Open Access

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

Fixed Point Theory and Applications20112011:21

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

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.

Recall that a mapping T : CH is said to be k-strict pseudo-contraction if there exists a constant k [0, 1) such that
(1.1)
These mappings are extensions of nonexpansive mappings which satisfy the inequality (1.1) with k = 0. That is, T : CH is nonexpansive if
We denote by F(T) the set of fixed points of the mapping T, that is

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

Let F : CH be a nonlinear operator, we consider the problem of finding a point x* C such that

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

Takahashi [1] introduced a classical CQ algorithm as follows:

where T is nonexpansive mapping, and {αn} [0, a] for some a [0, 1). Then they showed that {x n } converged strongly to PF(T)(x0) 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.

Take x0 H arbitrarily and define {x n } by
(1.2)
where T is a nonexpansive mapping on H, F is L-Lipschitzian and η-strongly monotone with k > 0, η > 0, 0 < μ < 2η/L2. If {λ n } is a sequence in (0, 1) satisfying the following conditions:
  1. (i)

    limn→∞ λ n = 0;

     
  2. (ii)

     
  3. (iii)

    either or ,

     
then the sequence {x n } converged strongly to the unique solution of the variational inequality
Besides, he also proposed cyclic algorithm:

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

On the other hand, Marino and Xu [3] considered the following general iterative method: an initial x0 is selected in H arbitrarily
(1.3)

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) limn→∞α n = 0;

(C2)

(C3) either or .

They proved that the sequence {x n } converged strongly to a fixed point of T which solves the variational inequality
Very recently, Tian [4] combined the iterative method (1.3) with the Yamada's method (1.2) and considered the following general iterative method
(1.4)

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 < 2η/k2.

He proved that if the sequence {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
(1.5)
In this paper we designed two algorithms for finding a common fixed point x* of finite strict pseudo-contractions which also solves the variational inequality
(1.6)

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

Let T be defined by
Where λ i > 0 such that . We will show that the sequence {x n } generated by the algorithm:
(1.7)

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

Another approach to the problem (1.6) is the cyclic algorithm. For each i = 1,..., N, let
where the constant β i satisfies k i < β i < 1. Beginning with x0 H, we define the sequence {x n } cyclically by
Indeed, the algorithm above can be written as
(1.8)

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

We will use the notations:
  1. 1.

    for weak convergence and → for strong convergence.

     
  2. 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 : CH 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

H, if there is a constant with property

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 : CC 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 .

Lemma 2.3. (see[7]) Assume {a n } is a sequence of nonnegative real numbers such that
where {γ n } is a sequence in (0, 1) and {δ n } is a sequence in such that:
  1. (i)

    limn→∞ γ n = 0 and ;

     
  2. (ii)

    limn→∞ δ n /γ n ≤ 0 or .

     

Then limn→∞a n = 0.

Lemma 2.4. (see[4]) Let H be a real Hilbert space, f : HH a contraction with coefficient 0 < α < 1, and F : HH 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 : CH is a k-strict pseudo-contraction. Define T : CH 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 ≤ iN, T i : CC 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 : HH is a k i -strict pseudo-contraction for some 0 ≤ k i < 1 (1 ≤ iN ): Let , k i < α i < 1 (1 ≤ iN), if , then

Lemma 2.8. Let F : HH be a η-strongly monotone and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < μ < 2η/L2, and 0 < t < 1. Then ||(I - μtF)x - (I - μtF)y|| ≤ (1 - ) ||x - y||.

Proof. Put g = I μtF, then
Therefore,
that is,

3. Synchronal algorithm

Theorem 3.1. Let H be a real Hilbert space and let T i : HH 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 : HH be a η-strongly monotone and L-Lipschitzian operator with L > 0, η > 0. Assume that 0 < μ < 2η/L2, . Given the initial guess x0 H chosen arbitrarily and given sequences {α n } and {β n } in (0, 1), satisfying the following conditions:

(3.1a) limn→∞α n = 0, ;

(3.1b) , ;

(3.1c) 0 ≤ max i k i β n < a < 1 for all n ≥ 0;

let {x n } be the sequences define d by the composite process (1.7), i.e.

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 ≤ iN} and .

We can rewrite the algorithm (1.7) as

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

Step 1. {x n } is bounded.

Take , from (1.7) and Lemma 2.9 we have
By simple induction, we have

Hence {x n } is bounded.

From , we have v F (T ), hence
It follows that
So, we have

Therefore, {Tx n } is bounded.

G is L-Lipschitzian, so

{Tx n } is bounded, so is bounded.

f is a contraction, so f(x n ) is bounded.

Step 2.
(3.1)
Observing that
we have
This in turn implies that
(3.2)
where M1 is an appropriate constant such that . On the other hand, we note that
(3.3)
where M2 is an appropriate constant such that M2 ≥ supn≥1{||x n - Tx n ||}. Substituting (3.3) into (3.2) yields

where M3 is an appropriate constant such that M3 ≥ max{M1, M2}. By conditions (3.1a) and (3.1b) and Lemma 2.3, we obtain that limn→∞||xn+1- x n || = 0.

From (1.7), we observe that
It follows from the condition (3.1a) and the boundedness of {f(x n )} and that
(3.4)
On the other hand,
Hence, by condition (3.1c), we have
From (3.1) and (3.4), we obtain
(3.5)
From the boundedness of {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
(3.6)

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

Step 3.
(3.7)
Indeed, there exists a subsequence such that
Without loss of generality, we may further assume that . It follows from (3.6) that x F(T). Since x* is the unique solution of (1.6), we obtain
Step 4.
(3.8)
From Lemma 2.9, we have
This implies that

where and . , from (3.1a), we have limn→∞γ n = 0; γ n ≥ 2α n (τ - γβ), from (3.1a), we have ; put M = sup {||x n - x*|| : n N}, we have . So, limn→∞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 : HH 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 : HH be a η-strongly monotone and L-Lipschitzian operator with L > 0, η > 0. Assume that . Given the initial guess x0 H chosen arbitrarily and given sequences {α n } and {β n } in (0, 1), satisfying the following conditions:

(4.1a) limn→∞α n = 0,

(4.1b) ;

(4.1c) , or ;

(4.1d) β[n] [k, 1), where k = max i {k i : 1 ≤ iN},

let {x n } be the sequences define d by the composite process (1.8), i.e.

where T[n]= T i , with i = n(modN ), 1 ≤ iN, namely, T[n]is one of T1, T2,..., T N circularly. Then {x n } converges strongly to a common fixed point of which solves the variational inequality (1.6).

Proof. Step 1. {x n } is bounded. Take , from (1.8) and Lemma 2.9 we have
By simple induction, we have

Hence {x n } is bounded.

From the proof of Step 1 in Section 3, we know that {T[n]x n }, {f (x n )}, {GA[n]x n } are bounded.

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

Step 2. limn→∞||xn+N- x n || = 0.

By (1.8) and Lemma 2.9, we have
(4.1)

where K1 is an appropriate constant such that K1 ≥ supn≥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 ||xn+N- x n || → 0 as n → ∞.

Step 3. limn→ ∞||x n - A[n+N]··· A[n+1]x n || = 0.

From (1.8), we observe that
It follows from the condition (4.1a) and the boundedness of {f(x n )} and {GA[n+1]x n } that
Recursively,
By condition (4.1d) and Lemma 2.5, we know that is nonexpansive, so we get
Proceeded accordingly, we have
Note that
From all the expressions above, we obtain
Since

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

Step 4.
(4.2)
Take a subsequence , by step 3, we get

Notice that, for each n j , is some permutation of the mappings A1A2 ··· A N , since A1, A2,···, A N are finite, all the finite permutation are N!, there must be some permutation appears infinite times.

Without loss of generality, suppose this permutation is A1A2···A N , we can take a subsequence such that and

By Lemma 2.5, we conclude that A1, A2,···, A N are all nonexpansive. It is easy to prove that is nonexpansive, so A1A2···A N is.

By Lemma 2.2, we have q = A1A2 ··· A N q. From Lemmas 2.5 and 2.7, we obtain
Step 5.
(4.3)
Indeed, there exists a subsequence such that
Without loss of generality, we may further assume that . It follows from (4.2) that x F(T). Since x* is the unique solution of (1.6), we obtain

Step 6. x n x*(n → ∞).

From Lemma 2.9, we have
This implies that

where and . , from (4.1a), we have limn→∞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, limsupn→∞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

Corollary 1 (see[4]) Let {x n } be generated by the following algorithm

Assume the sequence {α n } satisfies conditions:

(C1) limn→∞α n = 0;

(C2) ;

(C3) either or

then {x n } converged strongly to which solves the variational inequality

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

Corollary 2 (see[3]) Let {x n } be generated by the following algorithm:
Assume the sequence {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:

Corollary 3 (see[2]) Let {x n } be generated by the following algorithm
where T is a nonexpansive mapping on H, F is L-Lipschitzian and η-strongly monotone with k > 0, η > 0, 0 < μ < 2η/L2. If {λ n } is a sequence in (0, 1) satisfies the following conditions:
  1. (i)

    limn→∞ λ n = 0;

     
  2. (ii)

    ;

     
  3. (iii)

    either or

     
then the sequence {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

Corollary 4 (see[2]) Let {x n } be generated by the following algorithm
where T[n]= T n mod N. Assume {λ n } satisfies conditions (C1)-(C3) and C = F(T N ··· T 1) = F (T1T N ··· T3T2) = ··· = F (TN - 1TN - 2··· T1T N ), then {x n } converged strongly to the unique solution of the variational inequality

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

(1)
College of Science, Civil Aviation University of China

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© Tian and Di; licensee Springer. 2011

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