Abstract
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 kstrictly pseudocontractive mapping with F(T) ≠ ∅ for some 0 ≤ k < 1. Let F : C → C be a κLipschitzian and ηstrongly monotone operator with κ > 0 and η > 0 and f : C → C be a contraction with the contractive constant α ∈ (0, 1). Let , 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 : C → H 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
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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) ≤ αx  y, x, y ∈ C. A mapping T : C → H is said to be kstrictly pseudocontractive 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 kstrictly pseudocontractions includes the class of nonexpansive 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 0strictly pseudocontractive. The mapping T is also said to be pseudocontractive if k = 1 and T is said to be strongly pseudocontractive if there exists a constant λ ∈ (0, 1) such that T  λI is pseudocontractive. Clearly, the class of kstrictly pseudocontractive mappings falls into the one between classes of nonexpansive mappings and pseudocontractive mappings. Also we remark that the class of strongly pseudocontractive mappings is independent of the class of kstrictly pseudocontractive mappings (see [1–3]). The class of pseudocontraction 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 pseudocontractions, see, for example, [4–7] and references therein.
For nonexpansive mappings, one recent way to study them is to construct the iterative scheme, the socalled 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),
This iterative scheme was first introduced by Moudafi [8].
In particular, under the control conditions on {α_{ n } }
(C1) lim_{n→∞}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,
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 kstrictly pseudocontractive 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 kstrictly pseudocontractive 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 inf_{n→∞}β_{ n }≤ lim sup_{n→∞}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 κLipschitzian if there exists a positive constant κ such that
F is said to be ηstrongly monotone if there exists a positive constant η such that
From the definitions, we note that a strongly positive bounded linear operator A is a ALipschitzian and strongly monotone operator.
In 2001, Yamada [11] introduced the following hybrid iterative method for solving the variational inequality
where F : H → H is a κLipschitzian and ηstrongly 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 κLipschitzian and ηstrongly 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:
In this paper, motivated by the abovementioned results, we consider the following general iterative scheme for strictly pseudocontractive mapping: for C a closed convex subset of H such that C ± C ⊂ C, kstrictly pseudocontractive 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),
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 κLipschitzian and ηstrongly 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
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 kstrictly pseudocontractive mapping on C, then the fixed point set F(T) is closed convex, so that the projection P_{F(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 kstrictly pseudocontractive 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 kstrictly pseudocontractive 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 pseudocontractive mapping for some 0 ≤ k_{ i } < 1. Assume thatis a positive sequence such that . Then is a nonselfkstrictly pseudocontractive 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 nonnegative real numbers satisfying
where {λ_{ n } }, {δ_{ n } } and {r_{ n } } satisfy the following conditions:

(i)
{λ _{ n }} ⊂ [0, 1] and ,
(ii)lim sup_{n→∞}δ_{ n }≤ 0 or,

(iii)
r _{ n }≥ 0 (n ≥ 0), .
Then lim_{n→∞}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 x_{n+1}= γ_{ n }x_{ n }+ (1  γ_{ n })z_{ n }for all n ≥ 0 and
Then lim_{n→∞}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 κLipschitzian and ηstrongly 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 κLipschitzian and ηstrongly 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 kstrictly pseudocontractive mapping with F(T) ≠ ∅ for some 0 ≤ k < 1. Let F : C → C be a κLipschitzian and ηstrongly 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) lim_{n→∞}α_{ n }= 0;
(C2) ;

(B)
0 < lim inf_{n→∞} β _{ n }≤ lim sup_{n→∞} 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 lim_{n→∞}x_{n+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 lim_{n→∞}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 = lim_{t→0}x_{ 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
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 lim_{n→∞}x_{ n }  q = 0, where q = lim_{t→0}x_{ 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 }  q2 : n ≥ 0}, and
From the conditions (C1) and (C2) and Step 4, it is easy to see that λ_{ n } → 0, , and lim sup_{n→∞}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 pseudocontractive mapping together with removing the condition (C3) .

(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 κLipschitzian and ηstrongly 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 ± C ⊂ C, and T_{ i } : C → H be a k_{ i }strictly pseudocontractive mapping for some 0 ≤ k_{ i } < 1 and. Let F : C → C be a κLipschitzian and ηstrongly 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) lim_{n→∞}α_{ n }= 0;
(C2) ;

(B)
0 < lim inf_{n→∞} β _{ n }≤ lim sup_{n→∞} 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 kstrictly pseudocontractive 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, 0strictly pseudocontractive 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 κLipschitzian and ηstrongly 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) lim_{n→∞}α_{ n }= 0;
(C2) ;

(B)
0 < lim inf_{n→∞} β _{ n }≤ lim sup_{n→∞} 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 κLipschitzian and ηstrongly 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) together with using a κLipschitzian and ηstrongly monotone operator F.

(3)
As in [19], we also can establish the result for a countable family {T_{ i } } of k_{ i } strict pseudocontractive mappings with 0 ≤ k_{ i } < 1.
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This study was supported by research funds from DongA University.
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Jung, J.S. Some results on a general iterative method for kstrictly pseudocontractive mappings. Fixed Point Theory Appl 2011, 24 (2011). https://doi.org/10.1186/16871812201124
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DOI: https://doi.org/10.1186/16871812201124