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# Synchronal algorithm and cyclic algorithm for fixed point problems and variational inequality problems in hilbert spaces

- Ming Tian
^{1}Email author and - Lanyun Di
^{1}

**2011**:21

https://doi.org/10.1186/1687-1812-2011-21

© Tian and Di; licensee Springer. 2011

**Received:**22 October 2010**Accepted:**25 July 2011**Published:**25 July 2011

## Abstract

We design synchronal algorithm and cyclic algorithm based on the general iterative algorithm proposed by Tian in 2010 for finding the common fixed point *x** of finite family of strict pseudo-contractive mappings which is the solution of the variational inequality 〈(*γ f - μF*)*x**, *x - x**〉 ≤ 0,
.

**2000 Mathematics Subject Classification**: 58E35; 47H09; 65J15.

## Keywords

- strict pseudo-contractions
- nonexpansive mapping
- variational inequality
- synchronal algorithm
- cyclic algorithm
- fixed point

## 1. Introduction

Let *H* be a real Hilbert space with the inner product 〈·,·〉 and the norm || · ||, respectively. Let *C* be nonempty closed subset of *H*.

*T*:

*C*→

*H*is said to be

*k*-strict pseudo-contraction if there exists a constant

*k*∈ [0, 1) such that

*k*= 0. That is,

*T*:

*C*→

*H*is nonexpansive if

We assume that *F*(*T*) ≠ ∅ it is well known that *F*(*T*) is closed convex.

We denote by *V I*(*F*, *C*) the set of solutions of this variational inequality problem.

where *T* is nonexpansive mapping, and {α_{n}} ⊂ [0, *a*] for some *a* ∈ [0, 1). Then they showed that {*x*_{
n
}} converged strongly to *P*_{F(T)}(*x*_{0}) by the hybrid method in the mathematical programming. But it is hard to compute by this algorithm, because projection has to be used in every process.

The hybrid steepest descent method of Yamada [2] conquered this deficiency and proposed the following algorithm for solving the variational inequality.

*T*is a nonexpansive mapping on

*H*,

*F*is

*L*-Lipschitzian and

*η*-strongly monotone with

*k >*0,

*η >*0, 0

*< μ <*2

*η/L*

^{2}. If {

*λ*

_{ n }} is a sequence in (0, 1) satisfying the following conditions:

- (i)
lim

_{n→∞}*λ*_{ n }= 0; - (ii)
- (iii)
either or ,

*x*

_{ n }} converged strongly to the unique solution of the variational inequality

where *T*_{[n]}= *T*_{
n
}_{mod N}, he also got strong convergence theorems.

*x*

_{0}is selected in

*H*arbitrarily

where *T* is a nonexpansive mapping on *H*, *f* is a contraction, *A* is a linear bounded strongly positive operator, and {*α*_{
n
} } is a sequence in (0, 1) satisfying the following conditions:

(C1) lim_{n→∞}*α*_{
n
}= 0;

(C2)

(C3) either or .

*x*

_{ n }} converged strongly to a fixed point of

*T*which solves the variational inequality

where *T* is a nonexpansive mapping on *H*, *f* is a contraction, and *F* is *k*- Lipschitzian and *η*-strongly monotone with *k >* 0, *η >* 0, 0 *< μ <* 2*η/k*^{2}.

*α*

_{ n }} of parameters satisfies (C1)-(C3), then the sequence {

*x*

_{ n }} generated by (1.4) converged strongly to a fixed point of

*T*which solves the variational inequality

*x** of finite strict pseudo-contractions which also solves the variational inequality

where *N* ≥ 1 is a positive integer and
are *N* strict pseudo-contractions.

*λ*

_{ i }

*>*0 such that . We will show that the sequence {

*x*

_{ n }} generated by the algorithm:

will converge strongly to a solution to the problem (1.6).

*β*

_{ i }satisfies

*k*

_{ i }

*< β*

_{ i }

*<*1. Beginning with

*x*

_{0}∈

*H*, we define the sequence {

*x*

_{ n }} cyclically by

where *T*_{[n]}= *T*_{
i
} , with *i* = *n*(mod*N* ), 1 ≤ *i* ≤ *N*. We will show that this cyclic algorithm (1.8) is also strongly convergent if the sequences {*α*_{
n
} } and {*β*_{
n
} } are appropriately chosen.

- 1.
⇀ for weak convergence and → for strong convergence.

- 2.
denotes the weak !-limit set of {

*x*_{ n }}.

## 2. Preliminaries

We need some facts and tools which are listed as below.

**Definition 1**A mapping

*F*:

*C*→

*H*is called

*η*-strongly monotone if there exists a positive constant

*η >*0 such that

**Definition 2** *B* is called to be strongly positive bounded linear operator on

**Lemma 2.1**. *(see*[5]*) Let C be a nonempty closed convex subset of a real Hilbert space H and T* : *C ! C is a nonexpansive mapping. If a sequence* {*x*_{
n
} } *in C such that x*_{
n
} ⇀ *z and* (*I - T*)*x*_{
n
} → 0, *then z* = *Tz*.

**Lemma 2.2**. *(see*[6]*) Let C be a nonempty closed convex subset of a real Hilbert space H. If T* : *C* → *C is a κ-strict pseudo-contraction, then the mapping I - T is demiclosed at 0. That is, if* {*x*_{
n
} } *is a sequence in C such that*
*and* (*I - T*)*x*_{
n
} → 0, *then*
.

*where*{

*γ*

_{ n }}

*is a sequence in*(0, 1)

*and*{

*δ*

_{ n }}

*is a sequence in*ℝ

*such that:*

- (i)
lim

_{n→∞}*γ*_{ n }= 0*and*; - (ii)
lim

_{n→∞}*δ*_{ n }/*γ*_{ n }≤ 0*or*.

*Then* lim_{n→∞}*a*_{
n
}= 0.

**Lemma 2.4**.

*(see*[4]

*) Let H be a real Hilbert space, f*:

*H*→

*H a contraction with coefficient*0

*< α <*1,

*and F*:

*H*→

*H a k-Lipschitzian continuous operator and η-strongly monotone operator with k >*0,

*η >*0.

*Then for*0

*< γ < μη/α*,

*That is, μF - γf is strongly monotone with coefficient μη - γα*.

**Lemma 2.5**. *(see*[8]*) Suppose S* : *C* → *H is a k-strict pseudo-contraction*. *Define T* : *C* → *H by Tx* = *λx* + (1 - *λ*)*Sx for each ×* ∈ *C. Then, as λ* ∈ [*k*, 1), *T is a nonexpansive mapping such that F*(*T*) = *F*(*S*).

**Lemma 2.6**.

*(see*[6]

*) Assume C is a closed convex subset of a Hilbert space H. Given an integer N*≥ 1,

*assume for each*1 ≤

*i*≤

*N, T*

_{ i }:

*C*→

*C is a k*

_{ i }-

*strict pseudo-contraction for some*0 ≤

*k*

_{ i }< 1.

*Assume*

*is a positive sequence such that*.

*Suppose that*

*then*

**Lemma 2.7**.

*(see*[9]

*) Assume T*

_{ i }:

*H*→

*H is a k*

_{ i }-

*strict pseudo-contraction for some*0 ≤

*k*

_{ i }< 1 (1 ≤

*i*≤

*N*):

*Let*,

*k*

_{ i }<

*α*

_{ i }< 1 (1 ≤

*i*≤

*N*),

*if*,

*then*

**Lemma 2.8**. *Let F* : *H* → *H be a η-strongly monotone and L-Lipschitzian operator with L >* 0, *η >* 0. *Assume that* 0 *< μ <* 2*η/L*^{2},
*and* 0 *< t* < 1. *Then* ||(*I - μtF*)*x -* (*I - μtF*)*y*|| ≤ (1 - *tτ*) ||*x - y*||.

□

## 3. Synchronal algorithm

**Theorem 3.1**. *Let H be a real Hilbert space and let T*_{
i
} : *H* → *H be a k*_{
i
}*-strict pseudo-contraction for some k*_{
i
} ∈ (0, 1) (*i* = 1,..., *N* ) *such that*
, *f be a contraction with coefficient β* ∈ (0, 1) *and λ*_{
i
} *be a positive constant such that*
. *Let G* : *H* → *H be a η-strongly monotone and L-Lipschitzian operator with L >* 0, *η >* 0. *Assume that* 0 *< μ <* 2*η/L*^{2},
. *Given the initial guess x*_{0} ∈ *H chosen arbitrarily and given sequences* {*α*_{
n
}} *and* {*β*_{
n
}} *in* (0, 1), *satisfying the following conditions:*

(3.1a) lim_{n→∞}*α*_{
n
}= 0,
;

(3.1b) , ;

(3.1c) 0 ≤ max_{
i
}*k*_{
i
}≤ *β*_{
n
}< *a* < 1 *for all n* ≥ 0;

*Then* {*x*_{
n
} } *converges strongly to a common fixed point of*
*which solves the variational inequality (1.6)*.

*Proof*. Put
, then by Lemma 2.6, we conclude that *T* is a *k*-strict pseudo-contraction with *k* = max {*k*_{
i
} : 1 ≤ *i* ≤ *N*} and
.

Furthermore, by Lemma 2.5, we conclude that is a nonexpansive mapping and .

**Step 1**. {*x*_{
n
} } is bounded.

Hence {*x*_{
n
} } is bounded.

Therefore, {*Tx*_{
n
} } is bounded.

{*Tx*_{
n
} } is bounded, so
is bounded.

*f* is a contraction, so *f*(*x*_{
n
} ) is bounded.

*M*

_{2}is an appropriate constant such that

*M*

_{2}≥ sup

_{n≥1}{||

*x*

_{ n }

*- Tx*

_{ n }||}. Substituting (3.3) into (3.2) yields

where *M*_{3} is an appropriate constant such that *M*_{3} ≥ max{*M*_{1}, *M*_{2}}. By conditions (3.1a) and (3.1b) and Lemma 2.3, we obtain that lim_{n→∞}||*x*_{n+1}- *x*_{
n
}|| = 0.

*x*

_{ n }}, we deduced that {

*x*

_{ n }} converges weakly. Assume

*x*

_{ n }⇀

*p*, by Lemma 2.2 and (3.5), we obtain

*p*=

*Tp*. So, we have

Notice by Lemma 2.4, *μG - γ f* is strongly monotone, so the variational inequality (1.6) has a unique solution *x** ∈ *F*(*T*).

*x*∈

*F*(

*T*). Since

*x** is the unique solution of (1.6), we obtain

where
and
.
, from (3.1a), we have lim_{n→∞}*γ*_{
n
}= 0; *γ*_{
n
} ≥ 2*α*_{
n
} (*τ* - *γβ*), from (3.1a), we have
; put *M* = sup {||*x*_{
n
}- *x**|| : *n* ∈ *N*}, we have
. So, lim_{n→∞}*δ*_{
n
}/*γ*_{
n
}≤ 0. Hence, by Lemma 2.3, we conclude that *x*_{
n
} → *x** as *n* → ∞. □

## 4. Cyclic algorithm

**Theorem 4.1**. *Let H be a real Hilbert space and let T*_{
i
} : *H* → *H be a k*_{
i
}*-strict pseudo-contraction for some k*_{
i
} ∈ (0, 1) (*i* = 1,..., *N* ) *such that*
*and f be a contraction with coefficient β* ∈ (0, 1). *Let G* : *H* → *H be a η-strongly monotone and L-Lipschitzian operator with L >* 0, *η >* 0. *Assume that*
. *Given the initial guess x*_{0} ∈ *H chosen arbitrarily and given sequences* {*α*_{
n
} } *and* {*β*_{
n
} } *in* (0, 1), *satisfying the following conditions:*

(4.1a) lim_{n→∞}*α*_{
n
}= 0,

(4.1b) ;

(4.1c)
, *or*
;

(4.1d) *β*_{[n]}∈ [*k*, 1), *where k* = max_{
i
}{*k*_{
i
}: 1 ≤ *i* ≤ *N*},

*where T*_{[n]}= *T*_{
i
}*, with i* = *n*(mod*N* ), 1 ≤ *i* ≤ *N, namely, T*_{[n]}*is one of T*_{1}, *T*_{2},..., *T*_{
N
} *circularly. Then* {*x*_{
n
} } *converges strongly to a common fixed point of*
*which solves the variational inequality (1.6)*.

Hence {*x*_{
n
} } is bounded.

*T*

_{[n]}

*x*

_{ n }}, {

*f*(

*x*

_{ n })}, {

*GA*

_{[n]}

*x*

_{ n }} are bounded.

So, {*A*_{[n]}*x*_{
n
} } is bounded.

**Step 2**. lim_{n→∞}||*x*_{n+N}- *x*_{
n
}|| = 0.

where *K*_{1} is an appropriate constant such that *K*_{1} ≥ sup_{n≥1}{*μ*||*GA*_{[n+1]}*x*_{
n
} ||+ *γ* ||*f*(*x*_{
n
} )||}. By conditions (4.1a), (4.1b), (4.1c) and Lemma 2.3, we obtain ||*x*_{n+N}- *x*_{
n
}|| → 0 as *n* → ∞.

**Step 3**. lim_{n→ ∞}||*x*_{
n
}- *A*_{[n+N]}··· *A*_{[n+1]}*x*_{
n
} || = 0.

*f*(

*x*

_{ n })} and {

*GA*

_{[n+1]}

*x*

_{ n }} that

we conclude ||*x*_{
n
} *- A*_{[n+N]}··· *A*_{[n+1]}*x*_{
n
} || → 0(*n* → ∞).

Notice that, for each *n*_{
j
} ,
is some permutation of the mappings *A*_{1}*A*_{2} ··· *A*_{
N
} , since *A*_{1}, *A*_{2},···, *A*_{
N
} are finite, all the finite permutation are *N*!, there must be some permutation appears infinite times.

*A*

_{1}

*A*

_{2}···

*A*

_{ N }, we can take a subsequence such that and

By Lemma 2.5, we conclude that *A*_{1}, *A*_{2},···, *A*_{
N
} are all nonexpansive. It is easy to prove that
is nonexpansive, so *A*_{1}*A*_{2}···*A*_{
N
} is.

*x*∈

*F*(

*T*). Since

*x** is the unique solution of (1.6), we obtain

**Step 6**. *x*_{
n
} → *x**(*n* → ∞).

where
and
.
, from (4.1a), we have lim_{n→∞}*γ*_{
n
}= 0; *γ*_{
n
} ≥ 2α _{
n
} (*τ -γβ*), from (4.1b), we have
; put *M* = sup {||*x*_{
n
} - *x**||: *n* ∈ *N*}, we have
. So, limsup_{n→∞}*δ*_{
n
}/*γ*_{
n
}≤ 0. Hence, by Lemma 2.3, we conclude that *x*_{
n
} → *x** as *n* → ∞. . □

Taking *n* = 1, *β*_{
n
} = 0 and *T* is nonexpansive mapping in Theorems 3.1 and 4.1, we get

Assume the sequence {*α*_{
n
} } satisfies conditions:

(C1) lim_{n→∞}*α*_{
n
}= 0;

(C2) ;

(C3) either or

Taking *n* = 1, *β*_{
n
} = 0 and *T* is nonexpansive mapping, *G* = *A*, *μ* = 1 in Theorems 3.1 and 4.1, we get

*α*

_{ n }} satisfies conditions (C1)-(C3), then the sequence {

*x*

_{ n }} converged strongly to a fixed point of

*T*which solves the variational inequality

Taking *n* = 1, *β*_{
n
} = 0 and *T* is nonexpansive mapping, *γ* = 0 in Theorem 3.1 and Theorem 4.1, we get:

*T*is a nonexpansive mapping on

*H*,

*F*is

*L*-Lipschitzian and

*η*-strongly monotone with

*k >*0,

*η >*0, 0

*< μ <*2

*η/L*

^{2}. If {

*λ*

_{ n }} is a sequence in (0, 1) satisfies the following conditions:

- (i)
lim

_{n→∞}*λ*_{ n }= 0; - (ii)
;

- (iii)
either or

*x*

_{ n }} converged strongly to the unique solution of the variational inequality

Taking *n* = 1, *β*_{
n
} = 0 and *T* is nonexpansive mapping, *γ* = 0 in Theorem 4.1, we get

## Declarations

### Acknowledgements

The authors wish to thank the referees for their helpful comments, which notably improved the presentation of this manuscript. This work was supported by Fundamental Research Funds for the Central Universities (Grant no. ZXH2011C002).

## Authors’ Affiliations

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