 Research
 Open Access
 Published:
Synchronal algorithm and cyclic algorithm for fixed point problems and variational inequality problems in hilbert spaces
Fixed Point Theory and Applications volume 2011, Article number: 21 (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 pseudocontractive mappings which is the solution of the variational inequality 〈(γ f  μF)x*, x  x*〉 ≤ 0, .
2000 Mathematics Subject Classification: 58E35; 47H09; 65J15.
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 : C → H is said to be kstrict pseudocontraction if there exists a constant k ∈ [0, 1) such that
These mappings are extensions of nonexpansive mappings which satisfy the inequality (1.1) with k = 0. That is, T : C → H 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 : C → H 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 {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.
Take x_{0} ∈ H arbitrarily and define {x_{ n } } by
where T is a nonexpansive mapping on H, F is LLipschitzian and ηstrongly monotone with k > 0, η > 0, 0 < μ < 2η/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 ,
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 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 .
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
where T is a nonexpansive mapping on H, f is a contraction, and F is k Lipschitzian and ηstrongly monotone with k > 0, η > 0, 0 < μ < 2η/k^{2}.
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
In this paper we designed two algorithms for finding a common fixed point x* of finite strict pseudocontractions which also solves the variational inequality
where N ≥ 1 is a positive integer and are N strict pseudocontractions.
Let T be defined by
Where 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).
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 x_{0} ∈ H, we define the sequence {x_{ n } } cyclically by
Indeed, the algorithm above can be written as
where T_{[n]}= T_{ i } , with i = n(modN ), 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.
We will use the notations:

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 ηstrongly 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 : C → C is a κstrict pseudocontraction, then the mapping I  T is demiclosed at 0. That is, if {x_{ n } } is a sequence in C such thatand (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 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 kLipschitzian continuous operator and ηstrongly 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 kstrict pseudocontraction. 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 pseudocontraction for some 0 ≤ k_{ i }< 1. Assumeis a positive sequence such that. Suppose thatthen
Lemma 2.7. (see[9]) Assume T_{ i }: H → H is a k_{ i }strict pseudocontraction 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 ηstrongly monotone and LLipschitzian operator with L > 0, η > 0. Assume that 0 < μ < 2η/L^{2}, and 0 < t < 1. Then (I  μtF)x  (I  μtF)y ≤ (1  tτ) 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 } : H → H be a k_{ i }strict pseudocontraction 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 ηstrongly monotone and LLipschitzian operator with L > 0, η > 0. Assume that 0 < μ < 2η/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;
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 ofwhich solves the variational inequality (1.6).
Proof. Put , then by Lemma 2.6, we conclude that T is a kstrict pseudocontraction with k = max {k_{ i } : 1 ≤ i ≤ N} 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 LLipschitzian, so
{Tx_{ n } } is bounded, so is bounded.
f is a contraction, so f(x_{ n } ) is bounded.
Step 2.
Observing that
we have
This in turn implies that
where M_{1} is an appropriate constant such that . On the other hand, we note that
where 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.
From (1.7), we observe that
It follows from the condition (3.1a) and the boundedness of {f(x_{ n } )} and that
On the other hand,
Hence, by condition (3.1c), we have
From (3.1) and (3.4), we obtain
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
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.
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.
From Lemma 2.9, we have
This implies that
where and . , from (3.1a), we have lim_{n→∞}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, 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 pseudocontraction for some k_{ i } ∈ (0, 1) (i = 1,..., N ) such thatand f be a contraction with coefficient β ∈ (0, 1). Let G : H → H be a ηstrongly monotone and LLipschitzian 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},
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 ≤ 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 ofwhich 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. lim_{n→∞}x_{n+N} x_{ n } = 0.
By (1.8) and Lemma 2.9, we have
where K_{1} is an appropriate constant such that K_{1} ≥ sup_{n≥1}{μGA_{[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.
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.
Take a subsequence , by step 3, we get
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.
Without loss of generality, suppose this permutation is 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.
By Lemma 2.2, we have q = A_{1}A_{2} ··· A_{ N } q. From Lemmas 2.5 and 2.7, we obtain
Step 5.
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 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
Corollary 1 (see[4]) Let {x_{ n } } be generated by the following algorithm
Assume the sequence {α_{ n } } satisfies conditions:
(C1) lim_{n→∞}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 LLipschitzian and ηstrongly monotone with k > 0, η > 0, 0 < μ < 2η/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
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 (T_{1}T_{ N } ··· T_{3}T_{2}) = ··· = F (T_{N  1}T_{N  2}··· T_{1}T_{ N } ), then {x_{ n } } converged strongly to the unique solution of the variational inequality
References
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. J Math Anal Appl 2003, 279: 372–379. 10.1016/S0022247X(02)004584
Yamada I: The hybrid steepestdescent method for variational inequality problems over the intersection of the fixed point sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. NorthHolland, Amsterdam Edited by: Butnariu, D, Censor, Y, Reich S. 2001, 473–504.
Marino G, Xu HK: A general iterative method for nonexpansive mappings in Hilbert spaces. J Math Anal Appl 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028
Tian M: A general iterative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal 2010, 73: 689–694. 10.1016/j.na.2010.03.058
Geobel K, Kirk WA: Topics in Metric Fixed Point Theory. In Cambridge Studies in Advanced Mathematics. Volume 28. Cambridge University Press, Cambridge; 1990.
Acedo GL, Xu HK: Iterative methods for strict pseudocontractions in Hilbert spaces. Nonlinear Anal 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036
Xu HK: Iterative algorithms for nonlinear operators. J Lond Math Soc 2002, 66: 240–256. 10.1112/S0024610702003332
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl 1967, 20: 197–228. 10.1016/0022247X(67)900856
He SN, Liang XL: Hybrid steepestdescent methods for solving variational inequalities governed by boundedly Lipschitzian and strongly monotone operators. Fixed Point Theory Appl 2010.
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).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The authors declare that they have no completing interests.
Authors' contributions
All the authors read and approved the final manuscript.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Tian, M., Di, L. Synchronal algorithm and cyclic algorithm for fixed point problems and variational inequality problems in hilbert spaces. Fixed Point Theory Appl 2011, 21 (2011). https://doi.org/10.1186/16871812201121
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/16871812201121
Keywords
 strict pseudocontractions
 nonexpansive mapping
 variational inequality
 synchronal algorithm
 cyclic algorithm
 fixed point