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Convergence theorems of solutions of a generalized variational inequality
Fixed Point Theory and Applications volume 2011, Article number: 19 (2011)
Abstract
The convex feasibility problem (CFP) of finding a point in the nonempty intersection is considered, where r ≥ 1 is an integer and each C_{ m } is assumed to be the solution set of a generalized variational inequality. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A_{ m }, B_{ m } : C → H be relaxed cocoercive mappings for each 1 ≤ m ≤ r. It is proved that the sequence {x_{ n }} generated in the following algorithm:
where u ∈ C is a fixed point, {α_{ n }}, {β_{ n }}, {γ_{ n }}, {δ_{(1,n)}}, ..., and {δ_{(r,n)}} are sequences in (0, 1) and , are positive sequences, converges strongly to a solution of CFP provided that the control sequences satisfies certain restrictions.
2000 AMS Subject Classification: 47H05; 47H09; 47H10.
1. Introduction and Preliminaries
Many problems in mathematics, in physical sciences and in realworld applications of various technological innovations can be modeled as a convex feasibility problem (CFP). This is the problem of finding a point in the intersection of finitely many closed convex sets in a real Hilbert spaces H. That is,
where r ≥ 1 is an integer and each C_{ m } is a nonempty closed and convex subset of H. There is a considerable investigation on CFP in the setting of Hilbert spaces which captures applications in various disciplines such as image restoration [1, 2], computer tomography [3] and radiation therapy treatment planning [4].
Throughout this paper, we always assume that H is a real Hilbert space, whose inner product and norm are denoted by 〈·, ·〉 and ·. Let C be a nonempty closed and convex subset of H and A: C → H a nonlinear mapping. Recall the following definitions:

(a)
A is said to be monotone if

(b)
A is said to be ρstrongly monotone if there exists a positive real number ρ > 0 such that

(c)
A is said to be ηcocoercive if there exists a positive real number η > 0 such that

(d)
A is said to be relaxed ηcocoercive if there exists a positive real number η > 0 such that

(e)
A is said to be relaxed (η, ρ)cocoercive if there exist positive real numbers η, ρ > 0 such that
The main purpose of this paper is to consider the following generalized variational inequality. Given nonlinear mappings A : C → H and B : C → H, find a u ∈ C such that
where λ and τ are two positive constants. In this paper, we use GV I(C, B, A) to denote the set of solutions of the generalized variational inequality (1.2).
It is easy to see that an element u ∈ C is a solution to the variational inequality (1.2) if and only if u ∈ C is a fixed point of the mapping P_{ C }(τB  λA), where P_{ C } denotes the metric projection from H onto C. Indeed, we have the following relations:
Next, we consider a special case of (1.2). If B = I, the identity mapping and τ = 1, then the generalized variational inequality (1.1) is reduced to the following. Find u ∈ C such that
The variational inequality (1.4) emerging as a fascinating and interesting branch of mathematical and engineering sciences with a wide range of applications in industry, finance, economics, social, ecology, regional, pure and applied sciences was introduced by Stampacchia [5]. In this paper, we use V I(C, A) to denote the set of solutions of the variational inequality (1.4).
Let S : C → C be a mapping. We use F(S) to denote the set of fixed points of the mapping S. Recall that S is said to be nonexpansive if
It is well known that if C is nonempty bounded closed and convex subset of H, then the fixed point set of the nonexpansive mapping S is nonempty, see [6] more details. Recently, fixed point problems of nonexpansive mappings have been considered by many authors; see, for example, [7–16].
Recall that S is said to be demiclosed at the origin if for each sequence {x_{ n }} in C, x_{ n } ⇀ x_{0} and Sx_{ n } → 0 imply Sx_{0} = 0, where ⇀ and → stand for weak convergence and strong convergence.
Recently, many authors considered the variational inequality (1.4) based on iterative methods; see [17–32]. For finding solutions to a variational inequality for a cocoercive mapping, Iiduka et al. [22] proved the following theorem.
Theorem ITT. Let C be a nonempty closed convex subset of a real Hilbert space H and let A be an αcocoercive operator of H into H with V I(C, A) ≠ ∅. Let {x_{ n }} be a sequence defined as follows. x_{1} = x ∈ C and
for every n = 1, 2, ..., where C is the metric projection from H onto C, {α_{ n }} is a sequence in [1, 1], and {λ_{ n }} is a sequence in [0, 2α]. If {α_{ n }} and {λ_{ n }} are chosen so that {α_{ n }} ∈ [a, b] for some a, b with 1 < a < b < 1 and {λ_{ n }} ∈ [c, d] for some c, d with 0 < c < d < 2(1 + a)α, then {x_{ n }} converges weakly to some element of V I(C, A).
Subsequently, Iiduka and Takahashi [23] further studied the problem of finding solutions of the classical variational inequality (1.4) for cocoercive mappings (inversestrongly monotone mappings) and nonexpansive mappings. They obtained a strong convergence theorem. More precisely, they proved the following theorem.
Theorem IT. Let C be a closed convex subset of a real Hilbert space H. Let S : C → C be a nonexpanisve mapping and A an αcocoercive mapping of C into H such that F(S) ∩ V I(C, A) ≠ ∅. Suppose x_{1} = u ∈ C and {x_{ n }} is given by
for every n = 1, 2, ..., where {α_{ n }} is a sequence in [0, 1) and {λ_{ n }} is a sequence in [a, b].
If {α_{ n }} and {λ_{ n }} are chosen so that {λ_{ n }} ∈ [a, b] for some a, b with 0 < a < b < 2α,
then {x_{ n }} converges strongly to P_{F(S)∩V I(C,A)}x.
In this paper, motivated by research work going on in this direction, we study the CFP in the case that each C_{ m } is a solution set of generalized variational inequality (1.2). Strong convergence theorems of solutions are established in the framework of real Hilbert spaces.
In order to prove our main results, we need the following lemmas.
Lemma 1.1 [33]. Let {x_{ n }} and {y_{ n }} be bounded sequences in a Hilbert space H and {β_{ n }} a sequence in (0, 1) with
Suppose that x_{n+1}= (1  β_{ n })y_{ n } + β_{ n }x_{ n } for all integers n ≥ 0 and
Then lim_{n→∞}y_{ n }  x_{ n } = 0.
Lemma 1.2 [34]. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let S_{1} : C → C and S_{2} : C → C be nonexpansive mappings on C. Suppose that F(S_{1}) ∩ F (S_{2}) is nonempty. Define a mapping S : C → C by
where a is a constant in (0, 1). Then S is nonexpansive with F(S) = F(S_{1}) ∩ F (S_{2}).
Lemma 1.3 [35]. Let C be a nonempty closed and convex subset of a real Hilbert space H and S : C → C a nonexpansive mapping. Then I  S is demiclosed at zero.
Lemma 1.4 [36]. Assume that {α_{ n }} is a sequence of nonnegative real numbers such that
where {γ_{ n }} is a sequence in (0, 1) and {δ_{ n }} is a sequence such that

(a)
;

(b)
lim sup_{n→∞}δ_{ n }/γ_{ n } ≤ 0 or .
Then lim_{n→∞}α_{ n } = 0.
2. Main results
Theorem 2.1. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A_{ m } : C → H be a relaxed (η_{ m }, ρ_{ m })cocoercive and μ_{ m }Lipschitz continuous mapping and B_{ m } : C → H a relaxed cocoercive and Lipschitz continuous mapping for each 1 ≤ m ≤ r. Assume that . Let {x_{ n }} be a sequence generated in the following manner:
where u ∈ C is a fixed point, {α_{ n }}, {β_{ n }}, {γ_{ n }}, {δ_{(1,n)}}, ..., and {δ_{(r,n)}} are sequences in (0, 1) satisfying the following restrictions:

(a)
;

(b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;

(c)
lim_{n→∞}α_{ n } = 0 and ;

(d)
lim_{n→∞}δ_{(m,n)}= δ_{ m }∈ (0, 1), ∀1 ≤ m ≤ r,
And , are two positive sequences such that

(e)
.
Then the sequence {x_{ n }} generated in the iterative process (ϒ) converges strongly to a common element , which uniquely solves the following variational inequality.
Proof. First, we prove that the mapping P_{ C }(τ_{ m }B_{ m }  λ_{ m }A_{ m }) is nonexpansive for each 1 ≤ m ≤ r. For each x, y ∈ C, we have
It follows from the assumption that each A_{ m } is relaxed (η_{ m }, ρ_{ m })cocoercive and μ_{ m }Lipschitz continuous that
where . This shows that
In a similar way, we can obtain that
where . Substituting (2.2) and (2.3) into (2.1), we from the condition (e) see that P_{ C }(τ_{ m }B_{ m }  λ_{ m }A_{ m }) is nonexpansive for each 1 ≤ m ≤ r. Put
Fixing , we see that
It follows that
By mathematical inductions we arrive at
Since the mapping P_{ C }(τ_{ m }B_{ m }  λ_{ m }A_{ m }) is nonexpansive for each 1 ≤ m ≤ r, we see that
where M is an appropriate constant such that
Put , for all n ≥ 1. That is,
Now, we estimate l_{n+1} l_{ n }. Note that
which combines with (2.4) yields that
It follows from the conditions (b), (c) and (d) that
It follows from Lemma 1.1 that lim_{n→∞}l_{ n }  x_{ n } = 0. In view of (2.5), we see that x_{ n }+1 x_{ n } = (1  β_{ n })(l_{ n }  x_{ n }). It follows that
On the other hand, from the iterative algorithm (ϒ), we see that x_{ n }+1  x_{ n } = α_{ n }(u  x_{ n }) + γ_{ n }(y_{ n }  x_{ n }). It follows from (2.6) and the conditions (b), (c) that
Next, we show that . To show it, we can choose a subsequence of {x_{ n }} such that
Since is bounded, we obtain that there exists a subsequence of which converges weakly to q. Without loss of generality, we may assume that . Next, we show that . Define a mapping R : C → C by
where δ_{ m } = lim_{n→∞}δ_{(m,n)}. From Lemma 1.2, we see that R is nonexpansive with
Now, we show that Rx_{ n }  x_{ n } → 0 as n → ∞. Note that
From the condition (d) and (2.7), we obtain that lim_{n→∞}Rx_{ n }  x_{ n } = 0. From Lemma 1.3, we see that
In view of (2.8), we arrive at
Finally, we show that as n  ∞. Note that
which implies that
From the condition (c), (2.9) and applying Lemma 1.4 to (2.10), we obtain that
This completes the proof.
If B_{ m } ≡ I, the identity mapping and τ_{ m } ≡ 1, then Theorem 2.1 is reduced to the following result on the classical variational inequality (1.4).
Corollary 2.2. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A_{ m } : C → H be a relaxed (η_{ m }, ρ_{ m })cocoercive and μ_{ m }Lipschitz continuous mapping for each 1 ≤ m ≤ r. Assume that . Let {x_{ n }} be a sequence generated by the following manner:
where u ∈ C is a fixed point, {α_{ n }}, {β_{ n }}, {γ_{ n }}, {δ_{(1,n)}}, ..., and {δ_{(r,n)}} are sequences in (0, 1) satisfying the following restrictions.

(a)
;

(b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;

(c)
lim_{n→∞}α_{ n } = 0 and ;

(d)
lim_{n→∞}δ_{(m,n)}= δ_{ m } ∈ (0, 1), ∀1 ≤ m ≤ r, and is a positive sequence such that

(e)
, ∀1 ≤ m ≤ r.
Then the sequence {x_{ n }} converges strongly to a common element , which uniquely solves the following variational inequality
If r = 1, then Theorem 2.1 is reduced to the following.
Corollary 2.3. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C → H be a relaxed (η, ρ)cocoercive and μLipschitz continuous mapping and B : C → H a relaxed cocoercive and Lipschitz continuous mapping. Assume that GV I(C, B, A) is not empty. Let {x_{ n }} be a sequence generated in the following manner:
where u ∈ C is a fixed point, {α_{ n }}, {β_{ n }} and {γ_{ n }} are sequences in (0, 1) satisfying the following restrictions.

(a)
α_{ n } + β_{ n } + γ_{ n } = 1, ∀_{ n } ≥ 1;

(b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;

(c)
lim_{n→∞}α_{ n } = 0 and

(d)
.
Then the sequence {x_{ n }} converges strongly to a common element , which uniquely solves the following variational inequality
For the variational inequality (1.4), we can obtain from Corollary 2.3 the following immediately.
Corollary 2.4. Let C be a nonempty closed and convex subset of a real Hilbert space H. Let A : C → H be a relaxed (η, ρ)cocoercive and μLipschitz continuous mapping. Assume that V I(C, A) is not empty. Let {x_{ n }} be a sequence generated in the following manner:
where u ∈ C is a fixed point, {α_{ n }}, {β_{ n }} and {γ_{ n }} are sequences in (0, 1) satisfying the following restrictions.

(a)
α_{ n } + β_{ n } + γ_{ n } = 1, ∀n ≥ 1;

(b)
0 < lim inf_{n→∞}β_{ n } ≤ lim sup_{n→∞}β_{ n } < 1;

(c)
lim_{n→∞}α_{ n }= 0 and ;

(d)
.
Then the sequence {x_{ n }} converges strongly to a common element , which uniquely solves the following variational inequality
Remark 2.5. In this paper, the generalized variational inequality (1.2), which includes the classical variational inequality (1.4) as a special case, is considered based on iterative methods. Strong convergence theorems are established under mild restrictions imposed on the parameters. It is of interest to extend the main results presented in this paper to the framework of Banach spaces.
Abbreviations
 CFP:

convex feasibility problem.
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Acknowledgements
This work was supported by the National Natural Science Foundation of China under Grant no. 70871081 and Important Science and Technology Research Project of Henan province, China (102102210022).
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LY designed and performed all the steps of proof in this research and also wrote the paper. ML participated in the design of the study. All authors read and approved the final manuscript.
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Yu, L., Liang, M. Convergence theorems of solutions of a generalized variational inequality. Fixed Point Theory Appl 2011, 19 (2011). https://doi.org/10.1186/16871812201119
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Keywords
 nonexpansive mapping
 fixed point
 relaxed cocoercive mapping
 variational inequality