 Research
 Open Access
 Published:
An extragradientlike approximation method for variational inequalities and fixed point problems
Fixed Point Theory and Applications volume 2011, Article number: 22 (2011)
Abstract
The purpose of this paper is to investigate the problem of finding a common element of the set of fixed points of an asymptotically strict pseudocontractive mapping in the intermediate sense and the set of solutions of a variational inequality problem for a monotone and Lipschitz continuous mapping. We introduce an extragradientlike iterative algorithm that is based on the extragradientlike approximation method and the modified Mann iteration process. We establish a strong convergence theorem for two sequences generated by this extragradientlike iterative algorithm. Utilizing this theorem, we also design an iterative process for finding a common fixed point of two mappings, one of which is an asymptotically strict pseudocontractive mapping in the intermediate sense and the other taken from the more general class of Lipschitz pseudocontractive mappings.
1991 MSC: 47H09; 47J20.
1. Introduction
Let H be a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and  · , respectively, and let C be a nonempty closed convex subset of H. Corresponding to an operator A : C → H and set C, the variational inequality problem VIP(A, C) is defined as follows:
The set of solutions of VIP(A, C) is denoted by Ω. It is well known that if A is a strongly monotone and Lipschitzcontinuous mapping on C, then the VIP(A, C) has a unique solution. Not only the existence and uniqueness of a solution are important topics in the study of the VIP(A, C) but also how to compute a solution of the VIP(A, C) is important. For applications and further details on VIP(A, C), we refer to [1–4] and the references therein.
The set of fixed points of a mapping S is denoted by Fix(S), that is, Fix(S) = {x ∈ H : Sx = x}.
For finding an element of F(S) ∩ Ω under the assumption that a set C ⊂ H is nonempty, closed and convex, a mapping S : C → C is nonexpansive and a mapping A : C → H is βinversestrongly monotone, Takahashi and Toyoda [5] proposed an iterative scheme and proved that the sequence generated by the proposed scheme converges weakly to a point z ∈ F(S) ∩ Ω if F(S) ∩ Ω ≠ ∅.
Recently, motivated by the idea of Korpelevich's extragradient method [6], Nadezhkina and Takahashi [7] introduced an iterative scheme, called extragradient method, for finding an element of F(S) ∩ Ω and established the weak convergence result. Very recently, inspired by the work in [7], Zeng and Yao [8] introduced an iterative scheme for finding an element of F(S) ∩ Ω and obtained the weak convergence result. The viscosity approximation method for finding a fixed point of a given nonexpansive mapping was proposed by Moudafi [9]. He proved the strong convergence of the sequence generated by the proposed method to a unique solution of some variational inequality. Xu [10] extended the results of [9] to the more general version. Later on, Ceng and Yao [11] also introduced an extragradientlike approximation method, which is based on the above extragradient method and viscosity approximation method, and proved the strong convergence result under certain conditions.
An iterative method for the approximation of fixed points of asymptotically nonexpansive mappings was developed by Schu [12]. Iterative methods for the approximation of fixed points of asymptotically nonexpansive mappings have been further studied in [13, 14] and the references therein. The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [15]. The iterative methods for the approximation of fixed points of such types of nonLipschitzian mappings have been further studied in [16–18]. On the other hand, Kim and Xu [19] introduced the concept of asymptotically κstrict pseudocontractive mappings in a Hilbert space and studied the weak and strong convergence theorems for this class of mappings. Sahu et al. [20] considered the concept of asymptotically κstrict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian. They proposed modified Mann iteration process and proved its weak convergence for an asymptotically κstrict pseudocontractive mapping in the intermediate sense.
Very recently, Ceng et al. [21] established the strong convergence of viscosity approximation method for a modified Mann iteration process for asymptotically strict pseudocontractive mappings in intermediate sense and then proved the strong convergence of general CQ algorithm for asymptotically strict pseudocontractive mappings in intermediate sense. They extended the concept of asymptotically strict pseudocontractive mappings in intermediate sense to Banach space setting, called nearly asymptotically κstrict pseudocontractive mapping in intermediate sense.
They also established the weak convergence theorems for a fixed point of a nearly asymptotically κstrict pseudocontractive mapping in intermediate sense which is not necessarily Lipschitzian.
In this paper, we propose and study an extragradientlike iterative algorithm that is based on the extragradientlike approximation method in [11] and the modified Mann iteration process in [20]. We apply the extragradientlike iterative algorithm to designing an iterative scheme for finding a common fixed point of two nonlinear mappings. Here, we remind the reader of the following facts: (i) the modified Mann iteration process in [[20], Theorem 3.4] is extended to develop the extragradientlike iterative algorithm for finding an element of F(S) ∩ Ω; (ii) the extragradientlike iterative algorithm is very different from the extragradientlike iterative scheme in [11] since the class of mappings S in our scheme is more general than the class of nonexpansive mappings.
2. Preliminaries
Throughout the paper, unless otherwise specified, we assume that H is a real Hilbert space whose inner product and norm are denoted by 〈·,·〉 and  · , respectively, and C is a nonempty closed convex subset of H. The set of fixed points of a mapping S is denoted by Fix(S), that is, Fix(S) = {x ∈ H : Sx = x}. We write x_{ n } ⇀ x to indicate that the sequence {x_{ n } } converges weakly to x. The sequence {x_{ n } } converges strongly to x is denoted by x_{ n } → x.
Recall that a mapping S : C → C is said to be LLipschitzian if there exists a constant L ≥ 0 such that Sx  Sy ≤ Lx  y, ∀x, y ∈ C. In particular, if L ∈ [0, 1), then S is called a contraction on C; if L = 1, then S is called a nonexpansive mapping on C. The mapping S : C → C is called pseudocontractive if
A mapping A : C → H is called

(i)
monotone if

(ii)
βinversestrongly monotone [22, 23] if there exists a positive constant β such that
It is obvious that if A is βinversestrongly monotone, then A is monotone and Lipschitz continuous.
It is easy to see that if a mapping S : C → C is nonexpansive, then the mapping A = I  S is 1/2inversestrongly monotone; moreover, F(S) = Ω (see, e.g., [5]). At the same time, if a mapping S : C → C is pseudocontractive and LLipschitz continuous, then the mapping A = (I  S) is monotone and L + 1Lipschitz continuous; moreover, F(S) = Ω (see, e.g., [[24], proof of Theorem 4.5]).
Definition 2.1. Let C be a nonempty subset of a normed space X. A mapping S : C → C is said to be

(a)
asymptotically nonexpansive [25] if there exists a sequence {k_{ n } } of positive numbers such that lim_{n→∞} K_{ n } = 1 and

(b)
asymptotically nonexpansive in the intermediate sense [15] provided S is uniformly continuous and

(c)
uniformly Lipschitzian if there exists a constant L > 0 such that
It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian.
The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [25] as an important generalization of the class of nonexpansive mappings. The existence of fixed points of asymptotically nonexpansive mappings was proved by Goebel and Kirk [25] as below:
Theorem 2.1. [[25], Theorem 1] If C is a nonempty closed convex bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive mapping S : C → C has a fixed point in C.
Definition 2.2. [19] A mapping S : C → C is said to be an asymptotically κstrict pseudocontractive mapping with sequence {γ_{ n } } if there exist a constant κ ∈ [0, 1) and a sequence {γ_{ n } } in [0, ∞) with lim_{n→∞}γ_{ n } = 0 such that
It is important to note that every asymptotically κstrict pseudocontractive mapping with sequence {γ_{ n } } is a uniformly LLipschitzian mapping with .
Definition 2.3. [20] A mapping S : C → C is said to be an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } } if there exist a constant κ ∈ [0, 1) and a sequence {γ_{ n } } in [0, ∞) with lim_{n→∞}γ_{ n } = 0 such that
Put
Then, c_{ n } ≥ 0 (∀n ≥ 1), c_{ n } → 0 (n → ∞) and (2.2) reduces to the relation
Whenever c_{ n } = 0 for all n ≥ 1 in (2.3), then S is an asymptotically κstrict pseudocontractive mapping with sequence {γ_{ n } }.
For every point x ∈ H, there exists a unique nearest point in C, denoted by P_{ C }x, such that
P_{ C } is called the metric projection of H onto C. Recall that the inequality holds
Moreover, it is equivalent to
it is also equivalent to
It is easy to see that P_{ C } is a nonexpansive mapping from H onto C; see, e.g., [26] for further detail.
Lemma 2.1. Let A : C → H be a monotone mapping. Then,
Lemma 2.2. Let H be a real Hilbert space. Then, the following hold:
Lemma 2.3. [[20], Lemma 2.6] Let S : C → C be an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } }. Then,
for all x, y ∈ C and n ≥ 1.
Lemma 2.4. [[20], Lemma 2.7] Let S : C → C be a uniformly continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } }. Let {x_{ n } } be a sequence in C such that x_{ n } x_{n+1} → 0 and x_{ n }  S^{n}x_{ n }  → 0 as n → ∞. Then, x_{ n }  Sx_{ n }  → 0 as n → ∞.
Proposition 2.1 (Demiclosedness Principle). [[20], Proposition 3.1] Let S : C → C be a continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } }. Then, I  S is demiclosed at zero in the sense that if {x_{ n } } is a sequence in C such that x_{ n } ⇀ x ∈ C and lim sup_{m→ ∞}lim sup_{n→ ∞}x_{ n } S^{m}x_{ n }  = 0, then (I  S)x = 0.
Proposition 2.2. [[20], Proposition 3.2] Let S : C → C be a continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } } such that F (S) ≠ ∅. Then, F(S) is closed and convex.
Remark 2.1. Propositions 2.1 and 2.2 give some basic properties of an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } }. Moreover, Proposition 2.1 extends the demiclosedness principles studied for certain classes of nonlinear mappings in [19, 27–29].
Lemma 2.5. [30]Let (X, 〈·,·〉) be an inner product space. Then, for all x, y, z ∈ X and all α, β, γ ∈ [0, 1] with α + β + γ = 1, we have
Lemma 2.6. [[31], Lemma 2.5] Let {s_{ n } } be a sequence of nonnegative real numbers satisfying
where, , andsatisfy the conditions:
(i), or equivalently,;
(ii);
(iii).
Then, lim_{n→∞}s_{ n } = 0.
Lemma 2.7. [32]Let {x_{ n } } and {z_{ n } } be bounded sequences in a Banach space X and let {ϱ_{ n }} be a sequence in [0, 1] with 0 < lim inf_{n→∞}ϱ_{ n }≤ lim sup_{n→∞}ϱ_{ n }≤ 1. Suppose that x_{n+1}= ϱ _{ n }x_{ n } + (1  ϱ_{ n })z_{ n } for all integers n ≥ 1 and lim sup_{n→∞}(z_{n+1} z_{ n }   x_{n+1} x_{ n } ) ≤ 0. Then, lim_{n→∞}z_{ n }  x_{ n }  = 0.
The following lemma can be easily proved, and therefore, we omit the proof.
Lemma 2.8. In a real Hilbert space H, there holds the inequality
A setvalued mapping T : H → 2 ^{H} is called monotone if for all x, y ∈ H, f ∈ Tx and g ∈ Ty imply 〈x  y, f  g〉 ≥ 0. A monotone mapping T : H → 2 ^{H} is maximal if its graph G(T) is not properly contained in the graph of any other monotone mapping. It is known that a monotone mapping T is maximal if and only if for (x, f) ∈ H × H, 〈x  y, f  g〉 ≥ 0 for all (y, g) ∈ G(T) implies f ∈ Tx. Let A : C → H be a monotone, LLipschitz continuous mapping and let N_{ C }v be the normal cone to C at v ∈ C, i.e., N_{ C }v = {w ∈ H : 〈v  u, w〉 ≥ 0, ∀u ∈ C}. Define
It is known that in this case T is maximal monotone, and 0 ∈ Tv if and only if v ∈ Ω; see [33].
3. Extragradientlike approximation method and strong convergence results
Let A : C → H be a monotone and LLipschitz continuous mapping, f : C → C be a contraction with contractive constant α ∈ (0, 1) and S : C → C be an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } }. In this paper, we introduce an extragradientlike iterative algorithm that is based on the extragradientlike approximation method in [11] and the modified Mann iteration process in [20]:
where {λ_{ n } } is a sequence in (0, 1) with , and {α_{ n } }, {β_{ n } }, {μ_{ n } } and {ν_{ n } } are sequences in [0, 1] satisfying the following conditions:
(A1) α_{ n } + β_{ n } + ν_{ n } ≤ 1 for all n ≥ 1;
(A2) lim_{n→∞}α_{ n } = 0, ;
(A3) κ < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;
(A4) .
The following result shows the strong convergence of the sequences {x_{ n } }, {y_{ n } } generated by the scheme (3.1) to the same point q = P_{F(S)∩Ω}f (q) if and only if {Ax_{ n } } is bounded, (I  S^{n} )x_{ n }  → 0 and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C.
Theorem 3.1. Let A : C → H be a monotone and LLipschitz continuous mapping, f : C → C be a contraction with contractive constant α ∈ (0, 1) and S : C → C be a uniformly continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } } such that F(S) ∩ Ω ≠ ∅ and. Let {x_{ n } }, {y_{ n } } be the sequences generated by (3.1), where {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } }, {μ_{ n } } and{y_{ n } } are sequences in [0, 1] satisfying the conditions (A1)(A4). Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same point q = P_{F(S)∩Ω}f (q) if and only if {Ax_{ n } } is bounded, (I  S^{n} )x_{ n }  → 0 and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C.
Proof. "Necessity". Suppose that the sequences {x_{ n } }, {y_{ n } } converge strongly to the same point q = P_{F(S)∩Ω}f (q). Then from the LLipschitz continuity of A, it follows that {Ax_{ n } } is bounded, and for each y ∈ C:
which implies that
due to q ∈ Ω. Furthermore, utilizing Lemma 2.3, we have
due to x_{ n } → q, γ_{ n } → 0 and c_{ n } → 0. Consequently, we conclude that for each y ∈ C
That is, (I  S^{n} )x_{ n }  → 0.
"Sufficiency". Suppose that {Ax_{ n } } is bounded, (I  S^{n} )x_{ n }  → 0 and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C. Note that lim inf_{n→∞}β_{ n } > κ. Hence, we may assume, without loss of generality, that β_{ n } > κ for all n ≥ 1.
Next, we divide the proof of the sufficiency into several steps.
STEP 1. We claim that {x_{ n } } is bounded. Indeed, put t_{ n } = P_{ C } (x_{ n }  λ_{ n }Ay_{ n } ) for all n ≥ 1. Let x* ∈ F(S) ∩ Ω. Then, x* = P_{ C } (x*  λ_{ n }Ax*). Putting x = x_{ n }  λ_{ n }Ay_{ n } and y = x* in (2.5), we obtain
Since A is monotone and x* is a solution of VIP(A, C), we have
It follows from (3.2) that
Note that x_{ n } ∈ C for all n ≥ 1 and that y_{ n } = (1  μ_{ n } )x_{ n } + μ_{ n }P_{ C } (x_{ n }  λ_{ n }Ax_{ n } ). Hence, we have
Since {Ax_{ n } } is bounded and A is LLipschitz continuous, we have
and hence Ay_{ n }  ≤ (1+ L)Ax_{ n } , which implies that {Ay_{ n } } is bounded. Hence, we may assume that there exists a constant M ≥ sup{Ax_{ n }  + Ay_{ n }  + Ax*: n ≥ 1}. Then, it follows from (3.4) that
This together with (3.3) implies that
Observe that
Putting τ_{ n } = α_{ n } + β_{ n } + ν_{ n } and utilizing Lemma 2.5, we obtain from (3.5) and (3.6)
Now, let us show that for all n ≥ 1
As a matter of fact, whenever n = 1, from (3.7), we have
Assume that (3.8) holds for some n ≥ 1. Consider the case of n + 1. From (3.7), we obtain
This shows that (3.8) holds for the case of n + 1. By induction, we know that (3.8) holds for all n ≥ 1. Since , and , from (3.8) we deduce that for all n ≥ 1
This implies that {x_{ n } } is bounded.
STEP 2. We claim that lim_{n→∞}x_{n+1} x_{ n }  = 0. Indeed, observe that
and
Define a sequence {z_{ n } } by
where ϱ _{ n } = 1  α_{ n }  β_{ n }  ν_{ n } , ∀n ≥ 1. Then we have
From (3.9)(3.11), we get
which implies that
Note that the boundedness of {x_{ n } } implies that {f (x_{ n } )} is also bounded. Since
we know that {y_{ n } } is bounded and so is {f (y_{ n } )}. Moreover, {t_{ n } } is bounded by (3.5). Now, utilizing Lemma 2.3, we obtain that
Thus, from the boundedness of {t_{ n } }, it follows that {S^{n}t_{ n } } is bounded. Also, note that conditions
(ii), (iii) imply
and conditions (iii), (iv) lead to
Thus, we deduce from (3.12) that
Since ϱ _{ n } = 1  α_{ n }  β_{ n }  ν_{ n } , we know from conditions (ii), (iii), (iv) that
Thus, in terms of Lemma 2.7, we get lim_{n→∞}z_{ n }  x_{ n }  = 0. Consequently,
STEP 3. We claim that lim_{n→∞}Sx_{ n }  x_{ n }  = lim_{n→∞}St_{ n }  t_{ n }  = 0. Indeed, observe that
and hence
Note that the following condition holds:
Also, observe that
Utilizing Lemma 2.3 and t_{ n }  x_{ n } → 0, we have
Thus from (3.15)(3.17), we obtain
In addition, from (3.9) and x_{n+1} x_{n → 0}, it follows that t_{n+1} t_{n → 0}. Therefore, utilizing the uniform continuity of S and Lemma 2.4, we know that lim_{n→∞}Sx_{ n }  x_{ n }  = 0 and lim_{n→∞}St_{ n }  t_{ n }  = 0.
STEP 4. We claim that lim sup_{n→∞}〈f (q)  q, x_{ n }  q〉 ≤ 0. Indeed, we pick a subsequence of {x_{ n } } so that
Without loss of generality, let . Then, (3.19) reduces to
In order to show , it suffices to show that . Since S is uniformly continuous and x_{ n }  Sx_{ n }  → 0, we see that x_{ n }  S^{m}x_{ n }  → 0 for all m ≥ 1. By Proposition 2.1, we obtain . Now let us show that . Let
Then, T is maximal monotone and 0 ∈ Tv if and only if v ∈ Ω; see [33]. Let (v, w) ∈ G(T). Then, we have w ∈ Tv = Av + N_{ C }v and hence w  Av ∈ N_{ C }v. Therefore, we have 〈v  u, w  Av〉 ≥ 0 for all u ∈ C. In particular, taking , we get
and so . Since T is maximal monotone, we have and hence .
This shows that . Therefore by the property of the metric projection, we derive .
STEP 5. We claim that lim_{n→∞}x_{ n }  q = 0 where q = P_{F(S)∩Ω}f (q). Indeed, since {Ax_{ n } }, {Ay_{ n } }, {S^{n}t_{ n } } are bounded, we may assume that there exists a constant M ≥ sup{Ax_{ n }  +Ay_{ n }  + Aq + S^{n}t_{ n }  q: n ≥ 1 g. Then from (3.1), (3.5) and Lemma 2.8, we get
which implies that
Note that lim_{n→∞}α_{ n } = 0 and . Since lim sup_{n→∞}〈f (q)  q, x_{n+1} q〉 ≤ 0, lim_{n→∞}y_{ n }  x_{ n }  = 0 and {x_{ n }  q} is bounded, we know that
Also, since and , it is easy to see that
Therefore, according to Lemma 2.6, we deduce that from (3.20) that x_{ n }  q → 0. Further from y_{ n }  x_{ n }  → 0, we obtain y_{ n }  q → 0. This completes the proof. □
In Theorem 3.1, if we put ν_{ n } = 0 (∀n ≥ 1) and S = I the identity mapping. Then, the iterative scheme (3.1) reduces to the following scheme:
Moreover, it is easy to see that and (1  S^{n} )x_{ n }  → 0. Thus, we have following corollary.
Corollary 3.1. Let A : C → H be a monotone, LLipschitz continuous mapping, and f : C → C be a contraction with contractive constant α ∈ (0, 1). Let Ω ≠ ∅. Let {x_{ n } }, {y_{ n } } be the sequences generated by (3.21), where {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } } and {μ_{ n } } are three sequences in [0, 1] satisfying the conditions:
(B1) α_{ n } + β_{ n } ≤ 1 for all n ≥ 1,
(B2) lim_{n→∞}α_{ n } = 0, ;
(B3) 0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1.
Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same point q = P_{Ω}f (q) if and only if {Ax_{ n } } is bounded and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C.
If A^{1}0 = Ω and P_{ H } = I, the identity mapping of H, then the iterative scheme (3.1) reduces to the following iterative scheme:
The following corollary can be easily derived from Theorem 3.1.
Corollary 3.2. Let f : H → H be a contractive mapping with constant α ∈ (0, 1), A : H → H be a monotone, LLipschitz continuous mapping and S : H → H be a uniformly continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } } such that F(S) ∩ A^{ 1}0 ≠ ∅ and. Let {x_{ n } }, {y_{ n } } be the sequences generated by (3.22), where {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } }, {μ_{ n } } and {ν_{ n } } are four sequences in [0, 1] satisfying the conditions (A1)(A4). Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same pointif and only if {Ax_{ n } } is bounded, (I  S^{n} )x_{ n }  → 0 and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ H.
Let B : H → 2 ^{H} be a maximal monotone mapping. Then, for any x ∈ H and r > 0, consider . Such is called the resolvent of B and is denoted by .
If we put and P_{ H } = I, then the iterative scheme (3.1) reduces to the following scheme:
It is easy to see that κ = 0, γ_{ n } = 0 and c_{ n } = 0 for all n ≥ 1. Moreover, we have A^{1}0 = Ω and . Thus, utilizing Theorem 3.1, we obtain the following corollary.
Corollary 3.3. Let f : H → H be a contractive mapping with constant α ∈ (0, 1), A : H → H be a monotone, LLipschitz continuous mapping and B : H → 2 ^{H} be a maximal monotone mapping such that A^{1}0 ∩ B^{1} ≠ ∅. Letbe the resolvent of B for each r > 0. Let {x_{ n } }, {y_{ n } } be the sequences generated by (3.23), where {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } }, {μ_{ n } } and {ν_{ n } } are four sequences in [0, 1] satisfying the conditions (A1)(A4). Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same pointif and only if {Ax_{ n } } is bounded,and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ H.
Corollary 3.4. Let f : H → H be a contractive mapping with constant α ∈ (0, 1) and A : H → H be a monotone, LLipschitz continuous mapping such that A^{1}0 ≠ ∅.Let {x_{ n } }, {y_{ n } } be the sequences generated by
where {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } } and {μ_{ n } } are three sequences in [0, 1] satisfying the conditions (B1)(B3). Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same pointif and only if {Ax_{ n } } is bounded and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C.
Proof. In Theorem 3.1, put C = H, ν_{ n } = 0 (∀n ≥ 1) and S = I the identity mapping of H. Then, we know that κ = 0, γ_{ n } = 0 and c_{ n } = 0 for all n ≥ 1. Moreover, we have A^{1}0 = Ω. PH = I. In this case, it is easy to see that and (I  S^{n} )x_{ n }  → 0. Therefore, by Theorem 3.1, we obtain the desired conclusion. □
We also know one more definition of a pseudocontractive mapping, which is equivalent to the definition given in the preliminaries. A mapping S : C → C is called pseudocontractive [26] if
Obviously, the class of pseudocontractive mappings is more general than the class of nonexpansive mappings. For the class of pseudocontractive mappings, there are some nontrivial examples; see, e.g., [[24], p. 1239] for further details. In the following theorem, we introduce an iterative process that converges strongly to a common fixed point of two mappings, one of which is an asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } } and the other Lipschitz continuous and pseudocontractive.
Theorem 3.2. Let f : C → C be a contractive mapping with constant α ∈ (0, 1), T : C → C be a pseudocontractive, mLipschitz continuous mapping and S : C → C be a uniformly continuous asymptotically κstrict pseudocontractive mapping in the intermediate sense with sequence {γ_{ n } } such that F(S) ∩ F(T) ≠ ∅ and. Let {x_{ n } }, {y_{ n } } be the sequences generated by
where A = I  T, {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } }, {μ_{ n } } and {ν_{ n } } are four sequences in [0, 1] satisfying the conditions (A1)(A4). Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same point q = P_{F(S)∩F(T)}f (q) if and only if {Ax_{ n } } is bounded, (I  S^{n} )x_{ n }  → 0 and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C.
Proof. Let A = I  T. Let us show that the mapping A is monotone and (m + 1)Lipschitz continuous. Indeed, observe that
and
Now, let us show that F(T) = Ω. Indeed, we have, for fixed λ_{0} ∈ (0, 1),
By Theorem 3.1, we obtain the desired conclusion. □
Theorem 3.3. Let f : C → C be a contractive mapping with constant α ∈ (0, 1), T : C → C be a pseudocontractive, mLipschitz continuous mapping and S : C → C be a nonexpansive mapping such that F(S) ∩ F(T) ≠ ∅. Let {x_{ n } }, {y_{ n } } be the sequences generated by
where A = I  T, {λ_{ n } } is a sequence in (0, 1) with, and {α_{ n } }, {β_{ n } }, {μ_{ n } } and {ν_{ n } }
are sequences in [0, 1] satisfying the conditions (A1)(A4). Then, the sequences {x_{ n } }, {y_{ n } } converge strongly to the same point q = P_{F(S)∩F(T)}f (q) if and only if {Ax_{ n } } is bounded, (I  S^{n} )x_{ n }  → 0 and lim inf_{n→∞}〈Ax_{ n } , y  x_{ n } 〉 ≥ 0 for all y ∈ C.
Proof. Let A = I  T. In terms of the proof of Theorem 3.2, we know that A is a monotone and (m+1)Lipschitz continuous mapping such that F(T) = Ω. Since S is a nonexpansive mapping, we know that κ = 0, γ_{ n } = 0 and c_{ n } = 0 for all n ≥ 1. By Theorem 3.1, we obtain the desired conclusion. □
References
 1.
Baiocchi C, Capelo A: Variational and Quasivariational Inequalities, Application to Free Boundary Problems. Wiley, New York; 1984.
 2.
Kinderlehrer D, Stampacchia G: An Introduction to Variational Inequalities and their Applications. Academic Press, New York 1980.
 3.
Konnov IV: Combined Relaxation Methods for Variational Inequalities, Lecture Notes in Mathematical Economics. Springer, Berlin; 2000.
 4.
Patriksson M: Nonlinear Programming and Variational Inequality Problems. Kluwer Academic Publishers, Dordrecht; 1999.
 5.
Takahashi W, Toyoda M: Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2003, 118: 417–428. 10.1023/A:1025407607560
 6.
Korpelevich GM: The extragradient method for finding saddle points and other problems. Matecon 1976, 12: 747–756.
 7.
Nadezhkina N, Takahashi W: Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl 2006, 128: 191–201. 10.1007/s109570057564z
 8.
Zeng LC, Yao JC: Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwanese J Math 2006,10(5):1293–1303.
 9.
Moudafi A: Viscosity approximation methods for fixedpoints problems. J Math Anal Appl 2000, 241: 46–55. 10.1006/jmaa.1999.6615
 10.
Xu HK: Viscosity approximation methods for nonexpansive mappings. J Math Anal Appl 2004, 298: 279–291. 10.1016/j.jmaa.2004.04.059
 11.
Ceng LC, Yao JC: An extragradientlike approximation method for variational inequality problems and fixed point problems. Appl Math Comput 2007, 190: 205–215. 10.1016/j.amc.2007.01.021
 12.
Schu J: Iterative construction of fixed points of asymptotically nonexpansive mapping. J Math Anal Appl 1991, 159: 407–413.
 13.
Ceng LC, Xu HK, Yao JC: The viscosity approximation method for asymptotically nonexpansive mappings in Banach spaces. Nonlinear Anal 2008,69(4):1402–1412. 10.1016/j.na.2007.06.040
 14.
Ceng LC, Wong NC, Yao JC: Fixed point solutions of variational inequalities for a finite family of asymptotically nonexpansive mappings without common fixed point assumption. Comput Math Appl 2008, 56: 2312–2322. 10.1016/j.camwa.2008.05.002
 15.
Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloq Math 1993, 65: 169–179.
 16.
Agarwal RP, O'Regan D, Sahu DR: Iterative construction of fixed points of nearly asymptotically nonexpansive mappings. J Nonlinear Convex Anal 2007,8(1):61–79.
 17.
Chidume CE, Shahzad N, Zegeye H: Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense. Numer Funct Anal Optim 2004, 25: 239–257. 10.1081/NFA120039611
 18.
Kim GE, Kim TH: Mann and Ishikawa iterations with errors for nonLipschitzian mappings in Banach spaces. Comput Math Appl 2001, 42: 1565–1570. 10.1016/S08981221(01)002620
 19.
Kim TH, Xu HK: Convergence of the modified Mann's iteration method for asymptotically strict pseudocontractions. Nonlinear Anal 2008, 68: 2828–2836. 10.1016/j.na.2007.02.029
 20.
Sahu DR, Xu HK, Yao JC: Asymptotically strict pseudocontractive mappings in the intermediate sense. Nonlinear Anal 2009, 70: 3502–3511. 10.1016/j.na.2008.07.007
 21.
Ceng LC, Ansari QH, Yao JC: Strong and weak convergence theorems for asymptotically strict pseudocontraction mappings in intermediate sense. J Nonlinear Convex Anal 2010,11(2):283–308.
 22.
Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space. J Math Anal Appl 1967, 20: 747–756.
 23.
Liu F, Nashed MZ: Regularization of nonlinear illposed variational inequalities and convergence rates. SetValued Anal 1998, 6: 313–344. 10.1023/A:1008643727926
 24.
Nadezhkina N, Takahashi W: Strong convergence theorem by a hybrid method for nonexpansive mappings and Lipschitzcontinuous monotone mappings. SIAM J Optim 2006, 16: 1230–1241. 10.1137/050624315
 25.
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Am Math Soc 1972,35(1):171–174. 10.1090/S00029939197202985003
 26.
Takahashi W: Nonlinear Functional Analysis. Yokohama Publishers, Yokohama; 2000.
 27.
Gornicki J: Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces. Comment Math Univ Carolin 1989,30(2):249–252.
 28.
Marino G, Xu HK: Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces. J Math Anal Appl 2007, 329: 336–346. 10.1016/j.jmaa.2006.06.055
 29.
Xu HK: Existence and convergence for fixed points for mappings of asymptotically nonexpansive type. Nonlinear Anal 1991, 16: 1139–1146. 10.1016/0362546X(91)90201B
 30.
Osilike MO, Igbokwe DI: Weak and strong convergence theorems for fixed points of pseudocontractions and solutions of monotone type operator equations. Comput Math Appl 2000, 40: 559–567. 10.1016/S08981221(00)001796
 31.
Xu HK: Iterative algorithms for nonlinear operators. J London Math Soc 2002, 66: 240–256. 10.1112/S0024610702003332
 32.
Suzuki T: Strong convergence of Krasnoselskii and Mann's type sequences for oneparameter nonexpansive semigroups without Bochner integrals. J Math Anal Appl 2005, 305: 227–239. 10.1016/j.jmaa.2004.11.017
 33.
Rockafellar RT: On the maximality of sums of nonlinear monotone operators. Trans Am Math Soc 1970, 149: 75–88. 10.1090/S00029947197002822725
Acknowledgements
In this research, the first author, L.C. Ceng, was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), Science and Technology Commission of Shanghai Municipality Grant (075105118), and Shanghai Leading Academic Discipline Project (S30405). While, the third author, N.C. Wong, and the last author, J.C. Yao, were partially supported by the Taiwan NSC Grants 992115M110007MY3 and 992221E037007MY3, respectively.
Author information
Additional information
4. Competing interests
The authors declare that they have no competing interests.
5. Authors' contributions
All authors contribute equally and significantly in writing this paper. All authors read and approved the final manuscript.
Rights and permissions
About this article
Cite this article
Ceng, L., Ansari, Q.H., Wong, N. et al. An extragradientlike approximation method for variational inequalities and fixed point problems. Fixed Point Theory Appl 2011, 22 (2011). https://doi.org/10.1186/16871812201122
Received:
Accepted:
Published:
Keywords
 extragradientlike approximation method
 modified Mann iteration process
 variational inequality
 asymptotically strict pseudocontractive mapping in the intermediate sense
 fixed point
 monotone mapping
 strong convergence
 demiclosedness principle