Abstract
Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : C → C be an asymptotically nonexpansive map in the intermediate sense with the fixed point set F(S). Let A : C → H be a Lipschitz continuous map, and VI(C, A) be the set of solutions u ∈ C of the variational inequality
The purpose of this study is to introduce a hybrid extragradient-like approximation method for finding a common element in F(S) and VI(C, A). We establish some strong convergence theorems for sequences produced by our iterative method.
AMS subject classifications: 49J25; 47H05; 47H09.
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1 Introduction
Let H be a real Hilbert space with inner product (·, ·) and norm || · ||, respectively. Let C be a nonempty closed convex subset of H and let P C be the metric projection from H onto C. A mapping A : C → H is called monotone[1–3] if
and A is called k-Lipschitz continuous if there exists a positive constant k such that
Let S be a mapping of C into itself. Denote by F(S) the set of fixed points of S; that is F(S) = {u ∈ C : Su = u}. Recall that S is nonexpansive if
and S is asymptotically nonexpansive[4] if there exists a null sequence {γ n } in [0, + ∞) such that
We call S an asymptotically nonexpansive mapping in the intermediate sense[5] if there exists two null sequences {γ n } and {c n } in [0, + ∞) such that
Let A : C → H be a monotone and k-Lipschitz continuous mapping. The variational inequality problem [6] is to find the elements u ∈ C such that
The set of solutions of the variational inequality problem is denoted by VI(C, A). The idea of an extragradient iterative process was first introduced by Korpelevich in [7]. When S : C → C is a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense, a hybrid extragradient-like approximation method was proposed by Ceng et al. [8, Theorem 1.1] to ensure the weak convergence of some algorithms for finding a member of F(S) ∩ VI(C, A). Meanwhile, assuming S is nonexpansive, Ceng et al. in [9] introduced an iterative process and proved its strong convergence to a member of F(S) ∩ VI(C, A).
It is known that an asymptotically nonexpansive mapping in the intermediate sense is not necessarily nonexpansive. Extending both [8, Theorem 1.1, 9, Theorem 5], the main result, Theorem 1, of this article provides a technical method to show the strong convergence of an iterative scheme to an element of F(S) ∩ VI(C, A), under the weaker assumption on the asymptotical nonexpansivity in the intermediate sense of S.
2 Strong convergence theorems
Let C be a nonempty closed convex subset of a real Hilbert space H. For any x in H, there exists a unique element in C, which is denoted by P C x, such that ||x - P C x|| ≤ ||x - y|| for all y in C. We call P C the metric projection of H onto C. It is well-known that P C is a nonexpansive mapping from H onto C, and
see for example [10]. It is easy to see that (1) is equivalent to
Let A be a monotone mapping of C into H. In the context of variational inequality problems, the characterization of the metric projection (1) implies that
Theorem 1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : C → H be a monotone and k-Lipschitz continuous mapping. Let S : C → C be a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense with nonnegative null sequences {γ n } and {c n }. Suppose thatand F(S) ⋂ VI(C, A) is nonempty and bounded.
Assume that
(i) 0 < μ ≤ 1, and;
(ii) a ≤ λ n ≤ b, α n ≥ 0, β n ≥ 0, α n + β n ≤ 1, and 3/4 < δ n ≤ 1, for all n ≥ 0;
(iii) limn→∞α n = 0;
(iv) lim infn→∞β n > 0;
(v) limn→∞β n = 1.
Set, for all n ≥ 0,
Let {x n }, {y n } and {z n } be sequences generated by the algorithm:
Then, the sequences {x n }, {y n } and {z n } in (3) are well-defined and converge strongly to the same point q = PF(S)⋂VI(C,A)(x0).
Proof. First note that limn→∞γ n = limn→∞c n = 0. We will see that {Δ n } is bounded, and thus limn→∞d n = limn→∞w n = limn→∞v n = limn→∞ϑ n = 0.
We divide the proof into several steps.
Step 1. We claim that the following statements hold:
-
(a)
C n is closed and convex for all n ∈ ℕ;
-
(b)
||z n - u|| 2 ≤ ||x n - u||2 + d n ||Ay n || + w n ||Ax n ||2 + v n ||Ay n ||2 + ϑ n for all n ≥ 0 and u ∈ F(S) ⋂ VI(C, A);
-
(c)
F(S) ⋂ VI(C, A) ⊂ C n for all n ∈ ℕ.
It is obvious that C n is closed for all n ∈ ℕ. On the other hand, the defining inequality in C n is equivalent to the inequality
which is affine in z. Therefore, C n is convex.
Let t n = P C (x n - λ n Ay n ) for all n ≥ 0. Assume that u ∈ F(S) ⋂ VI(C, A) is arbitrary. In view of (3), the monotonicity of A, and the fact u ∈ VI(C, A), we conclude that
Now, using
we estimate the last term
It follows from the properties (1) and (2) of the projection P C (x n - λ n μAx n - λ n (1 - μ)Ay n ) that
In view of (4)-(6), λ n ≤ b, and the inequalities 2αβ ≤ α2 + β2 and (α + β)2 ≤ 2α2 + 2β2, we conclude that
Since and , we have from (7) for all n ∈ ℕ,
In view of the fact that u ∈ VI(A, C) and properties of P C , we obtain
Since S is asymptotically nonexpansive in the intermediate sense, in view of Snu = u, we conclude that
This implies that u ∈ C n . Therefore, F(S) ⋂ VI(C, A) ⊂ C n .
Step 2. We prove that the sequence {x n } is well-defined and F(S) ⋂ VI(C, A) ⊂ C n ⋂ Q n for all n ≥ 0.
We prove this assertion by mathematical induction. For n = 0 we get Q0 = C. Hence, by step 1, we deduce that F(S) ⋂ VI(C, A) ⊂ C1 ⋂ Q1. Assume that x k is defined and F(S) ⋂ VI(C, A) ⊂ C k ⋂ Q k for some k ≥ 1. Then, y k , z k are well-defined elements of C. We notice that C k is a closed convex subset of C since
It is easy to see that Q k is closed and convex. Therefore, C k ⋂ Q k is a closed and convex subset of C, since by the assumption we have F(S) ⋂ VI(C, A) ⊂ C k ⋂ Q k . This means that is well-defined.
By the definition of xk+1and of Qk+1, we deduce that C k ⋂ Q k ⊂ Qk+1. Hence, F(S) ⋂ VI(C, A) ⊂ Qk+1. Exploiting Step 1 we conclude that F(S) ⋂ VI(C, A) ⊂ Ck+1⋂ Qk+1.
Step 3. We claim that the following assertions hold:
-
(d)
limn→∞||x n - x 0|| exists and hence {x n }, as well as {Δ n }, is bounded.
-
(e)
limn→∞ ||x n+1- x n || = 0.
-
(f)
limn→∞ ||z n - x n || = 0.
Let u ∈ F(S) ⋂ VI(C, A). Since and u ∈ F(S) ⋂ VI(C, A) ⊂ C n ⋂ Q n , we conclude that
This means that {x n } is bounded, and so are {y n }, Ax n and {Ay n }, because of the Lipschitz-continuity of A. On the other hand, we have and xn+1∈ C n ⋂ Q n ⊂ Q n . This implies that
In particular, ||xn+1- x0|| ≥ ||x n - x0|| hence limn→∞||x n - x0|| exists. It follows from (12) that
Since xn+1∈ C n , we obtain
In view of limn→∞γ n = 0, limn→∞α n = 0, limn→∞δ n = 1 and from the boundedness of {Ax n } and {Ay n } we infer that limn→∞(xn+1- z n ) = 0. Combining with (13) we deduce that limn→∞(x n - z n ) = 0.
Step 4. We claim that the following assertions hold:
-
(g)
limn→∞||x n - y n || = 0.
-
(h)
limn→∞||Sx n - x n || = 0.
In view of (3), z n = (1 - α n - β n )x n + α n y n + β n Snt n , and Snu = u, we obtain from (9) and (8) that
Thus, we have
Since bkμ < 3/8 and 3/4 ≤ δ n ≤ 1 for all n ≥ 0, we have
In the same manner, from (14), we conclude that
Since A is k-Lipschitz continuous, we obtain ||Ay n - Ax n || → 0. On the other hand,
which implies that ||x n - t n || → 0. Since z n = (1 - α n - β n )x n + α n y n + β n Snt n , we have
From ||z n - x n || → 0, α n → 0, lim infn → 0β n > 0 and the boundedness of {x n , y n } we deduce that ||Snt n - x n || → 0. Thus, we get ||t n - Snt n || → 0. By the triangle inequality, we obtain
Hence, ||x n - Snx n || → 0. Since ||x n - xn+1|| → 0, it follows from Lemma 2.7 of Sahu et al. [5] that ||x n - Sx n || → 0. By the uniform continuity of S, we obtain ||x n - Smx n || → 0 as n → ∞ for all m ≥ 1.
Step 5. We claim that ω w (x n ) ⊂ F(S) ⋂ VI(C, A), where
The proof of this step is similar to that of [8, Theorem 1.1, step 5] and we omit it.
A similar argument as mentioned in [9, Theorem 5, Step 6] proves the following assertion.
Step 6. The sequences {x n }, {y n } and {z n } converge strongly to the same point q = PF(S)⋂VI(C,A)(x0), which completes the proof.
For α n = 0, β n = 1 and δ n = 1 for all n ∈ ℕ in Theorem 1, we get the following corollary.
Corollary 2. Let C be a nonempty closed convex subset of a real Hilbert spaces H. Let A : C → H be a monotone and k-Lipschitz continuous mapping and let S : C → C be a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense with nonnegative null sequences {γ n } and {c n }.
Suppose thatand F(S) ⋂ VI(C, A) is nonempty and bounded. Set ϑ n = γ n Δ n + c n . Let μ be a constant in (0, 1], and let {λ n } be a sequence in [a, b] with a > 0 and.
Let {x n }, {y n } and {z n } be sequences generated by
Then, the sequences {x n }, {y n } and {z n } in (15) are well-defined and converge strongly to the same point q = PF(S)⋂VI(C,A)(x0).
In Theorem 1, if we set α n = 0 and β n = 1 for all n ∈ ℕ then the following result concerning variational inequality problems holds.
Corollary 3. Let C be a nonempty closed convex subset of a real Hilbert spaces H. Let A : C → H be a monotone and k-Lipschitz continuous mapping and let S : C → C be a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense with null sequences {γ n } and {c n }.
Suppose thatand F(S) ⋂ VI(C, A) is nonempty and bounded. Let μ be a constant in (0, 1], let {λ n } be a sequence in [a, b] with a > 0 and, and let {δ n } be a sequence in [0, 1] such that limn→∞δ n = 1 andfor all n ≥ 0. Set Δ n = sup{||x n - u|| : u ∈ F(S) ⋂ VI(C, A)}, w n = 4b2μ2(1 + γ n )(1 - δ n ), ϑ n = γ n Δ n + c n for all n ≥ 0.
Let {x n }, {y n } and {z n } be sequences generated by
Then, the sequences {x n }, {y n } and {z n } in (16) are well-defined and converge strongly to the same point q = PF(s)⋂VI(C,A)(x0).
The following theorem is yet an other easy consequence of Theorem 1.
Corollary 4. Let H be a real Hilbert space. Let A : H → H be a monotone and k-Lipschitz continuous mapping and let S : H → H be a uniformly continuous asymptotically nonexpan-sive mapping in the intermediate sense with null sequences {γ n } and {c n }.
Suppose thatand F(S) ⋂ A-1(0) is nonempty and bounded. Let μ be a constant in (0, 1], let {λ n } be a sequence in [a, 3b/4] with, and let {α n }, {β n } and {δ n } be three sequences in [0, 1] satisfying the following conditions:
(i) α n + β n ≤ 1, ∀n ≥ 0;
(ii) limn→∞α n = 0;
(iii) lim infn→∞β n > 0;
(iv) limn→∞δ n = 1 andfor all n ≥ 0.
Set
for all n ≥ 0.
Let {x n }, {y n } and {z n } be sequences generated by
Then, the sequences {x n }, {y n } and {z n } in (17) are well-defined and converge strongly to the same point q = PF(S)⋂A-1(0)(x0).
Proof. Replace λ n by . Then, . For C = H, we have P C = I and VI(C, A) = A-1(0). In view of Theorem 1, the sequences {x n }, {y n } and {z n } are well-defined and converge strongly to the same point q = PF(S)⋂A-1(0)(x0).
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Acknowledgements
The first author of this study was conducted with a postdoctoral fellowship at the National Sun Yat-Sen University, Kaohsiung 804, Taiwan. The second and third authors of this research were partially supported by the Grant NSC 99-2115-M-110-007-MY3 and Grant NSC 99-2221-E-037-007-MY3, respectively.
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Naraghirad, E., Wong, NC. & Yao, JC. Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces. J Inequal Appl 2011, 119 (2011). https://doi.org/10.1186/1029-242X-2011-119
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DOI: https://doi.org/10.1186/1029-242X-2011-119