# Modified Mann-Halpern Algorithms for Pseudocontractive Mappings

Authors: Hong-bo Du
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Modified Mann-Halpern algorithms for finding the fixed points of pseudocontractive mappings are presented. Strong convergence theorems are obtained.


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## Body

## 1. Introduction

Finding the fixed points of nonlinear operators is an important topic in fixed point theory, due to the fact that many nonlinear problems can be reformulated as fixed point equations of nonlinear mappings. The research of this area dates back to Picard’s and Banach’s time. Now it is well known that the Picard iterates{Tnx} converge to the unique fixed point of T whenever T is a contraction of a complete metric space. However, if T is not a contraction (e.g., nonexpansive), then the Picard algorithm {Tnx} does not converge. Consequently, Mann’s algorithm was constructed by Mann [1] in 1953:
(1)xn+1=(1-αn)xn+αnTxn,n≥0.
There are a large number of papers on Mann’s algorithm in the literature. See [2–5]. Now we know that if T is nonexpansive, then Mann’s algorithm converges weakly to a fixed point of T. This algorithm however does not converge in the strong topology.In order to get the strong convergence, the following Halpern’s algorithm was introduced:(2)xn+1=αnu+(1-αn)Txn,n≥0.
The interest and importance of Halpern iterative method lie in the fact that strong convergence of the sequence {xn} is achieved under certain mild conditions on parameter {αn} in a general Banach space. Please refer to [6–12].In the present paper, we are devoted to find the fixed points of pseudocontractive mappings. For some related works, please see [13–23]. The interest of pseudocontractions lies in their connection with monotone operators. Browder and Petryshyn [24] studied weak convergence of Mann’s algorithm for the class of strict pseudocontractions. But Mann’s algorithm fails to converge for Lipschitzian pseudocontractions [25].Inspired by the results in the literature, the main purpose of this paper is to construct an iterative method for finding the fixed points of pseudocontractive mappings. Under some mild conditions, strong convergence results are given.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively. Let C be a nonempty closed convex subset of H. A mapping T:C→C is called pseudocontractive if
(3)〈Tx-Ty,x-y〉≤∥x-y∥2,x,y∈C.
A mapping T:C→C is called k-Lipschitzian if there exists k>0 such that
(4)∥Tx-Ty∥≤k∥x-y∥,
for all x,y∈C. In this case, if k<1, then T is a k-contraction.It is well known that in a real Hilbert spaceH the following inequality holds:
(5)∥x+y∥2≤∥x∥2+2〈y,x+y〉,
for all x,y∈H.In the present paper, we will use the following notations:(i)
we useFix(T) to denote the set of fixed points of T;(ii)
xn⇀x denotes the weak convergence of xn to x;(iii)
xn→x denotes the strong convergence of xn to x.Lemma 1 (see [26]).
LetC be a closed convex subset of a real Hilbert space H. Let T:C→C be a continuous pseudocontractive mapping. Then (I-T) is demiclosed at zero.Lemma 2 (see [27]).
Let{wn} be a sequence of real numbers. Assume {wn} does not decrease at infinity; that is, there exists at least a subsequence {wnk} of {wn} such that wnk≤wnk+1 for all k≥0. For every n≥M, define an integer sequence {W(n)} as
(6)W(n)=max{i≤n:wni<wni+1}.
Then W(n)→∞ as n→∞ and for all n≥M(7)max{wW(n),wn}≤wW(n)+1.Lemma 3 (see [28]).
Assume that{An} is a sequence of nonnegative real numbers such that
(8)An+1≤(1-γn)An+ξn,
where {γn} is a sequence in (0,1) and {ξn} is a sequence such that(1)
∑n=1∞‍γn=∞;(2)
limsupn→∞(ξn/γn)≤0 or ∑n=1∞‍|ξn|<∞.
Thenlimn→∞An=0.


## 3. Main Results

Now we present the statement of our algorithm.The Modified Mann-Halpern Algorithm. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T:C→C be a pseudocontractive mapping and S:C→H a ρ-contractive mapping. Let {αn}, {βn}, and {γn} be three real number sequences in [0,1]. We have the following steps.(1)
Initialization:(9)∀x0∈C.(2)
Mann step: for a givenxn, define a sequence yn by
(10)yn=(1-γn)xn+γnTxn,for all n≥0.(3)
Halpern step: for a givenxn and yn, define
(11)xn+1=αnS(xn)+(1-αn-βn)xn+βnTyn,for all n≥0.In the following, we assume that(i)
the mappingT:C→C is k-Lipschitzian;(ii)
the sequences{αn}, {βn}, and {γn} satisfy the following conditions (C1)–(C5):
(C1):
limn→∞αn=0;(C2):
∑n=1∞‍αn=∞;(C3):
αn+βn≤γn;(C4):
0<liminfn→∞βn;(C5):
0<limsupn→∞γn<1/(1+k2+1).Now, we prove our main result as follows.Theorem 4.
SupposeFix(T)≠∅. Then the sequence {xn} defined by (11) converges strongly to a fixed point of T.Proof.
SinceS is a ρ-condition, then ProjFix(T)S is a contractive mapping (where Proj is the metric projection). Hence, there exists a unique u such that u=ProjFix(T)S(u). In the sequel, we will show that the sequence {xn} defined by (11) converges strongly to u.
From (11), we get(12)∥xn+1-u∥=∥αnS(xn)+(1-αn-βn)xn+βnTyn-u∥≤∥αn(S(xn)-u)+(1-αn-βn)(xn-u)+βn(Tyn-u)∥=∥(1-αn-βn1-αn(xn-u)+βn1-αn(Tyn-u))αn(S(xn)-u)+(1-αn)×(1-αn-βn1-αn(xn-u)+βn1-αn(Tyn-u))∥≤αn∥S(xn)-u∥+(1-αn)×∥(1-αn-βn)(xn-u)1-αn+βn(Tyn-u)1-αn∥.It is well known that there holds the following inequality in Hilbert spaces:
(13)∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2
for all x,y∈H and t∈[0,1]. Hence, we have
(14)∥(1-αn-βn)(xn-u)1-αn+βn(Tyn-u)1-αn∥2≤1-αn-βn1-αn∥xn-u∥2+βn1-αn∥Tyn-u∥2-βn(1-αn-βn)(1-αn)2∥xn-Tyn∥2.
We know that T is pseudocontractive if and only if T satisfies the condition
(15)∥Tx-Ty∥2≤∥x-y∥2+∥(I-T)x-(I-T)y∥2
for all x,y∈C. Since u∈Fix(T), we have from (15) that
(16)∥Tx-u∥2≤∥x-u∥2+∥x-Tx∥2,
for all x∈C.
By using (13) and (16), we obtain
(17)∥Tyn-u∥2≤∥yn-u∥2+∥yn-Tyn∥2=∥(1-γn)xn+γnTxn-u∥2+∥(1-γn)xn+γnTxn-Tyn∥2=∥(1-γn)(xn-u)+γn(Txn-u)∥2+∥(1-γn)(xn-Tyn)+γn(Txn-Tyn)∥2=(1-γn)∥xn-u∥2+γn∥Txn-u∥2-γn(1-γn)∥xn-Txn∥2+(1-γn)∥xn-Tyn∥2+γn∥Txn-Tyn∥2-γn(1-γn)∥xn-Txn∥2≤(1-γn)∥xn-u∥2+γn(∥xn-u∥2+∥xn-Txn∥2)-γn(1-γn)∥xn-Txn∥2+(1-γn)∥xn-Tyn∥2+γn∥Txn-Tyn∥2-γn(1-γn)∥xn-Txn∥2.
Note that T is k-Lipschitzian and
(18)∥xn-yn∥=γn∥xn-Txn∥.
From (17), we have
(19)∥Tyn-u∥2≤(1-γn)∥xn-u∥2+γn(∥xn-u∥2+∥xn-Txn∥2)-γn(1-γn)∥xn-Txn∥2+(1-γn)∥xn-Tyn∥2+γnk2∥xn-yn∥2-γn(1-γn)∥xn-Txn∥2=(1-γn)∥xn-u∥2+γn(∥xn-u∥2+∥xn-Txn∥2)-γn(1-γn)∥xn-Txn∥2+(1-γn)∥xn-Tyn∥2+γn3k2∥xn-Txn∥2-γn(1-γn)∥xn-Txn∥2=∥xn-u∥2+(1-γn)∥xn-Tyn∥2-γn(1-2γn-γn2k2)∥xn-Txn∥2.
By condition (C5), without loss of generality, we may assume that γn≤a<1/(1+k2+1) for all n. Then, we have 1-2γn-γn2L2>0 for all n≥0. Substituting (19) to (14) and noting condition (C3), we have
(20)∥(1-αn-βn)(xn-u)1-αn+βn(Tyn-u)1-αn∥2≤1-αn-βn1-αn∥xn-u∥2+βn1-αn×(∥xn-u∥2+(1-γn)∥xn-Tyn∥2)-βn(1-αn-βn)(1-αn)2∥xn-Tyn∥2=∥xn-u∥2+βn(αn+βn-γn)(1-αn)2∥xn-Tyn∥2≤∥xn-u∥2.
Therefore,
(21)∥(1-αn-βn)(xn-u)1-αn+βn(Tyn-u)1-αn∥≤∥xn-u∥.
It follows from (12) and (21) that
(22)∥xn+1-u∥≤αn∥S(xn)-u∥+(1-αn)∥xn-u∥≤αn∥S(xn)-S(u)∥+αn∥S(u)-u∥+(1-αn)∥xn-u∥≤αnρ∥xn-u∥+αn∥S(u)-u∥+(1-αn)∥xn-u∥=αn∥S(u)-u∥+[1-(1-ρ)αn]∥xn-u∥≤max{∥xn-u∥,∥S(u)-u∥1-ρ}≤max{∥x0-u∥,∥S(u)-u∥1-ρ}.
This implies that the sequence {xn} is bounded.
From (5) and (11), we have
(23)∥xn+1-u∥2=∥(1-αn)(xn-u)-βn(xn-Tyn)+αn(S(xn)-u)∥2≤∥(1-αn)(xn-u)-βn(xn-Tyn)∥2+2αn〈S(xn)-u,xn+1-u〉=∥(1-αn)(xn-u)∥2-2βn(1-αn)〈xn-Tyn,xn-u〉+βn2∥xn-Tyn∥2+2αn〈S(xn)-u,xn+1-u〉.
Note that (19) is equivalent to
(24)2〈xn-Tyn,xn-u〉≥γn∥xn-Tyn∥2+γn(1-2γn-γn2k2)∥xn-Txn∥2.
Therefore,
(25)∥xn+1-u∥2≤(1-αn)∥xn-u∥2-βn(1-αn)γn∥xn-Tyn∥2+βn2∥xn-Tyn∥2-βn(1-αn)γn(1-2γn-γn2k2)∥xn-Txn∥2+2αn〈S(xn)-u,xn+1-u〉≤(1-αn)∥xn-u∥2-βn(1-αn)γn(1-2γn-γn2k2)∥xn-Txn∥2+2αn〈S(xn)-u,xn+1-u〉.
It follows that
(26)∥xn+1-u∥2-∥xn-u∥2+βn(1-αn)γn(1-2γn-γn2k2)∥xn-Txn∥2≤αn(〈2S(xn)-u,xn+1-u〉-∥xn-u∥2).
Since xn and S(xn) are bounded, there exists M>0 such that supn{2〈S(xn)-u,xn+1-u〉-∥xn-u∥2}≤M. So
(27)∥xn+1-u∥2-∥xn-u∥2+βn(1-αn)γn(1-2γn-γn2k2)∥xn-Txn∥2≤αnM.
Next, we prove two cases.
Assume there exists an integerm>0 such that {∥xn-u∥} is decreasing for all n≥m.
In this case, we know thatlimn→∞∥xn-u∥ exists. From (27), we deduce
(28)βn(1-αn)γn(1-2γn-γn2k2)∥xn-Txn∥2≤∥xn-u∥2-∥xn+1-u∥2+Mαn.
By conditions (C4) and (C5), we have liminfn→∞βn(1-αn)γn(1-2γn-γn2k2)>0. Thus, from (28), we get
(29)limn→∞∥xn-Txn∥=0.
Since {xn} is bounded, there exists a subsequence {xnk} of {xn} satisfying
(30)xnk⟶x~∈C,limsupn→∞〈S(u)-u,xn-u〉=limk→∞〈S(u)-u,xnk-u〉.
Thus, we use the demiclosed principle of T (Lemma 1) to deduce
(31)x~∈Fix(T).
So
(32)limsupn→∞〈S(u)-u,xn-u〉=limk→∞〈S(u)-u,xnk-u〉=〈S(u)-u,x~-u〉≤0.
Returning to (25) and using (5) we obtain
(33)∥xn+1-u∥2≤(1-αn)∥xn-u∥2+2αn〈S(xn)-u,xn+1-u〉=(1-αn)∥xn-u∥2+2αn〈S(xn)-S(u),xn+1-u〉+2αn〈S(u)-u,xn+1-u〉≤(1-αn)∥xn-u∥2+2αnρ∥xn-u∥∥xn+1-u∥+2αn〈S(u)-u,xn+1-u〉≤(1-αn)∥xn-u∥2+αnρ(∥xn-u∥2+∥xn+1-u∥2)+2αn〈S(u)-u,xn+1-u〉.
It follows that
(34)∥xn+1-u∥2≤[1-(1-ρ)αn]∥xn-u∥2+2αn1-αnρ〈Su-u,xn+1-u〉.
In Lemma 3, we take An=∥xn+1-u∥2, γn=(1-ρ)αn, and ξn=(2αn/(1-αnρ))〈Su-u,xn+1-u〉. We can check easily that ∑n=1∞‍γn=∞ and limsupn→∞(ξn/γn)≤0. Thus, we deduce that xn→u.
Assume there exists an integern0 such that ∥xn0-u∥≤∥xn0+1-u∥. In this case, we set ωn={∥xn-u∥}. Then, we have ωn0≤ωn0+1. Define an integer sequence {Wn} for all n≥n0 as follows:
(35)W(n)=max{l∈ℕ∣n0≤l≤n,ωl≤ωl+1}.
It is clear that W(n) is a nondecreasing sequence satisfying
(36)limn→∞W(n)=∞,ωτ(n)≤ωW(n)+1,
for all n≥n0. From (28), we get
(37)limn→∞∥xW(n)-TxW(n)∥=0.
This implies that ωw(xW(n))⊂Fix(T). Thus, we obtain
(38)limsupn→∞〈S(u)-u,xW(n)-u〉≤0.
Since ωW(n)≤ωW(n)+1, we have from (34) that
(39)ωW(n)2≤ωW(n)+12≤[1-(1-ρ)αW(n)]ωW(n)2+2αW(n)1-αW(n)ρ〈Su-u,xW(n)+1-u〉.
It follows that
(40)ωW(n)2≤2(1-αW(n)ρ)(1-ρ)〈Su-u,xW(n)+1-u〉.
Combining (38) and (40), we have
(41)limsupn→∞ωW(n)≤0,
and hence
(42)limn→∞ωW(n)=0.
From (34), we obtain
(43)∥xW(n)+1-u∥2≤[1-(1-ρ)αW(n)]∥xW(n)-u∥2+2αW(n)1-αW(n)ρ〈Su-u,xW(n)+1-u〉.
It follows that
(44)limsupn→∞ωW(n)+1≤limsupn→∞ωW(n).
This together with (42) implies that
(45)limn→∞ωW(n)+1=0.
Applying Lemma 2 we get
(46)0≤ωn≤max{ωW(n),ωW(n)+1}.
Therefore, ωn→0. That is, xn→u. The proof is completed.


## 4. Conclusions

It is now well known that Mann’s algorithm fails to converge for Lipschitzian pseudocontractions. Strong convergence of Ishikawa’s algorithm has not been achieved without compactness assumption. In the present paper, modified Mann-Halpern algorithms for finding the fixed points of pseudocontractive mappings are presented. Strong convergence theorems are obtained.



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