# An iterative Algorithm for Hemicontractive Mappings in Banach Spaces

Authors: Youli Yu; Zhitao Wu; Pei-Xia Yang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We First introduce a three-step iterative algorithm for approximating the fixed points of the hemicontractive mappings in Banach spaces. Consequently, we prove the strong convergence of the proposed algorithm under some assumptions. Since three-step iterations include Ishikawa iterations as special cases, our result continue to hold for these problems. Our main results can be viewed as an important refinement of the previously known results.


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## Body

## 1. Introduction

In recent years, several convergence results have been proved on iterative methods for approximating fixed points of pseudocontractive mappings (see, e.g., [1–10] and references therein). It is worth mentioning that such iterative type methods are known as Mann iterations and Ishikawa iterations. It is clear that a Lipschitz pseudocontractive mapping with a unique fixed point for which the Mann iteration sequence failed to converge, but it does converge for the sequence obtained by the Ishikawa iterations, see [11]. In 2000, Noor [12] suggested and analyzed three-step iterative methods for finding the approximate solution of a continuous mapping in the Hilbert space using the technique of updating the solution. Three-step iterations are also known as Noor iteration. It is well known [13] that three-step iterative schemes include one-step (Mann) and two-step (Ishikawa) iterations as special cases. It raises an interesting question. Is there exist any Lipschitz pseudo-contractive mapping with a unique fixed point for which Ishikawa iteration sequence fail to convergence, but Does convergence for the sequence obtained from Noor iteration? This is an open and challenging problem. To the best of our knowledge, main result of Ishikawa [14], see Theorem IS has never been extended to more general Banach spaces. Motivated and inspired by the recent research activities in this filed, we suggest and analyze a three-step iterative scheme associated with hemi-contractive mappings in Banach spaces. We also prove the strong convergence of the sequence generated by the three-step iterations under mild conditions. Since three-step iterations include Ishikawa iterations as special cases, our results continue to hold for these problems. It is worth mentioning that our results may be considered as very significant, interesting and important extensions of the previously known results concerning pseudo-contractive mappings.LetE be a real Banach space and E* be its dual space. The normalized duality mapping from E to 2E* is defined by
(1.1)J(x)={x*∈E*:〈x,x*〉=‖x‖‖x*‖,‖x*‖=‖x‖},∀x∈E,
where 〈·,·〉 denotes the generalized duality pairing.LetC be a nonempty subset of E, a mapping T:C→C is called pseudo-contractive if there exists j(x-y)∈J(x-y) such that
(1.2)〈Tx-Ty,j(x-y)〉≤‖x-y‖2,
for all x,y∈C. Let F(T):={x∈C:Tx=x}. A mapping T:C→C is called hemicontractive if F(T)≠∅ and
(1.3)〈Tx-x*,j(x-x*)〉≤‖x-x*‖2,∀x∈C,x*∈F(T).
It is easy to see that the class of pseudo-contractive mappings with fixed points is a subclass of the class of hemicontractions. A mapping T:C→C is called Lipschitzian if there exists a constant L≥0 such that ∥Tx-Ty∥≤L∥x-y∥ for each x,y∈C.In 1974, Ishikawa [14] proved the following result for the pseudo-contractive mappings.Theorem 1.1 (see [14]).
IfC is a compact convex subset of a Hilbert space H, T:C→C is a Lipschitzian pseudo-contractive mapping. For x0∈C, define the sequence {xn} iteratively by
(1.4)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥0,
where {αn}, {βn} are sequences of positive numbers satisfying the conditions
(i)0≤αn≤βn<1;
(ii)limn→∞βn=0;
(iii)∑n=1∞‍αnβn=∞.
Then the sequence{xn} defined by (1.4) converges strongly to a fixed point of T.Since its publication in 1974, Theorem IS, as far as we know, has never been extended to more general Banach spaces.In this paper, we suggest and analyze a three-step iteration below Algorithm1.2 associated with hemi-contractive mappings having a strong convergence in the setting of Banach spaces under some appropriate conditions.Algorithm 1.2.
For arbitraryx1∈C, let the sequence {xn} be generated by
(1.5)zn=(1-γn)xn+γnTxn,yn=(1-βn)xn+βnTzn,xn+1=(1-αn)xn+αnTyn,n≥1.It is clear that Algorithm1.2 includes Mann (one-step) and Ishikawa (two-step) iterations as special cases. For some related works, please refere to [15–19].


## 2.  Preliminaries

LetE be a Banach space, the modulus of convexity of E is the function δE:(0,2]→[0,1] defined by
(2.1)δE(ϵ)=inf⁡{1-12‖x+y‖:‖x‖=1,‖y‖=1,‖x-y‖≥ϵ}.
A Banach space E is called uniformly convex if and only if δE(ϵ)>0 for all ϵ∈(0,2]. For p>1, the (generalized) duality mapping Jp:E→2E* is defined as Jp(x):={x*∈E:〈x,x*〉=∥x∥p,∥x*∥=∥x∥p-1}. In particular, J=J2 is the normalized duality mapping on E. It is known that Jp(x)=∥x∥p-2J(x), x≠0. A Banach space E is called p-uniformly convex if there exists a constant c>0 such that δE(ϵ)≥cϵp, 0<ϵ≤2. It is known (see e.g., [12]) that Lp is
(2.2)2uniformlyconvex,if1<p≤2,puniformlyconvex,ifp≥2.For proving our main results, we shall need the following lemmas.Lemma 2.1 (see [20]).
Letp>1 be a given real number. Then the following statements about a Banach space E are equivalent:
(i)E is p-uniformly convex;
(ii)  there is a constantcp>0 such that for every x,y∈E, jp(x)∈Jp(x), the following inequality holds:
(2.3)‖x+y‖p≥‖x‖p+p〈y,jp(x)〉+cp‖y‖p.Remark 2.2.
Replacingx by (x+y), y by (-y) in inequality (2.3) and using the Cauchy-Schwarz inequality, we can obtain
(2.4)‖x+y‖p≤‖x‖p+p‖y‖⋅‖x+y‖p-1.Lemma 2.3 (see [20]).
Letp>1 be a given real number. Let E be a p-uniformly convex Banach space. Then, there exists a constant d>0 such that
(2.5)‖λx+(1-λ)y‖p≤λ‖x‖p+(1-λ)‖y‖p-Wp(λ)d‖x-y‖p,
for all λ∈[0,1] and x,y∈E, where Wp(λ)=λp(1-λ)+λ(1-λ)p.Lemma 2.4 (see [4]).
Let{ρn}:{σn} be two nonnegative sequences and for all integers n≥N0 (for some fixed N0), ρn+1≤ρn+σn.
(i)  if∑n=1∞σn<∞, then limn→∞ρn exists;
(ii)  if∑n=1∞σn<∞ and {ρn} has a sequence converging to zero, then limn→∞ρn=0.


## 3.  Main Results

In the sequel,cp and d will denote the constants appearing in inequalities (2.3) and (2.5), respectively. For the rest of this paper, we shall assume that E be a real p-uniformly convex Banach space such that 2-(p-2)dp>(p-1)cp-1 and p≤1+cp. For Lp spaces with 1<p≤2, the following inequalities hold (see [12, pages 1131-1132]):
(3.1)‖x+y‖2≥‖x‖2+2〈y,J(x)〉+cp‖y‖2,‖λx+(1-λ)y‖2≤λ‖x‖2+(1-λ)‖y‖2-W2(λ)(p-1)‖x-y‖2,
for λ∈[0,1] and for all x,y∈E, where cp=[1+tp(p-1)][(1+tp)-(p-1)], and for 0<tp<1, tp is the unique solution of the equation g(t)=(p-2)t(p-1)+(p-1)t(p-2)-1=0.Remark 3.1.
We observe that the functionh:[0,1]→[0,∞) defined by h(x)=(1+xp-1)/(1+x)p-1 is increasing on [0,1] (h′(x)=(1+x)p-2(p-1)(xp-2-1)/(1+x)2p-2≥0), hence for Lp(1<p≤2), we have cp≥1 and d=p-1. Therefore, the conditions 2-(p-2)dp>(p-1)cp-1 and p≤1+cp are satisfied.Lemma 3.2.
LetE be a real p-uniformly convex Banach space, ∅≠C⊂E nonempty closed convex and bounded, and T:C→C a hemi-contractive mapping with F(T)≠∅. Then, for each x∈C and for each integer n≥1, the following inequality holds:
(3.2)cp‖Tx-x*‖p≤(p-1)‖x-x*‖p+‖x-Tx‖p,∀x*∈F(T).Proof.
Replacingx by (1/2)(x-x*) and y by (-1/2)(Tx-x*) in inequality (2.3), we can get
(3.3)‖x-Tx‖p≥‖x-x*‖p-p2p-1〈Tx-x*,jp(12(x-x*))〉+cp‖Tx-x*‖p≥‖x-x*‖p-p‖x-x*‖p+cp‖Tx-x*‖p.
Since jp((1/2)(x-x*))∈Jp((1/2)(x-x*))=2-(p-1)∥x-x*∥(p-2)J(x-x*) so that
(3.4)cp‖Tx-x*‖p≤(p-1)‖x-x*‖p+‖x-Tx‖p.
This completes the proof.Remark 3.3.
We note that the functionf:[0,∞)→(-∞,+∞) defined by f(x)=Lpxp-dp(1-x)2-(p-2)+(p-1)cp-1 is strictly increasing on (0,∞). Hence, it has at most one zero on (0,∞), provided that f(0)=(p-1)cp-1-dp2-(p-2)<0. In this case, since f(1)=Lp+(p-1)cp-1>0, it follows that the zero tp∈(0,1).Lemma 3.4.
LetE be a real p-uniformly convex Banach space such that 2-(p-2)dp>(p-1)cp-1 and p≤1+cp. Let C be a nonempty closed convex and bounded subset of E, let T:C→C be a Lipschitz hemi-contractive mapping with Lipschitz constant L≥0 and F(T)≠∅. Let {αn}, {βn} and {γn} be three real sequences in [0,1] satisfying the following conditions:
(3.5)ϵ≤1-dcp(1-αn)2-(p-2)≤βn≤b,∑n=1∞‍γn<∞.
for all integers n≥1, some ϵ>0 and b∈(0,tp), where tp is the unique solution of the equation:
(3.6)Lpxp-dp(1-x)2-(p-2)+(p-1)cp-1=0,
on (0,∞).
For arbitraryx1∈C, let the sequence {xn} be generated by
(3.7)zn=(1-γn)xn+γnTxn,yn=(1-βn)xn+βnTzn,xn+1=(1-αn)xn+αnTyn,n≥1.
Then, limn→∞∥xn-Txn∥=0.Proof.
We shall useM to denote the possible different constants appearing in the following reasoning.
Letx*∈F(T). Using inequality (2.5), we have
(3.8)‖xn+1-x*‖p=‖(1-αn)(xn-x*)+αn(Tyn-x*)‖p≤(1-αn)‖xn-x*‖p+αn‖Tyn-x*‖p-Wp(αn)d‖xn-Tyn‖p.
From (3.2), we have
(3.9)cp‖Txn-x*‖p≤(p-1)‖xn-x*‖p+‖xn-Txn‖p,(3.10)cp‖Tyn-x*‖p≤(p-1)‖yn-x*‖p+‖yn-Tyn‖p.
Moreover, we also have
(3.11)‖yn-x*‖p=‖(1-βn)(xn-x*)+βn(Tzn-x*)‖p≤(1-βn)‖xn-x*‖p+βn‖Tzn-x*‖p-Wp(βn)d‖xn-Tzn‖p,(3.12)‖yn-Tyn‖p=‖(1-βn)(xn-Tyn)+βn(Tzn-Tyn)‖p≤(1-βn)‖xn-Tyn‖p+βn‖Tzn-Tyn‖p-Wp(βn)d‖xn-Tzn‖p.
At the same time, applying (2.4), we can obtain the following estimates:
(3.13)‖Tzn-x*‖p=‖Txn-x*+Tzn-Txn‖p≤‖Txn-x*‖p+p‖Tzn-Txn‖‖Tzn-x*‖p-1≤‖Txn-x*‖p+pL‖zn-xn‖‖Tzn-x*‖p-1=‖Txn-x*‖p+pLγn‖Txn-xn‖‖Tzn-x*‖p-1≤‖Txn-x*‖p+Mγn,(3.14)‖xn-Txn‖p=‖Tzn-Txn+xn-Tzn‖p≤‖xn-Tzn‖p+p‖Tzn-Txn‖‖xn-Txn‖p-1≤‖xn-Tzn‖p+Mγn,(3.15)‖Tzn-Tyn‖p=‖Txn-Tyn+Tzn-Txn‖p≤‖Txn-Tyn‖p+p‖Tzn-Txn‖‖Tzn-Tyn‖p-1≤‖Txn-Tyn‖p+Mγn.
Substitute (3.13) and (3.14) into (3.11) to get
(3.16)‖yn-x*‖p≤(1-βn)‖xn-x*‖p+βn‖Txn-x*‖p-Wp(βn)d‖xn-Txn‖p+Mγn,
this together with (3.9) implies that
(3.17)‖yn-x*‖p≤(1-βn)‖xn-x*‖p+βncp-1{(p-1)‖xn-x*‖p+‖xn-Txn‖p}-Wp(βn)d‖xn-Txn‖p+Mγn=[1+βncp-1(p-1-cp)]‖xn-x*‖p+[βncp-1-Wp(βn)d]‖xn-Txn‖p+Mγn.
Set tn=βncp-1(p-1-cp), rn=βncp-1-Wp(βn)d. Then,
(3.18)‖yn-x*‖p≤(1+tn)‖xn-x*‖p+rn‖xn-Txn‖p+Mγn.
From (3.12), (3.14), and (3.15), we have
(3.19)‖yn-Tyn‖p≤(1-βn)‖xn-Tyn‖p+βn‖Txn-Tyn‖p-Wp(βn)d‖xn-Txn‖p+Mγn.
Substitution of (3.18) and (3.19) into (3.10) yields
(3.20)cp‖Tyn-x*‖p≤(p-1)(1+tn)‖xn-x*‖p+(p-1)rn‖xn-Txn‖p+(1-βn)‖xn-Tyn‖p+βn‖Txn-Tyn‖p-Wp(βn)d‖xn-Txn‖p+Mγn=(p-1)(1+tn)‖xn-x*‖p+[(p-1)rn-Wp(βn)d]×‖xn-Txn‖p+(1-βn)‖xn-Tyn‖p+βn‖Txn-Tyn‖p+Mγn.
Substitution of this inequality into (3.8) now gives
(3.21)‖xn+1-x*‖p≤(1-αn)‖xn-x*‖p+αncp-1{(p-1)(1+tn)‖xn-x*‖p+[(p-1)rn-Wp(βn)d]×‖xn-Txn‖p+(1-βn)‖xn-Tyn‖p+βn‖Txn-Tyn‖p}-Wp(αn)d‖xn-Tyn‖p+Mγn=[(1-αn)+αncp-1(p-1)(1+tn)]‖xn-x*‖p+[αncp-1(1-βn)-Wp(αn)d]‖xn-Tyn‖p+αncp-1[(p-1)rn-Wp(βn)d]‖xn-Txn‖p+αnβncp-1‖Txn-Tyn‖p+Mγn,
that is,
(3.22)‖xn+1-x*‖p≤{1+αn[(p-1)cp-1(1+tn)-1]}‖xn-x*‖p-[Wp(αn)d-cp-1αn(1-βn)]‖xn-Tyn‖p-cp-1αn[Wp(βn)d-(p-1)rn]‖xn-Txn‖p+αnβncp-1‖Txn-Tyn‖p+Mγn.
Observe that cp-1(p-1)(1+tn)-1=cp-2(p-1-cp)[(p-1)βn+cp] and that by condition (3.5), since Wp(αn)≥αn(1-αn)2-(p-2), we get Wp(αn)d-cp-1αn(1-βn)≥0. so that
(3.23)‖xn+1-x*‖p≤{1+αncp-2(p-1-cp)[(p-1)βn+cp]}‖xn-x*‖p-αncp-1[Wp(βn)d-(p-1)rn]‖xn-Txn‖p+αnβncp-1‖Txn-Tyn‖p+Mγn.
Since T is Lipschitzian, we have
(3.24)‖Txn-Tyn‖p≤Lp‖xn-yn‖p=Lp‖βn(xn-Tzn)‖p≤Lpβnp‖xn-Tzn‖p=Lpβnp‖xn-Txn+Txn-Tzn‖p≤Lpβnp[‖xn-Txn‖p+p‖Txn-Tzn‖‖xn-Tzn‖p-1]≤Lpβnp‖xn-Txn‖p+Mγn.
By the assumption p≤1+cp, hence
(3.25)‖xn+1-x*‖p≤‖xn-x*‖p-αnβncp-1[dp(1-βn)2-(p-2)-(p-1)cp-1-βnpLp]×‖xn-Txn‖p+Mγn.
Since b∈(0,tp), it follows that δ=dp(1-b)2-(p-2)-(p-1)cp-1-bpLp>0. We can choose some ϵ such that ϵ′=1-(1-ϵ)2-(p-2)cpd>0. Then, condition (3.5) implies αn≥ϵ′>0. Furthermore, inequality (3.25) now yields the following estimates:
(3.26)‖xn+1-x*‖p≤‖xn-x*‖p-ϵϵ′cp-1δ‖xn-Txn‖p+Mγn≤‖xn-x*‖p+Mγn.
Since ∑n=0∞‍γn<∞, it follows from Lemma 2.3 that limn→∞∥xn-x*∥p exists. Let limn→∞∥xn-x*∥p=r. Inequality (3.26) also yields
(3.27)0<ϵϵ′cp-1δ‖xn-Txn‖p≤‖xn-x*‖p-‖xn+1-x*‖p+Mγn→0.
Hence, limn→∞∥xn-Txn∥=0. This completes the proof.Remark 3.5.
The interest and importance of Lemma3.2 lie in the fact that strong convergence of the sequence {xn} is achieved under certain mild compactness assumptions either on T or on its domain.Now, we give a strong convergence theorem as follows.Theorem 3.6.
LetE be a real p-uniformly convex Banach space such that 2-(p-2)dp>(p-1)cp-1 and p≤1+cp. Let C be a nonempty closed convex and bounded subset of E, T:C→C be a completely continuous Lipschitz hemi-contractive mapping with Lipschitz constant L≥0 and F(T)≠∅. Let {αn}, {βn}, and {γn} be three real sequences in [0,1] satisfying the following conditions:
(3.28)ϵ≤1-dcp(1-αn)2-(p-2)≤βn≤b,∑n=1∞‍γn<∞,
for all integers n≥1, some ϵ>0 and b∈(0,tp), where tp is the unique solution of the equation:
(3.29)Lpxp-dp(1-x)2-(p-2)+(p-1)cp-1=0,
on (0,∞).
For arbitraryx1∈C, let the sequence {xn} be defined by (3.7). Then, {xn} converges strongly to a fixed point of T.Proof.
By Lemma3.2  limn→∞∥xn-Txn∥=0. Since T is completely continuous, there exists a subsequence {Txni} of {Txn} such that Txni→y*. This implies, by Lemma 3.2 that
(3.30)xni⟶y*.
By the continuity of T and Lemma 3.2, we obtain Ty*=y*, that is, y* is a fixed point of T. Replacing the x* by y* in inequality (3.26), we obtain that
(3.31)‖xn+1-y*‖p≤‖xn-y*‖p-ϵϵ′cp-1δ‖xn-Txn‖p≤‖xn-y*‖p+Mγn.
From (3.30), we know that {∥xn-y*∥} has a sequence converging to zero. We note that the condition ∑n=0∞‍γn<∞. Hence from inequality (3.31) and Lemma 2.3, we can conclude that ∥xn-y*∥→∞ as n→∞, that is, {xn} converges to a fixed point of T. This completes the proof.From Theorem3.6, we can obtain the following result.Corollary 3.7.
LetE be a real p-uniformly convex Banach space such that 2-(p-2)dp>(p-1)cp-1 and p≤1+cp. Let C be a nonempty closed convex and bounded subset of E, T:C→C be a completely continuous Lipschitz hemi-contractive mapping with Lipschitz constant L≥0 and F(T)≠∅. Let {αn} and {βn} be two real sequences in [0,1] satisfying the following condition:
(3.32)ϵ≤1-dcp(1-αn)2-(p-2)≤βn≤b,
for all integers n≥1, some ϵ>0 and b∈(0,tp), where tp is the unique solution of the equation:
(3.33)Lpxp-dp(1-x)2-(p-2)+(p-1)cp-1=0
on (0,∞).
For arbitraryx1∈C, let the sequence {xn} be defined by (1.4). Then, {xn} converges strongly to a fixed points of T.



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