1 Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. Recall that a mapping f : C → C 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 : C → H is said to be k-strictly pseudo-contractive if there exists a constant k ∈ [0, 1) such that

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 [1–3]). 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, [4–7] 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),

(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) ;

(C3) ; or,

(C4) ,

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:

Recall that an operator A is strongly positive on H if there exists a constant with the property:

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 , a nonexpansive mapping T on H, a contraction f : H → H with the contractive constant α ∈ (0, 1), {α n } ⊂ (0, 1) and γ > 0,

(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

which is the optimality condition for the minimization problem

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 : C → H 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 constantand f : C → C be a contraction with the contractive constant α ∈ (0, 1) such that. 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

where S : C → H 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:

On the other hand, a mapping F : H → H is called κ-Lipschitzian if there exists a positive constant κ such that

(1.3)

F is said to be η-strongly monotone if there exists a positive constant η such that

(1.4)

From the definitions, we note that a strongly positive bounded linear operator A is a ||A||-Lipschitzian and -strongly monotone operator.

In 2001, Yamada [11] introduced the following hybrid iterative method for solving the variational inequality

(1.5)

where F : H → H is a κ-Lipschitzian and η-strongly monotone operator with κ > 0, η > 0, and S : H → H 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

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 : H → H be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0, and S : H → H be a nonexpansive mapping with F(S) ≠ ∅. Let f : H → H be a contraction with the contractive constant α ∈ (0, 1). Letand. 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

Then {x n } converges strongly to a fixed pointof S, which is the unique solution of the following variational inequality related to the operator F:

(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 : C → H with F(T) ≠ ∅, a contraction f : C → C with the contractive constant α ∈ (0, 1), μ > 0 and {α n }, {β n } ⊂ (0, 1),

(IS)

where S : C → H is a mapping defined by Sx = kx+(1 - k)Tx, P C is the metric projection of H onto C, and F : C → C 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) 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

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

(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

holds for every y ∈ H with y ≠ x.

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 : C → H 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 : C → H be a k-strictly pseudo-contractive mapping. Define a mapping S : C → H 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 ≤ i ≤ N, T i : C → H is a k i -strictly pseudo-contractive mapping for some 0 ≤ k i < 1. Assume thatis a positive sequence such that . Then is a nonself-k-strictly pseudo-contractive mapping with k= max{k i : 1 ≤ i ≤ N}.

Lemma 2.5. Letandbe given as in Lemma 2.4. Suppose thathas a common fixed point in C. Then.

Lemma 2.6[16, 17]. Let {s n } be a sequence of non-negative real numbers satisfying

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

  1. (i)

    {λ n } ⊂ [0, 1] and ,

(ii)lim supn→∞δ n ≤ 0 or,

  1. (iii)

    r n ≥ 0 (n ≥ 0), .

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:

Suppose that xn+1= γ n x n + (1 - γ n )z n for all n ≥ 0 and

Then limn→∞||z n - x n || = 0.

Lemma 2.8. In a Hilbert space H, the following inequality holds:

Lemma 2.9. Let C be a nonempty closed convex subset of a Hilbert space H such that C ± C ⊂ C. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Letand 0 < t < ρ < 1. Then S := ρI - tμF : C → C is a contraction with contractive constant ρ - tτ, wherewith.

Proof. From (1.3), (1.4) and (2.1), we have

where , and so

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 : C → C be a nonexpansive mapping with F(T) ≠ ∅. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let, 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. Then {x t } converges strongly to a fixed pointof T as t → 0, which solves the following variational inequality:

Equivalently, we have.

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 : C → H be a k-strictly pseudo-contractive mapping with F(T) ≠ ∅ for some 0 ≤ k < 1. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let, 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) ;

  1. (B)

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

Let {x n } be a sequence in C generated by

where S : C → H 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:

Proof. First, from the condition (C1), without loss of generality, we assume that α n τ < 1, and α n < (1 - β n ) for n ≥ 0.

We divide the proof several steps:

Step 1. We show that for all n ≥ 0 and all p ∈ F(T) = F(S). Indeed, let p ∈ F(T). Then from Lemma 2.9, we have

Using an induction, we have . 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

Observe that from the definition of z n ,

Thus, it follows that

From the condition (C1) and (B), it follows that

Hence, by Lemma 2.7, we have

Consequently,

Step 3. We show that limn→∞||x n - P C Sx n || = 0. Indeed, since

we have

that is,

So, from the conditions (C1) and (B) and Step 2, it follows that

Step 4. We show that

where q = limt→0x t being x t = tγf(x t ) + (I - tμF )P C Sx t for 0 < t < 1 and . 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

(3.1)

To show this, we can choose a subsequence of {x n } such that

Since {x n } is bounded, there exists a subsequence of which converges weakly to w. Without loss of generality, we can assume that . Since ||x n - P C Sx n || → 0 by Step 3, we obtain w = P C Sw. In fact, if w ≠ P C Sw, then, by Opial condition,

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

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 , and q is a solution of a variational inequality

Indeed, from (IS), we have

Applying Lemmas 2.8 and 2.9, we have

that is,

where M = sup{||x n - q||2 : n ≥ 0}, and

From the conditions (C1) and (C2) and Step 4, it is easy to see that λ n → 0, , 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) .

  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 : C → H be a k i -strictly pseudo-contractive mapping for some 0 ≤ k i < 1 and. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let, and τ < 1. Let {α n } and {βn} be sequences in (0, 1) which satisfy the conditions.

(C1) limn→∞α n = 0;

(C2) ;

  1. (B)

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

Let {x n } be a sequence in C generated by

where S : C → H is a mapping defined bywith k = max{k i : 1 ≤ i ≤ N} and {η i } is a positive sequence such thatand 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:

Proof. Define a mapping T : C → H by . By Lemmas 2.4 and 2.5, we conclude that T : C → H is a k-strictly pseudo-contractive mapping with k = max{k i : 1 ≤ i ≤ N} and . 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, be a finite family of nonexpansive mappings with. Let F : C → C be a κ-Lipschitzian and η-strongly monotone operator with κ > 0 and η > 0. Let f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let, and τ < 1. Let {α n } and {βn} be sequences in (0, 1) which satisfy the conditions.

(C1) limn→∞α n = 0;

(C2) ;

  1. (B)

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

Let {x n } be a sequence in C generated by

whereis a positive sequence such thatand P C is the metric projection of H onto C. Then {x n } converges strongly to a common fixed point q of, which solves the following variational inequality:

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) 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.