Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 129069, 7 pages http://dx.doi.org/10.1155/2014/129069

Research Article

A Modified Mixed Ishikawa Iteration for Common Fixed Points of Two Asymptotically Quasi Pseudocontractive Type Non-Self-Mappings

Yuanheng Wang and Huimin Shi

Department of Mathematics, Zhejiang Normal University, Zhejiang 321004, China Correspondence should be addressed to Yuanheng Wang; wangyuanhengmath@163.com Received 3 January 2014; Accepted 21 February 2014; Published 26 March 2014 Academic Editor: Rudong Chen

Copyright © 2014 Y. Wang and H. Shi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new modified mixed Ishikawa iterative sequence with error for common fixed points of two asymptotically quasi pseudocontrac-tive type non-self-mappings is introduced. By the flexible use of the iterative scheme and a new lemma, some strong convergence theorems are proved under suitable conditions. The results in this paper improve and generalize some existing results.

1. Introduction

Let E be a real Banach space with its dual E* and let C be a nonempty, closed, and convex subset of E. The mapping J : E ^ 2e is the normalized duality mapping defined by

J (x) = {x* e E* : (x, x*) = ||%|| • llx* II, ||%|| = ||x* 11},

Let T : C ^ E be a mapping. We denote the fixed point set of T by F(T); that is, F(T) = {x e C : x = Txj. Recall that a mapping T : C ^ E is said to be nonexpansive if, for each x,yeC,

\\Tx-Ty\\<\\x-y\\. (2)

T is said to be asymptotically nonexpansive if there exists asequence kn c [l,() with kn ^ 1 as n ^ ( suchthat

\\Tnx-Tny\\<kn\\x-y\\, Vx,yeC. (3)

A sequence of self-mappings {Ti}'^=1 on C is said to be uniform Lipschitzian with the coefficient L if, for any i = 1,2,..., the following holds:

WTfx-T^yWcLWx-yW, Wx,yeC. (4)

T is said to be asymptotically pseudocontractive if there exist kn c [1, (x) with kn as n ^ >x> and j(x - y) e

J(x - y) such that

(Tnx-Tny,j(x-y))<kn\\x-y\\2, Vx,yeC. (5)

It is obvious to see that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically pseudocontractive. Goebel and Kirk [1] introduced the class of asymptotically nonexpansive mappings in 1972. The class of asymptotically pseudocontractive mappings was introduced by Schu [2] and has been studied by various authors for its generalized mappings in Hilbert spaces, Banach spaces, or generalized topological vector spaces by using the modified Mann or Ishikawa iteration methods (see, e.g.,[3-21]).

In 2003, Chidume et al. [22] studied fixed points of an asymptotically nonexpansive non-self-mapping T : C ^ E and the strong convergence of an iterative sequence {xn} generated by

xn+1 =P((1- an) xn + anT(PT)n-1xn), n>l,Xl eC,

where P : E ^ C is a nonexpansive retraction.

In 2011, Zegeye et al. [23] proved a strong convergence of Ishikawa scheme to a uniformly L-Lipschitzian and asymptotically pseudocontractive mappings in the intermediate sense which satisfies the following inequality (see [24]):

lim sup sup ({T"x - Tny, x- y) - kn\\x - < 0,

Vx, y eC,

where kn c [1, m) with kn ^ 1 as n ^ m.

Motivated and inspired by the above results, in this paper, we introduce a new modified mixed Ishikawa iterative sequence with error for common fixed points of two more generalized asymptotically quasi pseudocontractive type non-self-mappings. By the flexible use of the iterative scheme and a new lemma (i.e., Lemma 6 in this paper), under suitable conditions, we prove some strong convergence theorems. Our results extend and improve many results of other authors to a certain extent, such as [6, 8,14-23].

2. Preliminaries

Definition 1. Let C be a nonempty closed convex subset of a real Banach space E. C is said to be a nonexpansive retract (with P) of E if there exists a nonexpansive mapping P : E ^ C such that, for all x e C, Px = x. And P is called a nonexpansive retraction.

Let T : C ^ E be a non-self-mapping (maybe self-mapping). T is called uniformly L-Lipschitzian (with P) if there exists a constant L> 0 such that

\\T(PT)n—1x - T(PT)n—1y\\ <L\\x - ,

Vx, y e C, n> 1.

T is said to be asymptotically pseudocontractive (with P) if there exist kn c [1, m) with kn ^ 1 as n ^ m and Vx, y eC, 3j(x - y) e J(x - y) such that

(T(PT)n-1x - T(PT)n—1y, j(x-y))< kn\x - y||2. (9)

T is said to be an asymptotically pseudocontractive type (with P) if there exist kn c [1, m) with kn ^ 1 as n ^ m and Vx, y e C, j(x - y) e J(x - y) such that

lim sup sup lim inf (^T(PT)n—1 x - T(PT)n-1 y,

n^rn x,yeC j(x—y)ij(x—y)

j(x- y)) -kjx- yf) <

T is said to be an asymptotically quasi pseudocontractive type (with P) if P(T) = 0, for p e F(T), there exist kn c [1,m) with kn ^ 1 as n ^ m, and, Vx e C, j(x - p) e J(x - p) such that

lim sup sup lim inf ((T(PT)n1x - p, j (x - y)\ -K\\x-p\\2)<0.

Remark 2. It is clear that every asymptotically pseudocontractive mapping (with P) is asymptotically pseudocontractive type (with P) and every asymptotically pseudocontractive type (with P) is asymptotically quasi pseudocontractive type (with P). If T : C ^ C is a self-mapping, then we can choose P = I as the identical mapping and we can get the usual definition of asymptotically pseudocontractive mapping, and so forth.

Definition 3. Let C be a nonexpansive retract (with P) of E, let T1,T2 : C ^ E be two uniformly L-Lipschitzian non-self-mappings and let T1 be an asymptotically quasi pseudocontractive type (with P).

The sequence [xn] is called the new modified mixed Ishikawa iterative sequence with error (with P), if [xn] is generated by

Xn+1 =P((1-Xn - Yn) *n + XnTi(PTi)n-

x ((1 - Pn) yn + PnT1(PT1)n-1yn) + ynun),

7n =P((l-< -Y'n)*n + «nT2(PT2)n-1

X((l-ß'n )Xn +ß'n T2(PT2 r1Xn)+Yn Vn),

where x1 e C is arbitrary, {un} and {vn} c C are bounded, and «n,Pn,Yn,«L,y'n e [0,1], n= 1,2, — If an = fin = y'n = 0, (12) turns to

= P{(l-an -yn)xn + *nTi(PTi)

X ((1 - Pn) Xn + PnT1(PT1)n 1Xn) + ynUn) ,

and it is called the new modified mixed Mann iterative sequence with error (with P). If Yn = Yn = 0, (12) becomes

Xn+1 =P((1-«n )xn + «nT^r1

X n— 1

*((l-ßn)yn + ßnT1(PT1 )n—1yn)),

7n =P((1-< )*n + < T2(PT2)"

x((1-& )Xn + P'n T2(PT2)n—1Xn)),

and it is called the new modified mixed Ishikawa iterative sequence (with P).

If Pn = &„ = 0,(14) turns to

Xn—1

Xn+1 = P((1-Xn )Xn + anT1(PT1)n 1yn), yn =P((l-*n )Xn + T2(PT2)n—1Xn)

and it is called the new mixed Ishikawa iterative sequence (with P).

If T1 = T2 = T : C ^ C is a self-mapping and P = I is the identical mapping, then (15) is just the modified Ishikawa iterative sequence

n+1 = (l-an)xn + anT"yn' yn = (l-a'n )xn + a'n T"Xn.

If an = 0, (15) becomes (6), obviously. So, iterative method (12) is greatly generalized.

The following lemmas will be needed in what follows to prove our main results.

Lemma 4 (see [19]). Let E be a real Banach space. Then, for all x,y e E, j(x + y) e J(x + y), thefollowinginequalityholds:

+ y\f < \\x\\2 + 2 {x, j (x + y)) .

Lemma 5 (see [6, 7]). Let {an}, {bn}, {cn} be three sequences of nonnegative numbers satisfying the recursive inequality:

<(l + bn)an + cn, Vn>n0

where n0 is some nonnegative integer. If < >x>, <

>x>, then lim.

n^œa„ exists.

Lemma 6. Suppose that $ : [0,+ot) ^ [0,+ot) is a strictly increasing function with 0) = 0. Let {an}, {bn}, {cn}, {Xn} (0 < Xn < 1) be four sequences of nonnegative numbers satisfying the recursive inequality:

<(! + bn) an - Xncp (an+1) + cn, Mn >

where n0 is some nonnegative integer. If Z'TO=1bn < >x>, Z'TO=1cn < m, Z'TO=1Xn = m, then lim„^TOan = 0.

Proof. From (19), we get

<(l + bn)an + cn, Mn>

By Lemma 5, we know that lim„^TOa„ = a > 0 exists. Let M = sup1<n<TO{fln} < >x>. Now we show a = 0. Otherwise, if a > 0, then 3n1 > n0, such that an+1 > (1/2)a > 0 when n > n1. Because 0 is a strictly increasing function, so 4>(an+1) > <p((1/2)a) > 0. From (19) again, we have

/ 1 \ œ n=l

( 1 \ nl /1\ œ = ï{-2a)ÏK + ZK

n=1 n=nl+1

1 ni œ

<$[-a)1ZXn + Z Xn ^(an+1)

n=1 n=n1 + 1

1 ni œ

<^[^a)ZXn + Z (an -an+1

n=n-i + 1

+ Z bn an + Z cn

n=ni+1 n=n-i + 1

/ 1 \ ni œ œ

< $ [2a) ZXn + Uni+1 + MZhn + Z° n=1 n=1 n=1

This is a contradiction with the given condition I^1Xn = œ>.

Therefore lim„^œa„ = 0.

Lemma 7. Suppose that $ : [0, +ot) ^ [0, +ot) is a strictly increasing function with 0) = 0. Let {an}, {bn}, {cn}, {Xn} (0 < Xn < 1), {en} be five sequences of nonnegative numbers satisfying the recursive inequality:

<(l + bn)an - Xncp (an+1 ) + cn + Xnen, Vn>n0,

where n0 is some nonnegative integer. If bn < >x>, Z'TO=1cn < <m, Z'^=1Xn = <m, lim„^TO£„ = 0, then limn^TOan = 0.

Proof. Firstly, we show lim inf n^TOan = a = 0. If a > 0, then, for arbitrary r e (0, a), 3n1 > n0, such that an+1 > r > 0 when n > n1. Because 0 is a strictly increasing function and lirn„^me„ = 0, so $(an+1) > <p(r) > 0 and en < (1/2)<p(r) when n>n1. From (22), we have

an+1 < (1 + bn) an - Xncp (an+1) + cn + Xn(an+1) = {\+bn)an -2XJ(an+1) + cn, Vn>n1.

By Lemma 6,weget 0 = limn^TOan = lim inf = a > 0.

This is contradictory. So, lim inf n^man = 0.

Secondly, Ve > 0, from the given conditions in Lemma 7, 3n2 > n0, when Vn > n2, we have

£n <$(£)> Zbn < ln 2, (24)

n=n2 n=n2

On the other hand, since lim infn^TOan = 0, 3N > n2 such that aN < e. Now we claim

ak <(e+Zc») exp (ZM> yk>N. (25)

V n=N ) \n=N )

In fact, when k = N, (25) holds. Suppose that (25) holds for k dose not for k + 1. Then

ak+1 > ( e + Z Cn ) exp ( Zbn).

Furthermore, ak+1 > e, $(ak+1) > $(e). But by (22), (24), and the inductive hypothesis, we have

an+1 <{1 + bn)an - XJ (an+1) + cn + xn£n

<(1 + bn) an - Xncp (e) +cn + Xncp (e)

< (1 + K)( Z^n ) exp ( ZK

\ n=N )

< [•B + Z Cn) exp \Zh"

\ n=N ) \n=N

< \ 'Z + ^ Cn ) exp \ ^ bnj.

+ C„

+ C„

This is a contradiction with (26). So, (25) holds. Whereupon,

/ œ \ / œ

lim sup ak < ( e + £ cn ) exp ( X b>

k^rn \ n=N ) \n=N

<2(e + e) = 4e.

Therefor limsupk ^œak = 0 = limn^œfln.

3. Main Results

Now, we are in a position to state and prove the main results of this paper.

Theorem 8. Let С be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2 ■ С ^ E are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers [kn] с [1,+от) ■ kn ^ 1 satisfying p = p(T1) n F(T2) = 0. Suppose that [un},{vn] с С are two bounded sequences; {an}, {pn}, {yn}, {an}, {fin}, {y'n} с [0,1] are six number sequences satisfying the following:

(C1) ZZi«n = ZZia2 < - 1) <

(C2) an + Yn < 1, a'n + y'n < 1, Z™=iYn <

(C3) K=1«nPn < K=1«n< < K=1«nY'n <

Ifx1 e C is arbitrary, then the iterative sequence [xn] generated by (12) converges strongly to the fixed point x* e F if and only if there exists a strictly increasing function $ : [0, ^

[0, +ot) with 0) = 0 such that

lim sup inf [(T1(PT1)

n^rn j(xn+1-x')ej(xn+1-x')

xn+1 - x '

j (Xn+1 X )) kJ\Xn+i .

+ Ф(\\хп+1 -*D] <o.

Proof. (Adequacy). Let

4 = ы \{T1(PT1)n

j(x„+1-x' )£j(xn+1-x' )

Xn+1 X

){Xn+1 X )) kn\\xn+l :

+ ф(\\хп+1 -**\\)],

£n = max [i,0}+1.

Then there exists j(xn+1 - x* ) e J(xn+1 - x*) such that (T1{PT1 )n-1Xn+i -X*,j(Xn+i -x*))

- K\Xn+1 - X \ + Ф {\Xn+1 - X D < £n.

From (29), we know that lim supn^œ e'n < 0. So,limn^msn = 0.

Now, from the given conditions and (12), we can let

^n = (l-ßn)yn + ßn T1(PT1)n-1yn,

S« =(l-ßn )Xn + ß'n T2 (РТ2Г1 Xn,

and M = supna1|||^n - x*||, \\vn - x*||} < œ. Then

\\8n - x* II < ß'n \\T2 (PT2) x« -x*\\ + (l-ß'n) \\x« - x* II

^ /yf T N * N N * N

<ßnL\\xn-x \\ + \\xn-x II;

\\у« -x*\\<{l-a.'n -y'n )\\x« -x*\\

/7- * N f N * N

+ anL\\Sn -X \\+Yn\\Vn -X II

-Il *W f Ы T 2 N * II

< \\x« -X \\+ (XnßnL \\x« - x II

f T N * N f n r

+ anL\\Xn -x \\+YnM = (l + a'nß'nL2 + a'nL) \\Xn -X*\\+ Y'nM

<(l + L + L2)\\xn-x*\\ + M-,

<ßn \^Ti(PTi)n-1yn-**||

+ (l-ßn)\\yn -**||

<ßn Ц\у« -х*\\ + \\у« -x*\\

<(1 + L)(l + L + L2) \\xn -x*\\ + (1+ L) M;

\\<J„ - X

n Xn+l\

< anL ^O« - x ^+<xn \\Xn - x

'rll^ *N 'N *N

+ anL\\Sn -X \\+an\\Xn -X II

+ (y« + Yn)\\x« -x*\\ + {y« + Yn)m

<anL[(l + L)(l+L + L2 )\\xn -x*\\ + (1+L)M]

+ a'n Ф+ß'n L)\K -*!]

+ n +<x'n + Yn + Yn) \\xn -x*\\ + (Yn + Yn) M < [a.nL(l + L)(l + L + L2) + anL(l+ß'nL)

I ми * 11

+ an + an + Yn + Yn \ \\Xn -X II

+ (anL(l+L)+yn + Y« )M;

Kn+i\\ < НУ« x«+i

+ ßn ||Ti (PTiTlyn -yn

<Sn\\Xn - X \\+tn

sn = anL(l + L)(l+L + L2)+a'n L(l +l) + an

+ a'n + Yn + Yn + Pn (1+L)(1 + L + L2); (34)

tn =[anL(l+L)+Yn + rn + pn (1+L)]M. So, by Lemma 4,

(T1(PT1)n-1an -T1(PT1)n-1Xn+1 ,j(xn+1 -x*)) <2anL\\Xn+1 - x*\11 | On -Xn+i\\ (35)

< 2(*nL \\Xn+1 - X* || \\Xn -x*\\+tn];

II _ *\\2

\\Xn+1 X II

< (l-an - Ynf\\Xn -x*\\2

+ 2an (Ti{PTi)n-1On - X*, j (Xn+1 - x*)) + 2Yn fan -x*j(xn+1 -XD

< (l-an -Ynf\\Xn -x*\\2 (36) + 2^ {Ti(PTi)n-1On -Ti (PTi)n-1 Xn+1,

j(Xn+i -X*)) + 2an (Ti{PTi)n-1Xn+i -X*,j(Xn+i -x*)) + 2YnM\\xn+i -x*||. For the third in (36), we have 2an (Ti{PTi)n-1Xn+i -x*,j(xn+i -x*))

= 2Undn + 2«n [kn\\Xn+i - x*\\2 - 0 {\Xn+i - X* ||)] (37)

< 2anzn + 2an [kn\\xn+i - x*\\2 - $ (\xn+i - x* ||)],

dn = (Ti(PTi)n-1Xn+i -x*,j(xn+i -x*)) - KWxn+i - x*\\2 + 0 {\xn+i - x ||) < en-Substituting (35) into (36), we get

\\Xn+1 -x*\\2 < (1 - an)2\\xn - X*\\2 + 2an£n

+ 2Unkn\\Xn+i - X*\\2 - 2(Xn$ (\Xn+i - X* + 2<XnL (Sn \\Xn -X*\\+ tn) \\Xn+i -X*\\ + 2YnM\\Xn+i -%*||.

Let an = \\xn - x*\\2, f(t) = 2$(^t), and %n = Lan sn

= L2a2l (1 + L)(l+L + L2) + ana'nL2 (l + ß'nL) + anL + ana'nL + Lanyn + Lany'n + Lanßn (1+L)(l + L + L2 ), pn = Lan tn + Myn

= [a^L2 (1 + L)+ Lanyn + Lany'n + anßn (L + L2)]M + ynM.

Then (39) becomes

an+1 < (1 - anfan + 2an£n + 2anKan+1 - anf (an+i) + 2($n\\Xn -X*\\+ Pn)lxn+1 -X*\\-

By using 2ab < a2 + b2,we have

an+1 < (1 - an)2an + 2an£n + 2anKan+1

- anf (an+1 ) + (an + an+i) + Pn (1 + an+i)

= {l-2an + an + Qan + (2ankn + $n + Pn) an+i

- anf (an+1 ) + 2an£n + Pn-

From (40), (41), and the given conditions, we know

^ < +œ, Y^n < Ypn < +(44)

Then, \imn^m(2ankn + + pn) = 0. Therefore 3n0, when n > nQ, 2ankn + + Pn ^ 1/2. Let

b = 1-2«n + <*2n + - i = 2«n (K -1) + <4 + 2$n + Pn , n 1 - 2anK -tn - Pn 1- 2anK -tn - Pn '

C„ =

1 2an^n Pn

So, when n> MQ,weget

0<bn <2 [2«n (kn -l) + «n + 2$n + Pn], 0<Cn < 2Pn.

From (44) and the given conditions, we have ^^^ bn < +ot, ^n=no cn < +œ|. On the other hand, from (43), we have

an+i < (1 + K) an - anf (an+x ) + \anzn + cn, Vn>nQ.

By Lemma 7, we at last get

lim аи = lim ||хи - x*||2 = 0; (48)

и^то " и^то" "II '

for example, lim„^TOx„ = x* e F = F(î\) П F(T2).

(Necessity). Suppose that lim„^TOx„ = x* e F. Then we can choose an arbitrary continuous strictly increasing function ф : [0, +œ) ^ [0, +œ) with ф(0) = 0, such as <£(f) = i. Wecangetlim„^œ0(yx„+1 -x* ||) = 0.

Because Г1 is an asymptotically quasi pseudocontractive type (with P), by (11) in Definition 1, for any p e F(T1) 2 F, we have

lim sup sup liminf uT(PT)"-1x - p, j (x - y))

И^ то xeC j(*-p)e/(*-p)

-fc„||x-pf)<0.

limsuP if [(Т1(РТ1)И-1хи+1 - x*,

"^то j(x„+i-x* )e/(x„+i-x')

= limsup if [(Т1(РТ1)И-1хи+1 - x*,

и^то j(x„+i-x*)e/(x„+i-x«)

- ^и||хи+1 - X || ]

+ „^(К^ -x*||)<0 + 0 = 0;

that is, (29) holds. This completes the proof of Theorem 8. □

Combining with Theorem 8 and Definition 3, we have some results as follows.

Theorem 9. Let С be nonexpansive retract (with P) of a real Banach space E. Assume that Г1,Г2 : С ^ E are two uniformly L-Lipschitzian non-self-mappings (with P) and Г1 is an asymptotically quasi pseudocontractive type with coefficient numbers с [1,+œ) : ^ 1 satisfying F = F(T1) n F(T2) = 0. Suppose that |а„}, |£J, }, } с [0,1] are four number sequences satisfying the following:

(C1) = +œ, «2 < +œ, a„(fc„ - 1) < +œ;

(C2) Z~1«»A < +œ, < +œ.

If x1 e С is arbitrary, then the iterative sequence |хи} generated by (14) converges strongly to the fixed point x* e F if and only if there exists a strictly increasing function ф : [0, +œ) ^ [0, +œ) with ф(0) = 0 such that (29) holds.

Theorem 10. Let С be nonexpansive retract (with P) ofa real Banach space E. Assume that Г1,Г2 : С ^ E are two

uniformly L-Lipschitzian non-self-mappings (with P) and Г1 is an asymptotically quasi pseudocontractive type with coefficient numbers |fc„} с [1,+œ) : ^ 1 satisfying F = F(T1 ) n F(T2) = 0. Suppose that |аи}, } с [0,1] are two number sequences satisfying the following:

(C1) = +œ, ZTO=1«2 < +œ, Z~1«»(*» - 1) < +œ;

(C2) ^«Х < +œ.

If x1 e С is arbitrary, then the iterative sequence |хи} generated by (15) converges strongly to the fixed point x* e F if and only if there exists a strictly increasing function ф : [0, +œ) ^ [0, +œ) with ф(0) = 0 such that (29) holds.

Theorem 11. Let С be a nonempty closed convex subset of a real Banach space E. Assume that T : С ^ С is uniformly L-Lipschitzian self-mappings and asymptotically quasi pseudocontractive type with coefficient numbers |fc„} с [1,+œ) : ^ 1 satisfying F = F(T) = 0. Suppose that |аи}, } с [0, 1] are two number sequences satisfying the following:

(C1) 2ТОЛ = +œ, Z~1«2 < +œ, 2=1«»(*» - 1) <

(C2) ZTO^A < +œ.

If x1 e С is arbitrary, then the iterative sequence |хи} generated by (16) converges strongly to the fixed point x* e F if and only if there exists a strictly increasing function ф : [0, +œ) ^ [0, +œ) with ф(0) = 0 such that (29) holds.

Remark 12. Our research and results in this paper have the following several advantaged characteristics. (a) The iterative scheme is the new modified mixed Ishikawa iterative scheme with error on two mappings Г1,Г2. (b) The common fixed point x* e F = F(T1) n F(T2) is studied. (c) The research object is the very generalized asymptotically quasi pseudocontractive type (with F) non-self-mapping. (d) The tool used by us is the very powerful tool: Lemma 7. So, our results here extend and improve many results of other authors to a certain extent, such as [6,8,14-23], and the proof methods are very different from the previous.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the editors and referees for their many useful comments and suggestions for the improvement of the paper. This work was supported by the National Natural Science Foundations of China (Grant no. 11271330) and the Natural Science Foundations of Zhejiang Province (Grant no. Y6100696).

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