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Strong convergence of a new general iterative method for variational inequality problems in Hilbert spaces
Fixed Point Theory and Applications volume 2012, Article number: 46 (2012)
Abstract
In this article, we introduce a new iterative scheme with Meir-Keeler contractions for strict pseudo-contractions in Hilbert spaces. We also discuss the strong convergence theorems of the new iterative scheme for variational inequality problems in Hilbert spaces. The methods in this article are interesting and are different from those given in many other articles. Our results improve and extend the corresponding results announced by many others.
MR(2000) Subject Classification: 47H09; 47H10.
1 Introduction
Let H be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥. Let C be a nonempty closed convex subset of H and let F : C → C be a nonlinear operator. The variational inequality problem is such that
Variational inequalities were introduced and studied by Stampacchia [1] in 1964. It is now well known that variational inequalities cover as diverse disciplines as partial differential equations, optimal control, optimization, mathematical programming, mechanics and finance, see [1–25].
It is known that if F is a strongly monotone and Lipschitzian mapping on C, then the VI(F, C) has a unique solution. It is also known that the VI(F, C) is equivalent to the fixed point equation
where P C is the projection of H onto the closed convex set C and μ > 0 is an arbitrarily fixed constant. So, fixed point methods can be implemented to find a solution of the VI(F,C) provided F satisfies some conditions and μ > 0 is chosen appropriately. A great deal of effort has gone into finding an approximate solution of the VI(F,C) see [3, 5, 15–19].
In 2001, Yamada [2] introduced the following hybrid iterative method for solving the variational inequality
On the other hand, Yao et al. [6] modified Mann's iterative scheme by using the so-called viscosity approximation method which was introduced by Moudafi [7]. More precisely, Yao et al. [6] introduced and studied the following iterative algorithm:
where T is a nonexpansive mapping of K into itself and f is a contraction on K. They obtained a strong convergence theorem under some mild restrictions on the parameters.
Zhou [8], Qin et al. [9] modified normal Mann's iterative process (1.3) for non-self-k-strictly pseudo-contractions to have strong convergence in Hilbert spaces. Qin et al. [9] introduced the following iterative algorithm scheme:
where T is non-self-k-strictly pseudo-contraction, f is a contraction and A is a strong positive linear bounded operator. They prove, under certain appropriate assumptions on the sequences {α n } and {β n }, that {x n } defined by (1.4) converges strongly to a fixed point of the k-strictly pseudo-contraction, which solves some variational inequality.
The following famous theorem is referred to as the Banach contraction principle.
Theorem 1. (Banach [10]) Let (X, d) be a complete metric space and let f be a contraction on X, i.e., there exists r ∈ (0,1) such that d(f(x), f(y)) ≤ rd(x, y) for all x, y ∈ X. Then f has a unique fixed point.
Theorem 2. (Meir and Keeler [11]) Let (X,d) be a complete metric space and let ϕ be a Meir-Keeler contraction (MKC) on X, i.e., for every ε > 0, there exists δ > 0 such that d(x, y) < ε + δ implies d(ϕ(x),ϕ(y)) < ε for all x, y ∈ X. Then ϕ has a unique fixed point.
Remark 1. Theorem 2 is one of generalizations of Theorem 1, because contractions are MKCs.
Question 1. Can Theorem 1 of Yao [6], Theorem 3.2 of Zhou [8], Theorem 2.1 of Qin [9], and so on be extended from one or finite k i -strictly pseudo-contraction to infinite k i -strictly pseudo-contraction?
Question 2. We know that the MKC is more general than the contraction. What happens if the contraction is replaced by the MKC?
Question 3. We know that the η-strongly monotone and L-Lipschitzian operator is more general than the strong positive linear bounded operator. What happens if the strong positive linear bounded operator is replaced by the η-strongly monotone and L-Lipschitzian operator?
Question 4. Can the restrictions imposed on the parameters {α n }, {β n } and {λ n } in [9] be relaxed?
The purpose of this article is to give the affirmative answers to these questions mentioned above. Motivated by the above works, in this article we suggest and analyze a hybrid iterative algorithm as follows:
where T i is a non-self-k i -strictly pseudo-contraction, ϕ is an MKC contraction and F : C → C is a L-Lipschitzian and η-strongly monotone mapping in Hilbert space. Under certain appropriate assumptions on the sequences {α n }, {β n }, {γ n }, and , that {x n } defined by (1.5) converges strongly to a common fixed point of an infinite family of k i -strictly pseudo-contractions, which solves some variational inequality.
2 Preliminaries
In this section, we first recall some notations. Let C be a nonempty closed convex subset of a real Hilbert space H. Let F : C → C be an operator. F is called L-Lipschitzian if there exists a positive constant L such that
for all x, y ∈ C, F is said to be η-strongly monotone if there exists a positive constant η such that
for all x, y ∈ C. Without loss of generality, we can assume that η ∈ (0, 1] and L ∈ [1, ∞). Under these conditions, it is well known that the variational inequality problem VI(F, C) has a unique solution x* ∈ C.
A self-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. We use Π C to denote the collection of all contractions on C. That is, Π C = {f|f : C → C a contraction}. We use F(T) to denote the fixed point set of the mapping T and P C to denote the metric projection of H onto its closed convex subset C.
A mapping T is said to be non-expansive, if
T is said to be a k-strict pseudo-contraction in the terminology of Browder and Petryshyn [12], if there exists a constant k ∈ [0,1) such that
It is clear that it is equivalent to
or is equivalent to
An operator A be a strongly positive bounded linear operator on H, that is, there exists a constant such that
Remark 2. From the definition of A, we note that a strongly positive bounded linear operator A is a ∥A∥-Lipschitzian and -strongly monotone operator.
In order to prove our main results, we need the following lemmas.
Lemma 2.1 (Zhou [8]). 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 (Zhou [8]). 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 . Then F(P C T) = F(T).
Lemma 2.3 (Browder and Petryshyn [12]). 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 J : C → H by Jx = αx +(1-α)Tx for all x ∈ C. Then, as α ∈ [k, 1), J is a non-expansive mapping such that F(J) = F(T).
Lemma 2.4. (see [13]). Let {x n }, {z n } be bounded sequences in a Banach space E and {β n } be a sequence in [0,1] which satisfies the following condition: 0 < lim infn→∞β n ≤ lim supn→∞β n < 1. Suppose that xn+1= (1 - β n )x n + β n z n for all n ≥ 0 and limsupn→∞(∥zn+1- z n ∥ - ∥xn+1-x n ∥) ≤ 0. then lim n→∞∥z n - x n ∥ = 0.
Lemma 2.5 (Xu [14]). Assume that {α n } is a sequence of non-negative real numbers such that αn + 1≤ (1 - γ n )α n + δ n , where γ n is a sequence in (0, 1) and δ n is a sequence in ℝ such that
-
(i)
,
-
(ii)
or
Then lim n → ∞α n = 0.
Lemma 2.6 ([23] Demiclosedness Principle). Let H be a Hilbert space, K a closed convex subset of H, and T:K → K a non-expansive mapping with . If {x n } is a sequence in K weakly converging to x and if {(I - T)x n } converges strongly to y, then (I - T)x = y.
Lemma 2.7 Let F be a L-Lipschitzian and η-strongly monotone operator on a nonempty closed convex subset C of a real Hilbert space H with 0 < η ≤ L and 0 < t < 2η/L2. Then S = (I - tF): C → C is a contraction with contraction coefficient .
Proof. From the definition of η-strongly monotone and L-Lipschitzian operator, we have
for all x, y ∈ C. From 0 < η ≤ L and 0 < t < 2η/L2, we have and
where . Hence, S is a contraction with contraction coefficient τ t .
Lemma 2.8 ([23] Lemma 2.3). Let ϕ be an MKC on a convex subset C of a Banach space E. Then for each ε > 0, there exists r ∈ (0,1) such that ∥x - y∥ ≥ ε implies ∥ϕx - ϕy∥ ≤ r ∥ x - y∥ for all x, y ∈ C.
Lemma 2.9. Let C be a closed convex subset of a Hilbert space H. Let S : C → C be a non-expansive mapping and ϕ be an MKC on C. Suppose F: C → C be a η-strongly monotone and L-Lipschitzian mapping with coefficient η and η > γ > 0. Then the sequence {x t } define by
converges strongly as t → 0 to a fixed point of S which solves the variational inequality:
Proof. The definition of {x t } is well definition. Indeed, From the definition of MKC, we can see MKC is also a non-expansive mapping. Consider a mapping S t on C defined by
It is easy to see that S t is a contraction, when Indeed, by Lemmas 2.7 and 2.8, we have
where θ t = tγ + τ t ∈ (0,1). Hence, S t has a unique fixed point, denoted by x t , which uniquely solves the fixed point equation
We next show the uniqueness of a solution of the variational inequality (2.1). Suppose both and are solutions to (2.1). Not lost generality, we may assume there is a positive number ε such that . Then, by Lemma 2.8, there is a number r ∈ (0,1) such that . From (2.1), we know
and
Adding up (2.2) and (2.3) gets
Noticing that
Therefore, and the uniqueness is proved. Below we use to denote the unique solution of (2.1).
We observe that {x t } is bounded. Indeed, We may assume . For ∀p ∈ F(S), fixed ε1, for each t
Case 1. ∥x t - p∥ < ε1; In this case, we can see easily that {x t } is bounded.
Case 2. ∥x t - p∥ ≥ ε1. In this case, by Lemma 2.8, there is a number r 1 ∈ (0,1) such that
therefore, . This implies the {x t } is bounded.
Next, we prove that as t → 0.
Since {x t } is bounded and H is reflexive, there exists a subsequence of {x t } such that . By x t - Sx t = t(γϕ(x t ) - FSx t ), we have , as t n → 0. It follows from Lemma 2.6 that x* ∈ F(S).
We claim
By contradiction, there is a number ε0 and a subsequence of such that . From Lemma 2.8, there is a number such that , we write
to derive that
It follows that
Therefore,
By (2.5), we get that . It is a contradiction. Hence, we have .
We next prove that x* solves the variational inequality (2.1). Since
we derive that
Notice
It follows that, for ∀z ∈ F(S),
Noticing
Hence, we have
Now replacing t in (2.6) with t n and letting n → ∞, noticing for x* ∈ F(S), we obtain 〈(F - γϕ)x*, x* - z〉 ≤ 0. That is, x* ∈ F(S) is a solution of (2.1); Hence, by uniqueness. We have shown that each cluster point of x t (at t → 0) equals . Therefore, as t → 0.
Lemma 2.10. Let H be a Hilbert space and C be a nonempty convex subset of H. Assume that T i : C → E is a countable family of k i -strict pseudo-contraction for some 0 ≤ k i < 1 and sup{k i : i ∈ ℕ} < 1 such that . Assume that {μ i } is a positive sequence such that . Then is a k-strict pseudo-contraction with k = sup{k i : i ∈ ℕ} and .
Proof. Let
and . Then, G n : C → E is a k i -strict pseudo-contraction with k = max{k i : 1 ≤ i ≤ n}. Indeed, we can firstly see the case of n = 2.
which shows that G2 : C → E is a k-strict pseudo-contraction with k = max{k i : i = 1,2}. By the same way, our proof method easily carries over to the general finite case.
Next, we prove the infinite case. From the definition of k-strict pseudo-contraction, we know
Hence, we can get
Taking p ∈ F(T n ), from 2.7 we have
Consequently, for ∀x ∈ H, if , μ i > 0 and , then strongly converges.
Let
we have
Hence,
So, we get T is k-strict pseudo-contraction.
Finally, we show . Suppose that , it is sufficient to show that . Indeed, for , we have
where k = sup{k i : i ∈ ℕ}. Hence, we get x = T i x, this means that .
3 Main results
Lemma 3.1. Let C be a closed convex subset of a real Hilbert space E such that C + C ⊂ C. Let ϕ be a MKC on C. Suppose F : C → C be a η-strongly monotone and L-Lipschitzian operator and 0 < γ < η and T i : C → E be k i -strictly pseudo-contractive non-self-mapping such that . Assume k = sup {k i : i ∈ ℕ} < 1. Let {x n } be a sequence of C generated by (1.5) with the sequences {α n }, {β n } and {γ n } in [0,1], assume for each n, is a infinity sequence of positive number such that .
The following control conditions are satisfied
(i)
(ii) k ≤ β n < 1,
(iii)
(iv) .
Then limn → ∞∥xn + 1- x n ∥ = 0.
Proof. Write, for each n ≥ 0, . By Lemma 2.10, each B n is a k-strict pseudo-contraction on C and for all n and the algorithm (1.5) can be rewritten as
The rest of the proof will now be split into two parts.
Step 1. First, we show that sequences {x n } and {y n } are bounded. Define a mapping
Then from the control condition (ii), Lemma 2.3, we obtain L n : C → C is non-expansive. Taking a point , by Lemma 2.2, we can get L n p = p. Hence, we have
Not lose generality, we can assume r n ≤ b < 1, and . From definition of MKC and Lemma 2.8, for any ε > 0 there is a number r ε ∈ (0, 1), if ∥x t - p∥ < ε then ∥ϕ(x t ) - ϕ(p)∥ < ε; If ∥x t -p∥ ≥ ε then ∥ϕ(x t ) - ϕ(p)∥ ≤ r ε ∥x t -p∥. It follows 3.1 and Lemma 2.7 that
By induction, we have
Hence, we have
which gives that the sequence {x n } is bounded, so are {y n } and {L n x n }.
Step 2. In this part, we shall claim that ∥xn + 1- x n ∥ → 0, as n → ∞. From(3.1), we get
Define
where
It follows that
which yields that
Next, we estimate ∥Ln + 1x n - L n x n ∥. Notice that
Substituting (3.5) into (3.4), we have
Hence, we have
Observing conditions (i), (iii), (iv), and the boundedness of {x n } it follows that
Thus by Lemma 2.4, we have limn → ∞∥l n - x n ∥ = 0.
From (3.3), we have
Therefore,
Theorem 3.2.. Let H be a real Hilbert space and C is a closed convex subset of H such that C + C ⊂ C. Let ϕ be an MKC on C. Suppose F: C → C is η-strongly monotone and L-Lipschitzian operator and η > γ > 0. Let T i : C → E be k i -strictly pseudo-contractive non-self-mapping such that . Assume k = sup{k i : i ∈ ℕ} < 1. Let {x n } be a sequence of C generated by (1.5) with the sequences {α n }, {β n } and {γ n } in [0,1]. Assume for each n, for all n and for all i ∈ ℕ. They satisfy the conditions (i), (ii), (iii), (iv) of Lemma 3.1 and (v) limn → ∞β n = α, and . Then {x n } converges strongly to , which also solves the following variational inequality
Proof. From (3.1), we obtain
So , which together with the condition (i), (iv) and Lemma 3.1 implies
Define , then B : C → E is a k-strict pseudo-contraction such that by Lemma 2.10, furthermore B n x → Bx as n → ∞ for all x ∈ C by (v). Defines T: C → E by
Then, T is non-expansive with F(T) = F(B) by Lemma 2.3. It follows from Lemma 2.2 that F(P C T) = F(T). Notice that
which combines with (3.7) yielding that
Next, we show that
where with x t being the fixed point of the contraction
To see this, we take a subsequence of {x n } such that
We may also assume that . Note that q ∈ F(T) in virtue of Lemmas 2.6, 2.2, and (3.8). It follows from Lemma 2.9, we can get that
Finally, we show . By contradiction, there is a number ε0 such that
Case 1. Fixed ε1 (ε1 < ε0), if for some n ≥ N ∈ ℕ such that , and for the other n ≥ N ∈ ℕ such that .
Let
From 3.9, we know lim supn → ∞M n ≤ 0. Hence, there are two numbers h and N, when n > N we have M n ≤ h, where . From the above introduction, we can extract a number n0 > N satisfying , then we estimate . From Lemma 2.7 and (3.1), we have
which implies that
Hence, we have
In the same way, we can get
It contradict the .
Case 2. Fixed ε1 (ε1 < ε0), if for all n ≥ N ∈ ℕ. In this case from Lemma 2.8, there is a number r ∈ (0,1), such that
It follow 3.1 that
which implies that
Apply Lemma 2.5 to (3.10) to conclude as n → ∞. It contradict the . This completes the proof.
Remark 3. We conclude the article with the following observations.
-
(i)
Theorem 3.2 improve and extend Theorem 3.4 of Marino and Xu [24], Theorem 3.2 of Zhou [8], Theorem 2.1 of Qin [9] and includes those results as special cases. Especially, our results extends above results form contractions to more general MKC. Our iterative scheme studied in this article can be viewed as a refinement and modification of the iterative methods in [8, 9, 24]. On the other hand, our iterative schemes concern an infinite countable family of k i -strict pseudo-contractions mappings, in this respect, they can be viewed as an another improvement.
-
(ii)
Our results extend above results form strong positive linear bounded operator to η-strongly monotone and L-Lipschitzian operator.
-
(iii)
The advantage of the results in this article is that less restrictions on the parameters {α n }, {β n }, {γ n } and are imposed. Our results unify many recent results including the results in [8, 9, 24].
-
(iv)
It is worth noting that we obtained two strong convergence results concerning an infinite countable family of λ i -strict pseudo-contractions mappings. Our result is new and the proofs are simple and different from those in [6, 8, 9, 24, 25].
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