Convergence and weaker control conditions for hybrid iterative algorithms

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

1 Introduction

Let 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}.

Yamada [2] introduced the following hybrid iterative method for solving the variational inequality:

(1.1)

where F is a k-Lipschitzian and η-strongly monotone operator with k > 0, η > 0 and 0 < μ < 2η /k². Let a sequence {λ_n} of real numbers in (0,1) satisfy the conditions below:

(C1) lim_n→∞λ _n = 0,

(C2) ,

(C3) .

He has proved that {x _n } generated by (1.1) converges strongly to the unique solution of the variational inequality:

An example of sequence {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 }.

Very recently, motivated by Xu and Kim [4], Yao et al. [5] considered the following algorithms: for x₀ ∈ H arbitrarily,

(1.2)

where 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)

then they proved that the sequences {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

Let 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

A mapping F : H ⇀ H is called k-Lipschitzian if there exists a positive constant k such that

(2.1)

F is said to be η-strongly monotone if there exists a positive constant η such that

(2.2)

It is known that 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.

Finally, it is known that in a Hilbert space, there holds the following equality

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.

Lemma 2.3. [10, 11]. Let {s _n } be a sequence of non-negative real numbers satisfying

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/k². Then S = (I - tF ) : H → H is a contraction with contraction coefficient .

Let {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:

(2.3)

Such a mapping W _n : H → H is called a W-mapping generated by T _n , T_n-1,..., T₁ and ξ _n , ξ_n-1,..., ξ₁.

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→1W _n x, and W _n x = lim_n→∞U_n,1x, 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

Let 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/k²) and , and consider a mapping S_t on H defined by

It is easy to see that S _t is a contraction. Indeed, from Lemma 2.5, we have

for all x, y ∈ H. Hence, it has a unique fixed point, denoted as x _t , which uniquely solves the fixed point equation

(3.1)

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/k²), 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:

(3.2)

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

(3.3)

and

(3.4)

Adding up (3.3) and (3.4) yields

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).

Next, we prove that {x _t } is bounded. Take u ∈ F (T ), from (3.1) and using Lemma 2.5, we have

that is,

(3.5)

Observe that

From 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

(3.6)

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

On the other hand, from (3.1), we obtain

(3.7)

To prove that x _t → x*. For a given u ∈ F (T ), using Lemma 2.5, we have

Therefore,

(3.8)

From , we have and . Observe that, if x _t ⇀ u, we have .

Since {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,

(3.9)

Now replacing t in (3.9) with t _n and letting n → _∞, 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 ≥ n₀ for some integer n₀ ≥ 0, and

(C 5) ': 0 < γ ≤ lim inf_n→∞α _n lim sup_n→∞α _n < 1 for some γ ∈ (0, 1).

For x₀ ∈ H arbitrarily, let the sequence {y _n } be generated by (1.2). Then,

where solves the variational inequality

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

(3.10)

∀n ≥ n₀ for some integer n₀ ≥ 0, where . Then, from (1.2) and (3.10), we obtain

By induction, we have

∀n ≥ n₀ for some integer n₀ ≥ 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

(3.11)

From (2.3), for , we have

(3.12)

where M₂ = sup{2 ||y _n - u||, n ≥ 0}. By (1.2) and (3.12), we have

(3.13)

Substituting (3.13) into (3.11), we have

that is,

Observing λ _n F(x _n ) → 0(n → ∞) and 0 < ξ _i ≤ b < 1, it follows that

(3.14)

By (C 5)' and using Lemma 2.2, we have lim_n→∞||z _n - x _n || = 0. Therefore,

Step 3. We claim that lim_n→∞||x _n - W _n x _n || = 0. Observe that

that is,

(3.15)

Step 4. We claim that lim_n→∞||x _n - Wx _n || = 0. Indeed, we have

(3.16)

By (3.15), (3.16) and using Lemma 2.6, we obtain

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

Step 6. We claim that {x _n } converges strongly to . From (1.2), we have

∀n ≥ n₀ for some integer n₀ ≥ 0, where M₃ = 2||Fx*||. For every n ≥ n₀, put and δ _n = 2M₁〈x* - x _n , Fx*〉 +M₁M₃ ||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 .

Observe that

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

On the other hand, suppose that as n → ∞, where solves the variational inequality:

From (1.2), we have

(3.17)

that is, . Again from (1.2), we obtain that

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→1λ _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 ≥ n₀ for some integer n₀ ≥ 0. Without loss of generality, we assume that , ∀n ≥ n₀ for some integer n₀ ≥ 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 - ξ₁)x, ∀x ∈ R. Let , we have , ∀x ∈ R. Given sequences {α _n } and , for all n ≥ 0. For an arbitrary x₀ ∈ H, let {x _n } defined as follows:

that is,

Observe that for all n ≥ 0,

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, .

Finally, we have

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

(B1) , ∀n ≥ n₀ for some integer n₀ ≥ 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 ≥ n₀ for some integer n₀ ≥ 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

(3.17a)

where M₄ = 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

(3.18)

Since λ _n → 0, and (A2), it follows from (3.18) that

Utilizing Lemma 2.6, 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 .

Next, we show that 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.