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Strong convergence theorems by a hybrid extragradient-like approximation method for asymptotically nonexpansive mappings in the intermediate sense in Hilbert spaces

Abstract

Let C be a nonempty closed convex subset of a real Hilbert space H. Let S : CC be an asymptotically nonexpansive map in the intermediate sense with the fixed point set F(S). Let A : CH be a Lipschitz continuous map, and VI(C, A) be the set of solutions u C of the variational inequality

A u , v - u 0 , v C .

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.

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 : CH is called monotone[13] if

A u - A v , u - v 0 , u , v C ;

and A is called k-Lipschitz continuous if there exists a positive constant k such that

A u - A v k u - v , u , v C .

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

S u - S v u - v , u , v C ;

and S is asymptotically nonexpansive[4] if there exists a null sequence {γ n } in [0, + ∞) such that

S n u - S n v ( 1 + γ n ) u - v , u , v C and n 1 .

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

S n x - S n y 2 ( 1 + γ n ) x - y 2 + c n , x , y C , n 1 .

Let A : CH be a monotone and k-Lipschitz continuous mapping. The variational inequality problem [6] is to find the elements u C such that

A u , v - u 0 , v C .

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 : CC 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

x - P C x , P C x - y 0 for all x H , y C ;
(1)

see for example [10]. It is easy to see that (1) is equivalent to

x - y 2 x - P C x 2 + y - P C x 2 for all x H , y C .
(2)

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

u V I ( C , A ) u = P C ( u - λ A u ) for some λ > 0 .

Theorem 1. Let C be a nonempty closed convex subset of a real Hilbert space H. Let A : CH be a monotone and k-Lipschitz continuous mapping. Let S : CC be a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense with nonnegative null sequences {γ n } and {c n }. Suppose that n = 1 λ n <and F(S) VI(C, A) is nonempty and bounded.

Assume that

(i) 0 < μ ≤ 1, and0<a<b< 3 8 k μ ;

(ii) aλ 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,

Δ n = sup { x n - u : u F ( S ) V I ( C , A ) } , d n = 2 b ( 1 - μ ) α n Δ n , w n = b 2 μ α n + 4 b 2 μ 2 β n ( 1 - δ n ) ( 1 + γ n ) , v n = b 2 ( 1 - μ ) α n + 4 b 2 ( 1 - μ ) 2 β n ( 1 - δ n ) ( 1 + γ n ) , a n d ϑ n = β n γ n Δ n 2 + β n c n .

Let {x n }, {y n } and {z n } be sequences generated by the algorithm:

x 0 C c h o s e n a r b i t r a r i l y , y n = ( 1 - δ n ) x n + δ n P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) , z n = ( 1 - α n - β n ) x n + α n y n + β n S n P C ( x n - λ n A y n ) , C n = { z C : z n - z 2 x n - z 2 + d n A y n + w n A x n 2 + v n A y n 2 + ϑ n } , Q n = { z C : x n - z , x 0 - x n 0 } , x n + 1 = P C n Q n ( x 0 ) , n 0 .
(3)

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:

  1. (a)

    C n is closed and convex for all n ;

  2. (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);

  3. (c)

    F(S) VI(C, A) C n for all n .

It is obvious that C n is closed for all n . On the other hand, the defining inequality in C n is equivalent to the inequality

2 ( x n - z n ) , z x n 2 - z n 2 + d n A y n + w n A x n 2 + v n A y n 2 + ϑ n ,

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

t n - u 2 x n - λ n A y n - u 2 - x n - λ n A y n - t n 2 = x n - u 2 - x n - t n 2 + 2 λ n A y n , u - t n = x n - u 2 - x n - t n 2 + 2 λ n [ A y n - A u , u - y n + A u , u - y n + A y n , y n - t n ] x n - u 2 - x n - t n 2 + 2 λ n A y n , y n - t n = x n - u 2 - x n - y n 2 - 2 x n - y n , y n - t n - y n - t n 2 + 2 λ n A y n , y n - t n = x n - u 2 - x n - y n 2 - y n - t n 2 + 2 x n - λ n A y n - y n , t n - y n .
(4)

Now, using

y n = ( 1 - δ n ) x n + δ n P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) ,

we estimate the last term

x n - λ n A y n - y n , t n - y n = x n - λ n μ A x n - λ n ( 1 - μ ) A y n - y n , t n - y n + λ n μ A x n - A y n , t n - y n x n - λ n μ A x n - λ n ( 1 - μ ) A y n - ( 1 - δ n ) x n - δ n P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) , t n - y n + λ n μ A x n - A y n t n - y n δ n x n - λ n μ A x n - λ n ( 1 - μ ) A y n - P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) , t n - y n - ( 1 - δ n ) λ n μ A x n + ( 1 - μ ) A y n , t n - y n + λ n μ k x n - y n t n - y n .
(5)

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

x n λ n μ A x n λ n ( 1 μ ) A y n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) , t n y n = x n λ n μ A x n λ n ( 1 μ ) A y n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) , t n ( 1 δ n ) x n δ n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) = ( 1 δ n ) x n λ n μ A x n λ n ( 1 μ ) A y n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) , t n x n + δ n x n λ n μ A x n λ n ( 1 μ ) A y n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) , t n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) ( 1 δ n ) x n λ n μ A x n λ n ( 1 μ ) A y n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n ) , t n x n ( 1 δ n ) x n λ n μ A x n λ n ( 1 μ ) A y n P C ( x n λ n μ A x n λ n ( 1 μ ) A y n t n x n ( 1 δ n ) λ n μ A x n + λ n ( 1 μ ) A y n t n x n ( 1 δ n ) λ n ( μ A x n + ( 1 μ ) A y n ) ( t n y n + y n x n ) .
(6)

In view of (4)-(6), λ n b, and the inequalities 2αβα2 + β2 and (α + β)2 ≤ 2α2 + 2β2, we conclude that

t n - u 2 x n - u 2 - x n - y n 2 - y n - t n 2 + 2 x n - λ n A y n - y n , t n - y n x n - u 2 - x n - y n 2 - y n - t n 2 + 2 λ n δ n ( 1 - δ n ) ( μ A x n + ( 1 - μ ) A y n ) ( t n - y n + y n - x n ) - 2 ( 1 - δ n ) λ n μ A x n + ( 1 - μ ) A y n , t n - y n + 2 λ n μ k x n - y n t n - y n x n - u 2 - x n - y n 2 - y n - t n 2 + 2 δ n ( 1 - δ n ) b ( μ A x n + ( 1 - μ ) A y n ) ( t n - y n + y n - x n ) + 2 ( 1 - δ n ) b ( μ A x n + ( 1 - μ ) A y n ) t n - y n + 2 b μ k x n - y n t n - y n = x n - u 2 - x n - y n 2 - y n - t n 2 + 2 δ n ( 1 - δ n ) ( b 2 μ 2 A x n 2 + b 2 ( 1 - μ ) 2 A y n 2 + t n - y n 2 + y n - x n 2 ) + ( 1 - δ n ) ( b 2 μ 2 A x n 2 + b 2 ( 1 - μ ) 2 A y n 2 + 2 t n - y n 2 ) + b μ k ( x n - y n 2 + t n - y n 2 ) = x n - u 2 - x n - y n 2 ( 1 - 2 δ n ( 1 - δ n ) - b k μ ) - t n - y n 2 ( 2 δ n 2 - δ n - b k μ ) + 2 ( 1 - δ n 2 ) b 2 μ 2 A x n 2 + 2 ( 1 - δ n 2 ) b 2 ( 1 - μ ) 2 A y n 2 .
(7)

Since 3 4 < δ n 1 and b< 3 8 k μ , we have from (7) for all n ,

t n - u 2 x n - u 2 + 4 ( 1 - δ n ) b 2 μ 2 A x n 2 + 4 ( 1 - δ n ) b 2 ( 1 - μ ) 2 A y n 2 .
(8)

In view of the fact that u VI(A, C) and properties of P C , we obtain

y n - u 2 = ( 1 - δ n ) ( x n - u ) + δ n ( P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) - u ) 2 ( 1 - δ n ) x n - u 2 + δ n P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) - P C ( u ) 2 ( 1 - δ n ) x n - u 2 + δ n x n - λ n μ A x n - λ n ( 1 - μ ) A y n - u 2 = ( 1 - δ n ) x n - u 2 + δ n x n - u 2 - 2 λ n μ A x n + λ n ( 1 - μ ) A y n , x n - u + λ n μ A x n + λ n ( 1 - μ ) A y n 2 = ( 1 - δ n ) x n - u 2 + δ n x n - u 2 - 2 λ n μ A x n , x n - u - 2 λ n ( 1 - μ ) A y n , x n - u + λ n μ A x n + λ n ( 1 - μ ) A y n 2 ( 1 - δ n ) x n - u 2 + δ n x n - u 2 + 2 λ n ( 1 - μ ) A y n x n - u + λ n 2 μ A x n 2 + λ n 2 ( 1 - μ ) A y n 2 ( 1 - δ n ) x n - u 2 + δ n x n - u 2 + 2 b ( 1 - μ ) Δ n A y n + b 2 μ A x n 2 + b 2 ( 1 - μ ) A y n 2 x n - u 2 + δ n 2 b ( 1 - μ ) Δ n A y n + b 2 μ 2 A x n 2 + b 2 ( 1 - μ ) 2 A y n 2 x n - u 2 + 2 b ( 1 - μ ) Δ n A y n + b 2 μ A x n 2 + b 2 ( 1 - μ ) A y n 2 .
(9)

Since S is asymptotically nonexpansive in the intermediate sense, in view of Snu = u, we conclude that

z n - u 2 = ( 1 - α n - β n ) x n + α n y n + β n S n t n - u 2 ( 1 - α n - β n ) x n - u 2 + α n y n - u 2 + β n S n t n - u 2 ( 1 - α n - β n ) x n - u 2 + α n x n - u 2 + 2 b ( 1 - μ ) Δ n A y n + b 2 μ A x n 2 + b 2 ( 1 - μ ) A y n 2 + β n ( 1 + γ n ) t n - u 2 + c n ( 1 - α n - β n ) x n - u 2 + α n x n - u 2 + 2 b ( 1 - μ ) Δ n A y n + b 2 μ A x n 2 + b 2 ( 1 - μ ) A y n 2 + β n ( 1 + γ n ) x n - u 2 + 2 ( 1 - δ n ) b 2 μ 2 A x n 2 + 2 ( 1 - δ n ) b 2 ( 1 - μ ) 2 A y n 2 + β n c n x n - u 2 + β n γ n Δ n 2 + 2 b ( 1 - μ ) α n Δ n A y n + ( b 2 μ α n + 2 b 2 μ 2 β n ( 1 - δ n ) ( 1 + γ n ) ) A x n 2 + ( b 2 ( 1 - μ ) α n + 2 b 2 ( 1 - μ ) 2 β n ( 1 - δ n ) ( 1 + γ n ) ) A y n 2 + β n c n .
(10)

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

C k = { z C : z k - x k 2 + 2 z k - x k , x k - z d n A y n + w n A x n 2 + v n A y n 2 + ϑ n } .

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 P C k Q k x 0 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:

  1. (d)

    limn→∞||x n - x 0|| exists and hence {x n }, as well as {Δ n }, is bounded.

  2. (e)

    limn→∞ ||x n+1- x n || = 0.

  3. (f)

    limn→∞ ||z n - x n || = 0.

Let u F(S) VI(C, A). Since x n + 1 = P C n Q n x 0 and u F(S) VI(C, A) C n Q n , we conclude that

x n + 1 - x 0 u - x 0 , n 0 .
(11)

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 x n = P Q n x 0 and xn+1 C n Q n Q n . This implies that

x n + 1 - x n 2 x n + 1 - x 0 2 - x n - x 0 2 , n 0 .
(12)

In particular, ||xn+1- x0|| ≥ ||x n - x0|| hence limn→∞||x n - x0|| exists. It follows from (12) that

lim n ( x n + 1 - x n ) = 0 .
(13)

Since xn+1 C n , we obtain

z n - x n + 1 2 x n - x n + 1 2 + d n A y n + w n A x n 2 + v n A y n 2 + ϑ n .

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:

  1. (g)

    limn→∞||x n - y n || = 0.

  2. (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

z n - u 2 = ( 1 - α n - β n ) x n + α n y n + β n S n t n - u 2 ( 1 - α n - β n ) x n - u 2 + α n y n - u 2 + β n S n t n - u 2 ( 1 - α n - β n ) x n - u 2 + α n x n - u 2 + 2 b ( 1 - μ ) Δ n A y n + b 2 μ A x n 2 + b 2 ( 1 - μ ) A y n 2 + β n ( 1 + γ n ) t n - u 2 + c n ( 1 - α n - β n ) x n - u 2 + α n x n - u 2 + 2 b ( 1 - μ ) Δ n A y n + b 2 μ A x n 2 ( 1 - μ ) A y n 2 + β n ( 1 + γ n ) x n - u 2 - ( 1 - 2 δ n ( 1 - δ n ) x n - y n 2 - b k μ ) - ( 2 δ n 2 - 1 - b k μ ) t n - y n 2 + 4 ( 1 - δ n ) b 2 μ 2 A x n 2 + 4 ( 1 - δ n ) b 2 ( 1 - μ ) 2 A y n 2 + β n c n x n - u 2 + β n γ n Δ n 2 + β n c n + 2 b ( 1 - μ ) α n Δ n A y n + b 2 μ α n + 4 b 2 μ 2 β n ( 1 + γ n ) ( 1 - δ n ) A x n 2 + b 2 ( 1 - μ ) α n + 4 b 2 ( 1 - μ ) 2 β n ( 1 + γ n ) ( 1 - δ n ) A y n 2 - β n ( 1 + γ n ) ( 1 - 2 δ n ( 1 - δ n ) - b k μ ) x n - y n 2 - β n ( 1 + γ n ) ( 2 δ n 2 - δ n - b k μ ) t n - y n 2 .
(14)

Thus, we have

β n ( 1 + γ n ) ( 1 - 2 δ n ( 1 - δ n ) - b k μ ) x n - y n 2 x n - u 2 - z n - u 2 + β n γ n Δ n 2 + β n c n + 2 b ( 1 - μ ) α n Δ n A y n + b 2 μ α n + 4 b 2 μ 2 β n ( 1 + γ n ) ( 1 - δ n ) A x n 2 + b 2 ( 1 - μ ) α n + 4 b 2 ( 1 - μ ) 2 β n ( 1 + γ n ) ( 1 - δ n ) A y n 2 ( x n - u + z n - u ) x n - z n + β n γ n Δ n 2 + β n c n + 2 b ( 1 - μ ) α n Δ n A y n + b 2 μ α n + 4 b 2 μ 2 β n ( 1 + γ n ) ( 1 - δ n ) A x n 2 + b 2 ( 1 - μ ) α n + 4 b 2 ( 1 - μ ) 2 β n ( 1 + γ n ) ( 1 - δ n ) A y n 2 .

Since bkμ < 3/8 and 3/4 ≤ δ n ≤ 1 for all n ≥ 0, we have

lim n x n - y n 2 = 0 .

In the same manner, from (14), we conclude that

lim n t n - y n 2 = 0 .

Since A is k-Lipschitz continuous, we obtain ||Ay n - Ax n || → 0. On the other hand,

x n - t n x n - y n + y n - t n ,

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

z n - x n = - α n x n + α n y n + β n ( s n t n - x n ) .

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

x n - S n x n x n - t n + t n - S n t n + S n t n - S n x n x n - t n + t n - S n t n + ( 1 + γ n ) t n - x n + c n .

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

ω w ( x n ) : = { x H : x n j x weakly for some subsequence { x n j } of { x n } } .

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 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 : CH be a monotone and k-Lipschitz continuous mapping and let S : CC be a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense with nonnegative null sequences n } and {c n }.

Suppose that n = 1 γ n <and 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 n } be a sequence in [a, b] with a > 0 andb< 3 8 k μ .

Let {x n }, {y n } and {z n } be sequences generated by

x 0 C c h o s e n a r b i t r a r i l y , y n = P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) , z n = S n P C ( x n - λ n A y n ) , C n = { z C : z n - z 2 x n - z 2 + ϑ n } , Q n = { z C : x n - z , x 0 - x n 0 } , x n + 1 = P C n Q n ( x 0 ) , n 0 .
(15)

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 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 : CH be a monotone and k-Lipschitz continuous mapping and let S : CC be a uniformly continuous asymptotically nonexpansive mapping in the intermediate sense with null sequences {γ n } and {c n }.

Suppose that n = 1 γ n <and F(S) VI(C, A) is nonempty and bounded. Let μ be a constant in (0, 1], let n } be a sequence in [a, b] with a > 0 andb< 3 8 k μ , and let {δ n } be a sequence in [0, 1] such that limn→∞δ n = 1 and δ n > 3 4 for 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

x 0 C c h o s e n a r b i t r a r i l y , y n = ( 1 - δ n ) x n + δ n P C ( x n - λ n μ A x n - λ n ( 1 - μ ) A y n ) , z n = S n P C ( x n - λ n A y n ) , C n = { z C : z n - z 2 x n - z 2 + w n A x n 2 + ϑ n } , Q n = { z C : x n - z , x 0 - x n 0 } , x n + 1 = P C n Q n ( x 0 ) , n 0 .
(16)

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 : HH be a monotone and k-Lipschitz continuous mapping and let S : HH be a uniformly continuous asymptotically nonexpan-sive mapping in the intermediate sense with null sequences {γ n } and {c n }.

Suppose that n = 1 γ n <and F(S) A-1(0) is nonempty and bounded. Let μ be a constant in (0, 1], let n } be a sequence in [a, 3b/4] with0<4a3<b< 3 8 k μ , 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 and δ n > 3 4 for all n ≥ 0.

Set

Δ n = sup { x n - u : u F ( S ) A - 1 ( 0 ) } , d n = 2 b ( 1 - μ ) α n Δ n , w n = b 2 μ α n + 4 b 2 μ 2 β n ( 1 - δ n ) ( 1 + γ n ) , v n = b 2 ( 1 - μ ) α n + 4 b 2 ( 1 - μ ) 2 β n ( 1 - δ n ) ( 1 + γ n ) , a n d ϑ n = β n γ n Δ n 2 + β n c n

for all n ≥ 0.

Let {x n }, {y n } and {z n } be sequences generated by

x 0 C c h o s e n a r b i t r a r i l y , y n = x n - λ n μ A x n - λ n ( 1 - μ ) A y n , z n = ( 1 - β n ) x n - α n μ A x n - α n λ n ( 1 - μ ) A y n + β n S n ( x n - λ n δ n A y n ) , C n = { z C : z n - z 2 x n - z 2 + d n A y n + w n A x n 2 + v n A y n 2 + ϑ n } , Q n = { z C : x n - z , x 0 - x n 0 } , x n + 1 = P C n Q n ( x 0 ) , n 0 .
(17)

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 λ n = λ n δ n . Then, a λ n < 4 3 λ n <b< 3 8 k μ . 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).

References

  1. Nadezhkina N, Takahashi W: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitz continuous monotone mappings. SIAM J Optim 2006, 16: 1230–1241. 10.1137/050624315

    MATH  MathSciNet  Article  Google Scholar 

  2. Rockafellar RT: Monotone operators and the proximal point algorithm. SIAM J Control Optim 1976, 14: 877–898. 10.1137/0314056

    MATH  MathSciNet  Article  Google Scholar 

  3. Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 1970, 149: 75–88. 10.1090/S0002-9947-1970-0282272-5

    MATH  MathSciNet  Article  Google Scholar 

  4. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3

    MATH  MathSciNet  Article  Google Scholar 

  5. Sahu DR, Xu H-K, Yao J-C: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal 2009, 70: 3502–3511. 10.1016/j.na.2008.07.007

    MATH  MathSciNet  Article  Google Scholar 

  6. Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and their Applications. Academic Press, New York; 1980.

    Google Scholar 

  7. Korpelevich GM: The extragradient method for finding saddle points and other problems. Matecon 1976, 12: 747–756.

    MATH  Google Scholar 

  8. Ceng LC, Sahu DR, Yao JC: Implicit iterative algorithms for asymptotically nonex-pansive mappings in the intermediate sense and Lipschitz continuous monotone mappings. J Comput Appl Math 2010, 233: 2902–2915. 10.1016/j.cam.2009.11.035

    MATH  MathSciNet  Article  Google Scholar 

  9. Ceng L-C, Hadjisavvas N, Wong N-C: Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Global Optim 2010, 46: 635–646. 10.1007/s10898-009-9454-7

    MATH  MathSciNet  Article  Google Scholar 

  10. Goebel K, Kirk WA: Topics on Metric Fixed-Point Theory. Cambridge University Press, Cambridge; 1990.

    Chapter  Google Scholar 

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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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Correspondence to Ngai-Ching Wong.

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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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Keywords

  • asymptotically nonexpansive mapping in the intermediate sense
  • variational inequality
  • hybrid extragradient-like approximation method
  • monotone mapping
  • fixed point
  • strong convergence