AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 129069 10.1155/2014/129069 129069 Research Article A Modified Mixed Ishikawa Iteration for Common Fixed Points of Two Asymptotically Quasi Pseudocontractive Type Non-Self-Mappings http://orcid.org/0000-0002-9846-3972 Wang Yuanheng Shi Huimin Chen Rudong 1 Department of Mathematics Zhejiang Normal University Zhejiang 321004 China zjnu.edu.cn 2014 2632014 2014 03 01 2014 21 02 2014 26 3 2014 2014 Copyright © 2014 Yuanheng Wang and Huimin 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 pseudocontractive 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:E2E* is the normalized duality mapping defined by (1)J(x)={x*E*:x,x*=x·x*,x=x*},00000000000000000000000000000000000000xE.

Let T:CE be a mapping. We denote the fixed point set of T by F(T); that is, F(T)={xC:x=Tx}. Recall that a mapping T:CE is said to be nonexpansive if, for each x,yC, (2)Tx-Tyx-y.

T is said to be asymptotically nonexpansive if there exists a sequence kn[1,) with kn1 as n such that (3)Tnx-Tnyknx-y,x,yC. A sequence of self-mappings {Ti}i=1 on C is said to be uniform Lipschitzian with the coefficient L if, for any i=1,2,, the following holds: (4)Tinx-TinyLx-y,x,yC.

T is said to be asymptotically pseudocontractive if there exist kn[1,) with kn1 as n and j(x-y)J(x-y) such that (5)Tnx-Tny,j(x-y)knx-y2,x,yC.

It is obvious to see that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is asymptotically pseudocontractive. Goebel and Kirk  introduced the class of asymptotically nonexpansive mappings in 1972. The class of asymptotically pseudocontractive mappings was introduced by Schu  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.,).

In 2003, Chidume et al.  studied fixed points of an asymptotically nonexpansive non-self-mapping T:CE and the strong convergence of an iterative sequence {xn} generated by (6)xn+1=P((1-αn)xn+αnT(PT)n-1xn),n1,x1C, where P:EC is a nonexpansive retraction.

In 2011, Zegeye et al.  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 ): (7)limsupnsupx,yC(Tnx-Tny,x-y-knx-y2)0,000000000000000000000000000000000x,yC, where kn[1,) with kn1 as n.

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, 1423].

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:EC such that, for all xC, Px=x. And P is called a nonexpansive retraction.

Let T:CE 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 (8)T(PT)n-1x-T(PT)n-1yLx-y,x,yC,n1.

T    is said to be asymptotically pseudocontractive (with P) if there exist kn[1,) with kn1 as n and x,yC, j(x-y)J(x-y) such that (9)T(PT)n-1x-T(PT)n-1y,j(x-y)knx-y2.

T is said to be an asymptotically pseudocontractive type (with P) if there exist kn[1,) with kn1 as n and x,yC, j(x-y)J(x-y) such that (10)limsupnsupx,yCliminfj(x-y)J(x-y)(T(PT)n-1x-T(PT)n-1y,xxxxxxxxxxxxxicxxxij(x-y)-knx-y2)0.

T is said to be an asymptotically quasi pseudocontractive type (with P) if F(T), for pF(T), there exist kn[1,) with kn1 as n, and, xC, j(x-p)J(x-p) such that (11)limsupnsupxCliminfj(x-p)J(x-p)(T(PT)n-1x-p,j(x-y)00000000000000000000-knx-p2)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:CC 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:CE 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 (12)xn+1=P((1-αn-γn)xn+αnT1(PT1)n-1cc×((1-βn)yn+βnT1(PT1)n-1yn)+γnun),yn=P((1-αn-γn)xn+αnT2(PT2)n-1cci×((1-βn)xn+βnT2(PT2)n-1xn)+γnvn), where x1C is arbitrary, {un} and {vn}C are bounded, and αn,βn,γn,αn,βn,γn[0,1], n=1,2,.

If αn=βn=γn=0, (12) turns to (13)xn+1=P((1-αn-γn)xn+αnT1(PT1)n-1cci×((1-βn)xn+βnT1(PT1)n-1xn)+γnun), and it is called the new modified mixed Mann iterative sequence with error (with P).

If γn=γn=0, (12) becomes (14)xn+1=P((1-αn)xn+αnT1(PT1)n-1cccci×((1-βn)yn+βnT1(PT1)n-1yn)),yn=P(((1-βn)xn+βnT2(PT2)n-1xn)(1-αn)xn+αnT2(PT2)n-1000000×((1-βn)xn+βnT2(PT2)n-1xn)), and it is called the new modified mixed Ishikawa iterative sequence (with P).

If βn=βn=0, (14) turns to (15)xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-αn)xn+αnT2(PT2)n-1xn), and it is called the new mixed Ishikawa iterative sequence (with P).

If T1=T2=T:CC is a self-mapping and P=I is the identical mapping, then (15) is just the modified Ishikawa iterative sequence (16)xn+1=(1-αn)xn+αnTnyn,yn=(1-αn)xn+αnTnxn. If αn=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 [<xref ref-type="bibr" rid="B19">19</xref>]).

Let E be a real Banach space. Then, for all x,yE, j(x+y)J(x+y), the following inequality holds: (17)x+y2x2+2x,j(x+y).

Lemma 5 (see [<xref ref-type="bibr" rid="B6">6</xref>, <xref ref-type="bibr" rid="B7">7</xref>]).

Let {an}, {bn}, {cn} be three sequences of nonnegative numbers satisfying the recursive inequality: (18)an+1(1+bn)an+cn,nn0, where n0 is some nonnegative integer. If Σn=1bn<, Σn=1cn<, then limnan exists.

Lemma 6.

Suppose that ϕ:[0,+)[0,+) is a strictly increasing function with ϕ(0)=0. Let {an},{bn},{cn},{λn}(0λn1) be four sequences of nonnegative numbers satisfying the recursive inequality: (19)an+1(1+bn)an-λnϕ(an+1)+cn,nn0, where n0 is some nonnegative integer. If Σn=1bn<, Σn=1cn<, Σn=1λn=, then limnan=0.

Proof.

From (19), we get (20)an+1(1+bn)an+cn,nn0. By Lemma 5, we know that limnan=a0 exists. Let M=sup1n{an}<. Now we show a=0. Otherwise, if a>0, then n1n0, such that an+1(1/2)a>0 when nn1. Because ϕ is a strictly increasing function, so ϕ(an+1)ϕ((1/2)a)>0. From (19) again, we have (21)0<ϕ(12a)n=1λn=ϕ(12a)n=1n1λn+ϕ(12a)n=n1+1λnϕ(12a)n=1n1λn+n=n1+1λnϕ(an+1)ϕ(12a)n=1n1λn+n=n1+1(an-an+1)+n=n1+1bnan+n=n1+1cnϕ(12a)n=1n1λn+an1+1+Mn=1bn+n=1cn<. This is a contradiction with the given condition Σn=1λn=. Therefore limnan=0.

Lemma 7.

Suppose that ϕ:[0,+)[0,+) is a strictly increasing function with ϕ(0)=0. Let {an},{bn},{cn},{λn}(0λn1),{εn} be five sequences of nonnegative numbers satisfying the recursive inequality: (22)an+1(1+bn)an-λnϕ(an+1)+cn+λnεn,nn0, where n0 is some nonnegative integer. If Σn=1bn<, Σn=1cn<, Σn=1λn=, limnεn=0, then limnan=0.

Proof.

Firstly, we show liminfnan=a=0. If a>0, then, for arbitrary r(0,a), n1n0, such that an+1r>0 when nn1. Because ϕ is a strictly increasing function and limnεn=0, so ϕ(an+1)ϕ(r)>0 and εn(1/2)ϕ(r) when nn1. From (22), we have (23)an+1(1+bn)an-λnϕ(an+1)+cn+λn12ϕ(an+1)=(1+bn)an-12λnϕ(an+1)+cn,nn1. By Lemma 6, we get 0=limnan=liminfnan=a>0. This is contradictory. So, liminfnan=0.

Secondly, ε>0, from the given conditions in Lemma 7, n2n0, when nn2, we have (24)εnϕ(ε),n=n2bnln2,n=n2cnε.

On the other hand, since liminfnan=0, Nn2 such that aNε. Now we claim (25)ak(ε+n=Nk-1cn)exp(n=Nk-1bn),kN. In fact, when k=N, (25) holds. Suppose that (25) holds for k dose not for k+1. Then (26)ak+1>(ε+n=Nkcn)exp(n=Nkbn). Furthermore, ak+1>ε, ϕ(ak+1)>ϕ(ε). But by (22), (24), and the inductive hypothesis, we have (27)an+1(1+bn)an-λnϕ(an+1)+cn+λnεn(1+bn)an-λnϕ(ε)+cn+λnϕ(ε)(1+bn)(ε+n=Nk-1cn)exp(n=Nk-1bn)+cn(ε+n=Nk-1cn)exp(n=Nkbn)+cn(ε+n=Nkcn)exp(n=Nkbn). This is a contradiction with (26). So, (25) holds. Whereupon, (28)limsupkak(ε+n=Ncn)exp(n=Nbn)2(ε+ε)=4ε. Therefore, limsupkak=0=limnan.

3. Main Results

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

Theorem 8.

Let C be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2:CE are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers {kn}[1,+):kn1 satisfying F=F(T1)F(T2). Suppose that {un},{vn}C are two bounded sequences; {αn},{βn},{γn},{αn},{βn},{γn}[0,1] are six number sequences satisfying the following:

Σn=1αn=+, Σn=1αn2<+,  Σn=1αn(kn-1)<+;

αn+γn1, αn+γn1, Σn=1γn<+;

Σn=1αnβn<+, Σn=1αnαn<+, Σn=1αnγn<+.

If x1C is arbitrary, then the iterative sequence {xn} generated by (12) converges strongly to the fixed point x*F if and only if there exists a strictly increasing function ϕ:[0,+)[0,+) with ϕ(0)=0 such that (29)limsupninfj(xn+1-x*)J(xn+1-x*)[T1(PT1)n-1xn+1-x*,00000000000000000000000j(xn+1-x*)T1(PT1)n-1-knxn+1-x*20000000000000000000000+ϕ(xn+1-x*)T1(PT1)n-1]0.

Proof.

(Adequacy). Let (30)εn=infj(xn+1-x*)J(xn+1-x*)[T1(PT1)n-1xn+1-x*,000000000000000000000j(xn+1-x*)(PT1)n-1-knxn+1-x*200000000000000000000+ϕ(xn+1-x*)(PT1)n-1],εn=max{εn,0}+1n. Then there exists j(xn+1-x*)J(xn+1-x*) such that (31)T1(PT1)n-1xn+1-x*,j(xn+1-x*)-knxn+1-x*2+ϕ(xn+1-x*)εn. From (29), we know that limsupnεn0. So, limnεn=0.

Now, from the given conditions and (12), we can let (32)σn=(1-βn)yn+βnT1(PT1)n-1yn,δn=(1-βn)xn+βnT2(PT2)n-1xn, and M=supn1{μn-x*,νn-x*}<. Then (33)δn-x*βnT2(PT2)xn-x*+(1-βn)xn-x*βnLxn-x*+xn-x*;yn-x*(1-αn-γn)xn-x*+αnLδn-x*+γnνn-x*xn-x*+αnβnL2xn-x*+αnLxn-x*+γnM=(1+αnβnL2+αnL)xn-x*+γnM(1+L+L2)xn-x*+M;σn-x*βnT1(PT1)n-1yn-x*+(1-βn)yn-x*βnLyn-x*+yn-x*(1+L)(1+L+L2)xn-x*+(1+L)M;yn-xn+1αnLσn-x*+αnxn-x*+αnLδn-x*+αnxn-x*+(γn+γn)xn-x*+(γn+γn)MαnL[(1+L)(1+L+L2)xn-x*+(1+L)M(1+L+L2)xn-x*]+αnL[(1+βnL)xn-x*]+(αn+αn+γn+γn)xn-x*+(γn+γn)M[αnL(1+L)(1+L+L2)+αnL(1+βnL)+αn+αn+γn+γn]xn-x*+(αnL(1+L)+γn+γn)M;σn-xn+1yn-xn+1+βnT1(PT1)n-1yn-ynsnxn-x*+tn, where (34)sn=αnL(1+L)(1+L+L2)+αnL(1+βnL)+αn+αn+γn+γn+βn(1+L)(1+L+L2);tn=[αnL(1+L)+γn+γn+βn(1+L)]M. So, by Lemma 4, (35)2αnT1(PT1)n-1σn-T1(PT1)n-1xn+1,j(xn+1-x*)2αnLxn+1-x*σn-xn+12αnLxn+1-x*[snxn-x*+tn];(36)xn+1-x*2(1-αn-γn)2xn-x*2+2αnT1(PT1)n-1σn-x*,j(xn+1-x*)+2γnμn-x*,j(xn+1-x*)(1-αn-γn)2xn-x*2+2αnT1(PT1)n-1σn-T1(PT1)n-1xn+1,j(xn+1-x*)(PT1)n-1+2αnT1(PT1)n-1xn+1-x*,j(xn+1-x*)+2γnMxn+1-x*. For the third in (36), we have (37)2αnT1(PT1)n-1xn+1-x*,j(xn+1-x*)=2αndn+2αn[knxn+1-x*2-ϕ(xn+1-x*)]2αnεn+2αn[knxn+1-x*2-ϕ(xn+1-x*)], where (38)dn=T1(PT1)n-1xn+1-x*,j(xn+1-x*)-knxn+1-x*2+ϕ(xn+1-x*)εn. Substituting (35) into (36), we get (39)xn+1-x*2(1-αn)2xn-x*2+2αnεn+2αnknxn+1-x*2-2αnϕ(xn+1-x*)+2αnL(snxn-x*+tn)xn+1-x*+2γnMxn+1-x*. Let an=xn-x*2, φ(t)=2ϕ(t), and (40)ξn=Lαnsn=L2αn2(1+L)(1+L+L2)+αnαnL2(1+βnL)+αn2L+αnαnL+Lαnγn+Lαnγn+Lαnβn(1+L)(1+L+L2),(41)ρn=Lαntn+Mγn=[αn2L2(1+L)+Lαnγn+Lαnγn+αnβn(L+L2)]M+γnM. Then (39) becomes (42)an+1(1-αn)2an+2αnεn+2αnknan+1-αnφ(an+1)+2(ξnxn-x*+ρn)xn+1-x*. By using 2aba2+b2, we have (43)an+1(1-αn)2an+2αnεn+2αnknan+1-αnφ(an+1)+ξn(an+an+1)+ρn(1+an+1)=(1-2αn+αn2+ξn)an+(2αnkn+ξn+ρn)an+1-αnφ(an+1)+2αnεn+ρn. From (40), (41), and the given conditions, we know (44)n=1αn2<+,n=1ξn<+,n=1ρn<+. Then, limn(2αnkn+ξn+ρn)=0. Therefore n0, when nn0, 2αnkn+ξn+ρn1/2. Let (45)bn=1-2αn+αn2+ξn1-2αnkn-ξn-ρn-1=2αn(kn-1)+αn2+2ξn+ρn1-2αnkn-ξn-ρn;cn=ρn1-2αnkn-ξn-ρn. So, when nn0, we get (46)0bn2[2αn(kn-1)+αn2+2ξn+ρn],0cn2ρn. From (44) and the given conditions, we have n=n0bn<+, n=n0cn<+. On the other hand, from (43), we have (47)an+1(1+bn)an-αnφ(an+1)+4αnεn+cn,nn0. By Lemma 7, we at last get (48)limnan=limnxn-x*2=0; for example, limnxn=x*F=F(T1)F(T2).

(Necessity). Suppose that limnxn=x*F. Then we can choose an arbitrary continuous strictly increasing function ϕ:[0,+)[0,+) with ϕ(0)=0, such as ϕ(t)=t. We can get limnϕ(xn+1-x*)=0.

Because T1 is an asymptotically quasi pseudocontractive type (with P), by (11) in Definition 1, for any pF(T1)F, we have (49)limsupnsupxCliminfj(x-p)J(x-p)(T(PT)n-1x-p,j(x-y)000000000000000000000-knx-p2)0. So, (50)limsupninfj(xn+1-x*)J(xn+1-x*)[T1(PT1)n-1xn+1-x*,00000000000000000000000j(xn+1-x*)(PT1)n-1-knxn+1-x*20000000000000000000000+ϕ(xn+1-x*)(PT1)n-1]=limsupninfj(xn+1-x*)J(xn+1-x*)[T1(PT1)n-1xn+1-x*,000000000000000000000000000j(xn+1-x*)(PT1)n-100000000000000000000000000-knxn+1-x*2]+limnϕ(xn+1-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 C be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2:CE are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers {kn}[1,+):kn1 satisfying F=F(T1)F(T2). Suppose that {αn},{βn},{αn},{βn}[0,1] are four number sequences satisfying the following:

Σn=1αn=+, Σn=1αn2<+, Σn=1αn(kn-1)<+;

Σn=1αnβn<+, Σn=1αnαn<+.

If x1C is arbitrary, then the iterative sequence {xn} generated by (14) converges strongly to the fixed point x*F if and only if there exists a strictly increasing function ϕ:[0,+)[0,+) with ϕ(0)=0 such that (29) holds.

Theorem 10.

Let C be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2:CE are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers {kn}[1,+):kn1 satisfying F=F(T1)F(T2). Suppose that {αn},{αn}[0,1] are two number sequences satisfying the following:

Σn=1αn=+, Σn=1αn2<+, Σn=1αn(kn-1)<+;

Σn=1αnαn<+.

If x1C is arbitrary, then the iterative sequence {xn} generated by (15) converges strongly to the fixed point x*F if and only if there exists a strictly increasing function ϕ:[0,+)[0,+) with ϕ(0)=0 such that (29) holds.

Theorem 11.

Let C be a nonempty closed convex subset of a real Banach space E. Assume that T:CC is uniformly L-Lipschitzian self-mappings and asymptotically quasi pseudocontractive type with coefficient numbers {kn}[1,+):kn1 satisfying F=F(T). Suppose that {αn},{αn}[0,1] are two number sequences satisfying the following:

Σn=1αn=+, Σn=1αn2<+, Σn=1αn(kn-1)<+;

Σn=1αnαn<+.

If x1C is arbitrary, then the iterative sequence {xn} generated by (16) converges strongly to the fixed point x*F if and only if there exists a strictly increasing function ϕ:[0,+)[0,+) with ϕ(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 T1,T2. (b) The common fixed point x*F=F(T1)F(T2) is studied. (c) The research object is the very generalized asymptotically quasi pseudocontractive type (with P) 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, 1423], 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).

Goebel K. Kirk W. A. A fixed point theorem for asymptotically nonexpansive mappings Proceedings of the American Mathematical Society 1972 35 171 174 10.1090/S0002-9939-1972-0298500-3 ZBL0256.47045 Schu J. Approximation of fixed points of asymptotically nonexpansive mappings Proceedings of the American Mathematical Society 1991 112 143 151 10.1090/S0002-9939-1991-1039264-7 ZBL0734.47037 Takahashi W. Wong N.-C. Yao J.-C. Attractive point and weak convergence theorems for new generalized hybrid mappings in Hilbert spaces Journal of Nonlinear and Convex Analysis 2012 13 745 757 ZBL1272.47068 Kurokawa Y. Takahashi W. Weak and strong convergence theorems for nonspreading mappings in Hilbert spaces Nonlinear Analysis, Theory, Methods and Applications 2010 73 6 1562 1568 2-s2.0-77953962996 10.1016/j.na.2010.04.060 ZBL1229.47117 Xu H. K. Vistosity approximation methods for nonexpansive mappings Journal of Mathematical Analysis and Applications 2004 298 279 291 10.1016/j.jmaa.2004.04.059 Sahu D. R. Xu H.-K. Yao J.-C. Asymptotically strict pseudocontractive mappings in the intermediate sense Nonlinear Analysis, Theory, Methods and Applications 2009 70 10 3502 3511 2-s2.0-61749099206 10.1016/j.na.2008.07.007 ZBL1221.47122 Kim J. K. Sahu D. R. Nam Y. M. Convergence theorem for fixed points of nearly uniformly L-Lipschitzian asymptotically generalized Φ-hemicontractive mappings Nonlinear Analysis, Theory, Methods and Applications 2009 71 12 e2833 e2838 2-s2.0-72149088458 10.1016/j.na.2009.06.091 ZBL1239.47055 Chang S. S. Some results for asymptotically pseudo-contractive mappings and asymptotically nonexpansive mappings Proceedings of the American Mathematical Society 2001 129 3 845 853 2-s2.0-0004316906 10.1090/S0002-9939-00-05988-8 ZBL0968.47017 Cholamjiak P. Suantai S. A new hybrid algorithm for variational inclusions, generalized equilibrium problems, and a finite family of quasi-nonexpansive mappings Fixed Point Theory and Applications 2009 2009 7 2-s2.0-70449699858 10.1155/2009/350979 350979 ZBL1186.47060 Klin-Eam C. Suantai S. Strong convergence of composite iterative schemes for a countable family of nonexpansive mappings in Banach spaces Nonlinear Analysis, Theory, Methods and Applications 2010 73 2 431 439 2-s2.0-77955421790 10.1016/j.na.2010.03.034 ZBL1226.47080 Plubtieng S. Ungchittrakool K. Approximation of common fixed points for a countable family of relatively nonexpansive mappings in a Banach space and applications Nonlinear Analysis, Theory, Methods and Applications 2009 72 6 2896 2908 2-s2.0-73249146044 10.1016/j.na.2009.11.034 Suzuki T. Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces Fixed Point Theory and Applications 2005 2005 1 103 123 2-s2.0-28244465340 10.1155/FPTA.2005.103 Zeng L.-C. Yao J.-C. Implicit iteration scheme with perturbed mapping for common fixed points of a finite family of nonexpansive mappings Nonlinear Analysis, Theory, Methods and Applications 2006 64 11 2507 2515 2-s2.0-33644965253 10.1016/j.na.2005.08.028 Marino G. Xu H.-K. A general iterative method for nonexpansive mappings in Hilbert spaces Journal of Mathematical Analysis and Applications 2006 318 1 43 52 2-s2.0-33644654188 10.1016/j.jmaa.2005.05.028 Wang Y.-H. Xia Y.-H. Strong convergence for asymptotically pseudocontractions with the demiclosedness principle in Banach spaces Fixed Point Theory and Applications 2012 2012 8 10.1186/1687-1812-2012-45 Wang Y.-H. Xuan W. F. Convergence theorems for common fixed points of a finite family of relatively nonexpansive mappings in Banach spaces Abstract and Applied Analysis 2013 2013 7 259470 Zeng L.-C. Yao J.-C. Stability of iterative procedures with errors for approximating common fixed points of a couple of q-contractive-like mappings in Banach spaces Journal of Mathematical Analysis and Applications 2006 321 2 661 674 2-s2.0-33646507074 10.1016/j.jmaa.2005.07.079 Zeng L.-C. Tanaka T. Yao J.-C. Iterative construction of fixed points of nonself-mappings in Banach spaces Journal of Computational and Applied Mathematics 2007 206 2 814 825 2-s2.0-34249796076 10.1016/j.cam.2006.08.028 Zeng L. C. Wong N. C. Yao J. C. Convergence analysis of iterative sequences for a pair of mappings in Banach spaces Acta Mathematica Sinica 2008 24 3 463 470 2-s2.0-42149091041 10.1007/s10114-007-1002-0 Wu X. Yao J. C. Zeng L. C. Uniformly normal structure and strong convergence theorems for asymptotically pseudocontractive mappings Journal of Nonlinear and Convex Analysis 2005 6 3 453 463 Ceng L.-C. Wong N.-C. Yao J.-C. Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption Computers and Mathematics with Applications 2008 56 9 2312 2322 2-s2.0-52049088041 10.1016/j.camwa.2008.05.002 Chidume C. E. Ofoedu E. U. Zegeye H. Strong and weak convergence theorems for asymptotically nonexpansive mappings Journal of Mathematical Analysis and Applications 2003 280 2 364 374 2-s2.0-0038730516 10.1016/S0022-247X(03)00061-1 Zegeye H. Robdera M. Choudhary B. Convergence theorems for asymptotically pseudocontractive mappings in the intermediate sense Computers and Mathematics with Applications 2011 62 1 326 332 2-s2.0-79959517134 10.1016/j.camwa.2011.05.013 Chang S.-S. Some problems and results in the study of nonlinear analysis Nonlinear Analysis, Theory, Methods and Applications 1997 30 7 4197 4208 2-s2.0-0000584295