# On Convergence Results for Lipschitz Pseudocontractive Mappings

Authors: Shin Min Kang; Arif Rafiq
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

We establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.


---

## Body

## 1.   Introduction and Preliminaries

LetE be a real Banach space and K be a nonempty convex subset of E. Let J denote the normalized duality mapping from E to 2E* defined by
(1.1)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 mapping by j.LetT:D(T)⊂E→E be a mapping with domain D(T) in E.Definition 1.1.
T is said to be Lipschitz if there exists a constant L>1 such that
(1.2)∥Tx-Ty∥≤L∥x-y∥,∀x,y∈D(T).Definition 1.2.
T is said to be nonexpansive if
(1.3)∥Tx-Ty∥≤∥x-y∥,∀x,y∈D(T).Definition 1.3.
T is said to be pseudocontractive if for all x,y∈D(T), there exists j(x-y)∈J(x-y) such that
(1.4)〈Tx-Ty,j(x-y)〉≤∥x-y∥2.Remark 1.4.
It is well known that every nonexpansive mapping is pseudocontractive. Indeed, ifT is nonexpansive mapping, then for all  x,y∈D(T), there exists  j(x-y)∈J(x-y)  such that
(1.5)〈Tx-Ty,j(x-y)〉≤∥Tx-Ty∥∥x-y∥≤∥x-y∥2.Rhoades [1] showed that the class of pseudocontractive mappings properly contains the class of nonexpansive mappings.The class of pseudocontractions is, perhaps, the most important generalization of the class of nonexpansive mappings because of its strong relationship with the class of accretive mappings. A mappingA:E→E  is accretive if and only if  I-A  is pseudocontractive.For a nonempty convex subsetK of a normed space E and a mapping  T:K→K.The Mann iteration scheme [2]: the sequence {xn} is defined by
(1.6)x1∈K,xn+1=(1-an)xn+anTxn,n≥1,
where  {an} is a sequence in  [0,1].The Ishikawa iteration scheme [3]: the sequence {xn} is defined by
(1.7)x1∈K,xn+1=(1-an)xn+anTyn,yn=(1-bn)xn+bnTxn,n≥1,
where  {an} and {bn} are sequences in [0,1].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 (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 (e.g.,  [4–6] and the references cited therein).In 1974, Ishikawa [3] introduced an iteration scheme which, in some sense, is more general than that of Mann and which converges, under this setting, to a fixed point of  T. He proved the following result.Theorem 1.5.
LetK be a compact convex subset of a Hilbert space H and let  T:K→K be a Lipschitz pseudocontractive mapping. For arbitrary x1∈K, let {xn}n=1∞ be a sequence defined iteratively by
(1.8)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥1,
where   {αn}n=1∞  and  {βn}n=1∞  are sequences satisfying conditions (i)
0≤αn≤βn<1;(ii)
limn→∞βn=0;(iii)
∑n=1∞‍αnβn=∞.
Then{xn}n=1∞ converges strongly to a fixed point of   T.In [4], Chidume extended the results of Schu [7] from Hilbert spaces to the much more general class of real Banach spaces and approximated the fixed points of strongly pseudocontractive mappings.In this paper, we establish the strong convergence for the Ishikawa iteration scheme associated with Lipschitz pseudocontractive mappings in real Banach spaces. Moreover, our technique of proofs is of independent interest.


## 2. Main Results

We will need the following results.Lemma 2.12.1 (see [8]).
LetJ:E→2E be the normalized duality mapping. Then, for any x,y∈E, one has
(2.1)∥x+y∥2≤∥x∥2+2〈y,j(x+y)〉,∀j(x+y)∈J(x+y).Lemma 2.22.2 (see [9]).
If there exists a positive integerN such that for all  n≥N,  n∈ℕ (the set of all positive integers)
(2.2)ρn+1≤(1-δn2)ρn+bn,
where δn∈[0,1),   ∑n=1∞‍δn2=∞ and bn=o(δn), then
(2.3)limn→∞ρn=0.We now prove our main results.Theorem 2.3.
LetK be a nonempty closed convex subset of a real Banach space E and T:K→K be a Lipschitz pseudocontractive mapping such that  p∈F(T):={x∈K:Tx=x}. Let {αn}n=1∞ and {βn}n=1∞ be sequences in  [0,1]  satisfying the conditions:(iv)
∑n=1∞αn2=∞;(v)
limn→∞αn=0;(vi)
limn→∞βn=0.
For arbitraryx1∈K, let {xn}n=1∞ be defined iteratively by
(2.4)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnTxn,n≥1.
Then the following conditions are equivalent:(a)
{xn}n=1∞ converges strongly to the fixed point p of T.(b)
{Txn}n=1∞ and {Tyn}n=1∞ are bounded.Proof.
Becausep is a fixed point of T, then the set F(T) of fixed points of T is nonempty.
Suppose thatlimn→∞xn=p, then since  T is Lipschitz, so
(2.5)limn→∞Txn=p,limn→∞yn=limn→∞[(1-βn)xn+βnTxn]=p,
which implies that limn→∞Tyn=p. Therefore {Txn}n=1∞ and {Tyn}n=1∞ are bounded.
Set(2.6)M1=∥x0-p∥+supn≥1∥Txn-p∥+supn≥1∥Tyn-p∥.
Obviously  M1<∞.
It is clear that∥x0-p∥≤M1. Let ∥xn-p∥≤M1. Next we will prove that  ∥xn+1-p∥≤M1.
Consider(2.7)∥xn+1-p∥=∥(1-αn)xn+αnTyn-p∥=∥(1-αn)(xn-p)+αn(Tyn-p)∥≤(1-αn)∥xn-p∥+αn∥Tyn-p∥≤M1.
So, from the above discussion, we can conclude that the sequence {xn-p}n=1∞ is bounded. Let M2=supn≥1∥xn-p∥.
DenoteM=M1+M2. Obviously  M<∞.
Now from Lemma2.1 we obtain for all  n≥1(2.8)∥xn+1-p∥2=∥(1-αn)xn+αnTyn-p∥2=∥(1-αn)(xn-p)+αn(Tyn-p)∥2≤(1-αn)2∥xn-p∥2+2αn〈Tyn-p,j(xn+1-p)〉=(1-αn)2∥xn-p∥2+2αn〈Txn+1-p,j(xn+1-p)〉+2αn〈Tyn-Txn+1,j(xn+1-p)〉≤(1-αn)2∥xn-p∥2+2αn∥xn+1-p∥2+2αn∥Tyn-Txn+1∥∥xn+1-p∥≤(1-αn)2∥xn-p∥2+2αn∥xn+1-p∥2+2αnλn,
where
(2.9)λn=M∥Tyn-Txn+1∥.
Using (2.4) we have
(2.10)∥yn-xn+1∥≤∥yn-xn∥+∥xn-xn+1∥=βn∥xn-Txn∥+αn∥xn-Tyn∥≤2M(αn+βn).
From the conditions  limn→∞αn=0=limn→∞βn  and (2.10), we obtain
(2.11)limn→∞∥yn-xn+1∥=0,
and since T is Lipschitz,
(2.12)limn→∞∥Tyn-Txn+1∥=0,
thus, we have
(2.13)limn→∞λn=0.
The real function f:[0,∞)→[0,∞) defined by f(t)=t2 is increasing and convex. For all λ∈[0,1]  and t1,  t2>0 we have
(2.14)((1-λ)t1+λt2)2≤(1-λ)t12+λt22.
Consider
(2.15)∥xn+1-p∥2=∥(1-αn)xn+αnTyn-p∥2=∥(1-αn)(xn-p)+αn(Tyn-p)∥2≤[(1-αn)∥xn-p∥+αn∥Tyn-p∥]2≤(1-αn)∥xn-p∥2+αn∥Tyn-p∥2≤(1-αn)∥xn-p∥2+M2αn.
Substituting (2.15) in (2.8), we get
(2.16)∥xn+1-p∥2≤[(1-αn)2+2αn(1-αn)]∥xn-p∥2+2αn(M2αn+λn)=(1-αn2)∥xn-p∥2+εnαn,
where εn=2(M2αn+λn). Now, with the help of  ∑n=1∞‍αn2=∞,   limn→∞αn=0, (2.13), and Lemma 2.2, we obtain from (2.16) that
(2.17)limn→∞∥xn-p∥=0.
This completes the proof.Remark 2.4.
Our technique of proofs is of independent interest.Corollary 2.5.
LetK be a nonempty closed convex subset of a real Hilbert space E and let  T:K→K  be a Lipschitz pseudocontractive mapping such that  p∈F(T). Let {αn}n=1∞  and  {βn}n=1∞ be sequences in [0,1] satisfying the conditions  (iv),  (v), and  (vi).
For arbitraryx1∈K, let {xn}n=1∞ be the sequence defined iteratively by (2.4). Then the following conditions are equivalent:(a)
{xn}n=1∞ converges strongly to the fixed point   p of   T.(b)
{Txn}n=1∞ and {Tyn}n=1∞ are bounded.The proof of the following result runs on the lines of proof of the Theorem2.3, so is omitted.Theorem 2.6.
LetK be a nonempty closed convex subset of a real Banach space E and let  T,S:K→K be two Lipschitz pseudocontractive mappings such that p∈F(T)∩F(S):={x∈K:Tx=x=Sx}. Let {αn}n=1∞ and  {βn}n=1∞  be sequences in [0,1] satisfying the conditions  (iv),  (v), and  (vi). For arbitrary x1∈K, let {xn}n=1∞ be a sequence defined iteratively by
(2.18)xn+1=(1-αn)xn+αnTyn,yn=(1-βn)xn+βnSxn,n≥1.
Then the following conditions are equivalent:(a)
{xn}n=0∞  converges strongly to the common fixed point p of   T and  S.(b)
{Txn}n=0∞  and  {Syn}n=0∞  are bounded.Corollary 2.7.
LetK be a nonempty closed convex subset of a real Hilbert space E and let  T,S:K→K be two Lipschitz pseudocontractive mappings such that p∈F(T)∩F(S).  Let {αn}n=1∞  and  {βn}n=1∞   be sequences in  [0,1]  satisfying conditions  (iv),  (v), and  (vi).
For arbitraryx1∈K,  let  {xn}n=1∞  be the sequence defined iteratively by  (2.18). Then the following conditions are equivalent:(a)
{xn}n=0∞ converges strongly to the common fixed point  p  of  T  and  S.(b)
{Txn}n=0∞ and {Syn}n=0∞ are bounded.Remark 2.8.
It is worth to mentioning that we have the following.(1)
The results of Chidume [4]  and Zhou and Jia [10] depend on the geometry of the Banach space, whereas in our case we do not need such geometry.(2)
We remove the boundedness assumption onK introduced in [4, 10].



---

*Source: 902601-2012-09-29.xml*