# Strong Convergence for Hybrid ImplicitS-Iteration Scheme of Nonexpansive and Strongly Pseudocontractive Mappings

Authors: Shin Min Kang; Arif Rafiq; Faisal Ali; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

LetK be a nonempty closed convex subset of a real Banach space E, let S:K→K be nonexpansive, and let  T:K→K be Lipschitz strongly pseudocontractive mappings such that p∈FS∩FT=x∈K:Sx=Tx=x and x-Sy≤Sx-Syandx-Ty≤Tx-Ty for all x,y∈K. Let βn be a sequence in 0,1 satisfying (i) ∑n=1∞βn=∞; (ii) limn→∞⁡βn=0. For arbitrary x0∈K, let xn be a sequence iteratively defined by xn=Syn,yn=1-βnxn-1+βnTxn,n≥1. Then the sequence xn converges strongly to a common fixed point p of S and T.


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## Body

## 1. Introduction and Preliminaries

LetE be a real Banach space and let K be a nonempty convex subset of E. Let J denote the normalized duality mapping from E to 2E* defined by
(1)J(x)={f*∈E*:〈x,f*〉=∥x∥2,∥f*∥=∥x∥},x∈E,
where E* denotes the dual space of E and 〈·,·〉 denotes the generalized duality pairing. We will denote the single-valued duality map by j.LetT:K→K be a mapping.Definition 1.
The mappingT is said to beLipschitzian if there exists L>1 such that
(2)∥Tx-Ty∥≤L∥x-y∥
for all x,y∈K.Definition 2.
The mappingT is said to benonexpansive if
(3)∥Tx-Ty∥≤∥x-y∥
for all x,y∈K.Definition 3.
The mappingT is said to bepseudocontractive if
(4)∥x-y∥≤∥x-y+t((I-T)x-(I-T)y)∥
for all x,y∈K and t>0.Remark 4.
As a consequence of a result of Kato [1], it follows from the inequality that T is pseudocontractive if and only if there exists j(x-y)∈J(x-y) such that
(5)〈Tx-Ty,j(x-y)〉≤∥x-y∥2
for all x,y∈K.Definition 5.
The mappingT is said to bestrongly pseudocontractive if there exists a constant t>1 such that
(6)∥x-y∥≤∥(1+r)(x-y)-rt(Tx-Ty)∥
for all x,y∈K and r>0. Or equivalently (see [2]) one has for 0<k<1(7)〈Tx-Ty,j(x-y)〉≤k∥x-y∥2
for all x,y∈K.For a nonempty convex subsetK of a normed space E, T:K→K is a mapping.(I) The sequence {xn}, defined by, for arbitrary x1∈K,
(8)xn+1=(1-an)xn+anTyn,yn=(1-bn)xn+bnTxn,n≥1,
where {an} and {bn} are sequences in [0,1], is known as the Ishikawa iteration process [3].Ifbn=0 for n≥1, then the Ishikawa iteration scheme becomes the Mann iteration process [4].(S) The sequence {xn}, defined by, for arbitrary x1∈K,
(9)xn+1=Tyn,yn=(1-bn)xn+bnTxn,n≥1,
where {bn} is a sequence in [0,1], is known as the S-iteration process [5, 6].In the last few years or so, numerous papers have been published on the iterative approximation of fixed points of Lipschitz strongly pseudocontractive mappings using the Ishikawa iteration scheme (see, e.g., [3]). Results which had been known only in Hilbert spaces and only for Lipschitz mappings have been extended to more general Banach spaces (see, e.g., [7–13] and the references cited therein).In 1974, Ishikawa [3] proved the following result.Theorem 6.
LetK be a compact convex subset of a Hilbert space H and let T:K→K be a Lipschitzian pseudocontractive mapping. For arbitrary x1∈K, let {xn} be a sequence defined iteratively by
(10)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥1,
where {αn} and {βn} are sequences satisfying(i)
0≤αn≤βn<1;(ii)
lim⁡n→∞βn=0;(iii)
∑n=1∞‍αnβn=∞.
Then the sequence{xn} converges strongly to a fixed point of T.In [7], Chidume extended the results of Schu [12] from Hilbert spaces to the much more general class of real Banach spaces and approximate the fixed points of pseudocontractive mappings. Also, in [14], he investigated the approximation of the fixed points of strongly pseudocontractive mappings.In [15], Zhou and Jia gave the answer of the question raised by Chidume [14] and proved the following.IfX is a real Banach space with a uniformly convex dual X*, K is a nonempty bounded closed convex subset of X, and T:K→K is a continuous strongly pseudocontractive mapping, then the Ishikawa iteration scheme converges strongly to the unique fixed point of T.In [16], Liu et al. introduced the following condition.Remark 7.
LetS,T:K→K be two mappings. The mappings S and T are said to satisfycondition(C1) if
(C1)∥x-Ty∥≤∥Sx-Ty∥
for all x,y∈K.In 2012, Kang et al. [17] established the strong convergence for the implicit S-iterative process associated with Lipschitzian hemicontractive mappings in Hilbert spaces.Theorem 8.
LetK be a compact convex subset of a real Hilbert space H and let T:K→K be a Lipschitzian hemicontractive mapping satisfying
(C2)∥x-Ty∥≤∥Tx-Ty∥
for all x,y∈K. Let {βn} be a sequence in [0,1] satisfying(i)
∑n=1∞‍βn=∞;(ii)
∑n=1∞‍βn2<∞.
For arbitraryx0∈K, let {xn} be a sequence iteratively defined by
(11)xn=Tyn,yn=(1-βn)xn-1+βnTxn,n≥1.
Then the sequence {xn} converges strongly to the fixed point x* of T.In 2013, Kang et al. [18] proved the following result.Theorem 9.
LetK be a nonempty closed convex subset of a real Banach space E, let S:K→K be a nonexpansive mapping, and let T:K→K be a Lipschitz strongly pseudocontractive mapping such that p∈F(S)∩F(T)={x∈K:Sx=Tx=x} and
(C3)∥x-Sy∥≤∥Sx-Sy∥,∥x-Ty∥≤∥Tx-Ty∥
for all x,y∈K. Let {βn} be a sequence in [0,1] satisfying(i)
∑n=1∞‍βn=∞;(ii)
lim⁡n→∞βn=0.
For arbitraryx1∈K, let {xn} be a sequence iteratively defined by
(12)xn+1=Syn,yn=(1-βn)xn+βnTxn,n≥1.
Then the sequence {xn} converges strongly to a common fixed point p of S and T.Keeping in view the importance of the implicit iteration schemes (see [17]) in this paper we establish the strong convergence theorem for the hybrid implicit S-iterative scheme associated with nonexpansive and Lipschitz strongly pseudocontractive mappings in real Banach spaces.


## 2. Main Results

We will need the following results.Lemma 10 (see [19, 20]).
LetJ:E→2E* be the normalized duality mapping. Then for any x,y∈E, one has
(13)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,wwwwwww∀j(x+y)∈J(x+y).Lemma 11 (see [13]).
Let{ρn} and {θn} be nonnegative sequences satisfying
(14)ρn+1≤(1-θn)ρn+bn,
where θn∈[0,1), ∑n=1∞‍θn=∞, and bn=o(θn). Then lim⁡n→∞ρn=0.The following is our main result.Theorem 12.
LetK be a nonempty closed convex subset of a real Banach space E, let S:K→K be a nonexpansive mapping, and let T:K→K be a Lipschitz strongly pseudocontractive mapping such that p∈F(S)∩F(T)={x∈K:Sx=Tx=x} and condition (C3).
Let{βn} be a sequence in [0,1] satisfying(i)
∑n=1∞‍βn=∞;(ii)
lim⁡n→∞βn=0.
For arbitraryx0∈K, let {xn} be a sequence iteratively defined by
(15)xn=Syn,yn=(1-βn)xn-1+βnTxn,n≥1.
Then the sequence {xn} converges strongly to a common fixed point p of S and T.Proof.
For strongly pseudocontractive mappings, the existence of a fixed point follows from Deimling [21]. It is shown in [15] that the set of fixed points for strongly pseudocontractions is a singleton.
By (ii), sincelim⁡n→∞βn=0, there exists n0∈N such that ∀n≥n0,
(16)βn≤min⁡{14k,1-k2(1+L)(1+2L)},
where k<1/2 and L is a Lipschitz constant of T. Consider
(17)∥xn-p∥2=〈xn-p,j(xn-p)〉=〈Syn-p,j(xn-p)〉=〈Txn-p,j(xn-p)〉+〈Syn-Txn,j(xn-p)〉≤k∥xn-p∥2+∥Syn-Txn∥∥xn-p∥,
where
(18)∥Syn-Txn∥≤∥Syn-Tyn∥+∥Tyn-Txn∥≤∥xn-Syn∥+∥xn-Tyn∥+∥Tyn-Txn∥≤∥Sxn-Syn∥+∥Txn-Tyn∥+∥Tyn-Txn∥=∥Sxn-Syn∥+2∥Txn-Tyn∥≤(1+2L)∥xn-yn∥,(19)∥xn-yn∥≤∥xn-xn-1∥+∥xn-1-yn∥=∥Syn-xn-1∥+∥xn-1-yn∥≤∥Sxn-1-Syn∥+∥xn-1-yn∥≤2∥xn-1-yn∥=2βn∥xn-1-Txn∥≤2βn(∥xn-1-p∥+∥p-Txn∥)≤2βn(∥xn-1-p∥+L∥xn-p∥),
and consequently from (18) and (19), we obtain
(20)∥Syn-Txn∥≤2(1+2L)βn∥xn-1-p∥+2L(1+2L)βn∥xn-p∥.
Substituting (20) in (17) and using (16), we get
(21)∥xn-p∥≤2(1+2L)βn1-k-2L(1+2L)βn∥xn-1-p∥≤∥xn-1-p∥forn≥n0.
So, from the above discussion, we can conclude that the sequence {xn-p} is bounded. Since T is Lipschitzian, so {Txn-p} is also bounded. Let M1=sup⁡n≥1∥xn-p∥+sup⁡n≥1∥Txn-p∥. Also by (ii), we have
(22)∥xn-1-yn∥=βn∥xn-1-Txn∥≤M1βn⟶0
as n→∞, which implies that {xn-1-yn} is bounded, so let M2=sup⁡n≥1∥xn-1-yn∥+M1. Further
(23)∥yn-p∥≤∥yn-xn-1∥+∥xn-1-p∥≤M2,
which implies that {yn-p} is bounded. Therefore {Tyn-p} is also bounded.
Set(24)M3=sup⁡n≥1∥yn-p∥+sup⁡n≥1∥Tyn-p∥.
DenoteM=M1+M2+M3. Obviously M<∞.
Now, from (15), for all n≥1, we obtain
(25)∥xn-p∥2=∥Syn-p∥2≤∥yn-p∥2,
and by Lemma 10,
(26)∥yn-p∥2=∥(1-βn)xn-1+βnTxn-p∥2=∥(1-βn)(xn-1-p)+βn(Txn-p)∥2≤(1-βn)2∥xn-1-p∥2+2βn〈Txn-p,j(yn-p)〉=(1-βn)2∥xn-1-p∥2+2βn〈Tyn-p,j(yn-p)〉+2βn〈Txn-Tyn,j(yn-p)〉≤(1-βn)2∥xn-1-p∥2+2kβn∥yn-p∥2+2βn∥Txn-Tyn∥∥yn-p∥≤(1-βn)2∥xn-1-p∥2+2kβn∥yn-p∥2+2MLβn∥xn-yn∥,∀j(yn-p)∈J(yn-p),
which implies that
(27)∥yn-p∥2≤(1-βn)21-2kβn∥xn-1-p∥2+2MLβn1-2kβn∥xn-yn∥≤(1-βn)∥xn-1-p∥2+4MLβn∥xn-yn∥forn≥n0.
Because of (16), we have (1-βn)/(1-2kβn)≤1 and 1/(1-2kβn)≤2. Also, by (ii) and (19), ∥xn-yn∥≤2M(1+L)βn→0 as n→∞.
Hence (25) and (27) give
(28)∥xn-p∥2≤(1-βn)∥xn-1-p∥2+4MLβn∥xn-yn∥.
For alln≥1, put
(29)ρn=∥xn-1-p∥,θn=βn,bn=4MLβn∥xn-yn∥;
then according to Lemma 11, we obtain from (28) that
(30)lim⁡n→∞∥xn-p∥=0.
This completes the proof.Corollary 13.
LetK be a nonempty closed convex subset of a real Hilbert space H, let S:K→K be a nonexpansive mapping, and let T:K→K be a Lipschitz strongly pseudocontractive mapping such that p∈F(S)∩F(T)={x∈K:Sx=Tx=x} and condition (C3). Let {βn} be a sequence in [0,1] satisfying conditions (i) and (ii) in Theorem 12.
For arbitraryx0∈K, let {xn} be a sequence iteratively defined by (15). Then the sequence {xn} converges strongly to a common fixed point p of S and T.Example 14.
As a particular case, we may choose, for instance,βn=1/n.Remark 15.
(
1
) Condition (C2) is due to Kang et al. [17] and condition (C1) with S=T becomes condition (C2).
(2) Condition(C3) is due to Kang et al. [18] and condition (C3) with S=T becomes condition (C2).



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*Source: 735673-2014-07-07.xml*