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 : 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

$$⟨Au,v-u⟩\ge 0,\forall v\in C.$$

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 : C → H is called monotone[1–3] if

$$⟨Au-Av,u-v⟩\ge 0,\forall u,v\in C;$$

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

$$∥Au-Av∥\le k∥u-v∥,\forall u,v\in C.$$

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

$$∥Su-Sv∥\le ∥u-v∥,\forall u,v\in C;$$

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

$$∥S^{n}u-S^{n}v∥\le \left(1+{\gamma }_n\right)∥u-v∥,\forall u,v\in C\mathsf{\text{and}}n\ge 1.$$

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

$${∥S^{n}x-S^{n}y∥}^2\le \left(1+{\gamma }_n\right){∥x-y∥}^2+c_n,\forall x,y\in C,\forall n\ge 1.$$

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

$$⟨Au,v-u⟩\ge 0,\forall v\in C.$$

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 : 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

$$⟨x-P_{C}x,P_{C}x-y⟩\ge 0\mathsf{\text{for}}\mathsf{\text{all}}x\in H,y\in C;$$

(1)

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

$${∥x-y∥}^2\ge {∥x-P_{C}x∥}^2+{∥y-P_{C}x∥}^2\mathsf{\text{for}}\mathsf{\text{all}}x\in H,y\in C.$$

(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

$$u\in VI\left(C,A\right)\iff u=P_C\left(u-\lambda Au\right)\mathsf{\text{for}}\mathsf{\text{some}}\lambda >0.$$

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 that${\sum }_{n=1}^{\infty }{\lambda }_n<\infty$and F(S) ⋂ VI(C, A) is nonempty and bounded.

Assume that

(i) 0 < μ ≤ 1, and$0<a<b<\frac{3}{8k\mu }$;

(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) lim_n→∞α _n = 0;

(iv) lim inf_n→∞β _n > 0;

(v) lim_n→∞β _n = 1.

Set, for all n ≥ 0,

$$\begin{array}{ll}\hfill {\Delta }_n& =sup\left\{∥x_n-u∥:u\in F\left(S\right)\cap VI\left(C,A\right)\right\},\\ \hfill d_n& =2b\left(1-\mu \right){\alpha }_n{\Delta }_n,\\ \hfill w_n& =b^2\mu {\alpha }_n+4b^2{\mu }^2{\beta }_n\left(1-{\delta }_n\right)\left(1+{\gamma }_n\right),\\ \hfill v_n& =b^2\left(1-\mu \right){\alpha }_n+4b^2{\left(1-\mu \right)}^2{\beta }_n\left(1-{\delta }_n\right)\left(1+{\gamma }_n\right),and\\ \hfill {\vartheta }_n& ={\beta }_n{\gamma }_n{\Delta }_n^2+{\beta }_n{c}_n.\end{array}$$

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

$$\left\{\begin{array}{c}{x}_0\in Cchosenarbitrarily,\\ y_n=\left(1-{\delta }_n\right)x_n+{\delta }_n{P}_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right),\\ z_n=\left(1-{\alpha }_n-{\beta }_n\right)x_n+{\alpha }_n{y}_n+{\beta }_n{S}^n{P}_C\left(x_n-{\lambda }_{n}A{y}_n\right),\\ C_n=\left\{z\in C:{∥z_n-z∥}^2\le {∥x_n-z∥}^2+d_n∥A{y}_n∥+w_n{∥A{x}_n∥}^2+v_n{∥A{y}_n∥}^2+{\vartheta }_n\right\},\\ Q_n=\left\{z\in C:⟨x_n-z,x_0-x_n⟩\ge 0\right\},\\ x_{n+1}=P_{C_n\cap Q_n}\left(x_0\right),\forall n\ge 0.\end{array}\right.$$

(3)

Then, the sequences {x _n }, {y _n } and {z _n } in (3) are well-defined and converge strongly to the same point q = P_{F(S)⋂VI(C,A)}(x₀).

Proof. First note that lim_n→∞γ _n = lim_n→∞c _n = 0. We will see that {Δ _n } is bounded, and thus lim_n→∞d _n = lim_n→∞w _n = lim_n→∞v _n = lim_n→∞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

$$⟨2\left(x_n-z_n\right),z⟩\le {∥x_n∥}^2-{∥z_n∥}^2+d_n∥A{y}_n∥+w_n{∥A{x}_n∥}^2+v_n{∥A{y}_n∥}^2+{\vartheta }_n,$$

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

$$\begin{array}{l}{∥t_n-u∥}^2\\ \le {∥x_n-{\lambda }_{n}A{y}_n-u∥}^2-{∥x_n-{\lambda }_{n}A{y}_n-t_n∥}^2\\ ={∥x_n-u∥}^2-{∥x_n-t_n∥}^2+2{\lambda }_n⟨A{y}_n,u-t_n⟩\\ ={∥x_n-u∥}^2-{∥x_n-t_n∥}^2+2{\lambda }_n\left[⟨A{y}_n-Au,u-y_n⟩+⟨Au,u-y_n⟩+⟨A{y}_n,y_n-t_n⟩\right]\\ \le {∥x_n-u∥}^2-{∥x_n-t_n∥}^2+2{\lambda }_n⟨A{y}_n,y_n-t_n⟩\\ ={∥x_n-u∥}^2-{∥x_n-y_n∥}^2-2⟨x_n-y_n,y_n-t_n⟩-{∥y_n-t_n∥}^2+2{\lambda }_n⟨A{y}_n,y_n-t_n⟩\\ ={∥x_n-u∥}^2-{∥x_n-y_n∥}^2-{∥y_n-t_n∥}^2+2⟨x_n-{\lambda }_{n}A{y}_n-y_n,t_n-y_n⟩.\end{array}$$

(4)

Now, using

$$y_n=\left(1-{\delta }_n\right)x_n+{\delta }_n{P}_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right),$$

we estimate the last term

$$\begin{array}{l}⟨x_n-{\lambda }_{n}A{y}_n-y_n,t_n-y_n⟩\\ =⟨x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n-y_n,t_n-y_n⟩+{\lambda }_n\mu ⟨A{x}_n-A{y}_n,t_n-y_n⟩\\ \le ⟨x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n-\left(1-{\delta }_n\right)x_n-{\delta }_n{P}_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right),t_n-y_n⟩\\ +{\lambda }_n\mu ∥A{x}_n-A{y}_n∥∥t_n-y_n∥\\ \le {\delta }_n⟨x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n-P_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right),t_n-y_n⟩\\ -\left(1-{\delta }_n\right){\lambda }_n⟨\mu A{x}_n+\left(1-\mu \right)A{y}_n,t_n-y_n⟩+{\lambda }_n\mu k∥x_n-y_n∥∥t_n-y_n∥.\end{array}$$

(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

$$\begin{array}{l}〈x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n),t_n-y_n〉\\ =〈x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n),\\ t_n-(1-{\delta }_n)x_n-{\delta }_n{P}_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n)〉\\ =(1-{\delta }_n)〈x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n),t_n-x_n〉\\ +{\delta }_n〈x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n),\\ t_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n)〉\\ \le (1-{\delta }_n)〈x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n),t_n-x_n〉\\ \le (1-{\delta }_n)\Vert x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n-P_C(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n(1-\mu )A{y}_n\Vert \Vert t_n-x_n\Vert \\ \le (1-{\delta }_n)\Vert {\lambda }_n\mu A{x}_n+{\lambda }_n(1-\mu )A{y}_n\Vert \Vert t_n-x_n\Vert \\ \le (1-{\delta }_n){\lambda }_n(\mu \Vert A{x}_n\Vert +(1-\mu )\Vert A{y}_n\Vert )(\Vert t_n-y_n\Vert +\Vert y_n-x_n\Vert ).\end{array}$$

(6)

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

$$\begin{array}{ll}\hfill {∥t_n-u∥}^2& \le {∥x_n-u∥}^2-{∥x_n-y_n∥}^2-{∥y_n-t_n∥}^2+2⟨x_n-{\lambda }_{n}A{y}_n-y_n,t_n-y_n⟩\\ \le {∥x_n-u∥}^2-{∥x_n-y_n∥}^2-{∥y_n-t_n∥}^2\\ +2{\lambda }_n\left[{\delta }_n\left(1-{\delta }_n\right)\left(\mu ∥A{x}_n∥+\left(1-\mu \right)∥A{y}_n∥\right)\left(∥t_n-y_n∥+∥y_n-x_n∥\right)\right.\\ \left.-2\left(1-{\delta }_n\right){\lambda }_n⟨\mu A{x}_n+\left(1-\mu \right)A{y}_n,t_n-y_n⟩+2{\lambda }_n\mu k∥x_n-y_n∥∥t_n-y_n∥\right]\\ \le {∥x_n-u∥}^2-{∥x_n-y_n∥}^2-{∥y_n-t_n∥}^2\\ +2{\delta }_n\left(1-{\delta }_n\right)b\left(\mu ∥A{x}_n∥+\left(1-\mu \right)∥A{y}_n∥\right)\left(∥t_n-y_n∥+∥y_n-x_n∥\right)\\ +2\left(1-{\delta }_n\right)b\left(\mu ∥A{x}_n∥+\left(1-\mu \right)∥A{y}_n∥\right)∥t_n-y_n∥+2b\mu k∥x_n-y_n∥∥t_n-y_n∥\\ ={∥x_n-u∥}^2-{∥x_n-y_n∥}^2-{∥y_n-t_n∥}^2\\ +2{\delta }_n\left(1-{\delta }_n\right)\left(b^2{\mu }^2{∥A{x}_n∥}^2+b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2+{∥t_n-y_n∥}^2+{∥y_n-x_n∥}^2\right)\\ +\left(1-{\delta }_n\right)\left(b^2{\mu }^2{∥A{x}_n∥}^2+b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2+2{∥t_n-y_n∥}^2\right)\\ +b\mu k\left({∥x_n-y_n∥}^2+{∥t_n-y_n∥}^2\right)\\ ={∥x_n-u∥}^2-{∥x_n-y_n∥}^2\left(1-2{\delta }_n\left(1-{\delta }_n\right)-bk\mu \right)\\ -{∥t_n-y_n∥}^2\left(2{\delta }_n^2-{\delta }_n-bk\mu \right)\\ +2\left(1-{\delta }_n^2\right)b^2{\mu }^2{∥A{x}_n∥}^2+2\left(1-{\delta }_n^2\right)b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2.\end{array}$$

(7)

Since $\frac{3}{4}<{\delta }_n\le 1$ and $b<\frac{3}{8k\mu }$, we have from (7) for all n ∈ ℕ,

$${∥t_n-u∥}^2\le {∥x_n-u∥}^2+4\left(1-{\delta }_n\right)b^2{\mu }^2{∥A{x}_n∥}^2+4\left(1-{\delta }_n\right)b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2.$$

(8)

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

$$\begin{array}{ll}\hfill {∥y_n-u∥}^2& ={∥\left(1-{\delta }_n\right)\left(x_n-u\right)+{\delta }_n\left(P_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right)-u\right)∥}^2\\ \le \left(1-{\delta }_n\right){∥x_n-u∥}^2+{\delta }_n{∥P_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right)-P_C\left(u\right)∥}^2\\ \le \left(1-{\delta }_n\right){∥x_n-u∥}^2+{\delta }_n{∥x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n-u∥}^2\\ =\left(1-{\delta }_n\right){∥x_n-u∥}^2+{\delta }_n\left[{∥x_n-u∥}^2-2⟨{\lambda }_n\mu A{x}_n+{\lambda }_n\left(1-\mu \right)A{y}_n,x_n-u⟩\right.\\ \left.+{∥{\lambda }_n\mu A{x}_n+{\lambda }_n\left(1-\mu \right)A{y}_n∥}^2\right]\\ =\left(1-{\delta }_n\right){∥x_n-u∥}^2+{\delta }_n\left[{∥x_n-u∥}^2-2{\lambda }_n\mu ⟨A{x}_n,x_n-u⟩-2{\lambda }_n\left(1-\mu \right)⟨A{y}_n,x_n-u⟩\right.\\ \left.+{∥{\lambda }_n\mu A{x}_n+{\lambda }_n\left(1-\mu \right)A{y}_n∥}^2\right]\\ \le \left(1-{\delta }_n\right){∥x_n-u∥}^2+{\delta }_n\left[{∥x_n-u∥}^2\right.+2{\lambda }_n\left(1-\mu \right)∥A{y}_n∥∥x_n-u∥\\ \left.+{\lambda }_n^2\mu {∥A{x}_n∥}^2+{\lambda }_n^2\left(1-\mu \right){∥A{y}_n∥}^2\right]\\ \le \left(1-{\delta }_n\right){∥x_n-u∥}^2+{\delta }_n\left[{∥x_n-u∥}^2+2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥\right.\\ \left.+b^2\mu {∥A{x}_n∥}^2+b^2\left(1-\mu \right){∥A{y}_n∥}^2\right]\\ \le {∥x_n-u∥}^2+{\delta }_n\left[2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥+b^2{\mu }^2{∥A{x}_n∥}^2+b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2\right]\\ \le {∥x_n-u∥}^2+2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥+b^2\mu {∥A{x}_n∥}^2+b^2\left(1-\mu \right){∥A{y}_n∥}^2.\end{array}$$

(9)

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

$$\begin{array}{ll}\hfill {∥z_n-u∥}^2& ={∥\left(1-{\alpha }_n-{\beta }_n\right)x_n+{\alpha }_n{y}_n+{\beta }_n{S}^n{t}_n-u∥}^2\\ \le \left(1-{\alpha }_n-{\beta }_n\right){∥x_n-u∥}^2+{\alpha }_n{∥y_n-u∥}^2+{\beta }_n{∥S^n{t}_n-u∥}^2\\ \le \left(1-{\alpha }_n-{\beta }_n\right){∥x_n-u∥}^2\\ +{\alpha }_n\left[{∥x_n-u∥}^2+2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥+b^2\mu {∥A{x}_n∥}^2+b^2\left(1-\mu \right){∥A{y}_n∥}^2\right]\\ +{\beta }_n\left[\left(1+{\gamma }_n\right){∥t_n-u∥}^2+c_n\right]\\ \le \left(1-{\alpha }_n-{\beta }_n\right){∥x_n-u∥}^2\\ +{\alpha }_n\left[{∥x_n-u∥}^2+2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥+b^2\mu {∥A{x}_n∥}^2+b^2\left(1-\mu \right){∥A{y}_n∥}^2\right]\\ +{\beta }_n\left(1+{\gamma }_n\right)\left[{∥x_n-u∥}^2+2\left(1-{\delta }_n\right)b^2{\mu }^2{∥A{x}_n∥}^2+2\left(1-{\delta }_n\right)b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2\right]\\ +{\beta }_n{c}_n\\ \le {∥x_n-u∥}^2+{\beta }_n{\gamma }_n{\Delta }_n^2+2b\left(1-\mu \right){\alpha }_n{\Delta }_n∥A{y}_n∥\\ +\left(b^2\mu {\alpha }_n+2b^2{\mu }^2{\beta }_n\left(1-{\delta }_n\right)\left(1+{\gamma }_n\right)\right){∥A{x}_n∥}^2\\ +\left(b^2\left(1-\mu \right){\alpha }_n+2b^2{\left(1-\mu \right)}^2{\beta }_n\left(1-{\delta }_n\right)\left(1+{\gamma }_n\right)\right){∥A{y}_n∥}^2\\ +{\beta }_n{c}_n.\end{array}$$

(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

$$C_k=\left\{z\in C:{∥z_k-x_k∥}^2+2⟨z_k-x_k,x_k-z⟩\le d_n∥A{y}_n∥+w_n{∥A{x}_n∥}^2+v_n{∥A{y}_n∥}^2+{\vartheta }_n\right\}.$$

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 $P_{C_k\cap Q_k}{x}_0$ is well-defined.

By the definition of x_k+1and of Q_k+1, we deduce that C _k ⋂ Q _k ⊂ Q_k+1. Hence, F(S) ⋂ VI(C, A) ⊂ Q_k+1. Exploiting Step 1 we conclude that F(S) ⋂ VI(C, A) ⊂ C_k+1⋂ Q_k+1.

Step 3. We claim that the following assertions hold:

• (d) lim_n→∞||x _n - x ₀|| exists and hence {x _n }, as well as {Δ _n }, is bounded.

• (e) lim_n→∞ ||x _n+1- x _n || = 0.

• (f) lim_n→∞ ||z _n - x _n || = 0.

Let u ∈ F(S) ⋂ VI(C, A). Since $x_{n+1}=P_{C_n\cap Q_n}{x}_0$ and u ∈ F(S) ⋂ VI(C, A) ⊂ C _n ⋂ Q _n , we conclude that

$$∥x_{n+1}-x_0∥\le ∥u-x_0∥,\forall n\ge 0.$$

(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 x_n+1∈ C _n ⋂ Q _n ⊂ Q _n . This implies that

$${∥x_{n+1}-x_n∥}^2\le {∥x_{n+1}-x_0∥}^2-{∥x_n-x_0∥}^2,\forall n\ge 0.$$

(12)

In particular, ||x_n+1- x₀|| ≥ ||x _n - x₀|| hence lim_n→∞||x _n - x₀|| exists. It follows from (12) that

$${lim}_{n\to \infty }\left(x_{n+1}-x_n\right)=0.$$

(13)

Since x_n+1∈ C _n , we obtain

$${∥z_n-x_{n+1}∥}^2\le {∥x_n-x_{n+1}∥}^2+d_n∥A{y}_n∥+w_n{∥A{x}_n∥}^2+v_n{∥A{y}_n∥}^2+{\vartheta }_n.$$

In view of lim_n→∞γ _n = 0, lim_n→∞α _n = 0, lim_n→∞δ _n = 1 and from the boundedness of {Ax _n } and {Ay _n } we infer that lim_n→∞(x_n+1- z _n ) = 0. Combining with (13) we deduce that lim_n→∞(x _n - z _n ) = 0.

Step 4. We claim that the following assertions hold:

• (g) lim_n→∞||x _n - y _n || = 0.

• (h) lim_n→∞||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

$$\begin{array}{l}{∥z_n-u∥}^2\\ ={∥\left(1-{\alpha }_n-{\beta }_n\right)x_n+{\alpha }_n{y}_n+{\beta }_n{S}^n{t}_n-u∥}^2\\ \le \left(1-{\alpha }_n-{\beta }_n\right){∥x_n-u∥}^2+{\alpha }_n{∥y_n-u∥}^2+{\beta }_n{∥S^n{t}_n-u∥}^2\\ \le \left(1-{\alpha }_n-{\beta }_n\right){∥x_n-u∥}^2\\ +{\alpha }_n\left[{∥x_n-u∥}^2+2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥+b^2\mu {∥A{x}_n∥}^2+b^2\left(1-\mu \right){∥A{y}_n∥}^2\right]\\ +{\beta }_n\left[\left(1+{\gamma }_n\right){∥t_n-u∥}^2+c_n\right]\\ \le \left(1-{\alpha }_n-{\beta }_n\right){∥x_n-u∥}^2\\ +{\alpha }_n\left[{∥x_n-u∥}^2+2b\left(1-\mu \right){\Delta }_n∥A{y}_n∥+b^2\mu {∥A{x}_n∥}^2\left(1-\mu \right){∥A{y}_n∥}^2\right]\\ +{\beta }_n\left(1+{\gamma }_n\right)\left[{∥x_n-u∥}^2-\left(1-2{\delta }_n\left(1-{\delta }_n\right){∥x_n-y_n∥}^2-bk\mu \right)\right.\\ -\left(2{\delta }_n^2-1-bk\mu \right){∥t_n-y_n∥}^2+4\left(1-{\delta }_n\right)b^2{\mu }^2{∥A{x}_n∥}^2\\ +4\left(1-{\delta }_n\right)b^2{\left(1-\mu \right)}^2{∥A{y}_n∥}^2+{\beta }_n{c}_n\\ \le {∥x_n-u∥}^2+{\beta }_n{\gamma }_n{\Delta }_n^2+{\beta }_n{c}_n\\ +2b\left(1-\mu \right){\alpha }_n{\Delta }_n∥A{y}_n∥+\left[b^2\mu {\alpha }_n+4b^2{\mu }^2{\beta }_n\left(1+{\gamma }_n\right)\left(1-{\delta }_n\right)\right]{∥A{x}_n∥}^2\\ +\left[b^2\left(1-\mu \right){\alpha }_n+4b^2{\left(1-\mu \right)}^2{\beta }_n\left(1+{\gamma }_n\right)\left(1-{\delta }_n\right)\right]{∥A{y}_n∥}^2\\ -\left[{\beta }_n\left(1+{\gamma }_n\right)\left(1-2{\delta }_n\left(1-{\delta }_n\right)-bk\mu \right)\right]{∥x_n-y_n∥}^2\\ -\left[{\beta }_n\left(1+{\gamma }_n\right)\left(2{\delta }_n^2-{\delta }_n-bk\mu \right)\right]{∥t_n-y_n∥}^2.\end{array}$$

(14)

Thus, we have

$$\begin{array}{l}{\beta }_n\left(1+{\gamma }_n\right)\left(1-2{\delta }_n\left(1-{\delta }_n\right)-bk\mu \right){∥x_n-y_n∥}^2\\ \le {∥x_n-u∥}^2-{∥z_n-u∥}^2+{\beta }_n{\gamma }_n{\Delta }_n^2+{\beta }_n{c}_n\\ +2b\left(1-\mu \right){\alpha }_n{\Delta }_n∥A{y}_n∥+\left[b^2\mu {\alpha }_n+4b^2{\mu }^2{\beta }_n\left(1+{\gamma }_n\right)\left(1-{\delta }_n\right)\right]{∥A{x}_n∥}^2\\ +\left[b^2\left(1-\mu \right){\alpha }_n+4b^2{\left(1-\mu \right)}^2{\beta }_n\left(1+{\gamma }_n\right)\left(1-{\delta }_n\right)\right]{∥A{y}_n∥}^2\\ \le \left(∥x_n-u∥+∥z_n-u∥\right)∥x_n-z_n∥+{\beta }_n{\gamma }_n{\Delta }_n^2+{\beta }_n{c}_n\\ +2b\left(1-\mu \right){\alpha }_n{\Delta }_n∥A{y}_n∥+\left[b^2\mu {\alpha }_n+4b^2{\mu }^2{\beta }_n\left(1+{\gamma }_n\right)\left(1-{\delta }_n\right)\right]{∥A{x}_n∥}^2\\ +\left[b^2\left(1-\mu \right){\alpha }_n+4b^2{\left(1-\mu \right)}^2{\beta }_n\left(1+{\gamma }_n\right)\left(1-{\delta }_n\right)\right]{∥A{y}_n∥}^2.\end{array}$$

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

$${lim}_{n\to \infty }{∥x_n-y_n∥}^2=0.$$

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

$${lim}_{n\to \infty }{∥t_n-y_n∥}^2=0.$$

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

$$∥x_n-t_n∥\le ∥x_n-y_n∥+∥y_n-t_n∥,$$

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

$$z_n-x_n=-{\alpha }_n{x}_n+{\alpha }_n{y}_n+{\beta }_n\left(s^n{t}_n-x_n\right).$$

From ||z _n - x _n || → 0, α _n → 0, lim inf_{n → 0}β _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

$$\begin{array}{ll}\hfill ∥x_n-S^n{x}_n∥& \le ∥x_n-t_n∥+∥t_n-S^n{t}_n∥+∥S^n{t}_n-S^n{x}_n∥\\ \le ∥x_n-t_n∥+∥t_n-S^n{t}_n∥+\sqrt{\left(1+{\gamma }_n\right)∥t_n-x_n∥+c_n}.\end{array}$$

Hence, ||x _n - Sⁿx _n || → 0. Since ||x _n - x_n+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 - S^mx _n || → 0 as n → ∞ for all m ≥ 1.

Step 5. We claim that ω _w (x _n ) ⊂ F(S) ⋂ VI(C, A), where

$${\omega }_w\left(x_n\right):=\left\{x\in H:x_{n_j}\to x\mathsf{\text{weakly}}\mathsf{\text{for}}\mathsf{\text{some}}\mathsf{\text{subsequence}}\left\{x_{n_j}\right\}\mathsf{\text{of}}\left\{x_n\right\}\right\}.$$

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 = P_{F(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 that${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$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 {λ _n } be a sequence in [a, b] with a > 0 and$b<\frac{3}{8k\mu }$.

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

$$\left\{\begin{array}{c}{x}_0\in Cchosenarbitrarily,\\ y_n=P_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right),\\ z_n=S^n{P}_C\left(x_n-{\lambda }_{n}A{y}_n\right),\\ C_n=\left\{z\in C:{∥z_n-z∥}^2\le {∥x_n-z∥}^2+{\vartheta }_n\right\},\\ Q_n=\left\{z\in C:⟨x_n-z,x_0-x_n⟩\ge 0\right\},\\ x_{n+1}=P_{C_n\cap Q_n}\left(x_0\right),\forall n\ge 0.\end{array}\right.$$

(15)

Then, the sequences {x _n }, {y _n } and {z _n } in (15) are well-defined and converge strongly to the same point q = P_{F(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 that${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$and 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$b<\frac{3}{8k\mu }$, and let {δ _n } be a sequence in [0, 1] such that lim_n→∞δ _n = 1 and${\delta }_n>\frac{3}{4}$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

$$\left\{\begin{array}{c}{x}_0\in Cchosenarbitrarily,\\ y_n=\left(1-{\delta }_n\right)x_n+{\delta }_n{P}_C\left(x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n\right),\\ z_n=S^n{P}_C\left(x_n-{\lambda }_{n}A{y}_n\right),\\ C_n=\left\{z\in C:{∥z_n-z∥}^2\le {∥x_n-z∥}^2+w_n{∥A{x}_n∥}^2+{\vartheta }_n\right\},\\ Q_n=\left\{z\in C:⟨x_n-z,x_0-x_n⟩\ge 0\right\},\\ x_{n+1}=P_{C_n\cap Q_n}\left(x_0\right),\forall n\ge 0.\end{array}\right.$$

(16)

Then, the sequences {x _n }, {y _n } and {z _n } in (16) are well-defined and converge strongly to the same point q = P_{F(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 that${\sum }_{n=1}^{\infty }{\gamma }_n<\infty$and 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$0<4a∕3<b<\frac{3}{8k\mu }$, 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) lim_n→∞α _n = 0;

(iii) lim inf_n→∞β _n > 0;

(iv) lim_n→∞δ _n = 1 and${\delta }_n>\frac{3}{4}$for all n ≥ 0.

Set

$$\begin{array}{ll}\hfill {\Delta }_n& =sup\left\{∥x_n-u∥:u\in F\left(S\right)\cap A^{-1}\left(0\right)\right\},\\ \hfill d_n& =2b\left(1-\mu \right){\alpha }_n{\Delta }_n,\\ \hfill w_n& =b^2\mu {\alpha }_n+4b^2{\mu }^2{\beta }_n\left(1-{\delta }_n\right)\left(1+{\gamma }_n\right),\\ \hfill v_n& =b^2\left(1-\mu \right){\alpha }_n+4b^2{\left(1-\mu \right)}^2{\beta }_n\left(1-{\delta }_n\right)\left(1+{\gamma }_n\right),and\\ \hfill {\vartheta }_n& ={\beta }_n{\gamma }_n{\Delta }_n^2+{\beta }_n{c}_n\end{array}$$

for all n ≥ 0.

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

$$\left\{\begin{array}{c}{x}_0\in Cchosenarbitrarily,\\ y_n=x_n-{\lambda }_n\mu A{x}_n-{\lambda }_n\left(1-\mu \right)A{y}_n,\\ z_n=\left(1-{\beta }_n\right)x_n-{\alpha }_n\mu A{x}_n-{\alpha }_n{\lambda }_n\left(1-\mu \right)A{y}_n+{\beta }_n{S}^n\left(x_n-\frac{{\lambda }_n}{{\delta }_n}A{y}_n\right),\\ C_n=\left\{z\in C:{∥z_n-z∥}^2\le {∥x_n-z∥}^2+dn∥A{y}_n∥+w_n{∥A{x}_n∥}^2+v_n{∥A{y}_n∥}^2+{\vartheta }_n\right\},\\ Q_n=\left\{z\in C:⟨x_n-z,x_0-x_n⟩\ge 0\right\},\\ x_{n+1}=P_{C_n\cap Q_n}\left(x_0\right),\forall n\ge 0.\end{array}\right.$$

(17)

Then, the sequences {x _n }, {y _n } and {z _n } in (17) are well-defined and converge strongly to the same point q = P_{F(S)⋂A-1(0)}(x₀).

Proof. Replace λ _n by ${\lambda }_n^{\prime }=\frac{{\lambda }_n}{{\delta }_n}$. Then, $a\le {\lambda }_n^{\prime }<\frac{4}{3}{\lambda }_n<b<\frac{3}{8k\mu }$. 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 = P_{F(S)⋂A-1(0)}(x₀).