On relaxed and contraction-proximal point algorithms in hilbert spaces

Shuyu Wang; Fenghui Wang

doi:10.1186/1029-242X-2011-41

Research
Open Access

On relaxed and contraction-proximal point algorithms in hilbert spaces

Shuyu Wang¹Email author and
Fenghui Wang¹

Journal of Inequalities and Applications20112011:41

https://doi.org/10.1186/1029-242X-2011-41

Received: 13 March 2011
Accepted: 25 August 2011
Published: 25 August 2011

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.

Keywords

maximal monotone operator
proximal point algorithm
firmly nonexpansive operator

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}} (x_{n} + e_{n}),

(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:

(I) ∥e_{n}∥ \leq ε_{n}, \sum_{n = 0}^{\infty} ε_{n} < \infty or

(II) ∥e_{n}∥ \leq η_{n} ∥{\tilde{x}}_{n} - x_{n}∥, \sum_{n = 0}^{\infty} η_{n} < \infty,

where

{\tilde{x}}_{n} = J_{c_{n}} (x_{n} + e_{n}) .

In 2001, Han and He [2] proved that in finite dimensional Hilbert space criterion (II) can be replaced by

(II’) ∥e_{n}∥ \leq η_{n} ∥{\tilde{x}}_{n} - x_{n}∥, \sum_{n = 0}^{\infty} η_{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} = (1 - ρ_{n}) x_{n} + ρ_{n} J_{c_{n}} (x_{n}) + 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} \geq \bar{c} > 0, 0 < δ \leq ρ_{n} \leq 2 - δ .

(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} = λ_{n} u + (1 - λ_{n}) J_{c_{n}} (x_{n}) + 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) $0 < \underset{}{c} \leq c_{n} \leq \bar{c} < \infty, \sum_{n} | c_{n + 1} - c_{n} | < \infty;$ (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} (\frac{t}{c} x + (1 - \frac{t}{c}) J_{c} x) .

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

0 < \underset{n \to \infty}{l i m i n f} ρ_{n} \leq \underset{n \to \infty}{l i m s u p} ρ_{n} < 1.

Assume that (x_n ) and (y_n ) are bounded sequences in H satisfying x_n +1 = (1 - ρ_n )x_n + ρ_ny_n . If

\underset{n \to \infty}{limsup} (∥y_{n + 1} - y_{n}∥ - ∥x_{n + 1} - x_{n}∥) \leq 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∥ \leq |1 - \frac{s}{r}| ∥x - T_{r} x∥ .

(2.1)

Proof. Using the resolvent identity, we have

\begin{align} ∥T_{s} x - T_{r} x∥ & = 2 ∥J_{s} x - J_{s} (\frac{s}{r} x + (1 - \frac{s}{r}) J_{r} x)∥ \\ \leq 2 ∥x - (\frac{s}{r} x + (1 - \frac{s}{r}) J_{r} x)∥ \\ = 2 |1 - \frac{s}{r}| ∥x - J_{r} x∥ \\ = |1 - \frac{s}{r}| ∥x - T_{r} x∥, \end{align}

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:

\{\begin{matrix} {\tilde{x}}_{n} = J_{c_{n}} (x_{n} + e_{n}), \\ x_{n + 1} = (1 - ρ_{n}) x_{n} + ρ_{n} {\tilde{x}}_{n}, \end{matrix}

(3.1)

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

c_{n} \geq \bar{c} > 0, 0 < δ \leq ρ_{n} \leq 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) $c_{n} \geq \bar{c} > 0;$

(b) 0 < δ ≤ ρ_n ≤ 2 - δ;

(c) $\sum_{n} ∥e_{n}∥ \leq η_{n} ∥{\tilde{x}}_{n} - x_{n}∥, \sum_{n} η_{n}^{2} < \infty .$

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}} (x_{n})∥ .

To see this, let z ∈ S be fixed. Since

J_{c_{n}}

is firmly nonexpansive and

z \in Fix (J_{c_{n}}),

applying Lemma 1 yields

⟨ {\tilde{x}}_{n} - z, {\tilde{x}}_{n} - x_{n} - e_{n} ⟩ \leq 0 .

This together with (3.1) enables us to get

\begin{align} {∥x_{n + 1} - z∥}^{2} - {∥x_{n} - z∥}^{2} & = {∥(x_{n} - z) + ρ_{n} ({\tilde{x}}_{n} - x_{n})∥}^{2} - {∥x_{n} - z∥}^{2} \\ = 2 ρ_{n} ⟨ x_{n} - z, {\tilde{x}}_{n} - x_{n} ⟩ + ρ_{n}^{2} {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ = 2 ρ_{n} ⟨ {\tilde{x}}_{n} - z, {\tilde{x}}_{n} - x_{n} ⟩ - ρ_{n} (2 - ρ_{n}) {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ \leq 2 ρ_{n} ⟨ {\tilde{x}}_{n} - z, e_{n} ⟩ - ρ_{n} (2 - ρ_{n}) {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ = 2 ρ_{n} ⟨ {\tilde{x}}_{n} - x_{n}, e_{n} ⟩ + 2 ρ_{n} ⟨ x_{n} - z, e_{n} ⟩ - ρ_{n} (2 - ρ_{n}) {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ \leq 2 ρ_{n} ∥e_{n}∥ ∥{\tilde{x}}_{n} - x_{n}∥ + 2 ρ_{n} ∥e_{n}∥ ∥x_{n} - z∥ - ρ_{n} (2 - ρ_{n}) {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ \leq 2 ρ_{n} η_{n} {∥{\tilde{x}}_{n} - x_{n}∥}^{2} + 2 ρ_{n} η_{n} ∥{\tilde{x}}_{n} - x_{n}∥ ∥x_{n} - z∥ \\ - ρ_{n} (2 - ρ_{n}) {∥{\tilde{x}}_{n} - x_{n}∥}^{2} . \end{align}

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

\begin{align} 2 ρ_{n} η_{n} ∥x_{n} - z∥ ∥{\tilde{x}}_{n} - x_{n}∥ & \leq \frac{2 ρ_{n}}{2 - ρ_{n}} {(η_{n} ∥x_{n} - z∥)}^{2} + \frac{2 - ρ_{n}}{2 ρ_{n}} {(ρ_{n} ∥{\tilde{x}}_{n} - x_{n}∥)}^{2} \\ = \frac{2 ρ_{n} η_{n}^{2}}{2 - ρ_{n}} {∥x_{n} - z∥}^{2} + \frac{ρ_{n} (2 - ρ_{n})}{2} {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ \leq \frac{2 (2 - δ) η_{n}^{2}}{δ} {∥x_{n} - z∥}^{2} + \frac{ρ_{n} (2 - ρ_{n})}{2} {∥{\tilde{x}}_{n} - x_{n}∥}^{2} \\ = ε_{n} {∥x_{n} - z∥}^{2} + \frac{ρ_{n} (2 - ρ_{n})}{2} {∥{\tilde{x}}_{n} - x_{n}∥}^{2}, \end{align}

where

ε_{n} = 2 (2 - δ) η_{n}^{2} ∕ δ

is a summable sequence. Substituting this into above yields

{∥x_{n + 1} - z∥}^{2} \leq (1 + ε_{n}) {∥x_{n} - z∥}^{2} - \frac{ρ_{n} (2 - ρ_{n} - 4 η_{n})}{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} \leq (1 + η_{n}) ∥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) $0 < \bar{c} \leq c_{n} \leq \tilde{c} < \infty;$

(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}} (x_{n})∥ .

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} = (1 - \frac{ρ_{n}}{2}) x_{n} + \frac{ρ_{n}}{2} T_{n} x_{n} + e_{n} .

In view of Lemma 4 and condition (c),

\begin{array}{l} ∥ T_{n + 1} x_{n + 1} - T_{n} x_{n} ∥ & \leq ∥ T_{n + 1} x_{n + 1} - T_{n + 1} x_{n} ∥ + ∥ T_{n + 1} x_{n} - T_{n} x_{n} ∥ \\ \leq ∥ x_{n} - x_{n + 1} ∥ + ∥ T_{n + 1} x_{n} - T_{n} x_{n} ∥ \\ \leq ∥ x_{n} - x_{n + 1} ∥ + |1 - \frac{c_{n + 1}}{c_{n}}| ∥ T_{n} x_{n} - x_{n} ∥ \\ \leq ∥ x_{n} - x_{n + 1} ∥ + \frac{| c_{n + 1} - c_{n} |}{\bar{c}} ∥ T_{n} x_{n} - x_{n} ∥ \\ \leq ∥ x_{n} - x_{n + 1} ∥ + M | c_{n + 1} - c_{n} |, \end{array}

where M > 0 is a suitable number. Consequently,

\begin{align} ∥x_{n + 1} - T_{n + 1} x_{n + 1}∥ & = ∥(1 - \frac{ρ_{n}}{2}) x_{n} + \frac{ρ_{n}}{2} T_{n} x_{n} + e_{n} - T_{n + 1} x_{n + 1}∥ \\ = ∥(1 - \frac{ρ_{n}}{2}) (x_{n} - T_{n} x_{n}) + (T_{n} x_{n} - T_{n + 1} x_{n + 1}) + e_{n}∥ \\ \leq (1 - \frac{ρ_{n}}{2}) ∥x_{n} - T_{n} x_{n}∥ + ∥T_{n} x_{n} - T_{n + 1} x_{n + 1}∥ + ∥e_{n}∥ \\ \leq (1 - \frac{ρ_{n}}{2}) ∥x_{n} - T_{n} x_{n}∥ + ∥x_{n} - x_{n + 1}∥ \\ + M | c_{n + 1} - c_{n} | + ∥e_{n}∥ \\ = (1 - \frac{ρ_{n}}{2}) ∥x_{n} - T_{n} x_{n}∥ + ∥\frac{ρ_{n}}{2} (x_{n} - T_{n} x_{n}) + e_{n}∥ \\ + M | c_{n + 1} - c_{n} | + ∥e_{n}∥ \\ \leq ∥x_{n} - T_{n} x_{n}∥ + M | c_{n + 1} - c_{n} | + 2 ∥e_{n}∥ . \end{align}

Using s_n = || x_n - T_nx_n ||/2, we therefore arrive at

s_{n + 1} \leq s_{n} + σ_{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} = λ_{n} u + r_{n} x_{n} + δ_{n} J_{c_{n}} (x_{n}) + 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} \geq \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}{l i m s u p} r_{n} <1 \Leftrightarrow \underset{n \to \infty}{l i m i n f} δ_{n} >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) $c_{n} \geq \bar{c} > 0, | c_{n + 1} - c_{n} | \to 0;$

(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{align} x_{n + 1} & = λ_{n} u + r_{n} x_{n} + \frac{δ_{n}}{2} (I + T_{n}) x_{n} + e_{n} \\ = (r_{n} + \frac{δ_{n}}{2}) x_{n} + λ_{n} u + \frac{δ_{n}}{2} T_{n} x_{n} + e_{n} . \end{align}

Let ρ_n = λ_n + (δ_n /2). Then the algorithm has the form:

x_{n + 1} = (1 - ρ_{n}) x_{n} + ρ_{n} y_{n},

(4.2)

where y_n = (2λ_nu + δ_nT_nx_n + 2e_n )/2ρ_n. Using nonexpansiveness of T_n and Lemma 4, we have

\begin{align} ∥T_{n + 1} x_{n + 1} - T_{n} x_{n}∥ & \leq ∥T_{n + 1} x_{n + 1} - T_{n + 1} x_{n}∥ + ∥T_{n + 1} x_{n} - T_{n} x_{n}∥ \\ \leq ∥x_{n + 1} - x_{n}∥ + |1 - \frac{c_{n + 1}}{c_{n}}| ∥T_{n} x_{n} - x_{n}∥ \\ \leq ∥x_{n + 1} - x_{n}∥ + \frac{| c_{n} - c_{n + 1} |}{\bar{c}} ∥T_{n} x_{n} - x_{n}∥ . \end{align}

(4.3)

On the other hand, it follows from the definition of y_n that

\begin{align} ∥y_{n + 1} - y_{n}∥ & = ∥\frac{1}{2 ρ_{n + 1}} (2 λ_{n + 1} u + δ_{n + 1} T_{n + 1} x_{n + 1} + 2 e_{n + 1}) \\ - \frac{1}{2 ρ_{n}} (2 λ_{n} u + δ_{n} T_{n} x_{n} + 2 e_{n})∥ \\ \leq |\frac{λ_{n + 1}}{ρ_{n + 1}} - \frac{λ_{n}}{ρ_{n}}| ∥u∥ + \frac{∥e_{n + 1}∥}{ρ_{n + 1}} + \frac{∥e_{n}∥}{ρ_{n}} \\ + ∥\frac{δ_{n + 1}}{2 ρ_{n + 1}} T_{n + 1} x_{n + 1} - \frac{δ_{n}}{2 ρ_{n}} T_{n} x_{n}∥ \\ \leq |\frac{λ_{n + 1}}{ρ_{n + 1}} - \frac{λ_{n}}{ρ_{n}}| ∥u∥ + \frac{∥e_{n + 1}∥}{ρ_{n + 1}} + \frac{∥e_{n}∥}{ρ_{n}} \\ + |\frac{δ_{n + 1}}{2 ρ_{n + 1}} - \frac{δ_{n}}{2 ρ_{n}}| ∥T_{n + 1} x_{n + 1}∥ \\ + \frac{δ_{n}}{2 ρ_{n}} ∥T_{n + 1} x_{n + 1} - T_{n} x_{n}∥ . \end{align}

(4.4)

Since (x_n ) is bounded and T_n is nonexpansive, we can find M > 0 so that (||T_nx_n || + ||x_n || + ||u||) ≤ M for all n ∈ ℕ Adding (4.3) and (4.4) and noting δ_n ≤ 2ρ_n yield

\begin{align} ∥y_{n + 1} - y_{n}∥ & \leq |\frac{λ_{n + 1}}{ρ_{n + 1}} - \frac{λ_{n}}{ρ_{n}}| ∥u∥ + \frac{∥e_{n + 1}∥}{ρ_{n + 1}} + \frac{∥e_{n}∥}{ρ_{n}} \\ + |\frac{δ_{n + 1}}{2 ρ_{n + 1}} - \frac{δ_{n}}{2 ρ_{n}}| ∥T_{n + 1} x_{n + 1}∥ \\ + ∥x_{n + 1} - x_{n}∥ + \frac{| c_{n} - c_{n + 1} |}{\bar{c}} ∥T_{n} x_{n} - x_{n}∥ \\ \leq ∥x_{n + 1} - x_{n}∥ + M (|\frac{λ_{n + 1}}{ρ_{n + 1}} - \frac{λ_{n}}{ρ_{n}}| + \frac{∥e_{n + 1}∥}{ρ_{n + 1}} \\ + \frac{∥e_{n}∥}{ρ_{n}} + |\frac{δ_{n + 1}}{2 ρ_{n + 1}} - \frac{δ_{n}}{2 ρ_{n}}| + \frac{| c_{n} - c_{n + 1} |}{\bar{c}}) . \end{align}

With the knowledge that ||e_n ||→ 0 and

\frac{λ_{n}}{ρ_{n}} = \frac{2 λ_{n}}{2 λ_{n} + δ_{n}} \to 0, \frac{δ_{n}}{2 ρ_{n}} = \frac{δ_{n}}{2 λ_{n} + δ_{n}} \to 1,

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

\begin{array}{l} \underset{n \to \infty}{l i m s u p} (‖ y_{n + 1} - y_{n} ‖ - ‖ x_{n + 1} - x_{n} ‖) \\ \leq \underset{n \to \infty}{l i m s u p} M (| \frac{λ_{n + 1}}{ρ_{n + 1}} - \frac{λ_{n}}{ρ_{n}} | + \frac{‖ e_{n + 1} ‖}{ρ_{n + 1}} + \frac{‖ e_{n} ‖}{ρ_{n}} \\ + | \frac{δ_{n + 1}}{2 ρ_{n + 1}} - \frac{δ_{n}}{2 ρ_{n}} | + \frac{| c_{n} - c_{n + 1} |}{\bar{c}}) \to 0. \end{array}

Note that lim inf _n ρ_n = lim inf_n(δ_n /2)> 0 and lim sup _nρ_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}∥ = ρ_{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) $c_{n} \geq \bar{c} > 0, | c_{n + 1} - c_{n} | \to 0;$

(c) Σ _n ||e_n || < ∞.

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

Abbreviations

CPPA:: contraction-proximal point algorithm

PPA:: proximal point algorithm

RPPA:: relaxed proximal point algorithm.

Declarations

Acknowledgements

The authors would like to express thier sincere thanks to the referees for their valuable suggestions. This study is supported by the Natural Science Foundation of Department of Education, Henan(2011B110023).

Authors’ Affiliations

(1)

Department Of Mathematics, Luoyang Normal University, Luoyang, 471022, China

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