On relaxed and contraction-proximal point algorithms in hilbert spaces

Abstract

We consider the relaxed and contraction-proximal point algorithms in Hilbert spaces. Some conditions on the parameters for guaranteeing the convergence of the algorithm are relaxed or removed. As a result, we extend some recent results of Ceng-Wu-Yao and Noor-Yao.

1. Introduction

Throughout, H denotes a real Hilbert space and A a multi-valued operator with domain D(A). We know that A is called monotone if 〈u - v, x - y〉 ≥ 0, for any u ∈ Ax, v ∈ Ay; maximal monotone if its graph G(A) = {(x,y): x ∈ D(A), y ∈ Ax} is not properly contained in the graph of any other monotone operator. Denote by S: = {x ∈ D(A): 0 ∈ Ax} the zero set and by J _c : = (I + cA)^-1 the resolvent of A. It is well known that J _c is single valued and D(J _c ) = H for any c > 0.

A fundamental problem of monotone operators is that of finding an element x so that 0 ∈ Ax. This problem is essential because it includes many concrete examples, such as convex programming and monotone variational inequalities. A successful and powerful algorithm for solving this problem is the well-known proximal point algorithm (PPA), which generates, for any initial guess, x₀ ∈ H, an iterative sequence as

$$x_{n+1}=J_{c_n}\left(x_n+e_n\right),$$

(1.1)

where (c _n ) is a positive real sequence and (e _n ) is the error sequence (see [1]). To guarantee the convergence of PPA, there are two kinds of accuracy criterion posed on the error sequence:

$$\mathsf{\text{(I)}}∥e_n∥\le {\epsilon }_n,\sum _{n=0}^{\infty }{\epsilon }_n<\infty \mathsf{\text{or}}$$

$$\mathsf{\text{(II)}}∥e_n∥\le {\eta }_n∥{\tilde{x}}_n-x_n∥,\sum _{n=0}^{\infty }{\eta }_n<\infty ,$$

where ${\tilde{x}}_n=J_{c_n}\left(x_n+e_n\right).$ In 2001, Han and He [2] proved that in finite dimensional Hilbert space criterion (II) can be replaced by

$$\mathsf{\text{(II’)}}∥e_n∥\le {\eta }_n∥{\tilde{x}}_n-x_n∥,\sum _{n=0}^{\infty }{\eta }_n^2<\infty .$$

The infinite version was obtained by Marino and Xu [3].

There are various generations or modifications on the PPA. Among them Eckstein and Bertsekas [4] proposed the relaxed proximal point algorithm (RPPA):

$$x_{n+1}=\left(1-{\rho }_n\right)x_n+{\rho }_n{J}_{c_n}\left(x_n\right)+e_n,$$

(1.2)

where (ρ _n ) ⊂ (0, 2) is a relaxation factor. The weak convergence of (1.2) is guaranteed provided that (e _n ) satisfies criterion (I),

$$c_n\ge \bar{c}>0,0<\delta \le {\rho }_n\le 2-\delta .$$

(1.3)

On the other hand, since the PPA does not necessarily converge strongly (see [5]), many authors have conducted worthwhile studies on modifying the PPA so that the strong convergence is guaranteed (see, for instance, [6–8]). In particular, Marino and Xu [3] proposed the contraction-proximal point algorithm (CPPA):

$$x_{n+1}={\lambda }_{n}u+\left(1-{\lambda }_n\right)J_{c_n}\left(x_n\right)+e_n,$$

(1.4)

where the parameters above satisfy (i) lim _n λ _n = 0, Σ _n λ _n = ∞; (ii) either Σ _n |λ _n +1- λ _n | < ∞; or lim _n λ_n/λ _n +1 = 1; (iii) $0<\underset{}{c}\le c_n\le \bar{c}<\infty ,{\sum }_n|c_{n+1}-c_n|<\infty ;$ (iv) Σ _n ||e _n || < ∞. Under these assumptions, the CPPA converges strongly to P _S (u), the projection of u onto S.

In this article, we shall focus on the RPPA and CPPA. We note that the resolvent is in fact the arithmetic mean of the identity and a nonexpansive operator. By using this fact, we relax or remove some sufficient conditions to guarantee the convergence of the algorithms. As a result, we extend and improve some recent results on the PPA.

2. Some lemmas

We know that an operator T : H → H is called (i) nonexpansive if ||Tx - Ty|| ≤ || x - y|| ∀x,y ∈ H; and (ii) firmly nonexpansive if 〈Tx - Ty, x - y〉 ≥ ||Tx - Ty||² ∀x,y ∈ H. Denote by Fix(T) = {x ∈ H : x = Tx} the fixed point set of T. It is well known that firmly nonexpansive operators have the following properties.

Lemma 1 (Goebel-Kirk [9]). Let T be firmly nonexpansive. Then (1) 2T - I is nonexpansive; (2) 〈Tx - x, Tx - z〉 ≤ 0 for all x ∈ H and for all z ∈ H Fix(T).

It is well known that J _c is firmly nonexpansive and consequently nonexpansive; moreover, S = Fix (J _c ). Since the fixed point set of nonexpansive operators is closed convex, the projection P _s onto the solution set S is well defined whenever S ≠ ∅. Hereafter, we assume that S is nonempty. The following lemmas play an important role in our convergence analysis.

Lemma 2 (resolvent identity [3]). Let c, t > 0. Then for any x ∈ H,

$$J_{c}x=J_t\left(\frac{t}{c}x+\left(1-\frac{t}{c}\right)J_{c}x\right).$$

Lemma 3 ([10]). Let (ρ _n ) be real sequence satisfying

$$\mathrm{0\; <}\underset{n\to \infty }{{\displaystyle liminf}}{\rho }_n\le \underset{n\to \infty }{{\displaystyle limsup}}{\rho }_n\mathrm{<\; 1.}$$

Assume that (x _n ) and (y _n ) are bounded sequences in H satisfying x _n +1 = (1 - ρ _n )x _n + ρ _n y _n . If

$$\underset{n\to \infty }{limsup}\left(∥y_{n+1}-y_n∥-∥x_{n+1}-x_n∥\right)\le 0,$$

then lim _n →∞||x _n -y _n || = 0.

Lemma 4 For r, s, > 0, let T _r = 2J _r - I. Then for any x ∈ H,

$$∥T_{s}x-T_{r}x∥\le \left|1-\frac{s}{r}\right|∥x-T_{r}x∥.$$

(2.1)

Proof. Using the resolvent identity, we have

$$\begin{array}{lll}\hfill ∥T_{s}x-T_{r}x∥& =2∥J_{s}x-J_s\left(\frac{s}{r}x+\left(1-\frac{s}{r}\right)J_{r}x\right)∥& \hfill \\ \le 2∥x-\left(\frac{s}{r}x+\left(1-\frac{s}{r}\right)J_{r}x\right)∥& \hfill \\ =2\left|1-\frac{s}{r}\right|∥x-J_{r}x∥& \hfill \\ =\left|1-\frac{s}{r}\right|∥x-T_{r}x∥,& \hfill \\ \hfill \end{array}$$

where the inequality uses the nonexpansive property of the resolvent.

Lemma 5 ([11]). Let (ε _n ) and (s _n ) be positive real sequences. Assume that Σ _n ε _n < ∞. If either (i) s _n+ 1≤ (1 + ε _n )s _n , or (ii) s _n+ 1≤ ε _n , then the limit of (s _n ) exists.

3. The relaxed proximal point algorithm

Under criterion (II'), Ceng et al. [12] considered another type, RPPA:

$$\left\{\begin{array}{c}\hfill {\tilde{x}}_n=J_{c_n}\left(x_n+e_n\right),\hfill \\ \hfill x_{n+1}=\left(1-{\rho }_n\right)x_n+{\rho }_n{\tilde{x}}_n,\hfill \end{array}\right.$$

(3.1)

and proved the weak convergence of (3.1) under the assumptions:

$$c_n\ge \bar{c}>0,0<\delta \le {\rho }_n\le 1.$$

We note that the choice of (ρ _n ) excludes the case whenever ρ _n ∈ (1,2), the overrelaxation. The overrelaxation, however, may indeed speed up the convergence of the algorithm (see [13]). Below, we shall improve their conditions on the relaxation factor from 0 < δ ≤ ρ _n ≤ 1 to 0 < δ ≤ ρ _n ≤ 2 - δ.

Theorem 6. Assume that the following conditions hold:

(a) $c_n\ge \bar{c}>0;$

(b) 0 < δ ≤ ρ _n ≤ 2 - δ;

(c) ${\sum }_n∥e_n∥\le {\eta }_n∥{\tilde{x}}_n-x_n∥,{\sum }_n{\eta }_n^2<\infty .$

Then the sequence generated by (3.1) converges weakly to a point in S.

Proof. The key point of our proof is to show lim _n s _n = 0, where $s_n=∥x_n-J_{c_n}\left(x_n\right)∥.$ To see this, let z ∈ S be fixed. Since $J_{c_n}$ is firmly nonexpansive and $z\in \mathsf{\text{Fix}}\left(J_{c_n}\right),$ applying Lemma 1 yields $⟨{\tilde{x}}_n-z,{\tilde{x}}_n-x_n-e_n⟩\le 0.$ This together with (3.1) enables us to get

$$\begin{array}{lll}\hfill {∥x_{n+1}-z∥}^2-{∥x_n-z∥}^2& ={∥\left(x_n-z\right)+{\rho }_n\left({\tilde{x}}_n-x_n\right)∥}^2-{∥x_n-z∥}^2& \hfill \\ =2{\rho }_n⟨x_n-z,{\tilde{x}}_n-x_n⟩+{\rho }_n^2{∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ =2{\rho }_n⟨{\tilde{x}}_n-z,{\tilde{x}}_n-x_n⟩-{\rho }_n\left(2-{\rho }_n\right){∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ \le 2{\rho }_n⟨{\tilde{x}}_n-z,e_n⟩-{\rho }_n\left(2-{\rho }_n\right){∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ =2{\rho }_n⟨{\tilde{x}}_n-x_n,e_n⟩+2{\rho }_n⟨x_n-z,e_n⟩-{\rho }_n\left(2-{\rho }_n\right){∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ \le 2{\rho }_n∥e_n∥∥{\tilde{x}}_n-x_n∥+2{\rho }_n∥e_n∥∥x_n-z∥-{\rho }_n\left(2-{\rho }_n\right){∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ \le 2{\rho }_n{\eta }_n{∥{\tilde{x}}_n-x_n∥}^2+2{\rho }_n{\eta }_n∥{\tilde{x}}_n-x_n∥∥x_n-z∥& \hfill \\ -{\rho }_n\left(2-{\rho }_n\right){∥{\tilde{x}}_n-x_n∥}^2.& \hfill \\ \hfill \end{array}$$

Using the basic inequality 2ab ≤ a² / ε + εb² (a,b ∈ ℝ, ε > 0), we arrive at

$$\begin{array}{lll}\hfill 2{\rho }_n{\eta }_n∥x_n-z∥∥{\tilde{x}}_n-x_n∥& \le \frac{2{\rho }_n}{2-{\rho }_n}{\left({\eta }_n∥x_n-z∥\right)}^2+\frac{2-{\rho }_n}{2{\rho }_n}{\left({\rho }_n∥{\tilde{x}}_n-x_n∥\right)}^2& \hfill \\ =\frac{2{\rho }_n{\eta }_n^2}{2-{\rho }_n}{∥x_n-z∥}^2+\frac{{\rho }_n\left(2-{\rho }_n\right)}{2}{∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ \le \frac{2\left(2-\delta \right){\eta }_n^2}{\delta }{∥x_n-z∥}^2+\frac{{\rho }_n\left(2-{\rho }_n\right)}{2}{∥{\tilde{x}}_n-x_n∥}^2& \hfill \\ ={\epsilon }_n{∥x_n-z∥}^2+\frac{{\rho }_n\left(2-{\rho }_n\right)}{2}{∥{\tilde{x}}_n-x_n∥}^2,& \hfill \\ \hfill \end{array}$$

where ${\epsilon }_n=2\left(2-\delta \right){\eta }_n^2∕\delta$ is a summable sequence. Substituting this into above yields

$${∥x_{n+1}-z∥}^2\le \left(1+{\epsilon }_n\right){∥x_n-z∥}^2-\frac{{\rho }_n\left(2-{\rho }_n-4{\eta }_n\right)}{2}{∥{\tilde{x}}_n-x_n∥}^2.$$

Since by Lemma 5 the limit of ||x _n - z ||² exists and lim inf _n ρ _n (2 - ρ _n -4η _n ) ≥ δ (2 - δ), this implies that $∥{\tilde{x}}_n-x_n∥\to 0.$ On the other hand, we note that for all n ∈ ℕ

$$s_n\le \left(1+{\eta }_n\right)∥x_n-{\tilde{x}}_n∥\to 0;$$

therefore, lim _n s _n = 0. The rest proof is similar to that of [12, Theorem 3.1].

We now turn to the RPPA (1.2). Under the criterion (I), the assumptions on relaxation factors can be relaxed to Σρ _n (2 - ρ _n ) = ∞ (see [3, Theorem 3.3]). Since the proof there is very technical, we wang to restate this result with a simple proof.

Theorem 7. Assume that the following conditions hold:

(a) Σ _n ||e _n || < ∞;

(b) Σ _n ρ_n(2 - ρ _n ) = ∞;

(c) $0<\bar{c}\le c_n\le \tilde{c}<\infty ;$

(d) Σ _n |c _n +1- c _n | < ∞.

Then the sequence generated by (1.2) converges weakly to a point in S.

Proof. The key step is to show lim _n s _n = 0, where $s_n=∥x_n-J_{c_n}\left(x_n\right)∥.$ It has been shown that Σ _n ρ _n (2 - ρ _n )s _n < ∞ (see [3, Lemma 3.2]). Therefore, it remains to show that lim _n s _n exists. By letting T _n = 2J _n - I, we rewrite (2) as

$$x_{n+1}=\left(1-\frac{{\rho }_n}{2}\right)x_n+\frac{{\rho }_n}{2}{T}_n{x}_n+e_n.$$

In view of Lemma 4 and condition (c),

$$\begin{array}{ll}\hfill \parallel T_{n+1}{x}_{n+1}-T_n{x}_n\parallel & \le \parallel T_{n+1}{x}_{n+1}-T_{n+1}{x}_n\parallel +\parallel T_{n+1}{x}_n-T_n{x}_n\parallel \\ \le \parallel x_n-x_{n+1}\parallel +\parallel T_{n+1}{x}_n-T_n{x}_n\parallel \\ \le \parallel x_n-x_{n+1}\parallel +\left|1-\frac{c_{n+1}}{c_n}\right|\parallel T_n{x}_n-x_n\parallel \\ \le \parallel x_n-x_{n+1}\parallel +\frac{|c_{n+1}-c_n|}{\bar{c}}\parallel T_n{x}_n-x_n\parallel \\ \le \parallel x_n-x_{n+1}\parallel +M|c_{n+1}-c_n|,\end{array}$$

where M > 0 is a suitable number. Consequently,

$$\begin{array}{lll}\hfill ∥x_{n+1}-T_{n+1}{x}_{n+1}∥& =∥\left(1-\frac{{\rho }_n}{2}\right)x_n+\frac{{\rho }_n}{2}{T}_n{x}_n+e_n-T_{n+1}{x}_{n+1}∥& \hfill \\ =∥\left(1-\frac{{\rho }_n}{2}\right)\left(x_n-T_n{x}_n\right)+\left(T_n{x}_n-T_{n+1}{x}_{n+1}\right)+e_n∥& \hfill \\ \le \left(1-\frac{{\rho }_n}{2}\right)∥x_n-T_n{x}_n∥+∥T_n{x}_n-T_{n+1}{x}_{n+1}∥+∥e_n∥& \hfill \\ \le \left(1-\frac{{\rho }_n}{2}\right)∥x_n-T_n{x}_n∥+∥x_n-x_{n+1}∥& \hfill \\ +M|c_{n+1}-c_n|+∥e_n∥& \hfill \\ =\left(1-\frac{{\rho }_n}{2}\right)∥x_n-T_n{x}_n∥+∥\frac{{\rho }_n}{2}\left(x_n-T_n{x}_n\right)+e_n∥& \hfill \\ +M|c_{n+1}-c_n|+∥e_n∥& \hfill \\ \le ∥x_n-T_n{x}_n∥+M|c_{n+1}-c_n|+2∥e_n∥.& \hfill \\ \hfill \end{array}$$

Using s _n = || x _n - T _n x _n ||/2, we therefore arrive at

$$s_{n+1}\le s_n+{\sigma }_n,$$

where σ _n = 2M |c _n +1- c _n | + 4||e _n || satisfying Σ _n σ _n < ∞ (due to (a) and (d)). By Lemma 5, we finally conclude that lim _n s _n = 0.

4. The contraction-proximal point algorithm

Recently, Yao and Noor [14] extended the CPPA to the following form:

$$x_{n+1}={\lambda }_{n}u+r_n{x}_n+{\delta }_n{J}_{c_n}\left(x_n\right)+e_n,$$

(4.1)

where (λ _n ),(r _n ),(δ _n )⊆ (0,1) and λ _n + r _n + δ _n = 1. They proved the strong convergence of the algorithm provided that (i) $c_n\ge \bar{c}>0,{lim}_n|c_{n+1}-c_n|=0;$ (ii) 0 < lim inf _n r _n ≤ lim sup _n r _n < 1; and (iii) Σ _n ||e _n || < ∞. Also, they claimed that their algorithm includes the CPPA as a special case. This is, however, not the case, because condition (ii) excludes the special case r _n ≡ 0. To overcome this drawback, we shall show the same result by replacing condition (ii) with the weak condition:

$$\underset{n\to \infty }{{\displaystyle limsup}}{r}_n\mathrm{<1}\iff \underset{n\to \infty }{{\displaystyle liminf}}{\delta }_n\mathrm{>0.}$$

In this situation, the CPPA is evidently a special case of algorithm (4.1). The idea of the following proof is followed by the second author [15].

Theorem 8. Let be (λ _n ), (r _n ) and (δ _n ) be parameters in (4.1). Assume that the following conditions hold:

(a) lim _n λ _n = 0, Σ _n λ _n = ∞;

(b) lim sup _n r _n < 1 ⇔ lim inf _n δ _n > 0;

(c) $c_n\ge \bar{c}>0,|c_{n+1}-c_n|\to 0;$

(d) Σ _n ||e _n || < ∞.

Then the sequence generated by (4.1) converges strongly to P _S (u).

Proof. All we need to do is to prove ||x _n +1- x _n || → 0, since the rest proof is similar to that of [14, Theorem 3.3]. To this end, set $J_n=J_{c_n}$ and T _n = 2J _n - I. It then follows from (4.1) that

$$\begin{array}{lll}\hfill x_{n+1}& ={\lambda }_{n}u+r_n{x}_n+\frac{{\delta }_n}{2}\left(I+T_n\right)x_n+e_n& \hfill \\ =\left(r_n+\frac{{\delta }_n}{2}\right)x_n+{\lambda }_{n}u+\frac{{\delta }_n}{2}{T}_n{x}_n+e_n.& \hfill \\ \hfill \end{array}$$

Let ρ _n = λ _n + (δ _n /2). Then the algorithm has the form:

$$x_{n+1}=\left(1-{\rho }_n\right)x_n+{\rho }_n{y}_n,$$

(4.2)

where y _n = (2λ _n u + δ _n T _n x _n + 2e _n )/2ρ _n. Using nonexpansiveness of T _n and Lemma 4, we have

$$\begin{array}{lll}\hfill ∥T_{n+1}{x}_{n+1}-T_n{x}_n∥& \le ∥T_{n+1}{x}_{n+1}-T_{n+1}{x}_n∥+∥T_{n+1}{x}_n-T_n{x}_n∥& \hfill \\ \le ∥x_{n+1}-x_n∥+\left|1-\frac{c_{n+1}}{c_n}\right|∥T_n{x}_n-x_n∥& \hfill \\ \le ∥x_{n+1}-x_n∥+\frac{|c_n-c_{n+1}|}{\bar{c}}∥T_n{x}_n-x_n∥.& \hfill \\ \hfill \end{array}$$

(4.3)

On the other hand, it follows from the definition of y _n that

$$\begin{array}{lll}\hfill ∥y_{n+1}-y_n∥& =∥\frac{1}{2{\rho }_{n+1}}\left(2{\lambda }_{n+1}u+{\delta }_{n+1}{T}_{n+1}{x}_{n+1}+2e_{n+1}\right)& \hfill \\ -\frac{1}{2{\rho }_n}\left(2{\lambda }_{n}u+{\delta }_n{T}_n{x}_n+2e_n\right)∥& \hfill \\ \le \left|\frac{{\lambda }_{n+1}}{{\rho }_{n+1}}-\frac{{\lambda }_n}{{\rho }_n}\right|∥u∥+\frac{∥e_{n+1}∥}{{\rho }_{n+1}}+\frac{∥e_n∥}{{\rho }_n}& \hfill \\ +∥\frac{{\delta }_{n+1}}{2{\rho }_{n+1}}{T}_{n+1}{x}_{n+1}-\frac{{\delta }_n}{2{\rho }_n}{T}_n{x}_n∥& \hfill \\ \le \left|\frac{{\lambda }_{n+1}}{{\rho }_{n+1}}-\frac{{\lambda }_n}{{\rho }_n}\right|∥u∥+\frac{∥e_{n+1}∥}{{\rho }_{n+1}}+\frac{∥e_n∥}{{\rho }_n}& \hfill \\ +\left|\frac{{\delta }_{n+1}}{2{\rho }_{n+1}}-\frac{{\delta }_n}{2{\rho }_n}\right|∥T_{n+1}{x}_{n+1}∥& \hfill \\ +\frac{{\delta }_n}{2{\rho }_n}∥T_{n+1}{x}_{n+1}-T_n{x}_n∥.& \hfill \\ \hfill \end{array}$$

(4.4)

Since (x _n ) is bounded and T _n is nonexpansive, we can find M > 0 so that (||T _n x _n || + ||x _n || + ||u||) ≤ M for all n ∈ ℕ Adding (4.3) and (4.4) and noting δ _n ≤ 2ρ _n yield

$$\begin{array}{lll}\hfill ∥y_{n+1}-y_n∥& \le \left|\frac{{\lambda }_{n+1}}{{\rho }_{n+1}}-\frac{{\lambda }_n}{{\rho }_n}\right|∥u∥+\frac{∥e_{n+1}∥}{{\rho }_{n+1}}+\frac{∥e_n∥}{{\rho }_n}& \hfill \\ +\left|\frac{{\delta }_{n+1}}{2{\rho }_{n+1}}-\frac{{\delta }_n}{2{\rho }_n}\right|∥T_{n+1}{x}_{n+1}∥& \hfill \\ +∥x_{n+1}-x_n∥+\frac{|c_n-c_{n+1}|}{\bar{c}}∥T_n{x}_n-x_n∥& \hfill \\ \le ∥x_{n+1}-x_n∥+M\left(\left|\frac{{\lambda }_{n+1}}{{\rho }_{n+1}}-\frac{{\lambda }_n}{{\rho }_n}\right|\right.+\frac{∥e_{n+1}∥}{{\rho }_{n+1}}& \hfill \\ \left.+\frac{∥e_n∥}{{\rho }_n}+\left|\frac{{\delta }_{n+1}}{2{\rho }_{n+1}}-\frac{{\delta }_n}{2{\rho }_n}\right|+\frac{|c_n-c_{n+1}|}{\bar{c}}\right).& \hfill \\ \hfill \end{array}$$

With the knowledge that ||e _n ||→ 0 and

$$\frac{{\lambda }_n}{{\rho }_n}=\frac{2{\lambda }_n}{2{\lambda }_n+{\delta }_n}\to 0,\frac{{\delta }_n}{2{\rho }_n}=\frac{{\delta }_n}{2{\lambda }_n+{\delta }_n}\to 1,$$

we therefore deduce from (b) and (c) that

$$\begin{array}{l}\underset{n\to \infty }{{\displaystyle limsup}}\mathrm{(}\Vert y_{n+1}-y_n\Vert -\Vert x_{n+1}-x_n\Vert \mathrm{)}\\ \le \underset{n\to \infty }{{\displaystyle limsup}}M(\left|\frac{{\lambda }_{n+1}}{{\rho }_{n+1}}-\frac{{\lambda }_n}{{\rho }_n}\right|+\frac{\Vert e_{n+1}\Vert }{{\rho }_{n+1}}+\frac{\Vert e_n\Vert }{{\rho }_n}\\ +\left|\frac{{\delta }_{n+1}}{2{\rho }_{n+1}}-\frac{{\delta }_n}{2{\rho }_n}\right|+\frac{\mathrm{|}{c}_n-c_{n+1}\mathrm{|}}{\overline{c}})\to 0.\end{array}$$

Note that lim inf _n ρ _n = lim inf _n (δ _n /2)> 0 and lim sup _n ρ _n = lim sup _n (δ _n /2) ≤ 1/2 < 1. On the other hand, it is easy to check that (x _n ) is bounded and so is (y _n ) We therefore apply Lemma 3 to yield lim _n ||x _n - y _n || = 0. By means of (4.2), we finally have

$$∥x_{n+1}-x_n∥={\rho }_n∥x_n-y_n∥\to ,$$

and thus the required result at once follows.

As a corollary, we improve [3, Theorem 4.1] as follows.

Theorem 9. Assume that the following conditions hold:

(a) lim _n λ _n = 0, Σ _n λ _n = ∞;

(b) $c_n\ge \bar{c}>0,|c_{n+1}-c_n|\to 0;$

(c) Σ _n ||e _n || < ∞.

Then the sequence generated by (1.4) converges strongly to P _S (u).