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 k-strict pseudo-contraction 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 L-Lipschitzian and η-strongly 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 η-strongly 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 pseudo-contractions which also solves the variational inequality

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:

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 }.

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

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 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 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 < 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,

□

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 < 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 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 .

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.

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 → ∞. □

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},

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 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. 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 {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.

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 L-Lipschitzian and η-strongly 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 ··· T1) = 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

The authors declare that they have no completing interests.

All the authors read and approved the final manuscript.