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 ± CC, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 x n+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 qF(T), which solves the variational inequality 〈μFq - γf(q), q - p〉 ≤ 0 for pF(T).

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

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, yC. A mapping T : CH is said to be k-strictly pseudo-contractive if there exists a constant k ∈ [0, 1) such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equa_HTML.gif

and F(T) denote the set of fixed points of the mapping T; that is, F(T) = {xC : 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, yC) 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),
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq3_HTML.gif ;

(C3) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq4_HTML.gif ; or,

(C4) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equb_HTML.gif
Recall that an operator A is strongly positive on H if there exists a constant http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq6_HTML.gif with the property:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equd_HTML.gif
which is the optimality condition for the minimization problem
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 ± CC, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq7_HTML.gif and f : CC be a contraction with the contractive constant α ∈ (0, 1) such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equ3_HTML.gif
(1.3)
F is said to be η-strongly monotone if there exists a positive constant η such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equ4_HTML.gif
(1.4)

From the definitions, we note that a strongly positive bounded linear operator A is a ||A||-Lipschitzian and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq9_HTML.gif -strongly monotone operator.

In 2001, Yamada [11] introduced the following hybrid iterative method for solving the variational inequality
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equ5_HTML.gif
(1.5)
where F : HH is a κ-Lipschitzian and η-strongly monotone operator with κ > 0, η > 0, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equi_HTML.gif
Then {x n } converges strongly to a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq10_HTML.gif of S, which is the unique solution of the following variational inequality related to the operator F:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 ± CC, 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),
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 xH, there exists a unique nearest point in C, denoted by P C (x), such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equj_HTML.gif

for all yC. 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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equk_HTML.gif

holds for every yH 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 xC. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq12_HTML.gif is a positive sequence such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq13_HTML.gif . Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq14_HTML.gif is a nonself-k-strictly pseudo-contractive mapping with k = max{k i : 1 ≤ iN}.

Lemma 2.5. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq15_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq12_HTML.gif be given as in Lemma 2.4. Suppose that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq15_HTML.gif has a common fixed point in C. Then http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq16_HTML.gif .

Lemma 2.6 [16, 17]. Let {s n } be a sequence of non-negative real numbers satisfying
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equl_HTML.gif

where {λ n }, {δ n } and {r n } satisfy the following conditions:

(i) {λ n } ⊂ [0, 1] and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq17_HTML.gif ,

(ii)lim supn→∞ δ n ≤ 0 or http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq18_HTML.gif ,

(iii) r n ≥ 0 (n ≥ 0), http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equm_HTML.gif
Suppose that xn+1 = γ n x n + (1 - γ n )z n for all n ≥ 0 and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equn_HTML.gif

Then limn→∞ ||z n - x n || = 0.

Lemma 2.8. In a Hilbert space H, the following inequality holds:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equo_HTML.gif

Lemma 2.9. Let C be a nonempty closed convex subset of a Hilbert space H such that C ± CC. Let F : CC be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif and 0 < t < ρ < 1. Then S := ρI - tμF : CC is a contraction with contractive constant ρ - tτ, where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq20_HTML.gif with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq21_HTML.gif .

Proof. From (1.3), (1.4) and (2.1), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equp_HTML.gif
where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq20_HTML.gif , and so
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 ± CC, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq2_HTML.gif and τ < 1. Let x t be a fixed point of a contraction Stx α tγf (x) + (I - tμF )Tx for t ∈ (0, 1) and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq21_HTML.gif . Then {x t } converges strongly to a fixed point http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq10_HTML.gif of T as t → 0, which solves the following variational inequality:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equr_HTML.gif

Equivalently, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 ± CC, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq3_HTML.gif ;

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

Let {x n } be a sequence in C generated by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 qF(T), which solves the following variational inequality:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equt_HTML.gif

Proof. First, from the condition (C1), without loss of generality, we assume that α n τ < 1, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq23_HTML.gif and α n < (1 - β n ) for n ≥ 0.

We divide the proof several steps:

Step 1. We show that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq24_HTML.gif for all n ≥ 0 and all pF(T) = F(S). Indeed, let pF(T). Then from Lemma 2.9, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equu_HTML.gif

Using an induction, we have http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equv_HTML.gif
Observe that from the definition of z n ,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equw_HTML.gif
Thus, it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equx_HTML.gif
From the condition (C1) and (B), it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equy_HTML.gif
Hence, by Lemma 2.7, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equz_HTML.gif
Consequently,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equaa_HTML.gif
Step 3. We show that limn→∞||x n - P C Sx n || = 0. Indeed, since
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equab_HTML.gif
we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equac_HTML.gif
that is,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equad_HTML.gif
So, from the conditions (C1) and (B) and Step 2, it follows that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equae_HTML.gif
Step 4. We show that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equaf_HTML.gif
where q = limt→0 x t being x t = tγf(x t ) + (I - tμF )P C Sx t for 0 < t < 1 and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq21_HTML.gif . We note that from Lemmas 2.2 and 2.3 and Theorem T2, qF(T) = F(S) and q is a solution of a variational inequality
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equ9_HTML.gif
(3.1)
To show this, we can choose a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq25_HTML.gif of {x n } such that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equag_HTML.gif
Since {x n } is bounded, there exists a subsequence http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq26_HTML.gif of http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq25_HTML.gif which converges weakly to w. Without loss of generality, we can assume that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 wF(T). Therefore, from (3.1), we conclude that
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equai_HTML.gif
Step 5. We show that limn→∞||x n - q|| = 0, where q = limt→0 x t being x t = tγf (xt) + (I - tμF)P C Sx t for 0 < t < 1 and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq21_HTML.gif , and q is a solution of a variational inequality
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equaj_HTML.gif
Indeed, from (IS), we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equak_HTML.gif
Applying Lemmas 2.8 and 2.9, we have
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equal_HTML.gif
that is,
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equam_HTML.gif
where M = sup{||x n - q||2 : n ≥ 0}, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq28_HTML.gif and
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equan_HTML.gif

From the conditions (C1) and (C2) and Step 4, it is easy to see that λ n → 0, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq4_HTML.gif .

(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.

(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 ± CC, and T i : CH be a k i -strictly pseudo-contractive mapping for some 0 ≤ k i < 1 and http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq3_HTML.gif ;

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

Let {x n } be a sequence in C generated by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equao_HTML.gif
where S : CH is a mapping defined by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq30_HTML.gif with k = max{k i : 1 ≤ iN} and {η i } is a positive sequence such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq13_HTML.gif and P C is the metric projection of H onto C. Then {x n } converges strongly to qF(T), which solves the following variational inequality:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equap_HTML.gif

Proof. Define a mapping T : CH by http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 ± CC, http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq33_HTML.gif be a finite family of nonexpansive mappings with http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq1_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq3_HTML.gif ;

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

Let {x n } be a sequence in C generated by
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_Equaq_HTML.gif
where http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq12_HTML.gif is a positive sequence such that http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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 http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq15_HTML.gif , which solves the following variational inequality:
http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_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.

(2) Theorems 3.2 and 3.3 also improve and complement the corresponding results of Cho et al. [5] by removing the condition (C3) http://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2011-24/MediaObjects/13663_2010_28_IEq4_HTML.gif together with using a κ-Lipschitzian and η-strongly monotone operator F.

(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

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© Jung; licensee Springer. 2011

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