# Strong Convergence Theorems for a Common Fixed Point of a Family of Asymptoticallyk-Strict Pseudocontractive Mappings

Authors: H. Zegeye; N. Shahzad
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We provide an iterative process which converges strongly to a common fixed point of finite family of asymptoticallyk-strict pseudocontractive mappings in Banach spaces. Our theorems improve and unify most of the results that have been proved for this important class of nonlinear operators.


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## Body

## 1. Introduction

LetE be a real normed linear space with dual E*. A gauge function φ:[0,∞]:=R+→R+ is a continuous and strictly increasing function satisfying φ(0)=0 and φ(t)→∞, as t→∞. The generalized duality mapping from E to 2E* associated with the gauge function φ (see, e.g., [1]) is defined by
(1)Jφ(x)≔{x*∈E*:〈x,x*〉=||x||φ(||x||),∥x*∥=φ(||x||)},∀x∈E,
where 〈·,·〉 denotes the duality pairing. In the case that φ(t)=t, the duality mapping Jφ=J is called the normalized duality mapping.Following Browder [2], we say that a Banach space E has a weakly continuous duality mapping if there exists a gauge φ for which the duality mapping Jφ is single valued and weak-to-weak* sequentially continuous (i.e., if {xn} is a sequence in E weakly convergent to a point x, then the sequence Jφ(xn) converges weak* to Jφ(x)). It is known that lp has a weakly continuous duality mapping with a gauge function φ(t)=tp-2, for all 1<p<∞.LetK be a nonempty subset of E. A mapping T:K→K is called asymptoticallyk-strict pseudocontractive, with sequence {ln}⊆[1,∞), limn→∞ln=1 (see, e.g., [3–6]) if for all x,y∈K, there exist j(x-y)∈J(x-y) and a constant k∈[0,1) such that
(2)〈Tnx-Tny,j(x-y)〉≤ln∥x-y∥2-k∥(I-Tn)x-(I-Tn)y∥2,
for all n≥1.IfI denotes the identity operator, then (2) can be equivalently written as
(3)〈(I-Tn)x-(I-Tn)y,j(x-y)〉≥k∥(I-Tn)x-(I-Tn)y∥2-(ln-1)∥x-y∥2,
for all n≥1.IfE=H, a real Hilbert space, it is shown by Osilike  et al. [4] that (2) (and hence (3)) is equivalent to the inequality
(4)∥Tnx-Tny∥2≤(1+2(ln-1))∥x-y∥2+λ∥(I-Tn)x-(I-Tn)y∥2,
where λ=(1-2k). T is called uniformly Lipschitz if there exists L≥0 such that ∥Tnx-Tny∥≤L∥x-y∥ for all x,y∈D(T). It is shown in [4] that an asymptotically k-strict pseudocontractive mapping is uniformly Lipschitz.The class of asymptoticallyk-strict pseudocontractive mappings was first introduced in Hilbert spaces by Liu [5]. He proved the following theorem.Theorem Q (see [5]).
LetK be a closed convex and bounded subset of a Hilbert space H. Let T:K→K be completely continuous asymptotically k-strict pseudocontractive mapping for some 0≤k<1 with sequence {ln}⊂[0,∞) such that ∑‍(ln-1)<∞ and F(T)≠∅. Let {xn} be a sequence generated by the modified Mann's iteration method:
(5)xn+1=αnxn+(1-αn)Tnxn,n≥1,
where {αn} is a real sequence satisfying ϵ≤αn≤1-k-ϵ for all n≥1 and some ϵ>0. Then, {xn} converges strongly to a fixed point of  T.The iteration scheme (5) is called modified Mann’s iterative processes which was introduced by Schu [7, 8] and has been used by several authors (see, e.g., [3–5, 9–17]). We observe that Liu [5] proved strong convergence of scheme (5) to a fixed point of asymptotically k-strict pseudocontractive mapping T with additional assumption that T is completely continuous, where T:C→C is said to be  completely continuous  if for every bounded sequence {xn}, there exists a subsequence, say {xnj} of {xn} such that the sequence {Txnj} converges strongly to some element of the range of T.In [12], Kim and Xu studied weak convergence theorem for the class of asymptotically k-strict pseudocontractive mappings in the frame work of Hilbert spaces. In fact, they proved the following.Theorem KX (see [12]).
LetK be a closed and convex subset of a Hilbert space H. Let T:K→K be an asymptotically k-strict pseudocontractive mapping for some 0≤k<1 with sequence {ln}⊂[0,∞) such that ∑‍(ln-1)<∞ and F(T)≠∅. Let {xn} be a sequence generated by the modified Mann's iteration method:
(6)xn+1=αnxn+(1-αn)Tnxn,n≥1,
where {αn} is a real sequence satisfying k+λ≤αn≤1-λ, for all n≥1, and λ∈(0,1). Then, {xn} converges weakly to a fixed point of  T.In 2007, Osilike  et al. [13] extended Theorem KX by proving weak convergence of scheme (6) to a fixed point of T in the frame work of q uniformly smooth Banach spaces which are also uniformly convex under suitable control conditions.In 2011, Zhang and Xie [17] extended Theorem of Osilike  et al. [13] to a more general real uniformly convex Banach space E with Fréchet differentiable norm. In addition, they proved strong convergence of scheme (5) to a fixed point of asymptotically k-strict pseudocontractive mapping provided that liminfn→∞d(xn,F(T))=0, where d(xn,F(T))=infp∈F(T)||xn-p||.However, we observe that the convergence obtained above is eitherweak or requiring additional assumption like liminfn→∞d(xn,F(T))=0 or T is completely continuous. But the requirement that liminfn→∞d(xn,F(T))=0 is not easy to verify, as F(T) is in general unknown, and there is also an example of asymptotically k-strict pseudocontractive mapping which is not completely continuous as shown below.An example of asymptoticallyk-strict pseudocontractive mapping which is not completely continuous.Example 1.
LetE=l2={x¯={xi}i=1∞,xi∈ℝ,∑i=1∞‍|xi|2<∞} and B¯={x¯∈l2:||x¯||≤1}. Define T:B¯→B¯ by Tx¯=(0,x12,a2x2,a3x3,…), where {ak}k=1∞ is a real sequence satisfying 0<ak<1, k≥2, and Πk=2∞ak=1/2. Then it is shown in [13] that T is asymptotically k-strict pseudocontractive mapping.Now, we show thatT is not completely continuous. Let {xn} be a sequence in B¯ defined by x1=(1,0,0,…), x2=(0,1,0,0,…), x3=(0,0,1,0,0,…),…. Then {xn}⊂B¯ and {Txn}={yn} is given by y1=(0,1,0,0,…), y2=(0,0,a2,0,0,…), y3=(0,0,0,a3,0,0,…),…. Hence, since ak→1, as k→∞, there is no subsequence {xni} of {xn} such that {Txni} converges strongly to a point in B¯, as ||Txni-Txnj||=||yni-ynj||=|ani|2+|anj|2↛0, as i,j→∞. Therefore, T is not completely continuous.Thus, one question is raised naturally:  can we obtain a scheme that converges  strongly to a fixed point of asymptotically k-strict pseudocontractive mappings without those additional assumptions?It is our purpose in this paper to provide an iterative scheme{xn} which converges strongly to a common fixed point of finite family of asymptotically k-strict pseudocontractive mappings in Banach spaces. The assumption that liminfn→∞d(xn,F(T))=0 or T is completely continuous is not required.


## 2. Preliminaries

We need the following definitions from [18]. The Banach space E is said to be uniformly convex if, given ε>0, there exists δ>0, such that, for all x,y∈E with ∥x∥≤1,∥y∥≤1 and ∥x-y∥≥ε, ∥(1/2)(x+y)∥≤1-δ. It is well known that Lp, ℓp, and Sobolev spaces Wmp, (1<p<∞), are uniformly convex.A Banach spaceE is said to have a Fréchet differentiable norm if for all x∈B={x∈E:||x||=1}(7)limt→0∥x+tγ∥-∥x∥t
exists and is attained uniformly in y∈B. It is well known that uniformly smooth Banach spaces has a Fréchet differentiable norm.In order to prove our results, we need the following lemmas.Lemma 2 (see [19]).
LetC be a nonempty close convex subset of a real Banach space E which has the Fréchet differentiable norm. For x∈E, let ρ be defined for 0<t<∞ by
(8)ρ(t)=supγ∈B|∥x+tγ∥2-∥x∥2t-2〈γ,j(x)〉|.
Then, limt→0ρ(t)=0 and
(9)∥x+h∥2≤∥x∥2+2〈h,j(x)〉+||h||ρ(∥h∥),∀h∈E∖{0}.It is shown in [19] that if E=H, a real Hilbert space, then ρ(t)=t, for t>0. In our general setting, throughout this paper we assume that ρ(t)≤2t.Lemma 3.
LetE be a real Banach space. Then the following inequality holds:
(10)∥x+y∥2≤∥x∥2+〈y,j(x+y)〉,a∀x,y∈H,j(x+y)∈J(x+y).Lemma 4 (see [20]).
LetE be a uniformly convex Banach space and BR(0) a closed ball of E. Then, there exists a continuous strictly increasing convex function g:[0,∞)→[0,∞) with g(0)=0 such that
(11)∥α0x0+α1x1+α2x2+⋯+αkxk∥2≤∑i=0k‍αi∥xi∥2-αiαjg(∥xi-xj∥),
for each αi∈(0,1) and for xi∈BR(0):={x∈E:||x||≤R}, i=0,1,2,…,k with ∑i=0k‍αi=1.Lemma 5 (see [21]).
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 6 (see [17]).
LetC be a nonempty closed convex subset of a real uniformly convex Banach space E which has the Fréchet differentiable norm. Let T:C→C be an asymptotically k-strict pseudocontractive mapping with fixed point of T, F(T):={x∈C:Tx=x}≠∅. Then (I-T) is demiclosed at zero, that is, if xn⇀x and Txn-xn→0, as n→∞, then x=T(x).Lemma 7 (see [22]).
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:
(13)amk≤amk+1,ak≤amk+1.
In fact, mk=max{j≤k:aj<aj+1}.


## 3. Main Results

We now prove our main theorem.Theorem 8.
LetC be a nonempty, closed, and convex subset of a real uniformly convex Banach space E which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from E into E*. Let Ti:C→C be asymptotically ki-strict pseudocontractive mappings for 0≤ki<1 with sequences {ln,i}⊂[1,∞), for i=1,2,…,N. Assume that F:=∩i=1NF(Ti) is nonempty. Let {xn} be a sequence defined by x1=u∈C and
(14)xn+1=αnu+(1-αn)((1-βn)xn+βnSnxn),n≥1,
where Sn:=θn,1T1n+θn,2T2n+⋯+θn,NTNn, such that θn,1+θn,2+⋯+θn,N=1, for each n≥1, {αn},{θn,i}⊂(0,c)⊂(0,1), satisfying liminfnθn,i>0, limn→∞αn=0, ∑‍αn=∞, limn→∞((ln,i-1)/αn)=0, for i=1,2,…,N and {βn}⊂[a,b]⊂(0,k) (a, b, and c constants), for k=min1≤i≤N{ki}, Then the sequence {xn} generated by (14) converges strongly to a common fixed point of {Ti:i=1,2,…,N}.Proof.
Fixx*∈F. Let yn=(1-βn)xn+βnSnxn and ln:=max{ln,i:i=1,2,…,N}. Then, using Lemma 2 and (3) we have that
(15)∥yn-x*∥2=∥(xn-x*)-βn(xn-Snxn)∥2=∥(xn-x*)-βn(xn-(θn,1T1n+θn,2T2n+⋯+θn,NTNn)xn)∥2≤∥xn-x*∥2-2βn〈θn,1(xn-T1nxn)+θn,2(xn-T2nxn)+⋯+θn,N(xn-TNnxn),j(xn-x*)〉+βn∥xn-Snxn∥ρ(βn∥xn-Snxn∥)≤∥xn-x*∥2-2βnθn,1〈xn-T1nxn,j(xn-x*)〉-2βnθn,2〈xn-T2nxn,j(xn-x*)〉-⋯-2βnθn,N〈xn-TNnxn,j(xn-x*)〉+2βn2∥xn-Snxn∥2≤∥xn-x*∥2-2βnθn,1[k∥xn-T1nxn∥2-(ln-1)∥xn-x*∥2]-2βnθn,2[k∥xn-T2nxn∥2-(ln-1)∥xn-x*∥2]-⋯-2βnθn,N[k∥xn-TNnxn∥2-(ln-1)∥xn-x*∥2]+2βn2∥xn-Snxn∥2≤[1+2βn(ln-1)]∥xn-x*∥2-2βnθn,1k∥xn-T1nxn∥2-2βnθn,2k∥xn-T2nxn∥2-⋯-2βnθn,Nk∥xn-TNnxn∥2+2βn2∥xn-Snxn∥2.
On the other hand using Lemma4 we get that
(16)∥xn-Snxn∥2=∥xn-(θn,1T1n+θn,2T2n+⋯+θn,NTNn)xn∥2=∥θn,1(xn-T1nxn)+θn,2(xn-T2nxn)+⋯+θn,N(xn-TNnxn)∥2≤θn,1∥xn-T1nxn∥2+θn,2∥xn-T2nxn∥2+⋯+θn,N∥xn-TNnxn∥2.
Now substituting (16) into (15) we obtain that
(17)∥yn-x*∥2≤[1+2βn(ln-1)]∥xn-x*∥2-2βnθn,1(k-βn)∥xn-T1nxn∥2-2βnθn,2(k-βn)∥xn-T2nxn∥2-⋯-2βnθn,N(k-βn)∥xn-TNnxn∥2(18)≤[1+2βn(ln-1)]∥xn-x*∥2,
since (k-βn)≥0 for each n≥1. Then now, from (14) and (18) we get that
(19)∥xn+1-x*∥2=∥αn(u-x*)+(1-αn)(yn-x*)∥2≤αn∥u-x*∥2+(1-αn)∥yn-x*∥2≤αn∥u-x*∥2+(1-αn)[1+2βn(ln-1)∥xn-x*∥2]≤αn∥u-x*∥2+(1-αn+ϵαn)∥xn-x*∥2,∀n≥N0,≤αn∥u-x*∥2+(1-αn(1-ϵ))∥xn-x*∥2,∀n≥N0,
where N0 is a positive integer such that 2(1-αn)βn(ln-1)/αn<ϵ, for all n≥N0, for some ϵ>0. Therefore, by induction,
(20)∥xn+1-x*∥2≤max{∥xN0-x*∥2,(1-ϵ)-1∥u-x*∥2},aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa∀n≥N0,
which implies that {xn} and hence {yn} are bounded.
Furthermore, from (14), Lemma 3, and (17) we get that
(21)∥xn+1-x*∥2=∥αn(u-x*)+(1-αn)(yn-x*)∥2≤(1-αn)∥yn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉≤(1-αn)[1+2βn(ln-1)]∥xn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉-2βnθn,1(k-βn)(1-αn)∥xn-T1nxn∥2-2βnθn,2(k-βn)(1-αn)∥xn-T2nxn∥2-⋯-2βnθn,N(k-βn)(1-αn)∥xn-TNnxn∥2≤(1-αn)∥xn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉+2βnM(ln-1)-2βnθn,1(1-αn)(k-βn)∥xn-T1nxn∥2-2βnθn,2(k-βn)(1-αn)∥xn-T2nxn∥2-⋯-2βnθn,N(k-βn)(1-αn)∥xn-TNnxn∥2(22)5ji≤(1-αn)∥xn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉5ji+2βnM(ln-1),
for some M>0.
Now, the rest of the proof is divided into two parts.
Case
1. Suppose that there exists N1≥0 such that {||xn-x*||} is decreasing for all n≥N1. Then we have that {||xn-x*||)} is convergent. Then from (21) and the assumptions on {βn}, {αn}, and {ln} we have that βnθn,i(1-c)(k-βn)||xn-Tinxn||2→0, as n→∞, which implies that
(23)xn-Tinxn⟶0,asn⟶∞,
for i=1,2,…,N. Then from (14) we obtain that
(24)xn+1-yn=αn(u-yn)⟶0,asn⟶∞.
Again, from (23) we get that
(25)∥yn-xn∥=∥βn(Snxn-xn)∥≤∥Snxn-xn∥≤θn,1∥T1nxn-xn∥+θn,2∥T2nxn-xn∥+⋯+θn,N∥TNnxn-xn∥⟶0,
as n→∞. Thus, (24) and (25) imply that
(26)xn+1-xn⟶0,asn⟶∞.
Therefore, since each Ti, for i=1,2,…,N, is uniformly L-Lipschitzian and
(27)∥xn-Tixn∥≤∥xn-xn+1∥+∥xn+1-Tin+1xn+1∥+∥Tin+1xn+1-Tin+1xn∥+∥Tin+1xn-Tixn∥≤∥xn-xn+1∥+∥xn+1-Tin+1xn+1∥+L∥xn+1-xn∥+∥Ti(Tinxn)-Tixn∥=(1+L)∥xn-xn+1∥+∥xn+1-Tin+1xn+1∥+∥Ti(Tinxn)-Tixn∥,
we have from (23), (26), and uniform continuity of Ti that
(28)∥xn-Tixn∥⟶0,asn⟶∞,
for each i=1,2,…,N. Furthermore, the fact that {xn} is bounded and E is reflexive implies that we can choose a subsequence {xni+1} of {xn+1} such that xni+1⇀z and(29)limsupn→∞〈u-x*,J(xn+1-x*)〉=limi→∞〈u-x*,J(xni+1-x*)〉.
Now, from (26) we get that xni⇀z and from Lemma 6 we have that z∈F(Ti), for each i=1,2,…,N. Hence, z∈∩i=1NF(Ti). Therefore, putting x*=z in (29) and using the fact that J is weakly sequentially continuous we immediately obtain that limsupn→∞〈u-z,J(xn+1-z)〉=limi→∞〈u-z,J(xni+1-z)〉=〈u-z,J(z-z)〉=0. Again, putting x*=z in inequality (22), we get that
(30)∥xn+1-z∥2≤(1-αn)∥xn-z∥2+2αn〈u-z,J(xn+1-z)〉+2βnM(ln-1),
and, hence, it follows from (30) and Lemma 5 that ||xn-z||→0, as n→∞. Consequently, xn→z.
Case
2. Suppose that there exists a subsequence {ni} of {n} such that
(31)∥xni-x*∥<∥xni+1-x*∥,
for all i∈N. Then, by Lemma 7, there exists a nondecreasing sequence {mj}⊂N such that mj→∞, ||xmj-x*||≤||xmj+1-x*|| and ||xj-x*||≤||xmj+1-x*|| for all j∈N. Then from (21) and following the method of Case 1, we get that
(32)∥xmj-Timjxmj∥⟶0,asj⟶∞,
for each i=1,2,…,N. Thus, again following the method of Case 1, we obtain that xmj+1-xmj→0 and xmj-Tixmj→0, as j→∞, for each i=1,2,…,N and there exists z*∈∩i=1NF(Ti) such that
(33)limsupj→∞〈u-z*,J(xmj+1-z*)〉=0.
Then now, putting x*=z* in (22) we have that
(34)∥xmj+1-z*∥2≤(1-αmj)∥xmj-z*∥2+2αmj〈u-z*,J(xmj+1-z*)〉+2βmjM(lmj-1).
Since ∥xmj-z*∥2≤∥xmj+1-z*∥2, (34) implies that
(35)αmj∥xmj-z*∥2≤∥xmj-z*∥2-∥xmj+1-z*∥2+2αmj〈u-z*,J(xmj+1-z*)〉+2βmjM(lmj-1).
Moreover, since αmj>0, inequality (35) gives that
(36)∥xmj-z*∥2≤2〈u-z*,J(xmj+1-z*)〉+2βmjM(lmj-1)αmj.
Then, from (33) and the fact that 2βmjM(lmj-1)/αmj→0, we obtain that ||xmj-z*||→0, as j→∞. This together with (34) gives that ||xmj+1-z*||→0, as j→∞. But ||xj-z*||≤||xmj+1-z*||, for all j∈N; thus we obtain that xj→z*. 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 Theorem8, we assume a single asymptotically k-strict pseudocontractive mapping we get the following corollary.Corollary 9.
LetC be a nonempty, closed, and convex subset of a real uniformly convex Banach space E which has Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from E into E*. Let T:C→C be an asymptotically k-strict pseudocontractive mapping for 0≤k<1 with sequences {ln}⊂[1,∞). Assume that F(T) is nonempty. Let {xn} be a sequence defined by x1=u∈C and
(37)xn+1=αnu+(1-αn)((1-βn)xn+βnTnxn),n≥1,
where {αn}⊂(0,c)⊂(0,1), satisfying limn→∞αn=0, ∑‍αn=∞, limn→∞((ln-1)/αn)=0, and {βn}⊂[a,b]⊂(0,k) (a, b, and c constants). Then the sequence {xn} generated by (37) converges strongly to a fixed point of T.Proof.
PuttingT=T1=T2=⋯=TN in (14), we get that Sn=Tn and the scheme reduces to scheme (37) and following the method of proof of Theorem 8 we get that (see (21) and (22))
(38)∥xn+1-x*∥2≤(1-αn)∥xn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉-2βn(k-βn)(1-αn)∥xn-Tnxn∥2+2βnM′(ln-1)≤(1-αn)∥xn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉-2βn(k-βn)(1-c)∥xn-Tnxn∥2+2βnM′(ln-1)≤(1-αn)∥xn-x*∥2+2αn〈u-x*,J(xn+1-x*)〉+2βnM′(ln-1),
for some M′>0. Now, considering cases, as in the proof of Theorem 8, we obtain the required result.Corollary 10.
LetK be a nonempty, closed, and convex subset of lp, 1<p<∞. Let Ti:C→C be asymptotically ki-strict pseudocontractive mappings for 0≤ki<1 with sequences {ln,i}⊂[1,∞), for i=1,2,…,N. Assume that F:=∩i=1nF(Ti) is nonempty. Let {xn} be a sequence defined by x1=u∈C and
(39)xn+1=αnu+(1-αn)((1-βn)xn+βnSnxn),n≥1,
where Sn:=θn,1T1n+θn,2T2n+⋯+θn,NTNn, such that θn,1+θn,2+⋯+θn,N=1, for each n≥1, {αn},{θn,i}⊂(0,c)⊂(0,1), satisfying liminfn→∞θn,i>0, limn→∞αn=0, ∑‍αn=∞, limn→∞((ln,i-1)/αn)=0 and {βn}⊂[a,b]⊂(0,k) (for a, b, and c constants), for k=min1≤i≤N{ki}. Then the sequence {xn} generated by (39) converges strongly to a common fixed point of {Ti:i=1,2,…,N}.Proof.
We note thatlp, 1<p<∞, spaces are uniformly convex which have Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping from E into E* (see, e.g., [18]). Thus, the result follows from Theorem 8.Corollary 11.
LetK be a nonempty, closed, and convex subset of lp, 1<p<∞. Let T:C→C be an asymptotically k-strict pseudocontractive mapping for some 0≤k<1 with sequences {ln}⊂[1,∞). Assume that F(T) is nonempty. Let {xn} be a sequence defined by x1=u∈C and
(40)xn+1=αnu+(1-αn)((1-βn)xn+βnTnxn),n≥1,
where {αn}⊂(0,c)⊂(0,1), and {βn}⊂[a,b]⊂(0,k) (for a, b, and c constants) satisfying limn→∞αn=0, ∑‍αn=∞ and limn→∞((ln-1)/αn)=0. Then the sequence {xn} converges strongly to a fixed point of  T.If in Theorem8 we have that E=H, a real Hilbert space, then E is uniformly convex with Fréchet differentiable norm possessing a weakly sequentially continuous duality mapping. Thus, we have the following corollary.Corollary 12.
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 0≤ki<1 with sequences {ln,i}⊂[1,∞), for i=1,2,…,N. Assume that F:=∩i=1nF(Ti) is nonempty. Let {xn} be a sequence defined by x1=u∈C and
(41)xn+1=αnu+(1-αn)((1-βn)xn+βnSnxn),n≥1,
where Sn:=θn,1T1n+θn,2T2n+⋯+θn,NTNn, such that θn,1+θn,2+⋯+θn,N=1, for each n≥1, {αn},{θn,i}⊂(0,c)⊂(0,1), satisfying liminfnθn,i>0, limn→∞αn=0, ∑‍αn=∞, limn→∞((ln,i-1)/αn)=0 and {βn}⊂[a,b]⊂(0,k) (for a, b, and c constants), for k=min1≤i≤N{ki}. Then the sequence {xn} generated by (41) converges strongly to a common fixed point of {Ti:i=1,2,…,N}.Corollary 13.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T:C→C be an asymptotically k-strict pseudocontractive mapping for some 0≤k<1 with sequences {ln}⊂[1,∞). Assume that F(T) is nonempty. Let {xn} be a sequence defined by x1=u∈C and
(42)xn+1=αnu+(1-αn)((1-βn)xn+βnTnxn),n≥1,
where {αn}⊂(0,c)⊂(0,1), and {βn}⊂[a,b]⊂(0,k) (for a, b, and c constants) satisfying limn→∞αn=0, ∑‍αn=∞ and limn→∞((ln-1)/αn)=0. Then the sequence {xn} converges strongly to a fixed point of T.Remark 14.
We note that Corollary9 generalizes several recent results of this nature. Particularly, it extends Theorem KX of [12], Theorem 2 of Liu [5], and corresponding theorem of Schu [7] in the sense that our convergence is strong in more general Banach spaces possessing weakly sequentially continuous duality mappings without the requirement that T be completely continuous.Remark 15.
Corollary9 is an improvement of Theorem 3.2 of Osilike  et al. [13] and Theorems 3.1 and 3.2 of Zhang and Xie [17] in the sense that our convergence is strong without the requirement that liminfn→∞d(xn,F(T))=0, provided that E possesses weakly sequentially continuous duality mappings.



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*Source: 142759-2013-03-27.xml*