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- Open Access

# Some results on a general iterative method for *k*-strictly pseudo-contractive mappings

- Jong Soo Jung
^{1}Email author

**2011**:24

https://doi.org/10.1186/1687-1812-2011-24

© Jung; licensee Springer. 2011

**Received:**30 September 2010**Accepted:**2 August 2011**Published:**2 August 2011

## Abstract

Let *H* be a Hilbert space, *C* be a closed convex subset of *H* such that *C* ± *C* ⊂ *C*, and *T* : *C* → *H* be a *k*-strictly pseudo-contractive mapping with *F*(*T*) ≠ ∅ for some 0 ≤ *k <* 1. Let *F* : *C* → *C* be a *κ*-Lipschitzian and *η*-strongly monotone operator with *κ >* 0 and *η >* 0 and *f* : *C* → *C* be a contraction with the contractive constant *α* ∈ (0, 1). Let
,
and *τ <* 1. Let {*α*_{
n
} } and {*β*_{
n
} } be sequences in (0, 1). It is proved that under appropriate control conditions on {*α*_{
n
} } and {*β*_{
n
} }, the sequence {*x*_{
n
} } generated by the iterative scheme *x*_{n+1}= *α*_{
n
}*γf*(*x*_{
n
}) + *β*_{
n
}*x*_{
n
}+ ((1 - *β*_{
n
})*I* - *α*_{
n
}*μF*)*P*_{
C
}*Sx*_{
n
}, where *S* : *C* → *H* is a mapping defined by *Sx* = *kx* + (1 - *k*)*Tx* and *P*_{
C
} is the metric projection of *H* onto *C*, converges strongly to *q* ∈ *F*(*T*), which solves the variational inequality 〈*μFq - γf*(*q*), *q - p*〉 ≤ 0 for *p* ∈ *F*(*T*).

**MSC:** 47H09, 47H05, 47H10, 47J25, 49M05

## Keywords

- Iterative schemes
*k*-strictly pseudo-contractive mapping- Nonexpansive mapping
- Fixed points
- Contraction
*κ*-Lipschitzian*η*-strongly monotone operator- Variational inequality
- Hilbert space

## 1 Introduction

*H*be a real Hilbert space and

*C*be a nonempty closed convex subset of

*H*. Recall that a mapping

*f*:

*C*→

*C*is a

*contraction*on

*C*if there exists a constant

*α*∈ (0, 1) such that ||

*f*(

*x*) -

*f*(

*y*)|| ≤

*α*||

*x - y*||,

*x*,

*y*∈

*C*. A mapping

*T*:

*C*→

*H*is said to be

*k-strictly pseudo-contractive*if there exists a constant

*k*∈ [0, 1) such that

and *F*(*T*) denote the set of fixed points of the mapping *T*; that is, *F*(*T*) = {*x* ∈ *C* : *Tx* = *x*}.

Note that the class of *k*-strictly pseudo-contractions includes the class of non-expansive mappings *T* on *C* (that is, ||*Tx - Ty*|| ≤ ||*x - y*||, *x*, *y* ∈ *C*) as a subclass. That is, *T* is nonexpansive if and only if *T* is 0-strictly pseudo-contractive. The mapping *T* is also said to be pseudo-contractive if *k* = 1 and *T* is said to be strongly pseudo-contractive if there exists a constant *λ* ∈ (0, 1) such that *T* - *λI* is pseudo-contractive. Clearly, the class of *k*-strictly pseudo-contractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mappings. Also we remark that the class of strongly pseudo-contractive mappings is independent of the class of *k*-strictly pseudo-contractive mappings (see [1–3]). The class of pseudo-contraction is one of the most important classes of mappings among nonlinear mappings. Recently, many authors have been devoting the studies on the problems of finding fixed points for pseudo-contractions, see, for example, [4–7] and references therein.

*T*, a contraction

*f*with the contractive constant

*α*∈ (0, 1), and

*α*

_{ n }∈ (0, 1),

This iterative scheme was first introduced by Moudafi [8].

In particular, under the control conditions on {*α*_{
n
} }

(C1) lim_{n→∞}*α*_{
n
}= 0;

(C2) ;

(C3) ; or,

(C4) ,

*x*

_{ n }} generated by (1.1) converges strongly to a fixed point

*q*of

*T*, which is the unique solution of the following variational inequality:

*A*on

*H*with constant , a nonexpansive mapping

*T*on

*H*, a contraction

*f*:

*H*→

*H*with the contractive constant

*α*∈ (0, 1), {

*α*

_{ n }} ⊂ (0, 1) and

*γ >*0,

*α*

_{ n }} satisfies the conditions (C1), (C2), and (C3) (or (C1), (C2), and (C4)), then the sequence {

*x*

_{ n }} generated by (1.2) converges strongly to the unique solution of the variational inequality

where *h* is a potential function for *γf*.

In 2010, in order to improve the corresponding results of Cho et al. [5] as well as Marino and Xu [10] by removing the condition (C3), Jung [6] studied the following composite iterative scheme for the class of *k*-strictly pseudo-contractive mappings.

**Theorem J**.

*Let H be a Hilbert space, C be a closed convex subset of H such that C*±

*C*⊂

*C, T*:

*C*→

*H be a k-strictly pseudo-contractive mapping with F*(

*T*) ≠ ∅,

*for some*0 ≤

*k <*1.

*Let A be a strongly positive bounded linear operator on C with constant*

*and f*:

*C*→

*C be a contraction with the contractive constant α*∈ (0, 1)

*such that*.

*Let*{

*α*

_{ n }}

*and*{

*β*

_{ n }}

*be sequences in*(0, 1)

*satisfying the conditions*(C1), (C2)

*and the condition*0

*<*lim inf

_{n→∞}

*β*

_{ n }≤ lim sup

_{n→∞}

*β*

_{ n }< 1.

*Let*{

*x*

_{ n }}

*be a sequence in C generated by*

*where S*:

*C*→

*H is a mapping defined by Sx*=

*kx*+ (1 -

*k*)

*Tx and P*

_{ C }

*is the metric projection of H onto C. Then*{

*x*

_{ n }}

*converges strongly to a fixed point q of T, which is the unique solution of the following variational inequality related to the linear operator A:*

*F*:

*H*→

*H*is called

*κ*-Lipschitzian if there exists a positive constant

*κ*such that

From the definitions, we note that a strongly positive bounded linear operator *A* is a ||*A*||-Lipschitzian and
-strongly monotone operator.

*F*:

*H*→

*H*is a

*κ*-Lipschitzian and

*η*-strongly monotone operator with

*κ >*0,

*η >*0, and

*S*:

*H*→

*H*is a nonexpansive mapping, and proved that if {

*λ*

_{ n }} satisfies appropriate conditions, then the sequence {

*x*

_{ n }} generated by (1.5) converges strongly to the unique solution of the variational inequality

In 2010, by combining the iterative method (1.2) with the Yamada's method (1.5), Tian [12] considered the following general iterative method.

**Theorem T1**.

*Let H be a Hilbert space, F*:

*H*→

*H be a κ-Lipschitzian and η-strongly monotone operator with κ >*0

*and η >*0,

*and S*:

*H*→

*H be a nonexpansive mapping with F*(

*S*) ≠ ∅.

*Let f*:

*H*→

*H be a contraction with the contractive constant α*∈ (0, 1).

*Let*

*and*.

*Let*{

*α*

_{ n }}

*be a sequence in*(0, 1)

*satisfying the conditions*(C1), (C2)

*and*(C3)

*(or*(C1), (C2)

*and*(C4)

*). Let*{

*x*

_{ n }}

*be a sequence in H generated by*

*Then*{

*x*

_{ n }}

*converges strongly to a fixed point*

*of S, which is the unique solution of the following variational inequality related to the operator F*:

*C*a closed convex subset of

*H*such that

*C*±

*C*⊂

*C*,

*k*-strictly pseudo-contractive mapping

*T*:

*C*→

*H*with

*F*(

*T*) ≠ ∅, a contraction

*f*:

*C*→

*C*with the contractive constant

*α*∈ (0, 1),

*μ*> 0 and {

*α*

_{ n }}, {

*β*

_{ n }} ⊂ (0, 1),

where *S* : *C* → *H* is a mapping defined by *Sx* = *kx*+(1 - *k*)*Tx*, *P*_{
C
} is the metric projection of *H* onto *C*, and *F* : *C* → *C* is a *κ*-Lipschitzian and *η*-strongly monotone operator with *κ >* 0 and *η >* 0. Under certain different control conditions on {*α*_{
n
} }, we establish the strong convergence of the sequence {*x*_{
n
} } generated by (IS) to a fixed point of *T*, which is a solution of the variational inequality (1.6) related to the operator *F*. By removing the condition (C3)
on {*α*_{
n
} }, the main results improve, develop and complement the corresponding results of Tian [12] as well as Cho et al. [5], Jung [6] and Marino and Xu [10]. Our results also improve the corresponding results of Halpern [13], Moudafi [8], Wittmann [14] and Xu [9].

## 2 Preliminaries and lemmas

Throughout this paper, when {*x*_{
n
} } is a sequence in *E*, then *x*_{
n
} → *x* (resp., *x*_{
n
} ⇀ *x*) will denote strong (resp., weak) convergence of the sequence {*x*_{
n
} } to *x*.

for all *y* ∈ *C*. *P*_{
C
} is called the *metric projection* of *H* onto *C*. It is well known that *P*_{
C
} is nonexpansive.

*H*satisfies the

*Opial condition*, that is, for any sequence {

*x*

_{ n }} with

*x*

_{ n }⇀

*x*, the inequality

holds for every *y* ∈ *H* with *y* ≠ *x*.

We need the following lemmas for the proof of our main results.

**Lemma 2.1**[15]. *Let H be a Hilbert space and C be a closed convex subset of H. If T is a k-strictly pseudo-contractive mapping on C, then the fixed point set F*(*T*) *is closed convex, so that the projection P*_{F(T)}*is well defined*.

**Lemma 2.2**[15]. *Let H be a Hilbert space and C be a closed convex subset of H. Let T* : *C* → *H be a k-strictly pseudo-contractive mapping with F*(*T*) ≠ ∅. *Then F*(*P*_{
C
}T) = *F* (*T* ).

**Lemma 2.3**[15]. *Let H be a Hilbert space, C be a closed convex subset of H, and T* : *C* → *H be a k-strictly pseudo-contractive mapping. Define a mapping S* : *C* → *H by Sx* = *λx* + (1 - *λ*) *Tx for all x* ∈ *C. Then, as λ* ∈ [*k*, 1), *S is a nonexpansive mapping such that F*(*S*) = *F*(*T*).

The following Lemmas 2.4 and 2.5 can be obtained from the Proposition 2.6 of Acedo and Xu [4].

**Lemma 2.4**. *Let H be a Hilbert space and C be a closed convex subset of H. For any N* ≥ 1, *assume that for each* 1 ≤ *i* ≤ *N, T*_{
i
} : *C* → *H is a k*_{
i
}*-strictly pseudo-contractive mapping for some* 0 ≤ *k*_{
i
} *<* 1. *Assume that*
*is a positive sequence such that*
. Then
*is a nonself-k-strictly pseudo-contractive mapping with k*= max{*k*_{
i
} : 1 ≤ *i* ≤ *N*}.

**Lemma 2.5**. *Let*
*and*
*be given as in Lemma 2.4. Suppose that*
*has a common fixed point in C. Then*
.

*where*{

*λ*

_{ n }}, {

*δ*

_{ n }}

*and*{

*r*

_{ n }}

*satisfy the following conditions*:

- (i)
{

*λ*_{ n }} ⊂ [0, 1]*and*,

_{n→∞}

*δ*

_{ n }≤ 0

*or*,

- (iii)
*r*_{ n }≥ 0 (*n*≥ 0), .

*Then* lim_{n→∞}*s*_{
n
}= 0.

**Lemma 2.7**[18].

*Let*{

*x*

_{ n }}

*and*{

*z*

_{ n }}

*be bounded sequences in a Banach space E and*{

*γ*

_{ n }}

*be a sequence in*[0, 1]

*which satisfies the following condition:*

*Then* lim_{n→∞}||*z*_{
n
} *- x*_{
n
} || = 0.

**Lemma 2.9**. *Let C be a nonempty closed convex subset of a Hilbert space H such that C* ± *C* ⊂ *C. Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let*
*and* 0 *< t < ρ <* 1. *Then S* := *ρI - tμF* : *C* → *C is a contraction with contractive constant ρ* - *tτ, where*
*with*
.

Hence *S* is a contraction with contractive constant *ρ - tτ*. □

## 3 Main results

We need the following result for the existence of solutions of a certain variational inequality, which is slightly an improvement of Theorem 3.1 of Tian [12].

**Theorem T2**.

*Let H be a Hilbert space, C be a closed convex subset of H such that C*±

*C*⊂

*C, and T*:

*C*→

*C be a nonexpansive mapping with F*(

*T*) ≠ ∅.

*Let F*:

*C*→

*C be a κ-Lipschitzian and η-strongly monotone operator with κ >*0

*and η >*0.

*Let f*:

*C*→

*C be a contraction with the contractive constant α*∈ (0, 1).

*Let*,

*and τ <*1.

*Let x*

_{ t }

*be a fixed point of a contraction St*∋

*x*α

*tγf*(

*x*) + (

*I - tμF*)

*Tx for t*∈ (0, 1)

*and*.

*Then*{

*x*

_{ t }}

*converges strongly to a fixed point*

*of T as t*→ 0,

*which solves the following variational inequality:*

*Equivalently, we have*
.

Now, we study the strong convergence result for a general iterative scheme (IS).

**Theorem 3.1**. *Let H be a Hilbert space, C be a closed convex subset of H such that C* ± *C* ⊂ *C, and T* : *C* → *H be a k-strictly pseudo-contractive mapping with F*(*T*) ≠ ∅ *for some* 0 ≤ *k <* 1. *Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let f* : *C* → *C be a contraction with the contractive constant α* ∈ (0, 1). *Let*
,
*and τ <* 1. *Let f*{*α*_{
n
} } *and* {*β*_{
n
} } *be sequences in* (0, 1) *which satisfy the conditions:*

(C1) lim_{n→∞}*α*_{
n
}= 0;

- (B)
0

*<*lim inf_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*βn*< 1.

*where S*:

*C*→

*H is a mapping defined by Sx*=

*kx*+ (1 -

*k*)

*Tx and P*

_{ C }

*is the metric projection of H onto C. Then*{

*x*

_{ n }}

*converges strongly to q*∈

*F*(

*T*),

*which solves the following variational inequality:*

**Proof**. First, from the condition (C1), without loss of generality, we assume that *α*_{
n
}*τ <* 1,
and *α*_{
n
} *<* (1 - *β*_{
n
} ) for *n* ≥ 0.

We divide the proof several steps:

**Step 1**. We show that for all

*n*≥ 0 and all

*p*∈

*F*(

*T*) =

*F*(

*S*). Indeed, let

*p*∈

*F*(

*T*). Then from Lemma 2.9, we have

Using an induction, we have
. Hence, {*x*_{
n
} } is bounded, and so are {*f*(*x*_{
n
} )}, {*P*_{
C
}*Sx*_{
n
} } and {*FP*_{
C
}*Sx*_{
n
} }.

*q*= lim

_{t→0}

*x*

_{ t }being

*x*

_{ t }=

*tγf*(

*x*

_{ t }) + (

*I - tμF*)

*P*

_{ C }

*Sx*

_{ t }for 0

*< t <*1 and . We note that from Lemmas 2.2 and 2.3 and Theorem T2,

*q*∈

*F*(

*T*) =

*F*(

*S*) and

*q*is a solution of a variational inequality

*x*

_{ n }} is bounded, there exists a subsequence of which converges weakly to

*w*. Without loss of generality, we can assume that . Since ||

*x*

_{ n }

*- P*

_{ C }

*Sx*

_{ n }|| → 0 by Step 3, we obtain

*w*=

*P*

_{ C }

*Sw*. In fact, if

*w*≠

*P*

_{ C }

*Sw*, then, by Opial condition,

*w*=

*P*

_{ C }

*Sw*. Since

*F*(

*P*

_{ C }

*S*) =

*F*(

*S*), from Lemma 2.3, we have

*w*∈

*F*(

*T*). Therefore, from (3.1), we conclude that

**Step 5**. We show that lim

_{n→∞}||

*x*

_{ n }

*- q*|| = 0, where

*q*= lim

_{t→0}

*x*

_{ t }being

*x*

_{ t }=

*tγf*(

*xt*) + (

*I - tμF*)

*P*

_{ C }

*Sx*

_{ t }for 0

*< t <*1 and , and

*q*is a solution of a variational inequality

From the conditions (C1) and (C2) and Step 4, it is easy to see that *λ*_{
n
} → 0,
, and lim sup_{n→∞}*δ*_{
n
}≤ 0. Hence, by Lemma 2.7, we conclude *x*_{
n
} → *q* as *n* → ∞. This completes the proof. □

**Remark 3.1**. (1) Theorem 3.1 extends and develops Theorem 3.2 of Tian [12] from a nonexpansive mapping to a strictly pseudo-contractive mapping together with removing the condition (C3) .

- (2)
Theorem 3.1 also generalizes Theorem 2.1 of Jung [6] as well as Theorem 2.1 of Cho et al. [5] and Theorem 3.4 of Marino and Xu [10] from a strongly positive bounded linear operator

*A*to a*κ*-Lipschitzian and*η*-strongly monotone operator*F*. - (3)
Theorem 3.1 also improves the corresponding results of Halpern [13], Moudafi [8], Wittmann [14] and Xu [9] as some special cases.

**Theorem 3.2**. *Let H be a Hilbert space, C be a closed convex subset of H such that C* ± *C* ⊂ *C, and T*_{
i
} : *C* → *H be a k*_{
i
}*-strictly pseudo-contractive mapping for some* 0 ≤ *k*_{
i
} *<* 1 *and*
. *Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let f* : *C* → *C be a contraction with the contractive constant α* ∈ (0, 1). *Let*
,
*and τ* < 1. *Let* {*α*_{
n
} } *and* {*βn*} *be sequences in* (0, 1) *which satisfy the conditions*.

(C1) lim_{n→∞}*α*_{
n
}= 0;

- (B)
0 < lim inf

_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*β*_{ n }< 1.

*where S*:

*C*→

*H is a mapping defined by*

*with k*= max{

*k*

_{ i }: 1 ≤

*i*≤

*N*}

*and*{

*η*

_{ i }}

*is a positive sequence such that*

*and P*

_{ C }

*is the metric projection of H onto C. Then*{

*x*

_{ n }}

*converges strongly to q*∈

*F*(

*T*),

*which solves the following variational inequality:*

**Proof**. Define a mapping *T* : *C* → *H* by
. By Lemmas 2.4 and 2.5, we conclude that *T* : *C* → *H* is a *k*-strictly pseudo-contractive mapping with *k* = max{*k*_{
i
} : 1 ≤ *i* ≤ *N*} and
. Then the result follows from Theorem 3.1 immediately. □

As a direct consequence of Theorem 3.2, we have the following result for nonexpansive mappings (that is, 0-strictly pseudo-contractive mappings).

**Theorem 3.3**. *Let H be a Hilbert space, C be a closed convex subset of H such that C* ± *C* ⊂ *C*,
*be a finite family of nonexpansive mappings with*
. *Let F* : *C* → *C be a κ-Lipschitzian and η-strongly monotone operator with κ >* 0 *and η >* 0. *Let f* : *C* → *C be a contraction with the contractive constant α* ∈ (0, 1). *Let*
,
*and τ <* 1. *Let* {*α*_{
n
} } *and* {*βn*} *be sequences in* (0, 1) *which satisfy the conditions*.

(C1) lim_{n→∞}*α*_{
n
}= 0;

- (B)
0 < lim inf

_{n→∞}*β*_{ n }≤ lim sup_{n→∞}*β*_{ n }< 1.

*where*

*is a positive sequence such that*

*and P*

_{ C }

*is the metric projection of H onto C. Then*{

*x*

_{ n }}

*converges strongly to a common fixed point q of*,

*which solves the following variational inequality:*

**Remark 3.2**. (1) Theorems 3.2 and 3.3 also generalize Theorems 2.2 and 2.4 of Jung [6] from a strongly positive bounded linear operator

*A*to a

*κ*-Lipschitzian and

*η*-strongly monotone operator

*F*.

- (2)
Theorems 3.2 and 3.3 also improve and complement the corresponding results of Cho et al. [5] by removing the condition (C3) together with using a

*κ*-Lipschitzian and*η*-strongly monotone operator*F*. - (3)
As in [19], we also can establish the result for a countable family {

*T*_{ i }} of*k*_{ i }-strict pseudo-contractive mappings with 0 ≤*k*_{ i }*<*1.

## Declarations

### Acknowledgements

This study was supported by research funds from Dong-A University.

## Authors’ Affiliations

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