Open Access

Some results on a general iterative method for k-strictly pseudo-contractive mappings

Fixed Point Theory and Applications20112011:24

DOI: 10.1186/1687-1812-2011-24

Received: 30 September 2010

Accepted: 2 August 2011

Published: 2 August 2011

Abstract

Let H be a Hilbert space, C be a closed convex subset of H such that C ± C C, and T : CH be a k-strictly pseudo-contractive mapping with F(T) ≠ for some 0 ≤ k < 1. Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0 and f : CC be a contraction with the contractive constant α (0, 1). Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq2_HTML.gif and τ < 1. Let {α n } and {β n } be sequences in (0, 1). It is proved that under appropriate control conditions on {α n } and {β n }, the sequence {x n } generated by the iterative scheme xn+1= α n γf(x n ) + β n x n + ((1 - β n )I - α n μF)P C Sx n , where S : CH is a mapping defined by Sx = kx + (1 - k)Tx and P C is the metric projection of H onto C, converges strongly to q F(T), which solves the variational inequality 〈μFq - γf(q), q - p〉 ≤ 0 for p F(T).

MSC: 47H09, 47H05, 47H10, 47J25, 49M05

Keywords

Iterative schemes k-strictly pseudo-contractive mapping Nonexpansive mapping Fixed points Contraction κ-Lipschitzian η-strongly monotone operator Variational inequality Hilbert space

1 Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Recall that a mapping f : CC is a contraction on C if there exists a constant α (0, 1) such that ||f(x) - f(y)|| ≤ α||x - y||, x, y C. A mapping T : CH is said to be k-strictly pseudo-contractive if there exists a constant k [0, 1) such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equa_HTML.gif

and F(T) denote the set of fixed points of the mapping T; that is, F(T) = {x C : Tx = x}.

Note that the class of k-strictly pseudo-contractions includes the class of non-expansive mappings T on C (that is, ||Tx - Ty|| ≤ ||x - y||, x, y C) as a subclass. That is, T is nonexpansive if and only if T is 0-strictly pseudo-contractive. The mapping T is also said to be pseudo-contractive if k = 1 and T is said to be strongly pseudo-contractive if there exists a constant λ (0, 1) such that T - λI is pseudo-contractive. Clearly, the class of k-strictly pseudo-contractive mappings falls into the one between classes of nonexpansive mappings and pseudo-contractive mappings. Also we remark that the class of strongly pseudo-contractive mappings is independent of the class of k-strictly pseudo-contractive mappings (see [13]). The class of pseudo-contraction is one of the most important classes of mappings among nonlinear mappings. Recently, many authors have been devoting the studies on the problems of finding fixed points for pseudo-contractions, see, for example, [47] and references therein.

For nonexpansive mappings, one recent way to study them is to construct the iterative scheme, the so-called viscosity iteration method: more precisely, for a nonexpansive mapping T, a contraction f with the contractive constant α (0, 1), and α n (0, 1),
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ1_HTML.gif
(1.1)

This iterative scheme was first introduced by Moudafi [8].

In particular, under the control conditions on {α n }

(C1) limn→∞α n = 0;

(C2) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq3_HTML.gif ;

(C3) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq4_HTML.gif ; or,

(C4) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq5_HTML.gif ,

Xu [9] proved that the sequence {x n } generated by (1.1) converges strongly to a fixed point q of T, which is the unique solution of the following variational inequality:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equb_HTML.gif
Recall that an operator A is strongly positive on H if there exists a constant https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq6_HTML.gif with the property:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equc_HTML.gif
In 2006, as the viscosity approximation method, Marino and Xu [10] considered the following iterative method: for a strongly positive bounded linear operator A on H with constant https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq6_HTML.gif , a nonexpansive mapping T on H, a contraction f : HH with the contractive constant α (0, 1), {α n } (0, 1) and γ > 0,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ2_HTML.gif
(1.2)
They proved that if the sequence {α n } satisfies the conditions (C1), (C2), and (C3) (or (C1), (C2), and (C4)), then the sequence {x n } generated by (1.2) converges strongly to the unique solution of the variational inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equd_HTML.gif
which is the optimality condition for the minimization problem
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Eque_HTML.gif

where h is a potential function for γf.

In 2010, in order to improve the corresponding results of Cho et al. [5] as well as Marino and Xu [10] by removing the condition (C3), Jung [6] studied the following composite iterative scheme for the class of k-strictly pseudo-contractive mappings.

Theorem J. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C C, T : CH be a k-strictly pseudo-contractive mapping with F(T) ≠ , for some 0 ≤ k < 1. Let A be a strongly positive bounded linear operator on C with constant https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq7_HTML.gif and f : CC be a contraction with the contractive constant α (0, 1) such that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq8_HTML.gif . Let {α n } and {β n } be sequences in (0, 1) satisfying the conditions (C1), (C2) and the condition 0 < lim infn→∞β n ≤ lim supn→∞β n < 1. Let {x n } be a sequence in C generated by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equf_HTML.gif
where S : CH is a mapping defined by Sx = kx + (1 - k)Tx and P C is the metric projection of H onto C. Then {x n } converges strongly to a fixed point q of T, which is the unique solution of the following variational inequality related to the linear operator A:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equg_HTML.gif
On the other hand, a mapping F : HH is called κ-Lipschitzian if there exists a positive constant κ such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ3_HTML.gif
(1.3)
F is said to be η-strongly monotone if there exists a positive constant η such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ4_HTML.gif
(1.4)

From the definitions, we note that a strongly positive bounded linear operator A is a ||A||-Lipschitzian and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq9_HTML.gif -strongly monotone operator.

In 2001, Yamada [11] introduced the following hybrid iterative method for solving the variational inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ5_HTML.gif
(1.5)
where F : HH is a κ-Lipschitzian and η-strongly monotone operator with κ > 0, η > 0, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif and S : HH is a nonexpansive mapping, and proved that if {λ n } satisfies appropriate conditions, then the sequence {x n } generated by (1.5) converges strongly to the unique solution of the variational inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equh_HTML.gif

In 2010, by combining the iterative method (1.2) with the Yamada's method (1.5), Tian [12] considered the following general iterative method.

Theorem T1. Let H be a Hilbert space, F : HH be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0, and S : HH be a nonexpansive mapping with F(S) ≠ . Let f : HH be a contraction with the contractive constant α (0, 1). Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq2_HTML.gif . Let {α n } be a sequence in (0, 1) satisfying the conditions (C1), (C2) and (C3) (or (C1), (C2) and (C4)). Let {x n } be a sequence in H generated by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equi_HTML.gif
Then {x n } converges strongly to a fixed point https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq10_HTML.gif of S, which is the unique solution of the following variational inequality related to the operator F:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ6_HTML.gif
(1.6)
In this paper, motivated by the above-mentioned results, we consider the following general iterative scheme for strictly pseudo-contractive mapping: for C a closed convex subset of H such that C ± C C, k-strictly pseudo-contractive mapping T : CH with F(T) ≠ , a contraction f : CC with the contractive constant α (0, 1), μ > 0 and {α n }, {β n } (0, 1),
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ7_HTML.gif
(IS)

where S : CH is a mapping defined by Sx = kx+(1 - k)Tx, P C is the metric projection of H onto C, and F : CC is a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Under certain different control conditions on {α n }, we establish the strong convergence of the sequence {x n } generated by (IS) to a fixed point of T, which is a solution of the variational inequality (1.6) related to the operator F. By removing the condition (C3) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq11_HTML.gif on {α n }, the main results improve, develop and complement the corresponding results of Tian [12] as well as Cho et al. [5], Jung [6] and Marino and Xu [10]. Our results also improve the corresponding results of Halpern [13], Moudafi [8], Wittmann [14] and Xu [9].

2 Preliminaries and lemmas

Throughout this paper, when {x n } is a sequence in E, then x n x (resp., x n x) will denote strong (resp., weak) convergence of the sequence {x n } to x.

For every point x H, there exists a unique nearest point in C, denoted by P C (x), such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equj_HTML.gif

for all y C. P C is called the metric projection of H onto C. It is well known that P C is nonexpansive.

In a Hilbert space H, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ8_HTML.gif
(2.1)
It is also well known that H satisfies the Opial condition, that is, for any sequence {x n } with x n x, the inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equk_HTML.gif

holds for every y H with yx.

We need the following lemmas for the proof of our main results.

Lemma 2.1[15]. Let H be a Hilbert space and C be a closed convex subset of H. If T is a k-strictly pseudo-contractive mapping on C, then the fixed point set F(T) is closed convex, so that the projection PF(T)is well defined.

Lemma 2.2[15]. Let H be a Hilbert space and C be a closed convex subset of H. Let T : CH be a k-strictly pseudo-contractive mapping with F(T) ≠ . Then F(P C T) = F (T ).

Lemma 2.3[15]. Let H be a Hilbert space, C be a closed convex subset of H, and T : CH be a k-strictly pseudo-contractive mapping. Define a mapping S : CH by Sx = λx + (1 - λ) Tx for all x C. Then, as λ [k, 1), S is a nonexpansive mapping such that F(S) = F(T).

The following Lemmas 2.4 and 2.5 can be obtained from the Proposition 2.6 of Acedo and Xu [4].

Lemma 2.4. Let H be a Hilbert space and C be a closed convex subset of H. For any N ≥ 1, assume that for each 1 ≤ iN, T i : CH is a k i -strictly pseudo-contractive mapping for some 0 ≤ k i < 1. Assume that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq12_HTML.gif is a positive sequence such that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq13_HTML.gif . Then https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq14_HTML.gif is a nonself-k-strictly pseudo-contractive mapping with k= max{k i : 1 ≤ iN}.

Lemma 2.5. Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq15_HTML.gif and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq12_HTML.gif be given as in Lemma 2.4. Suppose that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq15_HTML.gif has a common fixed point in C. Then https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq16_HTML.gif .

Lemma 2.6[16, 17]. Let {s n } be a sequence of non-negative real numbers satisfying
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equl_HTML.gif
where {λ n }, {δ n } and {r n } satisfy the following conditions:
  1. (i)

    {λ n } [0, 1] and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq17_HTML.gif ,

     
(ii)lim supn→∞δ n ≤ 0 or https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq18_HTML.gif ,
  1. (iii)

    r n ≥ 0 (n ≥ 0), https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq19_HTML.gif .

     

Then limn→∞s n = 0.

Lemma 2.7[18]. Let {x n } and {z n } be bounded sequences in a Banach space E and {γ n } be a sequence in [0, 1] which satisfies the following condition:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equm_HTML.gif
Suppose that xn+1= γ n x n + (1 - γ n )z n for all n ≥ 0 and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equn_HTML.gif

Then limn→∞||z n - x n || = 0.

Lemma 2.8. In a Hilbert space H, the following inequality holds:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equo_HTML.gif

Lemma 2.9. Let C be a nonempty closed convex subset of a Hilbert space H such that C ± C C. Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif and 0 < t < ρ < 1. Then S := ρI - tμF : CC is a contraction with contractive constant ρ - tτ, where https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq20_HTML.gif with https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq21_HTML.gif .

Proof. From (1.3), (1.4) and (2.1), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equp_HTML.gif
where https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq20_HTML.gif , and so
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equq_HTML.gif

Hence S is a contraction with contractive constant ρ - tτ. □

3 Main results

We need the following result for the existence of solutions of a certain variational inequality, which is slightly an improvement of Theorem 3.1 of Tian [12].

Theorem T2. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C C, and T : CC be a nonexpansive mapping with F(T) ≠ . Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : CC be a contraction with the contractive constant α (0, 1). Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq2_HTML.gif and τ < 1. Let x t be a fixed point of a contraction St x α tγf (x) + (I - tμF )Tx for t (0, 1) and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq21_HTML.gif . Then {x t } converges strongly to a fixed point https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq10_HTML.gif of T as t → 0, which solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equr_HTML.gif

Equivalently, we have https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq22_HTML.gif .

Now, we study the strong convergence result for a general iterative scheme (IS).

Theorem 3.1. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C C, and T : CH be a k-strictly pseudo-contractive mapping with F(T) ≠ for some 0 ≤ k < 1. Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : CC be a contraction with the contractive constant α (0, 1). Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq2_HTML.gif and τ < 1. Let f{α n } and {β n } be sequences in (0, 1) which satisfy the conditions:

(C1) limn→∞α n = 0;

(C2) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq3_HTML.gif ;
  1. (B)

    0 < lim infn→∞ β n ≤ lim supn→∞ βn < 1.

     
Let {x n } be a sequence in C generated by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equs_HTML.gif
where S : CH is a mapping defined by Sx = kx + (1 - k)Tx and P C is the metric projection of H onto C. Then {x n } converges strongly to q F(T), which solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equt_HTML.gif

Proof. First, from the condition (C1), without loss of generality, we assume that α n τ < 1, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq23_HTML.gif and α n < (1 - β n ) for n ≥ 0.

We divide the proof several steps:

Step 1. We show that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq24_HTML.gif for all n ≥ 0 and all p F(T) = F(S). Indeed, let p F(T). Then from Lemma 2.9, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equu_HTML.gif

Using an induction, we have https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq24_HTML.gif . Hence, {x n } is bounded, and so are {f(x n )}, {P C Sx n } and {FP C Sx n }.

Step 2. We show that limn→∞||xn+1- x n || = 0. To this show, define
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equv_HTML.gif
Observe that from the definition of z n ,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equw_HTML.gif
Thus, it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equx_HTML.gif
From the condition (C1) and (B), it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equy_HTML.gif
Hence, by Lemma 2.7, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equz_HTML.gif
Consequently,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equaa_HTML.gif
Step 3. We show that limn→∞||x n - P C Sx n || = 0. Indeed, since
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equab_HTML.gif
we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equac_HTML.gif
that is,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equad_HTML.gif
So, from the conditions (C1) and (B) and Step 2, it follows that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equae_HTML.gif
Step 4. We show that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equaf_HTML.gif
where q = limt→0x t being x t = tγf(x t ) + (I - tμF )P C Sx t for 0 < t < 1 and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq21_HTML.gif . We note that from Lemmas 2.2 and 2.3 and Theorem T2, q F(T) = F(S) and q is a solution of a variational inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equ9_HTML.gif
(3.1)
To show this, we can choose a subsequence https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq25_HTML.gif of {x n } such that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equag_HTML.gif
Since {x n } is bounded, there exists a subsequence https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq26_HTML.gif of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq25_HTML.gif which converges weakly to w. Without loss of generality, we can assume that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq27_HTML.gif . Since ||x n - P C Sx n || → 0 by Step 3, we obtain w = P C Sw. In fact, if wP C Sw, then, by Opial condition,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equah_HTML.gif
which is a contradiction. Hence w = P C Sw. Since F(P C S) = F(S), from Lemma 2.3, we have w F(T). Therefore, from (3.1), we conclude that
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equai_HTML.gif
Step 5. We show that limn→∞||x n - q|| = 0, where q = limt→0x t being x t = tγf (xt) + (I - tμF)P C Sx t for 0 < t < 1 and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq21_HTML.gif , and q is a solution of a variational inequality
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equaj_HTML.gif
Indeed, from (IS), we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equak_HTML.gif
Applying Lemmas 2.8 and 2.9, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equal_HTML.gif
that is,
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equam_HTML.gif
where M = sup{||x n - q||2 : n ≥ 0}, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq28_HTML.gif and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equan_HTML.gif

From the conditions (C1) and (C2) and Step 4, it is easy to see that λ n → 0, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq17_HTML.gif , and lim supn→∞δ n ≤ 0. Hence, by Lemma 2.7, we conclude x n q as n → ∞. This completes the proof. □

Remark 3.1. (1) Theorem 3.1 extends and develops Theorem 3.2 of Tian [12] from a nonexpansive mapping to a strictly pseudo-contractive mapping together with removing the condition (C3) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq4_HTML.gif .
  1. (2)

    Theorem 3.1 also generalizes Theorem 2.1 of Jung [6] as well as Theorem 2.1 of Cho et al. [5] and Theorem 3.4 of Marino and Xu [10] from a strongly positive bounded linear operator A to a κ-Lipschitzian and η-strongly monotone operator F.

     
  2. (3)

    Theorem 3.1 also improves the corresponding results of Halpern [13], Moudafi [8], Wittmann [14] and Xu [9] as some special cases.

     

Theorem 3.2. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C C, and T i : CH be a k i -strictly pseudo-contractive mapping for some 0 ≤ k i < 1 and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq29_HTML.gif . Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : CC be a contraction with the contractive constant α (0, 1). Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq2_HTML.gif and τ < 1. Let {α n } and {βn} be sequences in (0, 1) which satisfy the conditions.

(C1) limn→∞α n = 0;

(C2) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq3_HTML.gif ;
  1. (B)

    0 < lim infn→∞ β n ≤ lim supn→∞ β n < 1.

     
Let {x n } be a sequence in C generated by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equao_HTML.gif
where S : CH is a mapping defined by https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq30_HTML.gif with k = max{k i : 1 ≤ iN} and {η i } is a positive sequence such that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq13_HTML.gif and P C is the metric projection of H onto C. Then {x n } converges strongly to q F(T), which solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equap_HTML.gif

Proof. Define a mapping T : CH by https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq31_HTML.gif . By Lemmas 2.4 and 2.5, we conclude that T : CH is a k-strictly pseudo-contractive mapping with k = max{k i : 1 ≤ iN} and https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq32_HTML.gif . Then the result follows from Theorem 3.1 immediately. □

As a direct consequence of Theorem 3.2, we have the following result for nonexpansive mappings (that is, 0-strictly pseudo-contractive mappings).

Theorem 3.3. Let H be a Hilbert space, C be a closed convex subset of H such that C ± C C, https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq33_HTML.gif be a finite family of nonexpansive mappings with https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq29_HTML.gif . Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : CC be a contraction with the contractive constant α (0, 1). Let https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq1_HTML.gif , https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq2_HTML.gif and τ < 1. Let {α n } and {βn} be sequences in (0, 1) which satisfy the conditions.

(C1) limn→∞α n = 0;

(C2) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq3_HTML.gif ;
  1. (B)

    0 < lim infn→∞ β n ≤ lim supn→∞ β n < 1.

     
Let {x n } be a sequence in C generated by
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equaq_HTML.gif
where https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq12_HTML.gif is a positive sequence such that https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq13_HTML.gif and P C is the metric projection of H onto C. Then {x n } converges strongly to a common fixed point q of https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq15_HTML.gif , which solves the following variational inequality:
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_Equar_HTML.gif
Remark 3.2. (1) Theorems 3.2 and 3.3 also generalize Theorems 2.2 and 2.4 of Jung [6] from a strongly positive bounded linear operator A to a κ-Lipschitzian and η-strongly monotone operator F.
  1. (2)

    Theorems 3.2 and 3.3 also improve and complement the corresponding results of Cho et al. [5] by removing the condition (C3) https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_Article_28_IEq4_HTML.gif together with using a κ-Lipschitzian and η-strongly monotone operator F.

     
  2. (3)

    As in [19], we also can establish the result for a countable family {T i } of k i -strict pseudo-contractive mappings with 0 ≤ k i < 1.

     

Declarations

Acknowledgements

This study was supported by research funds from Dong-A University.

Authors’ Affiliations

(1)
Department of Mathematics, Dong-A University

References

  1. Browder FE: Fixed point theorems for noncompact mappings. Proc Natl Acad Sci USA 1965, 53: 1272–1276. 10.1073/pnas.53.6.1272MathSciNetView Article
  2. Browder FE: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch Ration Mech Anal 1967, 24: 82–90.MathSciNetView Article
  3. Browder FE, Petryshn WV: Construction of fixed points of nonlinear mappings Hilbert space. J Math Anal Appl 1967, 20: 197–228. 10.1016/0022-247X(67)90085-6MathSciNetView Article
  4. Acedo GL, Xu HK: Iterative methods for strictly pseudo-contractions in Hilbert space. Nonlinear Anal 2007, 67: 2258–2271. 10.1016/j.na.2006.08.036MathSciNetView Article
  5. Cho YJ, Kang SM, Qin X: Some results on k -strictly pseudo-contractive mappings in Hilbert spaces. Nonlinear Anal 2009, 70: 1956–1964. 10.1016/j.na.2008.02.094MathSciNetView Article
  6. Jung JS: Strong convergence of iterative methods for k -strictly pseudo-contractive mappings in Hilbert spaces. Appl Math Comput 2010, 215: 3746–3753. 10.1016/j.amc.2009.11.015MathSciNetView Article
  7. Morales CH, Jung JS: Convergence of paths for pseudo-contractive mappings in Banach spaces. Proc Am Math Soc 2000, 128: 3411–3419. 10.1090/S0002-9939-00-05573-8MathSciNetView Article
  8. Moudafi A: Viscosity approximation methods for fixed-points problems. J Math Anal Appl 2000, 241: 46–55. 10.1006/jmaa.1999.6615MathSciNetView Article
  9. Xu HK: Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059MathSciNetView Article
  10. Marino G, Xu HX: A general iterative method for nonexpansive mappings in Hilbert spaces. J Math Anal Appl 2006, 318: 43–52. 10.1016/j.jmaa.2005.05.028MathSciNetView Article
  11. Yamada I: The hybrid steepest descent for the variational inequality problems over the intersection of fixed points sets of nonexpansive mappings. In Inherently Parallel Algorithms in Feasibility and Optimization and Their Applications. Edited by: Butnariu D, Censor Y, Reich S. Elservier, New York; 2001:473–504.View Article
  12. Tian M: A general itewrative algorithm for nonexpansive mappings in Hilbert spaces. Nonlinear Anal 2010, 73: 689–694. 10.1016/j.na.2010.03.058MathSciNetView Article
  13. Halpern B: Fixed points of nonexpansive maps. Bull Am Math Soc 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0View Article
  14. Wittmann R: Approximation of fixed points of nonexpansive mappings. Arch Math 1992, 58: 486–491. 10.1007/BF01190119MathSciNetView Article
  15. Zhou H: Convergence theorems of fixed points for k -strict pseudo-contractions in Hilbert spaces. Nonlinear Anal 2008, 69: 456–462. 10.1016/j.na.2007.05.032MathSciNetView Article
  16. Liu LS: Iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces. J Math Anal Appl 1995, 194: 114–125. 10.1006/jmaa.1995.1289MathSciNetView Article
  17. Xu HK: Iterative algorithms for nonlinear operators. J Lond Math Soc 2002, 66: 240–256. 10.1112/S0024610702003332View Article
  18. Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for one parameter nonexpansive semigroups without Bochner integral. J Math Anal Appl 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017MathSciNetView Article
  19. Aoyama K, Kimura Y, Takahashi W, Toyoda M: Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space. Nonlinear Anal 2007, 67: 2350–2360. 10.1016/j.na.2006.08.032MathSciNetView Article

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© Jung; licensee Springer. 2011

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