# Approximation Analysis for a Common Fixed Point of Finite Family of Mappings Which Are Asymptoticallyk-Strict Pseudocontractive in the Intermediate Sense

Authors: H. Zegeye; N. Shahzad
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We introduce an iterative process which converges strongly to a common fixed point of a finite family of uniformly continuous asymptoticallyki-strict pseudocontractive mappings in the intermediate sense for i=1,2,…,N. The projection of x0 onto the intersection of closed convex sets Cn and Qn for each n≥1 is not required. Moreover, the restriction that the interior of common fixed points is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.


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## Body

## 1. Introduction and Preliminaries

LetC be a nonempty subset of a real Hilbert space H. A mapping T:C→H is called Lipschitzian if there exists L>0 such that ∥Tx-Ty∥≤L∥x-y∥, for all x, y∈C. If L=1, then T is called nonexpansive, and if L∈[0,1), T is called contraction. T is called uniformlyL-Lipschitzian if there exists L>0 such that ∥Tnx-Tny∥≤L∥x-y∥, for all x, y∈C and all n≥1. Clearly, every contraction mapping is nonexpansive and every nonexpansive mapping is uniformly L-Lipschitzian with L=1 and hence Lipschitzian.A mappingT:C→H is said to be asymptotically nonexpansive if there exists a sequence {γn}⊂[0,∞) with γn→0 such that ∥Tnx-Tny∥≤(1+γn)∥x-y∥ for all integers n≥1 and all x,y∈C. T is said to be asymptotically nonexpansive in the intermediate sense if there exist sequences {γn},{cn}⊂[0,∞) with γn→0, cn→0 such that ∥Tnx-Tny∥≤(1+γn)∥x-y∥+cn for all integers n≥1 and all x,y∈C.A mappingT:C→H is said to be asymptoticallyk-strict pseudocontractive if there exist a constant k∈[0,1) and a sequence {γn}⊂[0,∞) with γn→0, as n→∞, such that
(1)∥Tnx-Tny∥2≤(1+γn)∥x-y∥2+k∥(I-Tn)x-(I-Tn)y∥2,∀x,y∈C.
The class of asymptotically k-strict pseudocontractive mappings which includes the class of asymptotically nonexpansive, and hence the class of nonexpansive mappings was introduced by Liu [1] in 1996 (see, also [2]). Kim and Xu [3] proved that the fixed point set of asymptotically k-strict pseudocontractions is closed and convex. Recall that a fixed point of a map T:C→H is a set {x∈C:Tx=x}, and it is denoted by F(T). In addition, it is noted in [3] that every asymptotically k-strict pseudocontractive mapping with sequence {γn} is a uniformly L-Lipschitzian mapping with L:=sup{(k+1+(1-k)γn)/(1-k):n∈ℕ}.A mappingT is said to be an  asymptoticallyk-strict pseudocontractive in the intermediate sense if
(2)limsupn→∞supx,y∈C(∥Tnx-Tny∥2-(1+γn)∥x-y∥2-k∥(I-Tn)x-(I-Tn)y∥)≤0,
where k∈[0,1) and {γn}⊂[0,∞) such that γn→0, as n→∞. If we put
(3)cn:=max{0,supx,y∈C(∥Tnx-Tny∥2-(1+γn)∥x-y∥2-k∥(I-Tn)x-(I-Tn)y∥2){supx,y∈C}}.
It follows that cn→0, as n→∞, and (2) is reduced to the following:
(4)∥Tnx-Tny∥2≤(1+γn)∥x-y∥2+k∥(I-Tn)x-(I-Tn)y∥2+cn,∀x,y∈C.
We note that the class of asymptotically k-strict pseudocontractive mappings is properly contained in a class of asymptotically k-strict pseudocontractive mapping in the intermediate sense (see examples in [4]). The class of asymptotically k-strict pseudocontractive mappings in the intermediate sense was introduced by Sahu  et al. [4]. They obtained a weak convergence theorem of modified Mann iterative processes for these class of mappings. In [4], Sahu  et al. established the following classical result.Theorem SXY1 (see [4]).
LetH be a real Hilbert space, and let C⊂H be a nonempty closed and convex set. Let T be a uniformly continuous and asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequences {γn} and {cn} such that F(T) is nonempty and ∑n=1n‍γn<∞. Assume that {αn} is a sequence in (0,1) such that 0<δ≤αn≤1-k-δ<1 and ∑n=1n‍αncn<∞. Let {xn} be a sequence defined by x1=x∈C and
(5)xn+1=(1-αn)xn+αnTnxn,n≥1.
Then, {xn} converges weakly to a fixed point of T.But it is worth mentioning that the convergence obtained is aweak convergence. Furthermore, we observe from the proof of Theorem SXY1 that if, in addition, C or T has some compactness assumption, we obtain that the sequence {xn} given by (5) converges strongly to a fixed point of  T.Attempts to modify the Mann iteration method (5) so that strong convergence is guaranteed, without compactness assumption on C or T, have recently been made. Sahu  et al. [4] established the following hybrid Mann algorithm for approximating fixed points of asymptotically k-strict pseudocontractive mappings in the intermediate sense.Theorem SXY2 (see [4]).
LetH be a real Hilbert space, and let C⊂H be a nonempty, closed, and convex set. Let T be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequences {γn} and {cn} such that F(T) is nonempty and bounded. Assume that {αn} is a sequence in [0,1] such that 0<δ≤αn≤1-k, for all n∈N. Let {xn} be a sequence in C defined by u=x1∈C and
(6)yn=(1-αn)xn+αnTnxn,n≥1,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,u-xn〉≥0},xn+1=PCn∩Qn(u),n≥1,
where θn=cn+γnδn and δn=sup{∥xn-z∥:z∈F(T)}<∞. Then, {xn} converges strongly to PF(T)(u).Recently, Hu and Cai [5] studied the strong convergence of the modified Mann iteration process (5) for a finite family of asymptotically k-strict pseudocontractive mappings in the intermediate sense. More precisely, they obtained the following theorem.Theorem HC (see [5]).
LetH be a real Hilbert space, let C⊂H be a nonempty, closed, and convex set. Let Ti:C→C be uniformly continuous asymptotically ki-strict pseudocontractive mappings in the intermediate sense for some 0≤ki<1 with sequences {γn,i} and {cn,i} such that limn→∞γn,i=0 and limn→∞cn,i=0 for i=1, 2, …, N. Let k=max{ki:1≤i≤N}, γn=max{γn,i:1≤i≤N}, and cn=max{cn,i:1≤i≤N}. Assume that F:=⋂n=1NF(Ti) is nonempty and bounded. Let {βn} be a sequence in [0,1] such that 0<δ≤βn≤1-k for all n∈N. Let {xn} be a sequence defined by x0∈C and
(7)yn=(1-βn)xn+βnTi(n)k(n)xn,n≥1,Cn={z∈C:∥yn-z∥2≤∥xn-z∥2+θn},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qn(x0),n≥1,
where θn=γk(n)ρn2+ck(n)→0, as n→∞, for ρn=sup{∥xn-z∥:z∈F}<∞, n=(h-1)N+i, where i=i(n)∈{1,2,…,N}, h=h(n)≥1 a positive integer such that h(n)→∞, as n→∞. Then, {xn} converges strongly to PF(x0).But we observe that the iterative algorithms (6) and (7) generate a sequence {xn} by projecting x0 onto the intersection of closed convex sets Cn and Qn for each n≥1 which is not easy to compute.Attempts to remove projection mapping onto the intersection of closed convex setsCn and Qn for each n≥1 have recently been made. In [6], Zegeye  et al. studied the strong convergence of the modified Mann iteration process for the class of asymptotically k-strict pseudocontractive mappings in the intermediate sense. More precisely, they proved the following theorem.Theorem ZRS (see [6]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H, and let T:C→C be a uniformly L-Lipschitzian and asymptotically k-strict pseudocontractive mapping in the intermediate sense with sequences {γn}⊂[0,∞) and {cn}⊂[0,∞). Assume that the  interior of F(T) is nonempty. Let {xn} be a sequence defined by x1=x∈C and
(8)yn=βnTnxn+(1-βn)xn,xn+1=αnTnyn+(1-αn)xn,n≥1,
where {αn} and {βn} satisfy certain conditions. Then, {xn} converges strongly to a fixed point of T.But it is worth to mention that the assumptioninterior ofF(T)is nonempty is severe restriction.It is our purpose, in this paper, to construct an iteration scheme which converges strongly to a common fixed point of a finite family of uniformly continuous asymptoticallyki-strict pseudocontractive mappings in the intermediate sense for i=1, 2, …, N. The projection of x0 onto the intersection of closed convex sets Cn and Qn for each n≥1 is not required. Furthermore, the restriction that the interior of F(T) is nonempty is not required. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear mappings.In order to prove our results, we need the following lemmas.Lemma 1.
LetH be a real Hilbert space. Then, for any given x, y∈H, the following inequality holds:
(9)∥x+y∥2≤∥x∥2+2〈y,x+y〉.Lemma 2 (see [7]).
Let H be a real Hilbert space. Then, for allx, y∈H and αi,∈[0,1] for i=1, 2, …, n such that α0+α1+⋯+αn=1, the following equality holds:
(10)∥α0x0+α1x1+⋯+αnxn∥2=∑i=0n‍αi∥xi∥2-∑0≤i,j≤n‍αiαj∥xi-xj∥2.Lemma 3 (see [8]).
Let{an} be sequences of real numbers such that there exists a subsequence {ni} of {n} such that ani<ani+1 for all i∈N. Then, there exists a nondecreasing sequence {mk}⊂N such that mk→∞, and the following properties are satisfied by all (sufficiently large) numbers k∈N:
(11)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.Lemma 4 (see [9]).
Let{an} be a sequence of nonnegative real numbers satisfying the following relation:
(12)an+1≤(1-αn)an+αnδn,n≥n0,
where {αn}⊂(0,1) and {δn}⊂R satisfying the following conditions: limn→∞αn=0, ∑n=1∞‍αn=∞ and limsupn→∞δn≤0. Then, limn→∞an=0.Lemma 5 (see [4]).
LetC be a nonempty closed convex subset of a Hilbert space H, and let T:C→C be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense. Then, F(T) is closed and convex.Lemma 6 (see [4]).
LetC be a nonempty closed convex subset of a Hilbert space H, and let T:C→C be a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense. Let {xn} be a sequence in C such that ∥xn-Tnxn∥→0 and ∥xn-xn+1∥→0, as n→∞. Then, ∥xn-Txn∥→0, as n→∞.Lemma 7 (see [4]).
LetC be a nonempty closed convex subset of a Hilbert space H, and T:C→C be a continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense. Then if {xn} is a sequence in C such that xn converges weakly x∈C and limsupm→∞limsupn→∞∥xn-Tmxn∥=0, then (I-T)x=0.Lemma 8 (see [10]).
LetC be a nonempty closed, convex subset of a Hilbert space H and let PC be the metric projection mapping from H onto C. Given z=PCx if and only if 〈y-z,x-z〉≤0, for all y∈C.


## 2. Main Result

Theorem 9.
LetC be a nonempty, closed and convex subset of a real Hilbert space H. Let Ti:C→C be uniformly continuous asymptotically ki-strict pseudocontractive mappings in the intermediate sense for some 0≤ki<1 with sequences {γn,i} and {cn,i}, for i=1, 2, …, N. Assume that F:=⋂i=1NF(Ti) is nonempty. Let {xn} be a sequence generated by
(13)x0=w∈Cchosenarbitrarily,yn=βn,0xn+∑i=1N‍βn,iTinxn,xn+1=αnw+(1-αn)yn,
where 0<αn≤a0<1 such that limn→∞αn=0, ∑n=1∞‍αn=∞, limn→∞[γn,i/αn]=0=limn→∞[cn,i/αn], and 0<δ≤βn,i≤1-k-δ<1 satisfying βn,0+βn,1+⋯+βn,N=1 for each n≥1 and k:=max{ki:i=1,2,…,N}. Then, {xn} converges strongly to an element of F.Proof.
Letx*=PFw. Let γn:=max{γn,i:i=1,2,…,N} and cn:=max{cn,i:i=1,2,…,N}. Then, from (13), Lemma 2, and asymptotically ki-strict pseudocontractiveness of Ti, for each i∈{1,2,…,N}, we get that
(14)∥yn-x*∥2=∥βn,0xn+∑i=1N‍βn,iTinxn-x*∥2≤βn,0∥xn-x*∥2+∑i=1N‍βn,i∥Tinxn-x*∥2-∑i=1N‍βn,0βn,i∥Tinxn-xn∥2≤βn,0∥xn-x*∥2+(1-βn,0)(1+γn)∥xn-x*∥2+∑i=1N‍βn,ik∥Tinxn-xn∥2+(1-βn,0)cn-∑i=1N‍βn,0βn,i∥Tinxn-xn∥2≤(1+γn)∥xn-x*∥2-∑i=1N‍βn,i(1-k-δ)∥Tinxn-xn∥2+(1-βn,0)cn≤(1+γn)∥xn-x*∥2-δ2∑i=1N‍∥Tinxn-xn∥2+(1-βn,0)cn,(15)∥xn+1-x*∥2=∥αnw+(1-αn)yn-x*∥2≤αn∥w-x*∥2+(1-αn)∥yn-x*∥2≤αn∥w-x*∥2+(1-αn)×[{∑i=1N‍}(1+γn)∥xn-x*∥2-δ2×∑i=1N‍∥Tinxn-xn∥2+(1-βn,0)cn]≤αn∥w-x*∥2+(1-αn)(1+γn)∥xn-x*∥2-(1-αn)δ2∑i=1N‍∥Tinxn-xn∥2+(1-αn)(1-βn,0)cn≤αn[∥w-x*∥2+1]+(1-αn(1-ϵ))∥xn-x*∥2-(1-αn)δ2∑i=1N‍∥Tinxn-xn∥2≤αn[∥w-x*∥2+1]+(1-αn(1-ϵ))∥xn-x*∥2
since there exists N0>0 such that γn/αn≤ϵ and cn/αn≤1 for all n≥N0 and for some ϵ>0 satisfying (1-ϵ)αn≤1. Thus, by induction,
(16)∥xn+1-x*∥2≤max{(1-ϵ)-1[∥w-x*∥2+1],∥x0-x*∥2},≤max{(1-ϵ)-1[∥w-x*∥2+1]∥-x∥2}∀n≥N0,
which implies that {xn}, and hence {Tinxn} and {yn} are bounded. Moreover, from (13), (14) and Lemma 1, we obtain that
(17)∥xn+1-x*∥2=∥αn(w-x*)+(1-αn)(yn-x*)∥2≤(1-αn)∥yn-x*∥2+2αn〈w-x*,xn+1-x*〉≤(1-αn)[(1+γn)∥xn-x*∥2-δ2∑i=1N‍∥Tinxn-xn∥2+(1-βn,0)cn{∑i=1N‍}]+2αn〈w-x*,xn+1-x*〉≤(1-αn)∥xn-x*∥2+(1-αn)γn∥xn-x*∥2-δ2(1-αn)∑i=1N‍∥Tinxn-xn∥2+(1-αn)(1-βn,0)cn+2αn〈w-x*,xn+1-x*〉≤(1-αn)∥xn-x*∥2+2αn〈w-x*,xn+1-x*〉+(γn+cn)M-δ2(1-a0)∑i=1N‍∥Tinxn-xn∥2(18)≤(1-αn)∥xn-x*∥2+2αn〈w-x*,xn+1-x*〉+(γn+cn)M
for some M>0 and for all n∈N.
Now, following the method of proof of Lemma 3.2 of Maingé [8], we consider two cases.
Case  1. Suppose that there exists n0∈N such that {∥xn-x*∥} is nonincreasing for all n≥n0. In this situation, {∥xn-x*∥)} is convergent. Then, from (17), we have that
(19)xn-Tinxn⟶0,asn⟶∞,
for i=1, 2, …, N. Moreover, from (13) and (19) and the fact that αn→0, we get that
(20)∥yn-xn∥≤∑i=1N‍βn,i∥Tinxn-xn∥⟶0,xn+1-yn=αn(w-yn)⟶0,
as n→∞, and hence
(21)∥xn+1-xn∥⟶0,asn⟶∞.
Furthermore, from (19), (21), and Lemma 6, we obtain that
(22)xn-Tixn⟶0,asn⟶∞.
Let {xnk+1} be a subsequence of {xn+1} such that limsupn→∞〈w-x*,xn+1-x*〉 = limk→∞〈w-x*,xnk+1-x*〉 and {xnk+1} converges weakly to v. Then, from (21), we also get that {xnk} converges weakly to v. Moreover, since Ti is uniformly continuous and ∥xn-Tixn∥→0, for all i∈{1,2,…,N}, we get that ∥xn-Timxn∥→0, as n→∞, for all m∈N. Therefore, by Lemma 7, we obtain that v∈⋂i=1NF(Ti). Now, from Lemma 8, we have that
(23)limsupn→∞〈w-x*,xn+1-x*〉=〈w-x*,v-x*〉≤0.
Then, from (18), (23), and Lemma 4, we obtain that ∥xn-x*∥→0, as n→∞. Consequently, xn→x*.
Case  2. Suppose that there exists a subsequence {ni} of {n} such that
(24)∥xni-x*∥<∥xni+1-x*∥
for all i∈N. Then, by Lemma 3, there exist a nondecreasing sequence {mk}⊂N such that mk→∞,
(25)∥xmk-x*∥≤∥xmk+1-x*∥
and ∥xk-x*∥≤∥xmk+1-x*∥, for all k∈N. Then, from (17) and the fact that αn→0, we have
(26)δ2(1-a0)∑i=1N‍∥xmk-Timkxmk∥2≤∥xmk-x*∥2-∥xmk+1-x*∥2+αmk∥xmk-x*∥2+2αmk〈w-x*,xmk+1-x*〉+(γmk+cmk)M⟶0,ask⟶∞.
Then, we get that xmk-Timkxmk→0, as n→∞, for each i=1, 2, …, N. Thus, as in Case 1, we obtain that xmk-ymk→0 and xmk+1-xmk→0, as k→∞ and
(27)limsupk→∞〈w-x*,xmk+1-x*〉≤0.
Now, from (18), we have that
(28)∥xmk+1-x*∥2≤(1-αmk)∥xmk-x*∥+2αmk〈w-x*,xmk+1-x*〉+(γmk+cmk)M.
This implies that
(29)αmk∥xmk-x*∥2≤∥xmk-x*∥2-∥xmk+1-x*∥2+2αmk〈w-x*,xmk+1-x*〉+(γmk+cmk)M.
Then, using (25), inequality (29) implies that
(30)αmk∥xmk-x*∥2≤2αmk〈w-x*,xmk+1-x*〉+(γmk+cmk)M.
In particular, since αmk>0, we get
(31)∥xmk-x*∥2≤2〈w-x*,xmk+1-x*〉+γmk+cmkαmk.
Furthermore, using (27) and the fact that (γmk+cmk)/αmk→0, we obtain that ∥xmk-x*∥→0, as k→∞. This together with (28) give ∥xmk+1-x*∥→0, as k→∞. But ∥xk-x*∥≤∥xmk+1-x*∥ for all k∈N. Thus we obtain that xk→x*. Therefore, from the above two cases, we can conclude that {xn} converges strongly to an element of F, and the proof is complete.If in Theorem9, we assume that N=1, then we get the following corollary.Corollary 10.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T:C→C be  a uniformly continuous asymptotically k-strict pseudocontractive mapping in the intermediate sense for some 0≤k<1 with sequences {γn} and {cn}. Assume that F:=F(T) is nonempty. Let {xn} be a sequence generated by
(32)x0=w∈Cchosenarbitrarily,yn=βnxn+(1-βn)Tnxn,xn+1=αnw+(1-αn)yn,
where 0<αn≤a0<1 such that limn→∞αn=0, ∑n=1∞‍αn=∞, limn→∞[γn/αn]=0=limn→∞[cn/αn], and 0<δ≤βn≤1-k-δ<1 for each n≥1. Then, {xn} converges strongly to an element of F.If in Theorem9, we assume that each Ti is asymptotically ki-strict pseudocontractive mapping, then we get the following corollary.Corollary 11.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ti:C→C be asymptotically ki-strict pseudocontractive mappings for some 0≤ki<1 with sequences {γn,i}, for i=1, 2, …, N. Assume that F:=⋂i=1NF(Ti) is nonempty. Let {xn} be a sequence generated by
(33)x0=w∈Cchosenarbitrarily,yn=βn,0xn+∑i=1N‍βn,iTinxn,xn+1=αnw+(1-αn)yn,
where 0<αn≤a0<1 such that limn→∞αn=0, ∑n=1∞‍αn=∞, limn→∞[γn,i/αn]=0, and 0<δ≤βn,i≤1-k-δ<1 satisfying βn,0+βn,1+⋯+βn,N=1 for each n≥1 and k:=max{ki:i=1,2,…,N}. Then, {xn} converges strongly to an element of F.Proof.
Since every asymptoticallyki-strict pseudocontractive mapping is uniformly continuous and asymptotically ki-strict pseudocontractive mapping in the intermediate sense with cn,i≡0, for all n≥1 and each i=1,2,…,N, the conclusion follows from Theorem 9.If in Theorem9, we assume that each Ti is asymptotically nonexpansive in the intermediate sense we obtain the following corollary.Corollary 12.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ti:C→C be uniformly continuous asymptotically nonexpansive mappings in the intermediate sense with sequences {γn,i} and {cn,i}, for i=1, 2, …, N. Assume that F:=⋂i=1NF(Ti) is nonempty. Let {xn} be a sequence generated by
(34)x0=w∈Cchosenarbitrarily,yn=βn,0xn+∑i=1N‍βn,iTinxn,xn+1=αnw+(1-αn)yn,
where 0<αn≤a0<1 such that limn→∞αn=0, ∑n=1∞‍αn=∞, limn→∞[γn,i/αn] = 0 = limn→∞[cn,i/αn], and 0<δ≤βn,i≤1-δ<1 satisfying βn,0+βn,1+⋯+βn,N=1 for each n≥1. Then, {xn} converges strongly to an element of F.Proof.
Since every asymptotically nonexpansive mapping in the intermediate sense is asymptoticallyki-strict pseudocontractive mapping in the intermediate sense with ki=0, for each i=1, 2, …, N, the conclusion follows from Theorem 9.If in Theorem9, we assume that each Ti is asymptotically nonexpansive, we obtain the following corollary.Corollary 13.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let Ti:C→C be asymptotically nonexpansive mappings with sequences {γn,i}, for i=1, 2, …, N. Assume that F:=⋂i=1NF(Ti) is nonempty. Let {xn} be a sequence generated by
(35)x0=w∈Cchosenarbitrarily,yn=βn,0xn+∑i=1N‍βn,iTinxn,xn+1=αnw+(1-αn)yn,
where 0<αn≤a0<1 such that limn→∞αn=0, ∑n=1∞‍αn=∞, limn→∞[γn,i/αn]=0, and 0<δ≤βn,i≤1-δ<1 satisfying βn,0+βn,1+⋯+βn,N=1 for each n≥1. Then, {xn} converges strongly to an element of F.Proof.
Since every asymptotically nonexpansive mapping is uniformly continuous and asymptoticallyki-strict pseudocontractive mapping with ki=0 and cn,i=0, for all n≥1 and i=1, 2, …, N, the conclusion follows from Theorem 9.Remark 14.
Our results extend and unify most of the results that have been proved for this important class of nonlinear mappings. In particular, Theorem9 extends Theorem SXY1, SXY2, HC, and Theorem ZRS in the sense that our convergence is either strong, does not require computation of closed convex sets Cn and Qn for each n≥1, or does not require the assumption that interior of set of fixed points is nonempty.Remark 15.
We also remark that Corollary11 is more general than Theorem 3.1 of Kim and Xu [3] and Corollary 13 is more general than Theorem 2.2 of Kim and Xu [11] in the sense that our convergence is either strong, does not require computation of closed convex sets Cn and Qn for each n≥1, or does not require the assumption that interior of set of fixed points is nonempty.



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