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

Authors: Yuanheng Wang; Huimin Shi
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

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.


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## Body

## 1. Introduction

LetE 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
(1)J(x)={x*∈E*:〈x,x*〉=∥x∥·∥x*∥,∥x∥=∥x*∥},00000000000000000000000000000000000000x∈E.LetT:C→E be a mapping. We denote the fixed point set of T by F(T); that is, F(T)={x∈C:x=Tx}. Recall that a mapping T:C→E is said to be nonexpansive if, for each x,y∈C,
(2)∥Tx-Ty∥≤∥x-y∥.T is said to be asymptotically nonexpansive if there exists a sequence kn⊆[1,∞) with kn→1 as n→∞ such that
(3)∥Tnx-Tny∥≤kn∥x-y∥,∀x,y∈C.
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-Tiny∥≤L∥x-y∥,∀x,y∈C.T is said to be asymptotically pseudocontractive if there exist kn⊆[1,∞) with kn→1 as n→∞ and j(x-y)∈J(x-y) such that
(5)〈Tnx-Tny,j(x-y)〉≤kn∥x-y∥2,∀x,y∈C.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
(6)xn+1=P((1-αn)xn+αnT(PT)n-1xn),n≥1,x1∈C,
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]):
(7)limsup⁡n→∞sup⁡x,y∈C(〈Tnx-Tny,x-y〉-kn∥x-y∥2)≤0,000000000000000000000000000000000∀x,y∈C,
where kn⊆[1,∞) with kn→1 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., Lemma6 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.
LetC 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∈C, Px=x. And P is called a nonexpansive retraction.
LetT: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
(8)∥T(PT)n-1x-T(PT)n-1y∥≤L∥x-y∥,∀x,y∈C,n≥1.
Tis said to be asymptotically pseudocontractive (with P) if there exist kn⊆[1,∞) with kn→1 as n→∞ and ∀x,y∈C, ∃j(x-y)∈J(x-y) such that
(9)〈T(PT)n-1x-T(PT)n-1y,j(x-y)〉≤kn∥x-y∥2.
T is said to be an asymptotically pseudocontractive type (with P) if there exist kn⊆[1,∞) with kn→1 as n→∞ and ∀x,y∈C, j(x-y)∈J(x-y) such that
(10)limsup⁡n→∞sup⁡x,y∈Climinf⁡j(x-y)∈J(x-y)(〈T(PT)n-1x-T(PT)n-1y,xxxxxxxxxxxxxicxxxij(x-y)〉-kn∥x-y∥2)≤0.
T is said to be an asymptotically quasi pseudocontractive type (with P) if F(T)≠∅, for p∈F(T), there exist kn⊆[1,∞) with kn→1 as n→∞, and, ∀x∈C, j(x-p)∈J(x-p) such that
(11)lim⁡sup⁡n→∞sup⁡x∈Clim⁡inf⁡j(x-p)∈J(x-p)(〈T(PT)n-1x-p,j(x-y)〉00000000000000000000-kn∥x-p∥2)≤0.Remark 2.
It is clear that every asymptotically pseudocontractive mapping (withP) 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.
LetC 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
(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+αn′T2(PT2)n-1cci×((1-βn′)xn+βn′T2(PT2)n-1xn)+γn′vn),
where x1∈C is arbitrary, {un} and {vn}⊂C are bounded, and αn,βn,γn,αn′,βn′,γn′∈[0,1], n=1,2,….
Ifαn′=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).
Ifγn=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+βn′T2(PT2)n-1xn)(1-αn′)xn+αn′T2(PT2)n-1000000×((1-βn′)xn+βn′T2(PT2)n-1xn)),
and it is called the new modified mixed Ishikawa iterative sequence (with P).
Ifβn=βn′=0, (14) turns to
(15)xn+1=P((1-αn)xn+αnT1(PT1)n-1yn),yn=P((1-αn′)xn+αn′T2(PT2)n-1xn),
and it is called the new mixed Ishikawa iterative sequence (with P).
IfT1=T2=T:C→C 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+αn′Tnxn.
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 [19]).
LetE be a real Banach space. Then, for all x,y∈E, j(x+y)∈J(x+y), the following inequality holds:
(17)∥x+y∥2≤∥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:
(18)an+1≤(1+bn)an+cn,∀n≥n0,
where n0 is some nonnegative integer. If Σn=1∞bn<∞, Σn=1∞cn<∞, then lim⁡n→∞an exists.Lemma 6.
Suppose thatϕ:[0,+∞)→[0,+∞) is a strictly increasing function with ϕ(0)=0. Let {an},{bn},{cn},{λn}(0≤λn≤1) be four sequences of nonnegative numbers satisfying the recursive inequality:
(19)an+1≤(1+bn)an-λnϕ(an+1)+cn,∀n≥n0,
where n0 is some nonnegative integer. If Σn=1∞bn<∞, Σn=1∞cn<∞, Σn=1∞λn=∞, then lim⁡n→∞an=0.Proof.
From (19), we get
(20)an+1≤(1+bn)an+cn,∀n≥n0.
By Lemma 5, we know that lim⁡n→∞an=a≥0 exists. Let M=sup⁡1≤n≤∞{an}<∞. Now we show a=0. Otherwise, if a>0, then ∃n1≥n0, such that an+1≥(1/2)a>0 when n≥n1. 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=1∞λn=ϕ(12a)∑n=1n1λn+ϕ(12a)∑n=n1+1∞λn≤ϕ(12a)∑n=1n1λn+∑n=n1+1∞λnϕ(an+1)≤ϕ(12a)∑n=1n1λn+∑n=n1+1∞(an-an+1)+∑n=n1+1∞bnan+∑n=n1+1∞cn≤ϕ(12a)∑n=1n1λn+an1+1+M∑n=1∞bn+∑n=1∞cn<∞.
This is a contradiction with the given condition Σn=1∞λn=∞. Therefore lim⁡n→∞an=0.Lemma 7.
Suppose thatϕ:[0,+∞)→[0,+∞) is a strictly increasing function with ϕ(0)=0. Let {an},{bn},{cn},{λn}(0≤λn≤1),{εn} be five sequences of nonnegative numbers satisfying the recursive inequality:
(22)an+1≤(1+bn)an-λnϕ(an+1)+cn+λnεn,∀n≥n0,
where n0 is some nonnegative integer. If Σn=1∞bn<∞, Σn=1∞cn<∞, Σn=1∞λn=∞, lim⁡n→∞εn=0, then lim⁡n→∞an=0.Proof.
Firstly, we showliminf⁡n→∞an=a=0. If a>0, then, for arbitrary r∈(0,a), ∃n1≥n0, such that an+1≥r>0 when n≥n1. Because ϕ is a strictly increasing function and lim⁡n→∞εn=0, so ϕ(an+1)≥ϕ(r)>0 and εn≤(1/2)ϕ(r) when n≥n1. From (22), we have
(23)an+1≤(1+bn)an-λnϕ(an+1)+cn+λn12ϕ(an+1)=(1+bn)an-12λnϕ(an+1)+cn,∀n≥n1.
By Lemma 6, we get 0=lim⁡n→∞an=liminf⁡n→∞an=a>0. This is contradictory. So, liminf⁡n→∞an=0.
Secondly,∀ε>0, from the given conditions in Lemma 7, ∃n2≥n0, when ∀n≥n2, we have
(24)εn≤ϕ(ε),∑n=n2∞bn≤ln⁡2,∑n=n2∞cn≤ε.
On the other hand, sinceliminf⁡n→∞an=0, ∃N≥n2 such that aN≤ε. Now we claim
(25)ak≤(ε+∑n=Nk-1cn)exp⁡(∑n=Nk-1bn),∀k≥N.
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)lim⁡sup⁡k→∞ak≤(ε+∑n=N∞cn)exp⁡(∑n=N∞bn)≤2(ε+ε)=4ε.
Therefore, limsup⁡k→∞ak=0=lim⁡n→∞an.


## 3. Main Results

Now, we are in a position to state and prove the main results of this paper.Theorem 8.
LetC be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2:C→E are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers {kn}⊂[1,+∞):kn→1 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:(C1)
Σn=1∞αn=+∞, Σn=1∞αn2<+∞,Σn=1∞αn(kn-1)<+∞;(C2)
αn+γn≤1, αn′+γn′≤1, Σn=1∞γn<+∞;(C3)
Σn=1∞αnβn<+∞, Σn=1∞αnαn′<+∞, Σn=1∞αnγn′<+∞.If x1∈C 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)lim⁡sup⁡n→∞inf⁡j(xn+1-x*)∈J(xn+1-x*)[〈T1(PT1)n-1xn+1-x*,00000000000000000000000j(xn+1-x*)T1(PT1)n-1〉-kn∥xn+1-x*∥20000000000000000000000+ϕ(∥xn+1-x*∥)T1(PT1)n-1〉]≤0.Proof.
(Adequacy). Let
(30)εn′=inf⁡j(xn+1-x*)∈J(xn+1-x*)[〈T1(PT1)n-1xn+1-x*,000000000000000000000j(xn+1-x*)(PT1)n-1〉-kn∥xn+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*)〉-kn∥xn+1-x*∥2+ϕ(∥xn+1-x*∥)≤εn.
From (29), we know that limsup⁡n→∞εn′≤0. So, lim⁡n→∞εn=0.
Now, from the given conditions and (12), we can let
(32)σn=(1-βn)yn+βnT1(PT1)n-1yn,δn=(1-βn′)xn+βn′T2(PT2)n-1xn,
and M=sup⁡n≥1{∥μn-x*∥,∥νn-x*∥}<∞. Then
(33)∥δn-x*∥≤βn′∥T2(PT2)xn-x*∥+(1-βn′)∥xn-x*∥≤βn′L∥xn-x*∥+∥xn-x*∥;∥yn-x*∥≤(1-αn′-γn′)∥xn-x*∥+αn′L∥δn-x*∥+γn′∥νn-x*∥≤∥xn-x*∥+αn′βn′L2∥xn-x*∥+αn′L∥xn-x*∥+γn′M=(1+αn′βn′L2+αn′L)∥xn-x*∥+γn′M≤(1+L+L2)∥xn-x*∥+M;∥σn-x*∥≤βn∥T1(PT1)n-1yn-x*∥+(1-βn)∥yn-x*∥≤βnL∥yn-x*∥+∥yn-x*∥≤(1+L)(1+L+L2)∥xn-x*∥+(1+L)M;∥yn-xn+1∥≤αnL∥σn-x*∥+αn∥xn-x*∥+αn′L∥δn-x*∥+αn′∥xn-x*∥+(γn+γn′)∥xn-x*∥+(γn+γn′)M≤αnL[(1+L)(1+L+L2)∥xn-x*∥+(1+L)M(1+L+L2)∥xn-x*∥]+αn′L[(1+βn′L)∥xn-x*∥]+(αn+αn′+γn+γn′)∥xn-x*∥+(γn+γn′)M≤[αnL(1+L)(1+L+L2)+αn′L(1+βn′L)+αn+αn′+γn+γn′]∥xn-x*∥+(αnL(1+L)+γn+γn′)M;∥σn-xn+1∥≤∥yn-xn+1∥+βn∥T1(PT1)n-1yn-yn∥≤sn∥xn-x*∥+tn,
where
(34)sn=αnL(1+L)(1+L+L2)+αn′L(1+βn′L)+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)2αn〈T1(PT1)n-1σn-T1(PT1)n-1xn+1,j(xn+1-x*)〉≤2αnL∥xn+1-x*∥∥σn-xn+1∥≤2αnL∥xn+1-x*∥[sn∥xn-x*∥+tn];(36)∥xn+1-x*∥2≤(1-αn-γn)2∥xn-x*∥2+2αn〈T1(PT1)n-1σn-x*,j(xn+1-x*)〉+2γn〈μn-x*,j(xn+1-x*)〉≤(1-αn-γn)2∥xn-x*∥2+2αn〈T1(PT1)n-1σn-T1(PT1)n-1xn+1,j(xn+1-x*)(PT1)n-1〉+2αn〈T1(PT1)n-1xn+1-x*,j(xn+1-x*)〉+2γnM∥xn+1-x*∥.
For the third in (36), we have
(37)2αn〈T1(PT1)n-1xn+1-x*,j(xn+1-x*)〉=2αndn+2αn[kn∥xn+1-x*∥2-ϕ(∥xn+1-x*∥)]≤2αnεn+2αn[kn∥xn+1-x*∥2-ϕ(∥xn+1-x*∥)],
where
(38)dn=〈T1(PT1)n-1xn+1-x*,j(xn+1-x*)〉-kn∥xn+1-x*∥2+ϕ(∥xn+1-x*∥)≤εn.
Substituting (35) into (36), we get
(39)∥xn+1-x*∥2≤(1-αn)2∥xn-x*∥2+2αnεn+2αnkn∥xn+1-x*∥2-2αnϕ(∥xn+1-x*∥)+2αnL(sn∥xn-x*∥+tn)∥xn+1-x*∥+2γnM∥xn+1-x*∥.
Let an=∥xn-x*∥2, φ(t)=2ϕ(t), and
(40)ξn=Lαnsn=L2αn2(1+L)(1+L+L2)+αnαn′L2(1+βn′L)+αn2L+αnαn′L+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(ξn∥xn-x*∥+ρn)∥xn+1-x*∥.
By using 2ab≤a2+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=1∞‍αn2<+∞,∑n=1∞‍ξn<+∞,∑n=1∞‍ρn<+∞.
Then, lim⁡n→∞(2αnkn+ξn+ρn)=0. Therefore ∃n0, when n≥n0, 2αnkn+ξn+ρn≤1/2. 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 n≥n0, we get
(46)0≤bn≤2[2αn(kn-1)+αn2+2ξn+ρn],0≤cn≤2ρn.
From (44) and the given conditions, we have ∑n=n0∞bn<+∞, ∑n=n0∞cn<+∞. On the other hand, from (43), we have
(47)an+1≤(1+bn)an-αnφ(an+1)+4αnεn+cn,∀n≥n0.
By Lemma 7, we at last get
(48)lim⁡n→∞an=lim⁡n→∞∥xn-x*∥2=0;
for example, lim⁡n→∞xn=x*∈F=F(T1)∩F(T2).
(Necessity). Suppose that lim⁡n→∞xn=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 lim⁡n→∞ϕ(∥xn+1-x*∥)=0.
BecauseT1 is an asymptotically quasi pseudocontractive type (with P), by (11) in Definition 1, for any p∈F(T1)⊇F, we have
(49)limsup⁡n→∞sup⁡x∈Climinf⁡j(x-p)∈J(x-p)(〈T(PT)n-1x-p,j(x-y)〉000000000000000000000-kn∥x-p∥2)≤0.
So,
(50)limsup⁡n→∞inf⁡j(xn+1-x*)∈J(xn+1-x*)[〈T1(PT1)n-1xn+1-x*,00000000000000000000000j(xn+1-x*)(PT1)n-1〉-kn∥xn+1-x*∥20000000000000000000000+ϕ(∥xn+1-x*∥)(PT1)n-1]=limsup⁡n→∞inf⁡j(xn+1-x*)∈J(xn+1-x*)[〈T1(PT1)n-1xn+1-x*,000000000000000000000000000j(xn+1-x*)(PT1)n-1〉00000000000000000000000000-kn∥xn+1-x*∥2]+lim⁡n→∞ϕ(∥xn+1-x*∥)≤0+0=0;
that is, (29) holds. This completes the proof of Theorem 8.Combining with Theorem8 and Definition 3, we have some results as follows.Theorem 9.
LetC be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2:C→E are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers {kn}⊂[1,+∞):kn→1 satisfying F=F(T1)∩F(T2)≠∅. Suppose that {αn},{βn},{αn′},{βn′}⊂[0,1] are four number sequences satisfying the following:(C1)
Σn=1∞αn=+∞, Σn=1∞αn2<+∞, Σn=1∞αn(kn-1)<+∞;(C2)
Σn=1∞αnβn<+∞, Σn=1∞αnαn′<+∞.If x1∈C 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.
LetC be nonexpansive retract (with P) of a real Banach space E. Assume that T1,T2:C→E are two uniformly L-Lipschitzian non-self-mappings (with P) and T1 is an asymptotically quasi pseudocontractive type with coefficient numbers {kn}⊂[1,+∞):kn→1 satisfying F=F(T1)∩F(T2)≠∅. Suppose that {αn},{αn′}⊂[0,1] are two number sequences satisfying the following:(C1)
Σn=1∞αn=+∞, Σn=1∞αn2<+∞, Σn=1∞αn(kn-1)<+∞;(C2)
Σn=1∞αnαn′<+∞.If x1∈C 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.
LetC be a nonempty closed convex subset of a real Banach space E. Assume that T:C→C is uniformly L-Lipschitzian self-mappings and asymptotically quasi pseudocontractive type with coefficient numbers {kn}⊂[1,+∞):kn→1 satisfying F=F(T)≠∅. Suppose that {αn},{αn′}⊂[0,1] are two number sequences satisfying the following:(C1)
Σn=1∞αn=+∞, Σn=1∞αn2<+∞, Σn=1∞αn(kn-1)<+∞;(C2)
Σn=1∞αnαn′<+∞.If x1∈C 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 mappingsT1,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, 14–23], and the proof methods are very different from the previous.



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*Source: 129069-2014-03-26.xml*