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# Convergence and weaker control conditions for hybrid iterative algorithms

- Shuang Wang
^{1}Email author

**2011**:3

https://doi.org/10.1186/1687-1812-2011-3

© Wang; licensee Springer. 2011

**Received:**24 January 2011**Accepted:**20 June 2011**Published:**20 June 2011

## Abstract

Very recently, Yao et al. (Appl. Math. Comput. **216**, 822-829, 2010) have proposed a hybrid iterative algorithm. Under the parameter sequences satisfying some quite restrictive conditions, they derived a strong convergence theorem in a Hilbert space. In this article, under the weaker conditions, we prove the strong convergence of the sequence generated by their iterative algorithm to a common fixed point of an infinite family of nonexpansive mappings, which solves a variational inequality. It is worth pointing out that we use a new method to prove our results. An appropriate example, such that all conditions of this result that are satisfied and that other conditions are not satisfied, is provided. Furthermore, we also give a weak convergence theorem for their iterative algorithm involving an infinite family of nonexpansive mappings in a Hilbert space.

**MSC: 47H05, 47H09, 47H10**

## Keywords

- Strong convergence
- Variational inequality
- Fixed points
*k*-Lipschitzian*η*-strongly monotone- Hilbert space

## 1 Introduction

*H*be a real Hilbert space and

*C*be a nonempty, closed, convex subset of

*H*, let

*F*:

*H*→

*H*be a nonlinear operator. The variational inequality problem is formulated as finding a point

*x** ∈

*C*such that

In 1964, Stampacchia [1] introduced and studied variational inequality initially. It is now well known that variational inequalities cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics, and finance [1–5].

It is known that a mapping *T* : *H* → *H* is said to be nonexpansive if ||*Tx* - *Ty*|| ≤ ||*x* - *y*||, ∀*x*, *y* ∈ *H*. We use *F* (*T* ) to denote the set of fixed points of *T*, that is *F* (*T*) = {*x* ∈ *H* : *Tx* = *x*}.

where *F* is a *k*-Lipschitzian and *η*-strongly monotone operator with *k* > 0, *η* > 0 and 0 < *μ* < 2*η* /*k*^{2}. Let a sequence {*λ*_{n}} of real numbers in (0,1) satisfy the conditions below:

(C1) lim_{n→∞}*λ*_{
n
} = 0,

(C2) ,

(C3) .

*x*

_{ n }} generated by (1.1) converges strongly to the unique solution of the variational inequality:

*λ*

_{ n }} which satisfies conditions (C1)-(C3) is given by

*λ*

_{n}= 1/

*n*

^{ σ }, where 0 <

*σ*< 1. We note that the condition (C3) was first used by Lions [3]. It was observed that Lion's conditions on the sequence {

*λ*

_{ n }} excluded the canonical choice

*λ*

_{ n }= 1/n. This was overcome in 2003 by Xu and Kim [4], if {

*λ*

_{ n }} satisfies conditions (C1), (C2), and (C4)

who proved the strong convergence of {*x*_{
n
}} to the unique solution *u** of the variational inequality 〈*Fu**, *v* - *u**〉 ≥ 0, ∀*v* ∈ *C*. It is easy to see that the condition (C4) is strictly weaker than condition (C3), coupled with conditions (C1) and (C2). Moreover, (C4) includes the important and natural choice {1/n} of {*λ*_{
n
}}.

*x*

_{0}∈

*H*arbitrarily,

*F*is a

*k*-Lipschitzian and

*η*-strongly monotone operator on

*H*, and

*W*

_{ n }is a

*W*-mapping defined by (2.3) cited later. Take

*k*∈ [1, ∞),

*η*∈ (0, 1), and {

*λ*

_{ n }} satisfying the conditions (C1) and (C2). If a sequence {

*α*

_{ n }} satisfying (C5)

*x*

_{ n }} and {

*y*

_{ n }} defined by (1.2) converge strongly to , which solves the following variational inequality:

We remind the reader of the fact that in order to guarantee the strong convergence of the iterative sequence {*x*_{
n
}}, there is at least one parameter sequence converging to zero (i.e., *λ*_{
n
} → 0) as a result of Yamada [2], Xu and Kim [4, Theorem 3.1, and Theorem 3.2] and Yao et al. [5, Theorem 3.2]. In addition, *η* ∈ (0, 1) and (C5) are quite restrictive assumptions in Yao et al. [5].

In this article, under the convergence of no parameter sequences to zero and the weaker conditions on *α*_{n} and *η*, we prove that the sequence {*y*_{
n
}} generated by the iterative algorithm (1.2) converges to a common fixed point of an infinite family of nonexpansive mappings, which solves the variational inequality 〈*Fx**, *u* - *x**〉 ≥ 0,
. In the meantime, we illustrate that this result is more general than Theorem 3.2 of Yao et al. [5]. That is, we give an appropriate example such that all conditions of this result are satisfied and the conditions *η* ∈ (0, 1), (C1), and (C5) in Yao et al. [5, Theorem 3.2] are not satisfied. Furthermore, we also give a weak convergence theorem for hybrid iterative algorithm (1.2) involving an infinite family of nonexpansive mappings in a Hilbert space *H*. It is worth pointing out that we use a new method to prove our main results. The results presented in this article can be viewed as the improvement, supplement, and extension of the results obtained in [2–5].

## 2 Preliminaries

*H*be a real Hilbert space with inner product 〈·,·〉 and norm ||·||. For the sequence {

*x*

_{ n }} in

*H*, we write

*x*

_{ n }⇀

*x*to indicate that the sequence {

*x*

_{ n }} converges weakly to

*x*.

*x*

_{ n }→

*x*implies that {

*x*

_{ n }} converges strongly to

*x*. We denote by

*ω*

_{ w }(

*x*

_{ n }) the weak

*ω*-limit set of {

*x*

_{ n }}, that is

*X*satisfies Opial's property [6] provided, for each sequence {

*x*

_{ n }} in

*X*, the condition

*x*

_{ n }⇀

*x*implies

It is known that each *λ*^{
p
}(1 ≤ *p* < ∞) enjoys this property, while *L*^{
p
} does not unless *p* = 2.

for all *x*, *y* ∈ *H* and *λ* ∈ [0,1] (see Takahashi [7]).

In order to prove our main results, we need the following lemmas:

**Lemma 2.1**. [8]. *Let H be a Hilbert space, C a closed, convex subset of H, and T* : *C* → *C a nonexpansive mapping with F* (*T* ) ≠ ∅; *if* {*x*_{
n
}} *is a sequence in C weakly converging to × and if* {(*I* - *T* )*x*_{
n
}} *converges strongly to y*, *then* (*I* - *T* )*x* = *y*.

**Lemma 2.2**. [9].

*Let*{

*x*

_{ n }}

*and*{

*z*

_{ n }}

*be bounded sequences in Banach space E and*{

**γ**

_{ n }}

*be a sequence in*[0,1]

*which satisfies the following condition:*

*Suppose that* *x*_{n+1}= γ_{
n
}*x*_{
n
} + (1 - γ_{
n
})*z*_{
n
}, *n* ≥ 0, *and* lim sup_{n→∞}(||*z*_{n+1}- *z*_{
n
}|| - ||*x*_{n+1}- *x*_{
n
}||) ≤ 0. *Then*, lim_{n→∞}||*z*_{
n
}- *x*_{
n
}|| = 0.

*where* {*λ*_{
n
}} *and* {γ_{
n
}} *satisfy the following conditions: (i)* {*λ*_{
n
}} ⊂ [0,1] *and*
, *(ii)* lim sup_{n→∞}or
, *(iii)* γ_{
n
}≥ 0(*n* ≥ 0),
. *Then* lim_{n→∞}*S*_{
n
}= 0.

**Lemma 2.4**. [12]. *Let* {*a*_{
n
}} *and* {*b*_{
n
}} *be sequences of nonnegative real numbers such that*
*and* *a*_{n+1}≤ *a*_{
n
}+ *b*_{
n
}*for all* *n* ≥ 0. *Then* lim_{n→∞}*a*_{
n
}exists.

**Lemma 2.5**. [13]. *Let F be a k-Lipschitzian and η-strongly monotone operator on a Hilbert space H with* 0 < *η* ≤ *k* *and* 0 < *t* < *η*/*k*^{2}. *Then S* = (*I* - *tF* ) : *H* → *H is a contraction with contraction coefficient*
.

*T*

_{ i }:

*H*→

*H*}. be a family of infinitely nonexpansive mappings, {

*ξ*

_{ i }}be a real sequence such that 0 <

*ξ*

_{ i }≤

*b*< 1, ∀

*i*≥ 1. For any

*n*≥ 1, define a mapping

*W*

_{ n }:

*H*→

*H*as follows:

Such a mapping *W*_{
n
}: *H* → *H* is called a *W*-mapping generated by *T*_{
n
}, *T*_{n-1},..., *T*_{1} and *ξ*_{
n
}, *ξ*_{n-1},..., *ξ*_{1}.

We have the following crucial conclusion concerning *W*_{
n
}. We can find them in [14–17]. Now we only need the following similar version in Hilbert spaces:

**Lemma 2.6**. *Let H be a real Hilbert space*, {*T*_{
i
}: *H* → *H*} *be a family of infinitely nonexpansive mappings with*
, {*ξ*_{
i
}} *be a real sequence such that* 0 < *ξ*_{
i
} ≤ *b* < 1, ∀*i* ≥ 1. *Then*,

*(1) W*_{
n
} *is a nonexpansive and*
*for each n* ≥ 1;

*(2) For every x* ∈ *H and k* ∈ *N, the limit* lim_{n→∞}*U*_{
n, k
}*x exists;*

*(3) If we define a mapping W* : *H* → *H* as *Wx* = lim_{n→1}*W*_{
n
}x, *and W*_{
n
}*x* = lim_{n→∞}*U*_{n,1}*x*, *for every* ∈ *H*, *then*,
;

*(4) For any bounded sequence* {*x*_{
n
}} ⊂ *H*, *we have* lim_{n→∞}||*Wx*_{
n
}- *W*_{
n
}x_{
n
}|| = 0.

## 3 Main results

*F*be a

*k*-Lipschitzian and

*η*-strongly monotone operator on

*H*with 0 <

*k*,

*T*:

*H*→

*H*be a nonexpansive mapping. Let

*t*∈ (0,

*η*/

*k*

^{2}) and , and consider a mapping

*S*

_{t}on

*H*defined by

*x*,

*y*∈

*H*. Hence, it has a unique fixed point, denoted as

*x*

_{ t }, which uniquely solves the fixed point equation

**Theorem 3.1**.

*Let H be a real Hilbert space. Let T*:

*H*→

*H be a nonexpansive mapping such that F*(

*T*) ≠ ∅,.

*Let F be a k-Lipschitzian and η-strongly monotone operator on H with*0 <

*η*≤

*k*.

*For each t*∈ (0,

*η*/

*k*

^{2}),

*let the net*{

*x*

_{ t }}

*be generated by (3.1). Then, as t*→ 0,

*the net*{

*x*

_{ t }}

*converges strongly to a fixed point x** of

*T which solves the variational inequality:*

*Proof*. We first show the uniqueness of a solution of the variational inequality (3.2), which is indeed a consequence of the strong monotonicity of

*F*. Suppose

*x** ∈

*F*(

*T*) and both are solutions to (3.2), then

The strong monotonicity of *F* implies that
and the uniqueness is proved. Later, we use *x** ∈ *F* (*T* ) to denote the unique solution of (3.2).

*t*→ 0, we may assume, without loss of generality, that , where is a arbitrarily small positive number. Thus, we have is continuous, . Therefore, we obtain

From (3.5) and (3.6), we have that {*x*_{
t
}} is bounded and so is {*Fx*_{
t
}}.

From
, we have
and
. Observe that, if *x*_{
t
} ⇀ *u*, we have
.

*x*

_{ t }} is bounded, we see that if {

*t*

_{ n }} is a sequence in such that

*t*

_{ n }→ 0 and , then by (3.8), we see . Moreover, by (3.7) and using Lemma 2.1, we have . We next prove that solves the variational inequality (3.2). From (3.1) and

*u*∈

*F*(

*T*), we have

That is
is a solution of (3.2), hence
by uniqueness. In a nutshell, we have shown that each cluster point of {*x*_{
t
}} (at t → 0) equals *x**. Therefore, *x*_{
t
} → *x** as *t* → 0.

**Theorem 3.2**. *Let H be a real Hilbert space*. *Let F be a k-Lipschitzian and η-strongly monotone operator on H with* 0 < *η* ≤ *k*. *Let*
*be an infinite family of nonexpansive mappings such that*
*and W*_{
n
} *be a W-mapping defined by (2.3). Let* {*λ*_{
n
}} *be a sequence in* [0, ∞) *and* {*α*_{
n
}} *be a sequence in* [0,1], ε *be a arbitrarily small positive number. Assume that the control conditions (C2)*, (*C* 1)', and (*C* 5)' hold for {*λ*_{
n
}} and {*α*_{
n
}},

(*C* 1)':
, ∀*n* ≥ *n*_{0} *for some integer n*_{0} ≥ 0, *and*

(*C* 5) ': 0 < γ ≤ lim inf_{n→∞}*α*_{
n
} lim sup_{n→∞}*α*_{
n
} < 1 *for some* γ ∈ (0, 1).

*Proof*. On the one hand, suppose that *λ*_{
n
}*F*(*x*_{
n
}) → 0(*n* → ∞). We proceed with the following steps:

*Step 1*. We claim that {

*x*

_{ n }} is bounded. In fact, let , from (1.2), (

*C*1)' and using Lemma 2.5, we have

∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0, where
.Therefore, {*x*_{
n
}} is bounded. We also obtain that {*y*_{
n
}}, {*W*_{
n
}*y*_{
n
}} and {*Fx*_{
n
}} are bounded.

*Step 2*. We claim that lim

_{n→∞}||

*x*

_{n+1}-

*x*

_{ n }|| = 0. To this end, define

*x*

_{n+1}= (1 -

*α*

_{ n })

*x*

_{ n }+

*α*

_{ n }

*z*

_{ n }. We observe that

*Step 5*. We claim that lim sup

_{n→∞}〈

*Fx**,

*x** -

*x*

_{ n }〉 ≤ 0, where

*x** = lim

_{n→∞}

*x*

_{ t }and

*x*

_{ t }defined by

*x*

_{ t }=

*W*[(1 -

*tF*)

*x*

_{ t }]. Since

*x*

_{ n }is bounded, there exists a subsequence of {

*x*

_{ n }} which converges weakly to

*ω*. From Step 4, we obtain . From Lemma 2.1, we have

*ω*∈

*F*(

*W*). Hence, by Theorem 3.1, we have

*n*≥

*n*

_{0}for some integer

*n*

_{0}≥ 0, where

*M*

_{3}= 2||

*Fx**||. For every

*n*≥

*n*

_{0}, put and

*δ*

_{ n }= 2

*M*

_{1}〈

*x** -

*x*

_{ n },

*Fx**〉 +

*M*

_{1}

*M*

_{3}||

*λ*

_{ n }

*Fx*

_{ n }||. It follows that

It is easy to see that
and lim sup_{n→∞}*δ*_{
n
} ≤ 0. Hence, by Lemma 2.3, the sequence {*x*_{
n
}} converges strongly to
.

It follows that the sequence {*y*_{
n
}} converges strongly to
. From *x** = lim_{t→0}*x*_{
t
}and Theorem 3.1, we have *x** is the unique solution of the variational inequality: 〈*Fx**, *x** - *u*〉 ≤ 0,
.

Since
and
, we get *λ*_{
n
}*F*(*x*_{
n
}) → 0. This completes the proof.

**Remark 3.3**. *It is clear that condition (C1)' is strictly weaker than condition (C1). In the meantime, condition (C5)' is also strictly weaker than condition (C5)*.

**Corollary 3.4**. *(Yao et al. [5, Theorem 3.2]). Let H be a real Hilbert space. Let F : H* → *H be k-Lipschitzian and η-strongly monotone operator with k* ∈ [1, ∞) *and η* ∈ (0, 1). *Let*
*be an infinite family of nonexpansive mappings such that*
and {*W*_{
n
}} *be W-mapping defined by (2.3). Let* {*λ*_{
n
}} *be a sequence in* [0, ∞) *and* {*α*_{
n
}} *be a sequence in* [0,1]. *Assume that*

*(C1)* lim_{n→1}*λ*_{
n
} = 0;

*(C2)*
;

*(C5)*
*for some γ* > 0.

*Then, the sequence* {*x*_{
n
}} *and* {*y*_{
n
}} *generated by (1.2) converge strongly to*
, *which solves the following variational inequality* 〈*Fx**, *x** - *x*〉 ≤ 0,
.

*Proof*. Since lim_{n→∞}*λ*_{
n
} = 0, it is easy to see that
, ∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0. Without loss of generality, we assume that
, ∀*n* ≥ *n*_{0} for some integer *n*_{0} ≥ 0. Repeating the same argument as in the proof of Theorem 3.2, we know that {*x*_{
n
}} is bounded, and so are the sequence {*y*_{
n
}} and {*F*(*x*_{
n
})}. Therefore, we have *λ*_{
n
}*F*(*x*_{
n
}) → 0.

From
for some *γ* > 0, we have 0 < *γ* ≤ lim inf_{n→∞}*α*_{
n
} ≤ lim sup_{n→∞}*α*_{
n
} < 1 for some *γ* ∈ (0, 1). Therefore, all conditions of Theorem 3.2 are satisfied. Hence, using Theorem 3.2, we have that {*y*_{
n
}} converges strongly to
which solves the following variational inequality 〈*Fx**, *x** - *x*〉 ≤ 0,
. It follows from (3.17) that {*x*_{
n
}} also converges strongly to
. This completes the proof.

**Remark 3.5**. *Theorem 3.2 is more general than Theorem 3.2 of Yao et al.* [5]. *The following example shows that all conditions of Theorem 3.2 are satisfied. However, the conditions λ*_{
n
} → 0, *η* ∈ (0, 1) *and*
*for some γ* > 0 *in [5, Theorem 3.2] are not satisfied*.

**Example 3.6**.

*Let H = R the set of real numbers and T*

_{ n }≡

*T. Define a nonexpansive mapping T*:

*H*→

*H and an operator F*:

*H*→

*H as follows:*

*It is easy to see that F*(

*T*) = {0},

*and W*

_{ n }

*x*= (1 -

*ξ*

_{1})

*x*, ∀

*x*∈

*R*.

*Let*,

*we have*, ∀

*x*∈

*R*.

*Given sequences*{

*α*

_{ n }}

*and*,

*for all n*≥ 0.

*For an arbitrary x*

_{0}∈

*H*,

*let*{

*x*

_{ n }}

*defined as follows:*

*Hence, we have*
*for all n* ≥ 0. *This implies that* {*x*_{
n
}} *converges strongly to*
. *Since*
, *we have that* {*y*_{
n
}} *converges strongly to*
.

*Observe that* 〈*F*(0), 0 - *u*〉 ≤ 0,
, *that is*, 0 *is the solution of the variational inequality* 〈*Fx**, *x** - *u*〉 ≤ 0,
.

*By F*(*x*) = *x, we have η* = *k* = 1. *Furthermore, it is easy to see that the following hold true:*

*(B1)*
, ∀*n* ≥ *n*_{0} *for some integer n*_{0} ≥ 0;

*(B2)*
;

*(B3)*
*for some constant*
.

*Hence, there is no doubt that all conditions of Theorem 3.2 are satisfied. Since*
, *η* = 1 *and*
, *the conditions that λ*_{
n
} → 0,
*for some γ* > 0 *and η* ∈ (0, 1) *of Yao et al. [5, Theorem 3.2] are not satisfied*.

Next, we give a weak convergence theorem for hybrid iterative algorithm (1.2) involving an infinite family of nonexpansive mappings in a Hilbert space.

**Theorem 3.7**. *Let H be a real Hilbert space. Let F* : *H* → *H be k-Lipschitzian and η-strongly monotone operator with* 0 < η ≤ *k*. *Let*
*be an infinite family of nonexpansive mappings such that*
, *and* {*W*_{
n
}} *be W-mapping defined by (2.3). Let* {*λ*_{
n
}} *and* {*α*_{
n
}} *be two sequences in* (0, 1). *Assume that*

*(A1)*
;

*(A2)* 0 < lim inf_{n→∞}*α*_{
n
} ≤ lim sup_{n→∞}*α*_{
n
} < 1.

*Then, the sequence* {*x*_{
n
}} *and* {*y*_{
n
}} *generated by (1.2) converge weakly to*
.

*Proof*. From (A1), we have , ∀

*n*≥

*n*

_{0}for some integer

*n*

_{0}≥ 0. Repeating the same argument as in the proof of Theorem 3.2, we know that {

*x*

_{ n }} is bounded, and so are the sequences {

*y*

_{ n }} and {

*F*(

*x*

_{ n })}. Assuming , we have

*M*

_{4}= sup{||

*F*(

*x*

_{ n })||

^{2}, 2 ||

*x*

_{ n }-

*p*|| ||

*F*(

*x*

_{ n })||,

*n*≥ 0}. Since , we have . Therefore, . Utilizing Lemma 2.4, we deduce that lim

_{n→∞}||

*x*

_{ n }-

*p*|| exists. Further-more, from(3.17), we have

Now, we show that *ω*_{
w
}(*y*_{
n
}) ⊂ *F*(*T*). Indeed, let *x** ∈ *ω*_{
w
}(*y*_{
n
}). Then, there exists a subsequence
of {*y*_{
n
}} such that
. Since ||*y*_{
n
} - *Wy*_{
n
}|| → 0, by Lemma 2.1, we have
.

*ω*

_{ w }(

*y*

_{ n }) is a singleton. Indeed, let be another subsequence of {

*y*

_{ n }} such that . Then, . If , then, by Opial's property of

*H*, we have

This is a contradiction. Therefore, *ω*_{
w
}(*y*_{
n
}) is a singleton. Consequently, {*y*_{
n
}} converges weakly to
. From (1.2), we have that {*x*_{
n
}} converges weakly to
. This completes the proof.

**Remark 3.8**. *It is worth pointing out that the conditions (C1) and (C2) in [5, Theorem 3.2] are replaced by the one (A1) in Theorem 3.7. It is also worth pointing out that condition (A2) is strictly weaker than the condition (C5). The advantages of there results in this study are that weaker and fewer restrictions are imposed on parameters α*_{
n
}, *λ*_{
n
} *and η*.

## Declarations

### Acknowledgements

This study was supported by the Natural Science Foundation of Yancheng Teachers University under Grant (10YCKL022).

## Authors’ Affiliations

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