# On the Convergence of Implicit Picard Iterative Sequences for Strongly Pseudocontractive Mappings in Banach Spaces

Authors: Shin Min Kang; Arif Rafiq; Sun Young Cho
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. (2009).


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## Body

## 1. Introduction and Preliminaries

LetE be a real Banach space with dual E*. The symbol D(T) stands for the domain of T.LetT:D(T)→E be a mapping.Definition 1.
The mappingT is said to be Lipschitzian if there exists a constant L>0 such that
(1)∥Tx-Ty∥≤L∥x-y∥
for all x,y∈D(T).Definition 2.
The mappingT is called strongly pseudocontractive if there exists t>1 such that
(2)∥x-y∥≤∥(1+r)(x-y)-rt(Tx-Ty)∥
for all x,y∈D(T) and r>0. If t=1 in inequality (2), then T is called pseudocontractive.We will denote byJ the normalized duality mapping from E to 2E* defined by
(3)J(x)={f*∈E*:〈x,f*〉=∥x∥2=∥f*∥2},
where 〈·,·〉 denotes the generalized duality pairing. It follows from inequality (2) that T is strongly pseudocontractive if and only if there exists j(x-y)∈J(x-y) such that
(4)〈(I-T)x-(I-T)y,j(x-y)〉≥k∥x-y∥2
for all x,y∈D(T), where k=(t-1)/t∈(0,1). Consequently, from inequality (4) it follows easily that T is strongly pseudocontractive if and only if
(5)∥x-y∥≤∥x-y+s[(I-T-kI)x-(I-T-kI)y]∥
for all x,y∈D(T) and s>0.Closely related to the class of pseudocontractive maps is the class of accretive operators.LetA:D(A)→E be an operator.Definition 3.
The operatorA is called accretive if
(6)∥x-y∥≤∥x-y+s(Ax-Ay)∥
for all x,y∈D(A) and s>0.Also, as a consequence of Kato [1], this accretive condition can be expressed in terms of the duality mapping as follows.For eachx,y∈D(A), there exists j(x-y)∈J(x-y) such that
(7)〈Ax-Ay,j(x-y)〉≥0.
Consequently, inequality (2) with t=1 yields that A is accretive if and only if T:=(I-A) is pseudocontractive. Furthermore, from setting A:=(I-T), it follows from inequality (5) that T is strongly pseudocontractive if and only if (A-kI) is accretive, and, using (7), this implies that T=(I-A) is strongly pseudocontractive if and only if there exists k∈(0,1) such that
(8)〈Ax-Ay,j(x-y)〉≥k∥x-y∥2
for all x,y∈D(A). The operator A satisfying inequality (8) is called strongly accretive. It is then clear that A is strongly accretive if and only if T=(I-A) is strongly pseudocontractive. Thus, the mapping theory for strongly accretive operators is closely related to the fixed point theory of strongly pseudocontractive mappings. We will exploit this connection in the sequel.The notion of accretive operators was introduced independently in 1967 by Kato [1] and Browder [2]. An early fundamental result in the theory of accretive operators, due to Browder, states that the initial value problem
(9)dudt+Au=0,u(0)=u0
is solvable if A is locally Lipschitzian and accretive on E. If u is independent of t, then Au=0 and the solution of this equation corresponds to the equilibrium points of the system (9). Consequently, considerable research efforts have been devoted, especially within the past 15 years or so, to developing constructive techniques for the determination of the kernels of accretive operators in Banach spaces (see, e.g., [3–19]. Two well-known iterative schemes, the Mann iterative method (see, e.g., [20]) and the Ishikawa iterative scheme (see, e.g., [21]), have successfully been employed.The Mann and Ishikawa iterative schemes are global and their rate of convergence is generally of the orderO(n-1/2). It is clear that if, for an operator U, the classical iterative sequence of the form, xn+1=Uxn, x0∈D(U) (the so-called Picard iterative sequence) converges, then it is certainly superior and preferred to either the Mann or the Ishikawa sequence since it requires less computations and, moreover, its rate of convergence is always at least as fast as that of a geometric progression.In [22, 23], Chidume proved the following results.Theorem 4.
LetE be an arbitrary real Banach space and A:E→E Lipschitz (with constant L>0) and strongly accretive with a strong accretive constant k∈(0,1). Let x* denote a solution of the equation Ax=0. Set ϵ:=(1/2)(k/(1+L(3+L-k))) and define Aϵ:E→E by Aϵx:=x-ϵAx for each x∈E.
For arbitraryx0∈E, define the sequence {xn}n=0∞ in E by
(10)xn+1=Aϵxn,n≥0.
Then {xn}n=0∞ converges strongly to x* with
(11)∥xn+1-x*∥≤δn∥x0-x*∥,
where δ=(1-(1/2)kϵ)∈(0,1). Moreover, x* is unique.Corollary 5.
LetE be an arbitrary real Banach space and K a nonempty convex subset of E. Let T:K→K be Lipschitz (with constant L>0) and strongly pseudocontractive (i.e., T satisfies inequality (5) for all x,y∈K). Assume that T has a fixed point x*∈K. Set ϵ0:=(1/2)(k/(1+L(3+L-k))) and define Tϵ0:K→K by Tϵ0x=(1-ϵ0)x+ϵ0Tx for each x∈K. For arbitrary x0∈K, define the sequence {xn}n=0∞ in K by
(12)xn+1=Tϵ0xn,n≥0.
Then{xn}n=0∞ converges strongly to x* with
(13)∥xn+1-x*∥≤δn∥x0-x*∥,
where δ=(1-(1/2)kϵ0)∈(0,1). Moreover, x* is unique.Recently, Ćirić et al. [24] improved the results of Chidume [22, 23], Liu [14], and Sastry and Babu [18] as in the following results.Theorem 6.
LetE be an arbitrary real Banach space and A:E→E a Lipschitz (with constant L>0) and strongly accretive with a strong accretive constant k∈(0,1). Let x* denote a solution of the equation Ax=0. Set ε:=(k-η)/L(2+L), η∈(0,k) and define Aε:E→E by Aεx:=x-εAx for each x∈E. For arbitrary x0∈E, define the sequence {xn}n=0∞ in E by
(14)xn+1=Aεxn,n≥0.
Then {xn}n=0∞ converges strongly to x* with
(15)∥xn+1-x*∥≤θn∥x0-x*∥,
where θ=(1-((k-η)/(k(k-η)+L(2+L)))η)∈(0,1). Thus the choice η=k/2 yields θ=1-k2/2[k+2L(2+L)]. Moreover, x* is unique.Corollary 7.
LetE be an arbitrary real Banach space and K a nonempty convex subset of E. Let T:K→K be Lipschitz (with constant L>0) and strongly pseudocontractive (i.e., T satisfies inequality (5) for all x,y∈K). Assume that T has a fixed point x*∈K. Set ε0:=(k-η)/L(2+L), η∈(0,k) and define Tε0:K→K by Tε0x=(1-ε0)x+ε0Tx for each x∈K. For arbitrary x0∈K, define the sequence {xn}n=0∞ in K by
(16)xn+1=Tε0xn,n≥0.
Then {xn}n=0∞ converges strongly to x* with
(17)∥xn+1-x*∥≤θn∥x0-x*∥,
where θ=(1-((k-η)/(k(k-η)+L(2+L)))η)∈(0,1). Moreover, x* is unique.In this paper, we study the convergence of implicit Picard iterative sequences for strongly accretive and strongly pseudocontractive mappings. We have also improved the results of Ćirić et al. [24].


## 2. Main Results

In the following theorems,L>1 will denote the Lipschitz constant of the operator A and k will denote the strong accretive constant of the operator A as in inequality (8). Furthermore, ϵ>0 is defined by
(18)ϵ:=k-ηL+(1+L)(k-η),η∈(0,k).
With these notations, we prove the following theorem.Theorem 8.
LetE be an arbitrary real Banach space and A:E→E Lipschitz and strongly accretive with a strong accretive constant k∈(0,1). Let x* denote a solution of the equation Ax=0. Define Aϵ:E→E by Aϵxn=(1-ϵ)xn-1+ϵxn-ϵAxn for each xn∈E. For arbitrary x0∈E, define the sequence {xn}n=0∞ in E by
(19)xn=Aϵxn,n≥1.
Then {xn}n=0∞ converges strongly to x* with
(20)∥xn+1-x*∥≤ρn∥x0-x*∥,
where ρ=(1-((k-η)/(L+((k-η)(1+L+k)))η)∈(0,1). Thus the choice η=k/2 yields ρ=1-k2/2[2L+k(1+L+k)]. Moreover, x* is unique.Proof.
Existence ofx* follows from [5, Theorem 13.1]. Define T=(I-A) where I denotes the identity mapping on E. Observe that Ax*=0 if and only if x* is a fixed point of T. Moreover, T is strongly pseudocontractive since A is strongly accretive, and so T also satisfies inequality (5) for all x,y∈E and s>0. Furthermore, the recursion formula xn=Aϵxn becomes
(21)xn=(1-ϵ)xn-1+ϵTxn,n≥1.
Observe that
(22)x*=(1+ϵ)x*+ϵ(I-T-kI)x*-(1-k)ϵx*,
and from the recursion formula (21)
(23)xn-1=(1+ϵ)xn+ϵ(I-T-kI)xn-(1-k)ϵxn+ϵ2(xn-1-Txn),
which implies that
(24)xn-1-x*=(1+ϵ)(xn-x*)+ϵ[(I-T-kI)xn-(I-T-kI)x*]-(1-k)ϵ(xn-x*)+ϵ2(xn-1-Txn).
This implies using inequality (5) with s=ϵ/(1+ϵ) and y=x* that
(25)∥xn-1-x*∥≥(1+ϵ)(∥(xn-x*)+ϵ1+ϵ×[(I-T-kI)xn-(I-T-kI)x*]ϵ1+ϵ∥)-(1-k)ϵ∥xn-x*∥-ϵ2∥xn-1-Txn∥≥(1+ϵ)∥xn-x*∥-(1-k)ϵ∥xn-x*∥-ϵ2∥xn-1-Txn∥=(1+kϵ)∥xn-x*∥-ϵ2∥xn-1-Txn∥.
Observe that
(26)∥xn-1-Txn∥≤∥xn-1-Txn-1∥+∥Txn-1-Txn∥≤∥Axn-1∥+∥xn-1-xn∥+∥Axn-1-Axn∥≤L∥xn-1-x*∥+(1+L)∥xn-1-xn∥=L∥xn-1-x*∥+(1+L)ϵ∥xn-1-Txn∥,
and so
(27)∥xn-1-Txn∥≤L1-(1+L)ϵ∥xn-1-x*∥,
so that from (25) we obtain
(28)∥xn-1-x*∥≥(1+kϵ)∥xn-x*∥∥xn-1-x*∥-Lϵ21-(1+L)ϵ∥xn-1-x*∥.
Therefore
(29)∥xn-x*∥≤1+Lϵ2/(1-(1+L)ϵ)1+kϵ∥xn-1-x*∥,
where
(30)ρ=1+Lϵ2/(1-(1+L)ϵ)1+kϵ=1-ϵ1+kϵ(k-Lϵ1-(1+L)ϵ)=1-ϵ1+kϵη=1-k-ηL+(k-η)(1+L+k)η.
From (29) and (30), we get
(31)∥xn-x*∥≤ρ∥xn-1-x*∥≤⋯≤ρn∥x0-x*∥⟶0
as n→∞. Hence xn→x* as n→∞. Uniqueness follows from the strong accretivity property of A.The following is an immediate corollary of Theorem8.Corollary 9.
LetE be an arbitrary real Banach space and K a nonempty convex subset of E. Let T:K→K be Lipschitz (with constant L>1) and strongly pseudocontractive (i.e., T satisfies inequality (5) for all x,y∈K). Assume that T has a fixed point x*∈K. Set ϵ0:=(k-η)/(L+(1+L)(k-η)), η∈(0,k) and define Aϵ0:K→K by Aϵ0xn=(1-ϵ0)xn-1+ϵ0xn-ϵ0Axn for each xn∈K. For arbitrary x0∈K, define the sequence {xn}n=0∞ in K by
(32)xn=Aϵ0xn,n≥1.
Then {xn}n=0∞ converges strongly to x* with
(33)∥xn+1-x*∥≤ρ0n∥x0-x*∥,
where ρ0=(1-((k-η)/(L+(k-η)(1+L+k)))η)∈(0,1). Thus the choice η=k/2 yields ρ0=1-k2/2[2L+k(1+L+k)]. Moreover, x* is unique.Proof.
Observe thatx* is a fixed point of T if and only if it is a fixed point of Tϵ0. Furthermore, the recursion formula (32) is simplified to the formula
(34)xn=(1-ϵ0)xn-1+ϵ0Txn,
which is similar to (21). Following the method of computations as in the proof of the Theorem 8, we obtain
(35)∥xn-x*∥≤1+Lϵ02/(1-(1+L)ϵ0)1+kϵ0∥xn-1-x*∥=(1-k-ηL+(k-η)(1+L+k)η)∥xn-1-x*∥.
Set ρ0=1-((k-η)/(L+(k-η)(1+L+k)))η. Then from (35) we obtain
(36)∥xn-x*∥≤ρ0∥xn-1-x*∥≤⋯≤ρ0n∥x0-x*∥⟶0
as n→∞. This completes the proof.Remark 10.
SinceL>1 and k<L, we have
(37)L>k-η.
So we can easily obtain
(38)1L+(k-η)(1+L+k)>1k(k-η)+L(2+L).
Now(39)ρ=1-(k-η)L+(k-η)(1+L+k)η<1-(k-η)k(k-η)+L(2+L)η=θ.
Thus the relation between Ćirić et al. [24] and our parameter of convergence, that is, between θ and ρ, respectively, is the following:
(40)ρ<θ.Our convergence parameterρ shows the overall improvement for θ, and consequently the results of Ćirić et al. [24] are improved.



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