Some comments on Noor’s iterations in Banach spaces

Applied Mathematics and Computation 206 (2008) 12–15

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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Some comments on Noor’s iterations in Banach spaces q Zhiqun Xue *, Ruiqin Fan Department of Mathematics and Physics, Shijiazhuang Railway Institute, Shijiazhuang 050043, P.R. China

a r t i c l e

i n f o

a b s t r a c t In this paper, the proofs of theorem concerning Rafiq [Arif Rafiq, Modified Noor iterations for nonlinear equations in Banach spaces, Appl. Math. Comput. 182 (2006) 589–595] are proved incorrect. And his results are revised. Ó 2008 Elsevier Inc. All rights reserved.

Keywords: Modified three-step iterative process Uniformly continuous Strongly pseudocontractive map Strongly accretive operator Banach spaces

1. Introduction The paper by Rafiq [8] contains two mistakes. In the proof of Theorem 2 of [8] the following problems occured. (i) In formula (2.7) (i.e. kyn  xnþ1 k ¼ k  bn ðxn  T 2 zn Þ þ an ðxn  T 1 yn Þk 6 bn kxn  T 2 zn k þ an kxn  T 1 yn k 6 2Mðan þ bn Þ), the author estimated kxn  T 2 zn k 6 M which cannot be assured in the 15th line of page 593; (ii) The uniform continuity of T 1 and T 3 leads to limn!1 kT 1 yn  T 3 xnþ1 k ¼ 0, which is wrong in the 19th line of page 593. Hence the final results are unable to be obtained under these conditions. The following is his main results. Theorem R. Let E be a real Banach space and K be a nonempty closed convex subset of E. Let T 1 ; T 2 ; T 3 be strongly pseudocontractive self maps of K with T 1 ðKÞ bounded and T 1 ; T 3 be uniformly continuous. Let fxn g1 n¼0 be the sequence defined by

8 > < xnþ1 ¼ ð1  an Þxn þ an T 1 yn ; yn ¼ ð1  bn Þxn þ bn T 2 zn ; > : zn ¼ ð1  cn Þxn þ cn T 3 xn ; n P 0;

ð1:1Þ

1 1 where fan g1 n¼0 , fbn gn¼0 and fcn gn¼0 are the three real sequences in [0, 1] satisfying the conditions

lim an ¼ lim bn ¼ 0;

n!1

n!1

1 X

an ¼ 1:

n¼0

If FðT 1 Þ \ FðT 2 Þ \ FðT 3 Þ–;, then the sequence fxn g1 n¼0 converges strongly to the common fixed point of T 1 ; T 2 ; T 3 . In order to correct above mentioned errors, it is highly advisable in this remark to provide their correct versions. In this  paper, suppose that E is an arbitrary real Banach space with a dual E . The normalized duality mapping from E to 2E is defined by

Jx ¼ ff 2 E : hx; f i ¼ jjxjj  jjf jj ¼ jjf jj2 g; where h; i denotes the generalized duality pairing. We denote the single-valued normalized duality mapping by j.

q

Project supported by Hebei province science and technology foundation (No. 072056197D). * Corresponding author. E-mail address: [email protected] (Z. Xue).

0096-3003/$ - see front matter Ó 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2008.09.001

ð1:2Þ

Z. Xue, R. Fan / Applied Mathematics and Computation 206 (2008) 12–15

13

A mapping T with domain DðTÞ and range RðTÞ in E is said to be strongly pseudocontractive if for all x; y 2 DðTÞ, there exist jðx  yÞ 2 Jðx  yÞ and a constant k 2 ð0; 1Þ so that

hTx  Ty; jðx  yÞi 6 kkx  yk2 :

ð1:3Þ

Closely related to the class of strongly pseudocontractive is the important class of strongly accretive. It is well known that T is strongly pseudocontractive if and only if ðI  TÞ is strongly accretive, where I denotes the identity map E. Therefore, an operator T : E ! E is called strongly accretive if there exists a constant k > 0 so that

hTx  Ty; jðx  yÞi P kkx  yk2

ð1:4Þ

holds for all x; y 2 E and some jðx  yÞ 2 Jðx  yÞ. Without the loss of generality, we assume that 0 < k < 1. Such operators have been studied and used extensively by several researchers. The concept of accretive operator was first introduced independently by Browder [1] and Kato [2] in 1967. Interest in accretive operators stems mainly from their connection with the existing theory of solutions for nonlinear evolution equations in Banach spaces. It is known that many physically significant problems can be modeled by an initial value problem of the form

dxðtÞ þ TxðtÞ ¼ 0; xð0Þ ¼ x0 ; dt

ð1:5Þ

where T is an accretive operator in an appropriate Banach space. The examples where such evolution equations occur can be seen in the heat, wave or Schrodinger equations. The solutions to the equation Tx ¼ 0 are precisely the equilibrium points of the system (1.5). At present, several research papers are published on the methods of approximating these equilibrium point. The Mann [3] iterative scheme, invented in 1953, was used to prove the convergence of the sequence to a fixed point of many mappings to which the Banach principle is not appliable. Later, in 1974, Ishikawa [4] devised a new iteration scheme to establish the convergence of a Lipschitzian pseudocontractive map when Mann iteration process failed to converge. In 2000, Noor [5] gave the following three-step iteration process for approximating fixed point and solving the nonlinear equation. Let D be a nonempty closed convex subset of E and T : D ! D be a mapping. For an arbitrary x0 2 D. The sequence fxn g1 n¼0  D, defined by

8 > < zn ¼ ð1  cn Þxn þ cn Txn ; yn ¼ ð1  bn Þxn þ bn Tzn ; > : xnþ1 ¼ ð1  an Þxn þ an Tyn ; n P 0;

ð1:6Þ

1 1 where fan g1 n¼0 , fbn gn¼0 and fcn gn¼0 are three real sequences satisfying an ; bn ; cn 2 ½0; 1, is called the three-step iteration (or Noor iteration). When cn ¼ 0, then the three-step iteration reduces to Ishikawa iterative sequence, fxn g1 n¼0  D defined by



yn ¼ ð1  bn Þxn þ bn Txn ;

ð1:7Þ

xnþ1 ¼ ð1  an Þxn þ an Tyn ; n P 0: If bn ¼ cn ¼ 0, then (1.6) becomes Mann iteration. It is the sequence fxn g1 n¼0  D defined by

xnþ1 ¼ ð1  an Þxn þ an Txn ; n P 0:

ð1:8Þ

Recently, Rafiq [8] introduced a new type of iteration – the modified three-step iterative process. Let T 1 ; T 2 ; T 3 : D ! D be three mappings. For any given x0 2 D, the modified three-step iteration fxn g1 n¼0  D is defined by

8 > < xnþ1 ¼ ð1  an Þxn þ an T 1 yn ; yn ¼ ð1  bn Þxn þ bn T 2 zn ; > : zn ¼ ð1  cn Þxn þ cn T 3 xn ; n P 0

ð1:9Þ

1 1 with fan g1 n¼0 , fbn gn¼0 and fcn gn¼0 being three real sequences satisfying some conditions. It is clear that the iteration schemes (1.6)–(1.8) are (1.9) in special cases. To provide the correct results for the proof in [8], we need to introduce the following lemmas.

Lemma 1.1 [10]. Let E be a real Banach space, then for all x; y 2 E, there exists jðx þ yÞ 2 Jðx þ yÞ so that

kx þ yk2 6 kxk2 þ 2hy; jðx þ yÞi: Lemma 1.2 [11]. Let fqn g1 n¼0 be a nonnegative sequence which satisfies the following inequality:

qnþ1 6 ð1  kn Þqn þ rn ; n P 0; where kn 2 ð0; 1Þ, n ¼ 0; 1; 2; . . .,

P1

n¼0 kn

¼ 1 and

rn ¼ oðkn Þ. Then qn ! 0 as n ! 1.

14

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2. Main results Theorem 2.1. Let E be a real Banach space and D be a nonempty closed convex subset of E. Let T 1 ; T 2 and T 3 be strongly pseudocontractive self maps of D with T 1 ðDÞ bounded and, T 1 ; T 2 and T 3 uniformly continuous. Let fxn g1 n¼0 be defined by (1.9), , fbn g1 and fcn g1 are three real sequences in ½0; 1 which satisfy the conditions: an ; bn ! 0 as n ! 1 and where fan g1 n¼0 n¼0 n¼0 P1 1 n¼0 an ¼ 1. If FðT 1 Þ \ FðT 2 Þ \ FðT 3 Þ–;, then the sequence fxn gn¼0 converges strongly to the common fixed point of T 1 ; T 2 and T 3 . Proof. Since T i (i ¼ 1; 2; 3) is strongly pseudocontractive, then there exists a common constant k ¼ maxfk1 ; k2 ; k3 g so that

hT i x  T i y; jðx  yÞi 6 kkx  yk2 ;

i ¼ 1; 2; 3;

where ki ði ¼ 1; 2; 3Þ is the strongly pseudocontractive constant of T i ði ¼ 1; 2; 3Þ, respectively. Let q 2 FðT 1 Þ \ FðT 2 Þ \ FðT 3 Þ, by the definition of strongly pseudocontractive T i , we have FðT 1 Þ \ FðT 2 Þ \ FðT 3 Þ ¼ q. Let M1 ¼ kx0  qk þ supnP0 kT 1 yn  qk. By induction, we obtain

kxn  qk 6 M 1

8n:

Indeed, it is clear that kx0  qk 6 M 1 . Assume that kxn  qk 6 M 1 hold. Next we will prove that kxnþ1  qk 6 M 1 . From (1.9), it follows that

kxnþ1  qk 6 kð1  an Þðxn  qÞ þ an ðT 1 yn  qÞk 6 ð1  an Þkxn  qk þ an kT 1 yn  qk 6 ð1  an ÞM 1 þ an M1 ¼ M 1 :

ð2:1Þ

Using uniform continuity of T 3 , we get that fT 3 xn g is bounded. Denote M 2 ¼ maxfM1 ; supfkT 3 xn  qkgg, then

kzn  qk 6 cn kxn  qk þ ð1  cn ÞkT 3 xn  qk 6 cn M 1 þ ð1  cn ÞM 2 6 M2 : Thus, by virtue of uniform continuity of T 2 , we get that fT 2 zn g is bounded. Set M ¼ supnP0 kT 2 zn  qk þ M 2 . Applying Lemma 1.1 and (1.9), we have

kxnþ1  qk2 ¼ kð1  an Þðxn  qÞ þ an ðT 1 yn  T 1 qÞk2 6 ð1  an Þ2 kxn  qk2 þ 2an hT 1 yn  T 1 q; jðxnþ1  qÞi 6 ð1  an Þ2 kxn  qk2 þ 2an hT 1 xnþ1  T 1 q; jðxnþ1  qÞi þ 2an hT 1 yn  T 1 xnþ1 ; jðxnþ1  qÞi 6 ð1  an Þ2 kxn  qk2 þ 2an kkxnþ1  qk2 þ 2an kT 1 yn  T 1 xnþ1 k  kxnþ1  qk 6 ð1  an Þ2 kxn  qk2 þ 2an kkxnþ1  qk2 þ 2an dn M 1 ;

ð2:2Þ

where dn ¼ kT 1 yn  T 1 xnþ1 k ! 0 as n ! 1. Indeed, since

kyn  xnþ1 k 6 bn kxn  T 2 zn k þ an kxn  T 1 yn k 6 bn ðkxn  qk þ kT 2 zn  qkÞ þ an ðkxn  qk þ kT 1 yn  qkÞ 6 bn ðM1 þ MÞ þ an ð2M 1 Þ 6 2Mðan þ bn Þ ! 0

ð2:3Þ

as n ! 1. It is easily seen that, in view of the uniform continuity of T 1 , dn ! 0 as n ! 1. Since an ! 0 as n ! 1, then there is a positive integer N so that

(

1 1k an < min ; 2k ð1  kÞ2 þ k2

)

for all n P N. It follows from (2.2) that

ð1  an Þ2 2an dn M 1 1  2kan þ 2kan  2an þ a2n 2an dn M 1 kxn  qk2 þ 6 kxn  qk2 þ 1  2kan 1  2kan 1  2kan 1  2kan   2  2k  an 2an dn M 1 2an dn M1 6 1 an kxn  qk2 þ 6 ð1  ð1  kÞan Þkxn  qk2 þ : 1  2kan 1  2kan 1  2kan

kxnþ1  qk2 6

ð2:4Þ

n dn M 1 Set qn ¼ kxn  qk2 , kn ¼ ð1  kÞan and rn ¼ 2a . Applying Lemma 1.2, we obtain kxn  qk ! 0 as n ! 1. This completes 12kan the proof. h

Remark 1. Our Theorem 2.1 revises the corresponding conditions and proofs of Theorem 2 of Rafiq [8] in the following way. (i) The uniform continuity of T 1 and T 3 [8, Theorem 2] is turned into the uniform continuity of T 1 ; T 2 and T 3 . (ii) In the reasoning course of Theorem 2.1, we obtain that fT 3 xn g; fzn g and fT 2 zn g are bounded, and by the uniform continuity of T 1 , get the correct form of dn ! 0 as n ! 1. Theorem 2.2. Let E be a real Banach space, T 1 ; T 2 ; T 3 : E ! E be uniformly continuous and strongly accretive operators with RðI  T 1 Þ bounded, where I is the identity mapping on E. Let q denote the unique common solution to the equation T i x ¼ f ði ¼ 1; 2; 3Þ. For a given f 2 E, define the operator Hi : E ! E by Hi x ¼ f þ x  T i x, i ¼ 1; 2; 3. For any x0 2 E, the sequence fxn g1 n¼0 is defined by

Z. Xue, R. Fan / Applied Mathematics and Computation 206 (2008) 12–15

8 > < xnþ1 ¼ ð1  an Þxn þ an H1 yn ; yn ¼ ð1  bn Þxn þ bn H2 zn ; > : zn ¼ ð1  cn Þxn þ cn H3 xn ; n P 0;

15

ð2:5Þ

1 1 where fan g1 n¼0 , fbn gn¼0 and fc n gn¼0 are three real sequences in [0, 1] satisfying the conditions: an ; bn ! 0 as n ! 1 and P1 1 n¼0 an ¼ 1. Then the sequence fxn gn¼0 converges strongly to the unique common solution to T i x ¼ f ði ¼ 1; 2; 3Þ.

Proof. We observe first that, if q is the unique common solution to the equation T i x ¼ f ði ¼ 1; 2; 3Þ, then q is the unique common fixed point of H1 ; H2 ; H3 . Since T 1 ; T 2 ; T 3 are all strongly accretive operators, then Hi ði ¼ 1; 2; 3Þ is a strongly pseudocontractive map. We observe also the fact that T i ði ¼ 1; 2; 3Þ is uniformly continuous with RðI  T 1 Þ bounded implies that Hi ði ¼ 1; 2; 3Þ is uniformly continuous with RðH1 Þ bounded, then the conclusion of Theorem 2.2 from Theorem 2.1 may be exactly obtained. h Remark 2. Theorem 2.2 extends the result of Theorem 3 of Rafiq [8] from an operator H to three different operators T 1 ; T 2 and T 3 . Remark 3. The following examples reveal that Theorem 2.1 is applicable. x for all x 2 D. Then Example. Let E ¼ ð1; þ1Þ with the usual norm and let D ¼ ½0; þ1Þ. Define T 1 : D ! D by T 1 x ¼ 2ð1þxÞ FðT 1 Þ ¼ f0g; RðT 1 Þ ¼ ½0; 12Þ, and T 1 is a uniformly continuous and strongly pseudocontractive mapping. Define T 2 : D ! D by T 2 x ¼ 2x for all x 2 D. Then FðT 2 Þ ¼ f0g and T 2 is a uniformly continuous and strongly pseudocontractive mapping. Define 2 T 3 : D ! D by T 3 x ¼ sin4 x for all x 2 D. Then FðT 3 Þ ¼ f0g and T 3 is a uniformly continuous and strongly pseudocontractive 1 1 1 ffi pffiffiffiffiffiffi for each n P 0. For an arbitrary x0 2 D, the sequence fxn g1 mapping. Set an ¼ nþ1, bn ¼ ðnþ1Þþðnþ1Þ 2 , cn ¼ n¼0  D defined nþ1 by (1.9) converges strongly to the common fixed point {0} of T 1 ; T 2 and T 3 .

References [1] [2] [3] [4] [5] [8] [10] [11]

F.E. Browder, Nonlinear mappings of nonexpansive and accretive in Banach spaces, Bull. Am. Math. Soc. 73 (1967) 875–882. T. Kato, Nonlinear semigroups and evolution equations, J. Math. Soc. Jpn. 19 (1967) 508–520. W.R. Mann, Mean value methods in iteration, Proc. Am. Math. Soc. 4 (1953) 506–510. S. Ishikawa, Fixed points by a new iteration method, Proc. Am. Math. Soc. 44 (1974) 147–150. M.A. Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000) 217–229. Arif Rafiq, Modified Noor iterations for nonlinear equations in Banach spaces, Appl. Math. Comput. 182 (2006) 589–595. Klaus Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin, 1985. X. Weng, Fixed point iteration for local strictly pseudocontractive mapping, Proc. Am. Math. Soc. 113 (3) (1991) 727–731.

Further reading [6] M.A. Noor, T.M. Rassias, Z.Y. Huang, Three-step iterations for nonlinear accretive operator equations, J. Math. Anal. Appl. 274 (2002) 59–68. [7] M.A. Noor, Some developments in general variational inequalities, Appl. Math. Comput. 152 (2004) 199–277. [9] J. Bogin, On strict pseudo-contractions and a fixed point theorem, Technion Preprint MT-29, Haifa, 1974.