators somehow more general than the strongly accretive ones, which are defined as follows. Let a: ½0; 1Þ ! ½0; 1Þ be a function for which að0Þ ¼ 0, lim inf r!1 aðrÞ > 0 and the function hðrÞ ¼ raðrÞ is nondecreasing on ½0; 1Þ. An operator
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A: DðAÞ X ! 2X is called a-strongly accretive if for each x; y 2 DðAÞ there exists jðx yÞ 2 J ðx yÞ such that hu v; jðx yÞi P aðkx ykÞkx yk for all u 2 Ax and v 2 Ay. We also say that T is a-strongly pseudo-contractive if I T is a-strongly accretive (see [24,25]). In the last 10 years or so, the theory of single- (multi-)valued accretive, maccretive and strongly accretive operators in connection with the Ishikawa [12] and Mann [20] iterative processes have been studied by many authors in the attempt to approximate fixed points of some nonlinear mappings and solutions of some nonlinear operator equations in Banach spaces (see [4– 6,15,19,35,36]). Some further extensions of these iterative methods by adding an error term have also been explored (see [10,13,14,18,26,32,34]). In particular, Liu [18] and Xu [32] introduced the iterative processes which they called Ishikawa and Mann iterative processes ‘‘with errors’’ for nonlinear strongly accretive mappings. But we notice here that Xu’s process with errors is a special case of Liu’s process with errors. Very recently, Xue et al. [33] also introduced the Ishikawa iterative process with ‘‘with mixed errors’’ for m-accretive operators. On the other hand, recently, Jung and Morales [15] investigated the convergences of the Ishikawa and Mann iterative processes to approximate the solution of nonlinear operator equations of the type z 2 Sx þ kAx
ðEÞ
for all x 2 DðAÞ, z 2 X and k > 0, where A : DðAÞ X ! 2X is an m-accretive operator and S : X ! X is a continuous and a-strongly accretive operator, and generalized the previous corresponding results. Jung [13,14] gave some strong convergence theorems of the Ishikawa and Mann iterative processes with errors for nonlinear operator equation (E). The main purpose of this paper is to study the convergences of the Ishikawa and Mann iterative processes with mixed errors to approximate the solution of nonlinear operator equation (E). Our study can be viewed as a continuation of [15] in order to extend the main results of [15] to the case with mixed errors. First, as a more general case, we study convergences of the Ishikawa and Mann iterative processes with mixed errors for approximating the unique fixed point of a-strongly pseudo-contractive operator. Then we establish some convergence theorems of the Ishikawa and Mann iterative processes with mixed errors for approximating the unique solution of Eq. (E) in uniformly smooth Banach spaces of which the dual spaces have the Frechet differentiable norm. Our results improve, generalize and unify most of the known previous results in [7,10,13,19,26,32,35] as well as the results in [15].
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2. Preliminaries and lemmas Let X be a real Banach space and X the dual space of X . Let U ¼ fx 2 X : kxk ¼ 1g be the unit sphere of X . The norm of X is said to be G^ateaux differentiable (and X is said to be smooth) if the limit lim t!0
kx þ tyk kxk t
exists for each x and y in U . It is said to be Frechet differentiable if, for each x 2 U , the limit is obtained uniformly for y 2 U . Finally, the space X is said to have a uniformly Frechet differentiable norm (and X is said to be uniformly smooth) if the limit is attained uniformly for ðx; yÞ 2 U U . Since the dual X of X is uniformly convex if and only if the norm of X is uniformly Frechet differentiable, it follows that if X is uniformly convex, then X has a Frechet differentiable norm. The converse is false [8]. In the sequel, we need the following lemmas for the proof of our main results. The first lemma is actually Lemma 1 of Petryshyn [27]. Also Asplund [1] proved a general result for single-valued duality mappings, which can be used to derive the lemma below. Lemma 1. Let X be a real Banach space and let J be the normalized duality mapping. Then for any given x; y 2 X , we have 2
2
kx þ yk 6 kxk þ 2hy; jðx þ yÞi
ð2Þ
for all jðx þ yÞ 2 J ðx þ yÞ. Proof. Let x; y 2 X and jðx þ yÞ 2 J ðx þ yÞ. Then 2
kx þ yk ¼ hx þ y; jðx þ yÞi ¼ hx; jðx þ yÞi þ hy; jðx þ yÞi 6
1 2 2 ðkxk þ kjðx þ yÞk Þ 2
þ hy; jðx þ yÞi: Therefore we have 2
2
kx þ yk 6 kxk þ 2hy; jðx þ yÞi: This completes the proof.
The next lemma is indeed Theorem 1 of [24], where the notion of aexpansiveness is used in the following sense: a mapping A : DðAÞ X ! 2X is said to be a-expansive if for every x; y 2 DðAÞ, u 2 Ax and v 2 Ay, ku vk P aðkx ykÞ;
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where the function a maps ½0; 1Þ into itself with að0Þ ¼ 0 and aðrÞ > 0 for all r > 0. Lemma 2 (Theorem 1 of (Morales, 1985 [24])). Let X be a uniformly smooth Banach space. Suppose A: DðAÞ X ! 2X is m-accretive and a-expansive for which lim inf r!1 aðrÞ > 0. Then A is surjective. Lemma 3 (Xue et al., 2000 [33]). Let fan g, fbn g, fcn g and fwn g be four nonnegative real sequences satisfying anþ1 6 ð1 tn Þan þ wn an þ bn þ cn ;
n P n0 ; P1 P1 where nP 0 is some positive integer, 0 6 tn < 1, n¼0 tn ¼ 1, bn ¼ oðtn Þ, n¼0 cn < 1 and 1 w < 1. Then lim a ¼ 0. n n!1 n n¼0
3. Main results Now, we give our main results in this paper. Theorem 1. Let X be a uniformly smooth Banach space and T : DðT Þ X ! 2X a-strongly pseudo-contractive with a fixed point x in DðT Þ. Suppose there exists a nonexpansive retraction Q of X onto DðT Þ. Let fun g, fvn g be two sequences in X and fan g, fbn g be real sequences in ½0; 1 satisfying the followingPconditions: 1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 00 and kun k ¼ oðan Þ, 2. limn!1 kvn k ¼ 0, P1 3. limn!1 an ¼ limn!1 bn ¼ 0 and n¼0 an ¼ 1. Then, for an arbitrary initial value x0 in DðT Þ, the Ishikawa scheme with errors xnþ1 ¼ Qpn ; n P 0; pn 2 ð1 an Þxn þ an TQyn þ un ; yn 2 ð1 bn Þxn þ bn Txn þ vn ;
nP0
ð3Þ
n P 0;
converges strongly to the unique fixed point of T provided that there exist bounded selections fwn g and fzn g with wn 2 TQyn and zn 2 Txn . Proof. Since ku00n k ¼ oðan Þ, we see that u00n ¼ en an , where en ! 0 as n ! 1. Due to the choice of wn and zn , Eq. (3) can be re-written as xnþ1 ¼ Qpn ; n P 0; pn ¼ ð1 an Þxn þ an ðwn þ en Þ þ un ; yn ¼ ð1 bn Þxn þ bn zn þ vn ;
n P 0:
n P 0;
ð4Þ
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Since the sequences fen g, fwn g and fzn g are bounded, we may denote d ¼ sup kwn þ en x k þ sup kzn x k þ kx0 x k nP0
nP0
and M ¼dþ
1 X
ku0n k:
n¼0
This implies that kp1 x k ¼ kð1 a1 Þðx1 x Þ þ a1 ðw1 þ e1 x Þ þ u01 k 6ð1 a1 ÞkQp0 Qx k þ a1 kw1 þ e1 x k þ ku01 k 6ð1 a1 Þkp0 x k þ a1 d þ ku01 k 6ð1 a1 Þðð1 a0 Þkx0 x k þ a0 kw0 þ e0 x kÞ þ ku00 k þ a1 d þ ku01 k 6ð1 a1 Þðð1 a0 Þd þ a0 dÞ þ a1 d þ ku00 k þ ku01 k ¼ d þ ku00 k þ ku01 k: By induction, we obtain kpn x k 6 d þ
n X
ku0i k;
n P 0;
i¼0
and hence kpn x k 6 M;
ð5Þ
n P 0;
as well as kxn x k ¼ kQpn1 Qx k 6 kpn1 x k 6 M and kyn x k ¼ kð1 bn Þðxn x Þ þ bn ðzn x Þ þ vn k 6 M þ kvn k and so kJ ðQyn x Þk ¼ kQyn Qx k 6 kyn x k 6 M þ kvn k: On the other hand, we also have 2
2
kQyn x k 6 kyn x k ¼ kð1 bn Þðxn x Þ þ bn ðzn x Þ þ vn k
2
6 ð1 bn Þ2 kxn x k2 þ 2bn hzn x ; J ðyn x Þi þ 2hvn ; J ðyn x Þi 2
6 kxn x k þ 2ðbn kzn x k þ kvn kÞkyn x k 2
6 kxn x k þ 2ðbn M þ kvn kÞðM þ kvn kÞ; n P 0: ð6Þ
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By using (2), we have kpn x k2 ¼ kð1 an Þðxn x Þ þ an ðwn þ en x Þ þ u0n k2 2
2
6 ð1 an Þ kxn x k þ 2an hwn þ en x ; J ðpn x u0n Þi þ 2hu0n ; J ðpn x Þi:
ð7Þ
Now observe that 2
2hu0n ; J ðpn x Þi 6 2ku0n kkpn x k 6 ku0n kð1 þ kpn x k Þ
ð8Þ
and hwn þ en x ;J ðpn x u0n Þi wn þ en x pn x u0n xn x ;J ¼ J ð1þ kxn x kÞ2 1þ kxn x k 1þ kxn x k 1þ kxn x k w n þ en x xn x Qyn x 2 ;J þ J ð1 þ kxn x kÞ 1þ kxn x k 1 þ kxn x k 1þ kxn x k þ hwn þ en x ;J ðQyn x Þi 2
6Ln ðAn þ Bn Þð1þ kxn x kÞ þ hwn þ en x ;J ðQyn x Þi;
ð9Þ
where kwn þ en x k ; 1 þ kxn x k pn x un xn x ; An ¼ J J 1 þ kx x k 1 þ kxn x k n xn x Qyn x : Bn ¼ J J 1 þ kx x k 1 þ kxn x k n Ln ¼
1
Now we show that fLn gn¼0 is bounded and An ; Bn ! 0 as n ! 1. Indeed, kwn þ en x k=ð1 þ kxn x kÞ 6 M. Since fkwn þ en x k=ð1 þ kxn x kÞg1 n¼0 and fkxn x k=ð1 þ kxn x kÞg1 n¼0 are bounded and pn x u0n xn x kwn þ en x k kxn x k 1 þ kx x k 1 þ kx x k 6 an 1 þ kx x k þ 1 þ kx x k n
n
n
n
!0 as n ! 1, by uniform continuity of J on bounded subsets of X , we have An ! 0 as n ! 1. Also, since
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xn x Qyn x 1 þ kx x k 1 þ kx x k n
n
kxn yn k bn ðkxn x k þ kzn x kÞ þ kvn k 6 6 !0 1 þ kxn x k 1 þ kxn x k as n ! 1, we obtain Bn ! 0 as n ! 1. By using (6)–(9), we have kpn x k2 6ð1an Þ2 kxn x k2 þ2an hwn þen x ;J ðQyn x Þi 2
2
þ2an Ln ðAn þBn Þð1þkxn x kÞ þku0n kð1þkpn x k Þ 2
2
2
6ð1an Þ kxn x k þ2an ðkQyn x k aðkQyn x kÞkQyn x kÞ þ2an ken kðM þkvn kÞþ2an Ln ðAn þBn Þð1þkxn x kÞ
2
2
þku0n kð1þkpn x k Þ
aðkQyn x kÞ 6ð1an Þ2 kxn x k2 þ2an 1 kyn x k2 kQyn x k þ2an ken kðM þkvn kÞþ2an Ln ðAn þBn Þð1þkxn x kÞ
2
2
þku0n kð1þkpn x k Þ 2
2
6ð1an Þ kxn x k þ2an
aðkQyn x kÞ 2 1 kxn x k kQyn x k
þ4an ðbn M þken kþkvn kÞðM þkvn kÞ 2
2
þ2an Ln ðAn þBn Þð1þkxn x kÞ þku0n kð1þkpn x k Þ aðkQyn x kÞ 2 2 kpn1 x k 6ð1þa2n Þkpn1 x k 2an kQyn x k þ4an ðbn M þken kþkvn kÞðM þkvn kÞ 2
2
þ2an Ln ðAn þBn Þð1þkpn1 x kÞ þku0n kð1þkpn x k Þ: ð10Þ If lim inf n!1 kQyn x k > 0, then there exists k > 0 such that k
0 for all n P N , we have
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397
1 ku0n k þ a2n 2kan þ 4an Ln ðAn þ Bn Þ þ ku0n k 2 kpn1 x k 1 ku0n k 4an Ln ðAn þ Bn Þ þ 1 ku0n k 4an ðbn M þ ken k þ kvn kÞðM þ kvn kÞ ku0n k þ þ 1 ku0n k 1 ku0n k 2k an 4Ln ðAn þ Bn Þ an kpn1 x k2 6 1 1 ku0n k ku0n k 2 kpn1 x k þ 1 ku0n k 4an Ln ðAn þ Bn Þ þ 4an ðbn M þ ken k þ kvn kÞðM þ kvn kÞ þ 1 ku0n k 0 kun k : ð11Þ þ 1 ku0n k
kpn x k 6
In (11), put an ¼ kpn1 x k2 ;
wn ¼
ku0n k ; 1 ku0n k
bn ¼
4an Ln ðAn þ Bn Þ þ 4an ðbn M þ ken k þ kvn kÞðM þ kvn kÞ ; 1 ku0n k
tn ¼
2k an 4Ln ðAn þ Bn Þ : 1 ku0n k 1
Then, since fLn gn¼0 is bounded and An ; Bn ! 0 as n ! 1 and bn ; en ; vn ! 1 as n ! 1, we see that bn ¼ oðan Þ and there exists N 0 > N such that tn ¼
2k an 4Ln ðAn þ Bn Þ Pk 1 ku0n k
for all n P N 0 . Then the inequality (11) reduces to anþ1 6 ð1 kan Þan þ wn an þ bn þ wn ; P1 where n¼N 0 wn < 1 and bn ¼ oðan Þ. It follows from Lemma 3 that an ! 0 as n ! 1, that is, fpn g converges strongly to x as n ! 1. Since x 2 DðT Þ and Q is a nonexpansive retraction, we obtain kxnþ1 x k ¼ kQpn Qx k 6 kpn x k ! 0 as n ! 1, that is, fxn g converges strongly to x . Suppose now that lim inf n!1 kQyn x k ¼ 0. Then there exists a subsequence fynj g of fyn g such that limj!1 kQynj x k ¼ 0. For a given e > 0 we may choose j0 2 N so that
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e kxj0 x k < pffiffiffi ; 2
e kpj0 Qyj0 k < ; 2
sj < ea
e
; 2
ku0j k
e ¼ : 2 2 Since we can also derive 2
2
2
kpn x k 6 ð1 an Þ kxn x k þ 2an hwn þ en x ; J ðpn x Þi þ 2hu0n ; J ðpn x Þi 6 ð1 an Þ2 kxn x k2 þ 2an hwn þ en x ; J ðQyn x Þi þ 2an hwn þ en x ; J ðpn x Þ J ðQyn x Þi þ 2ku0n kkpn x k 6 ð1 an Þ2 kxn x k2 2
þ 2an ðkQyn x k aðkQyn x kÞkQyn x kÞ þ 2an ½dn þ ken kðM þ kvn kÞ þ 2ku0n kM 2
6 ð1 þ a2n Þkxn x k 2an aðkQyn x kÞkQyn x k þ 4an ðbn M þ kvn kÞðM þ kvn kÞ þ 2an ½dn þ ken kðM þ kvn kÞ þ 2ku0n kM 6 kxn x k2 2an aðkQyn x kÞkQyn x k þ an ½an M 2 þ 4ðbn M þ ken k þ kvn kÞðM þ kvn kÞ þ 2dn þ 2ku0n kM for all n P 0, it follows that 2
2
kxj0 þ1 x k 6 kxj0 x k 2aj0 aðkQyj0 x kÞkQyj0 x k þ aj0 sj0 þ 2ku0j0 kM e e e e2 e2 þ aj0 ea þ ¼ e2 ; < 2aj0 a 2 2 2 2 2
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which is a contradiction. Therefore kxj0 þ1 x k < e and, inductively, we have kxn x k < e for all n P j0 . Therefore the sequence fxn g converges strongly to the unique fixed point of T . This completes the proof. By using Theorem 1, we have the following: Theorem 2. Let X be a uniformly smooth Banach space and A : DðAÞ X ! 2X be m-accretive with bounded range RðAÞ. Suppose that S : DðAÞ ! X is continuous and a-strongly accretive with RðI SÞ being bounded. Suppose that there exists a nonexpansive retraction Q of X onto DðAÞ. Let fun g, fvn g be two sequences in X and fan g, fbn g be real sequences in ½0; 1 satisfying the following conditions: P1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ, 2. limn!1 kvn k ¼ 0, P1 3. limn!1 an ¼ limn!1 bn ¼ 0 and n¼0 an ¼ 1. Then, for any x0 2 DðAÞ and z 2 X , the Ishikawa scheme with errors xnþ1 ¼ Qpn ;
n P 0;
pn 2 ð1 an Þxn þ an ðz þ ðI S kAÞQyn Þ þ un ; n P 0; yn 2 ð1 bn Þxn þ bn ðz þ ðI S kAÞxn Þ þ vn ; n P 0; converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0. Proof. Since the duality mapping J is single-valued, the operator S þ kA is accretive and hence by Theorem 5.3 of [17], it is also m-accretive. In addition, since S is a-strongly accretive, so is S þ kA, and in particular is a-expansive. Therefore, by Lemma 2, Eq. (E) has a unique solution x for each z 2 X and k > 0. Set Tx ¼ z þ x Sx kAx. Then T is a-strongly pseudo-contractive with x as its unique fixed point. Therefore, by Theorem 1, we conclude that the sequence fxn g converges strongly to x . This completes the proof. About the existence of a nonexpansive retraction Q, we first observe that if X is a Banach space whose dual X has a Frechet differentiable norm and A: DðAÞ X ! 2X is m-accretive, then DðAÞ is convex (see [30]). Secondly, if X is a reflexive and strictly convex Banach space and A : DðAÞ X ! 2X is maccretive, then there exists a nonexpansive retraction Q of X onto coðDðAÞÞ, the closed convex hull of the domain of A. In particular, if X is a uniformly smooth Banach space of which the dual X has a Frechet differentiable norm, then such a nonexpansive retraction Q is defined by Qx ¼ limr!0þ Jr ðxÞ for each x 2 X , where Jr ¼ ðI þ rAÞ1 and r > 0 (see [28,31]).
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Using Theorems 1 and 2, we obtain the following: Corollary 1. Let X be a uniformly smooth Banach space of which the dual X has a Frechet differentiable norm. Let A, S, RðAÞ, RðI SÞ, fun g, fu0n g, fu00n g, fvn g, fan g and fbn g be as in Theorem 2. Let DðAÞ be closed. Then, for any x0 2 DðAÞ and z 2 X , the Ishikawa scheme with errors xnþ1 ¼ Qpn ; n P 0; pn 2 ð1 an Þxn þ an ðz þ ðI S kAÞQyn Þ þ un ; n P 0; yn 2 ð1 bn Þxn þ bn ðz þ ðI S kAÞxn Þ þ vn ; n P 0; converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0, where Q is a nonexpansive retraction of X onto DðAÞ. Proof. Since X has a Frechet differentiable norm, X is reflexive and strictly convex. Thus the result follows from the above observations and Theorem 2. From Theorem 2 and Corollary 1, we have the following: Corollary 2. Let X , A, RðAÞ, S, and RðI SÞ be as in Theorem 2. Let fun g be a bounded sequence in X and fan g a real sequence in ½0; 1 satisfying the following conditions: P1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ, P 1 2. limn!1 an ¼ 0 and n¼0 an ¼ 1. If there exists a nonexpansive retraction Q of X onto DðAÞ, then for any x0 2 DðAÞ and z 2 X , the Mann scheme fxn g with errors xnþ1 ¼ Qpn ; n P 0; pn 2 ð1 an Þxn þ an ðz þ ðI S kAÞxn Þ þ un ;
n P 0;
converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0. Proof. The conclusion follows from Theorem 2 with vn ¼ 0 and bn ¼ 0 for all n P 0. Corollary 3. Let X be a Banach space which is both uniformly convex and uniformly smooth. Let A, DðAÞ, RðAÞ, S, RðI SÞ, fun g, fu0n g, fu00n g, fvn g, fan g and fbn g be as in Theorem 2. Then, for any x0 2 DðAÞ and z 2 X , the Ishikawa scheme fxn g with errors xnþ1 ¼ Qpn ; n P 0; pn 2 ð1 an Þxn þ an ðz þ ðI S kAÞQyn Þ þ un ; yn 2 ð1 bn Þxn þ bn ðz þ ðI S kAÞxn Þ þ vn ;
n P 0; n P 0;
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converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0, where Q is a nonexpansive retraction of X onto DðAÞ. Proof. Since X is uniformly convex, X has a Frechet differentiable norm. Hence the result follows from Corollary 1. Corollary 4. Let X be a Banach space which is both uniformly convex and uniformly smooth. Let A, DðAÞ, RðAÞ, S, and RðI SÞ be as in Theorem 2. Let fun g be a bounded sequence in X and fan g a real sequences in ½0; 1 satisfying the following conditions: P 0 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with 1 n¼0 kun k < 1 00 and kun k ¼ oðan Þ, P 2. limn!1 an ¼ 0 and 1 n¼0 an ¼ 1. If there exists a nonexpansive retraction Q of X onto DðAÞ, then for any x0 2 DðAÞ and z 2 X , the Mann scheme fxn g with errors xnþ1 ¼ Qpn ;
n P 0;
pn 2 ð1 an Þxn þ an ðz þ ðI S kAÞxn Þ þ un ;
n P 0;
converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0. Remark 1. Theorem 2 and Corollary 1 make use of simple inequality (2) rather than Reich’s inequality in [29], which was used in Theorem 3.1 of [10], and so the assumption limn!1 an bðan Þ ¼ 0 was excluded. With mixed errors, Corollaries 1 and 3 improve Theorem 3.1 of [10] and Theorem 2.1 of [19] to a more general nonlinear operator equation of the form z 2 Sx þ kAx, k > 0, involving the multi-valued m-accretive operator (in particular, without the Lipschitz condition in contrast to [19]) in more general Banach spaces. Further, with un ¼ 0 and vn ¼ 0, n P 0, Corollary 1 generalizes Theorem 3 of [7] in several aspects. Remark 2. Theorem 5 in [7] and Theorem 3 in [36] are the special cases of Corollary 2 in more general Banach spaces. Using Theorem 1 and the above observations, we can also obtain the following: Theorem 3. Let X be a uniformly smooth Banach space of which the dual X has a Frechet differentiable norm and T : DðT Þ X ! 2X a-strongly pseudo-contractive with a fixed point x in DðT Þ. Let fun g and fvn g be two sequences in X and fan g and fbn g two real sequences in ½0; 1 satisfying the following P1 conditions: 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ,
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2. limn!1 kvn k ¼ 0, P1 3. limn!1 an ¼ limn!1 bn ¼ 0 and n¼0 an ¼ 1. Then, for an arbitrary initial value x0 in DðT Þ, the Ishikawa scheme with errors xnþ1 ¼ Qpn ;
n P 0;
pn 2 ð1 an Þxn þ an TQyn þ un ; n P 0; yn 2 ð1 bn Þxn þ bn Txn þ vn ; n P 0; converges strongly to the unique fixed point of T , provided that there exist bounded selections fwn g and fzn g with wn 2 TQyn and zn 2 Txn , where Q is a nonexpansive retraction of X onto DðT Þ. Corollary 5. Let X and T be as in Theorem 3. Let fun g be a sequence in X and fan g a real sequences in ½0; 1 satisfying: P1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ, P 1 2. limn!1 an ¼ 0 and n¼0 an ¼ 1. Then, for any x0 2 DðT Þ and z 2 X , the Mann scheme with errors xnþ1 ¼ Qpn ;
nP0
pn 2 ð1 an Þxn þ an Txn þ un ;
n P 0;
converges strongly to the unique fixed point of T provided that there exists a bounded selection fwn g with wn 2 Txn , where Q is a nonexpansive retraction of X onto DðT Þ. Proof. The conclusion follows from Theorem 3 with vn ¼ 0, bn ¼ 0 for all n P 0. Corollary 6. Let X be a Banach space which is both uniformly convex and uniformly smooth. Let T : DðT Þ X ! 2X be a-strongly pseudo-contractive with a fixed point x in DðT Þ. Let fun g and fvn g be two sequences in X and fan g and fbn g two real sequences in ½0; 1 satisfying the following conditions: P 0 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with 1 n¼0 kun k < 1 00 and kun k ¼ oðan Þ, 2. limn!1 kvn k ¼ 0, P1 3. limn!1 an ¼ limn!1 bn ¼ 0 and n¼0 an ¼ 1. Then, for an arbitrary initial value x0 in DðT Þ, the Ishikawa scheme with errors xnþ1 ¼ Qpn ; n P 0; pn 2 ð1 an Þxn þ an TQyn þ un ; yn 2 ð1 bn Þxn þ bn Txn þ vn ;
n P 0; n P 0;
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converges strongly to the unique fixed point of T provided that there exist bounded selections fwn g and fzn g with wn 2 TQyn and zn 2 Txn , where Q is a nonexpansive retraction of X onto DðT Þ. Corollary 7. Let X be a Banach space which is both uniformly convex and uniformly smooth. Let T be as in Theorem 3. Let fun g be a sequence in X and fan g a real sequence in ½0; 1 satisfying the following conditions: P1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ, P 1 2. limn!1 an ¼ 0 and n¼0 an ¼ 1. Then, for any x0 2 DðT Þ and z 2 X , the Mann scheme with errors xnþ1 ¼ Qpn ; n P 0; pn 2 ð1 an Þxn þ an Txn þ un ;
n P 0;
converges strongly to the unique fixed point of T provided that there exists a bounded selection fwn g with wn 2 Txn , where Q is a nonexpansive retraction of X onto DðT Þ. Remark 3. Theorem 3.3 of [32] can be also considered as a special case of Theorem 3 and Corollary 5. If, in Theorems 1 and 2, the domain of A happens to be the whole space X , then the assumption on the existence of a retraction Q will not be needed. Therefore the following results can be stated. Theorem 4. Let X be a uniformly smooth Banach space and A : DðAÞ ¼ X ! 2X m-accretive with bounded range RðAÞ. Suppose that S : X ! X is continuous and a-strongly accretive with RðI SÞ being bounded. Let fun g and fvn g be two sequences in X and fan g and fbn g two real sequences in ½0; 1 satisfying the following conditions: P1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ, 2. limn!1 kvn k ¼ 0, P1 3. limn!1 an ¼ limn!1 bn ¼ 0 and n¼0 an ¼ 1. Then, for any x0 2 X and z 2 X , the Ishikawa scheme with errors xnþ1 2 ð1 an Þxn þ an ðz þ ðI S kAÞyn Þ þ un ; yn 2 ð1 bn Þxn þ bn ðz þ ðI S kAÞxn Þ þ vn ;
n P 0; n P 0;
converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0. Theorem 5. Let X be a uniformly smooth Banach space and T : DðT Þ ¼ X ! 2X a-strongly pseudo-contractive with a fixed point x in DðT Þ. Let fun g and fvn g be
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two sequences in X and fan g and fbn g two real sequences in ½0; 1 satisfying the following conditions: P 0 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with 1 n¼0 kun k < 1 00 and kun k ¼ oðan Þ, 2. limn!1 kvn k ¼ 0, P1 3. limn!1 an ¼ limn!1 bn ¼ 0 and n¼0 an ¼ 1. Then, for an arbitrary initial value x0 in DðT Þ, the Ishikawa scheme with errors xnþ1 2 ð1 an Þxn þ an Tyn þ un ; n P 0; yn 2 ð1 bn Þxn þ bn Txn þ vn ; n P 0; converges strongly to the unique fixed point of T , provided that there exist bounded selections fwn g and fzn g with wn 2 Tyn and zn 2 Txn . With vn ¼ 0 and bn ¼ 0, n P 0, in Theorems 4 and 5, we have the following: Corollary 8. Let X , A, RðAÞ, S, and RðI SÞ be as in Theorem 4. Let fun g be a sequence in X and fan g a real sequences in ½0; 1 satisfying the following conditions: P1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 and ku00n k ¼ oðan Þ, P 2. limn!1 an ¼ 0 and 1 n¼0 an ¼ 1. Then, for any x0 2 X and z 2 X , the Mann scheme with errors xnþ1 2 ð1 an Þxn þ an ðz þ ðI S kAÞxn Þ þ un ;
n P 0;
converges strongly to the unique solution of the equation z 2 Sx þ kAx for k > 0. Corollary 9. Let X and T be as in Theorem 5. Let fun g be a sequence in X and fan g a real sequences in ½0; 1 satisfying the following conditions:P 1 1. un ¼ u0n þ u00n for any sequences fu0n g, fu00n g in X and n P 0 with n¼0 ku0n k < 1 00 and kun k ¼ oðan Þ, P 1 2. limn!1 an ¼ 0 and n¼0 an ¼ 1. Then for any x0 2 X , the Mann scheme with errors xnþ1 2 ð1 an Þxn þ an Txn þ un ;
n P 0;
converges strongly to the unique fixed point of T provided that there exists a bounded selection fwn g with wn 2 Txn . Remark 4. Theorem 4 and Corollary 8 extend Theorem 3.2 and Corollary 3.2 of [10] to the case that A is multi-valued and to the case of the more general operator equation of the form z 2 Sx þ kAx, k > 0, with mixed errors, respectively. With un ¼ 0, n P 0, Corollary 8 also generalizes the corollary of [36] not only to uniformly smooth Banach space without uniform convexity but also to the case of a more general operator equation.
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Remark 5. Theorem 5 and Corollary 9 are also significant generalizations of Theorem 3.3 and Corollary 3.4 of [32] and Corollary 3 of [26] to the more general class of operators than the strong pseudo-contractive ones. Remark 6. By letting un ¼ vn ¼ 0, n P 0, in Theorem 1, Corollary 1 and Theorems 3–5, we also obtain the main results of [15]. Remark 7. It is clear that the hypothesis that the ranges of A and ðI SÞ are bounded imposed in Theorem 2, Corollary 1 and Theorem 4, respectively, can be replaced by the assumption that the sequences fðI S kAÞxn g, fðI S kAÞQyn g are bounded. Remark 8. By putting u00n ¼ 0, n P 0, in iterative processes with mixed errors defined in our main results, we also obtain Liu’s processes with errors. Remark 9. In the case that S : DðSÞ X ! 2X is an a-strongly accretive operator with a solution of the equation z 2 Sx, from Theorem 1, Corollary 1 and Theorem 5, we can also obtain some convergence theorems of the Ishikawa and Mann iteration processes with mixed errors for approximating the unique solution of the equation z 2 Sx. References [1] E. Asplund, Positivity of duality mappings, Bull. Am. Math. Soc. 73 (1967) 200–203. [2] F.E. Browder, Nonlinear mappings of nonexpansive and accretive type in Banach spaces, Bull. Am. Math. Soc. 73 (1967) 875–882. [3] R.E. Bruck Jr., The iterative solution of the equation y 2 x þ Tx for a monotone operator T in Hilbert space, Bull. Am. Math. Soc. 79 (1973) 1258–1262. [4] S.S. Chang, Y.J. Cho, B.S. Lee, J.S. Jung, S.M. Kang, Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudo-contractive mappings in Banach spaces, J. Math. Anal. Appl. 224 (1998) 149–165. [5] C.E. Chidume, Iterative solution of nonlinear equations with strongly accretive operators, J. Math. Anal. Appl. 192 (1995) 502–518. [6] C.E. Chidume, Convergence theorems for strongly pseudo-contractive and strongly accretive maps, J. Math. Anal. Appl. 228 (1998) 254–264. [7] C.E. Chidume, M.O. Osilike, Approximation methods for nonlinear operator equations of the m-accretive type, J. Math. Anal. Appl. 189 (1995) 225–239. [8] M.M. Day, Normed Linear Spaces, third ed., Springer, Berlin, 1973. [9] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985. [10] X.P. Ding, Iterative process with errors of nonlinear equations involving m-accretive operators, J. Math. Anal. Appl. 209 (1997) 191–201. [11] W.G. Doston, An iterative process for nonlinear monotone nonexpansive operators in Hilbert space, Math. Comput. 32 (1978) 223–225. [12] S. Ishikawa, Fixed point by a new iteration method, Proc. Am. Math. Soc. 44 (1974) 147–150. [13] J.S. Jung, Iterative approximation for perturbed m-accretive operator equations in arbitrary Banach spaces, Commun. Appl. Nonlinear Anal. 8 (1) (2001) 51–62.
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