Open Access

Convergence analysis of projection methods for a new system of general nonconvex variational inequalities

Fixed Point Theory and Applications20122012:59

https://doi.org/10.1186/1687-1812-2012-59

Received: 23 July 2011

Accepted: 13 April 2012

Published: 13 April 2012

Abstract

In this article, we introduce and consider a new system of general nonconvex variational inequalities defined on uniformly prox-regular sets. We establish the equivalence between the new system of general nonconvex variational inequalities and the fixed point problems to analyze an explicit projection method for solving this system. We also consider the convergence of the projection method under some suitable conditions. Results presented in this article improve and extend the previously known results for the variational inequalities and related optimization problems.

MSC (2000): 47J20; 47N10; 49J30.

Keywords

system of general nonconvex variational inequalitiesexplicit projection methodsuniform prox-regular setr-strongly monotone mappingsμ-Lipschitz continuous.

1 Introduction

Variational inequalities theory, which was introduced by Stampacchia [1], has emerged as an interesting and fascinating branch of mathematical and engineering sciences. The ideas and techniques of variational inequalities are being applied in structural analysis, economics, optimization, operations research fields. It has been shown that variational inequalities provide the most natural, direct, simple, and efficient framework for a general treatment of some unrelated problems arising in various fields of pure and applied sciences. In recent years, there have been considerable activities in the development of numerical techniques including projection methods, Wiener-Hopf equations, auxiliary principle, and descent framework for solving variational inequalities; see [217] and the references therein. These activities have motivated us to generalize and extend the variational inequalities and related optimization problems in several directions using novel techniques.

Projection technique has played a significant role in the numerical solution of variational inequalities based on the convergence analysis. It is worth mentioning that almost all the results regarding the existence and iterative schemes for variational inequalities, which have been investigated and considered, if the underlying set is a convex set. This is because all the techniques are based on the properties of the projection operator over convex sets, which may not hold in general, when the sets are nonconvex. Recently, Clarke et al. [8] and Poliquin et al. [9] have introduced and studied a new class of nonconvex sets, which are called uniformly prox-regular sets. It is known that uniformly prox-regular sets are nonconvex sets and include convex sets as a special case. This class of uniformly prox-regular sets has played an important part in many nonconvex applications such as optimization, dynamical systems, and differential inclusions.

In 2009, Noor [10] introduced a nonconvex variational inequalities based on the uniformly prox-regular sets. Moreover, he discussed the existence and algorithm of the solution for the nonconvex variational inequalities, which shows projection technique can be extended to nonconvex sets. Noor [11] proposed some iterative methods for solving a general nonconvex variational inequalities with projection methods and Wiener-Hopf equations technique. On the other hand, Verma [13] and Noor and Noor [14] proposed explicit projection methods for solving systems of variational inequalities and general variational inequalities on a closed convex subset of Hilbert space, respectively. Very recently, Wen [15] modified projection methods to a generalized system of nonconvex variational inequalities with different nonlinear operators. However, only iterative sequences {g(x n )}, {g(y n )} come from the projection methods, which requires that mapping g must be injective in order to arrive at a solution of the generalized system. Furthermore, the property defined on the underlying operator T depends on the mapping g in convergence analysis. These strict conditions rule out many applications of the projection type methods for the generalized system of nonconvex variational inequalities.

In this article, motivated and inspired by the research going on in this direction, we introduce and consider a more general system, which is called a new general nonconvex variational inequalities. The new system includes the system of variational inequalities involving two different nonlinear operators, the general nonconvex variational inequalities and the systems of variational inequalities defined on closed convex sets as special cases.

The purpose of this article is not only to show that projection technique can be extended to the new system of general nonconvex variational inequalities on uniformly prox-regular sets, but also to get rid of the dependence of T on the mapping g and the injective property defined on g in convergence analysis of the projection method for solving the new system of general nonconvex variational inequalities. Our results extend and improve the corresponding results of [7, 1015].

2 Preliminaries

Let H be a real Hilbert space whose inner product and norm are denoted by 〈.,.〉 and ., respectively. Let K be a nonempty and convex subset in H.

First of all, we recall the following well-known concepts from nonlinear convex analysis and non-smooth analysis [8, 9].

Definition 2.1. The proximal normal cone of K at u H is given by
N K P ( u ) := { ξ H : u P K [ u + α ξ ] } ,
where α > 0 is a constant and
P K [ u ] = { u * K : d K ( u ) = u - u * } .

Here d K (.) is the usual distance function to the subset K, that is d K (u) = infvKv - u. The proximal normal cone N K P ( u ) has the following characterization.

Lemma 2.1. Let K be a nonempty, closed, and convex subset in H. Then ς N K P ( u ) if and only if there exists a constant α = α(ζ, u) > 0 such that
ς , v - u α v - u 2 , v K .
Definition 2.2. The Clarke normal cone, denoted by N K C ( u ) , is defined as
N K C ( u ) = c o ¯ [ N K P ( u ) ] ,

where c o ¯ means the closure of the convex hull. Clearly N K P ( u ) N K C ( u ) , but the converse is not true. Note that N K C ( u ) is always closed and convex, whereas N K P ( u ) is convex, but may not be closed [9].

Definition 2.3. For a given r (0, ∞], a subset K r is said to be normalized uniformly r-prox-regular if and only if every nonzero proximal normal to K r can be realized by an r-ball, that is, u K r and 0 ξ N K r P ( u ) , one has
ξ ξ , v - u 1 2 r v - u 2 , v K r .

It is clear that the class of normalized uniformly prox-regular sets is sufficiently large to include the class of convex sets, p-convex sets, C1,1 submanifolds (possibly with boundary) of H, the images under a C1,1 diffeomorphism of convex sets and many other nonconvex sets; see [8, 9]. It is known that if K r is a uniformly prox-regular set, then the proximal normal cone N K r P ( u ) is closed as a set-valued mapping. Thus, we have N K r P ( u ) = N K r C ( u ) .

Remark 2.1. It is clear that if r = ∞, then uniformly prox-regularity of K r is equivalent to the convexity of K, that is, K r = K.

Let K r be a uniformly r-prox-regular (nonconvex) set, and T1, T2 : K r × K r K r and g, h : HK r be different nonlinear operators, respectively. For any given constants ρ > 0 and η > 0, we consider the problem of finding x*, y* K r such that
ρ T 1 ( y * , x * ) + x * - g ( y * ) , g ( x ) - x * + 1 2 r g ( x ) - x * 2 0 , x H : g ( x ) K r ,
(2.1a)
η T 2 ( x * , y * ) + y * - h ( x * ) , h ( x ) - y * + 1 2 r h ( x ) - y * 2 0 , x H : h ( x ) K r ,
(2.1b)

which is called a new system of general nonconvex variational inequalities.

If g = h = I, the identity operator, then problem (2.1) is equivalent to finding x*, y* K r such that
ρ T 1 ( y * , x * ) + x * - y * , x - x * + 1 2 r x - x * 2 0 , x K r , ρ > 0 ,
(2.2a)
η T 2 ( x * , y * ) + y * - x * , x - y * + 1 2 r x - y * 2 0 , x K r , η > 0 ,
(2.2b)

which appears to be the other new system of nonconvex variational inequalities.

We note that, if r = ∞, K r = K, the convex subset in H, then problem (2.1) is equivalent to finding x*, y* K such that
ρ T 1 ( y * , x * ) + x * - g ( y * ) , g ( x ) - x * 0 , x H : g ( x ) K , ρ > 0 ,
(2.3a)
η T 2 ( x * , y * ) + y * - h ( x * ) , h ( x ) - y * 0 , x H : h ( x ) K , η > 0 ,
(2.3b)

which is known as the system of general variational inequalities involving four different nonlinear operators, introduced, and studied by Noor and Noor [14].

If g = h = I, T1, T2 : KK are two univariate nonlinear operators, then problem (2.3) is equivalent to finding x*, y* K such that
ρ T 1 ( y * ) + x * - y * , x - x * 0 , x K , ρ > 0 ,
(2.4a)
η T 2 ( x * ) + y * - x * , x - y * 0 , x K , η > 0 ,
(2.4b)

which is known as the system of nonlinear variational inequalities involving two different nonlinear operators. If T1 = T2, problem (2.4) reduces to the system of variational inequalities, which was introduced and studied by Verma [13].

It is worth mentioning that if T1 = T2 = T : K r K r is a univariate nonlinear operator, and x* = y* = u, then problem (2.2) reduces to finding u K r such that
T u , v - u + 1 2 r v - u 2 0 , v K r ,
(2.5)
which is more general than the normal nonconvex variational inequality, introduced and studied by Bounkhel et al. [3] and Noor [7, 10], that is
T u , v - u 0 , v K r .
(2.6)
It is well known that problem (2.6) is equivalent to finding u K r such that
0 T u + N K r P ( u ) ,
(2.7)

where N K r P ( u ) denotes the normal cone of K r at u in the sense of nonconvex analysis. Problem (2.7) is called the variational inclusion associated with nonconvex variational inequality (2.6), which implies that the nonconvex variational inequality is equivalent to finding a zero of the sum of two monotone operators. This equivalent formulation plays a crucial and basic part in this article, which allows us to use the projection operator technique for solving the general system of nonconvex variational inequalities (2.1).

We now recall the well-known properties of the uniform prox-regular sets [810, 15].

Lemma 2.2. Let K be a nonempty closed subset of H, r (0, ∞] and set K r = {u H : d(u, K) < r}. If K r is uniformly prox-regular, then
  1. (i)

    u K r , P K r ( u ) 0 . .

     
  2. (ii)

    r ( 0 , r ) , P K r is Lipschitz continuous with constant δ = r r - r on K r' .

     
  3. (iii)

    The proximal normal cone is closed as a set-valued mapping.

     
Lemma 2.3[18]. Assume {α n } is a sequence of nonnegative real numbers such that
α n + 1 ( 1 - γ n ) α n + σ n , n 0 ,
where {γ n } is a sequence in (0,1) and {σ n } is a sequence in R such that
  1. (i)

    n = 0 γ n = ;

     
  2. (ii)

    lim sup n σ n / γ n 0 or n = 0 σ n < .

     

Then limn→∞a n = 0.

Definition 2.4. An operator T : HH is said to be r-strongly monotone, if there exists a constant r > 0 such that
T x - T y , x - y r x - y 2 , x , y H .
Definition 2.5. An operator T : HH is said to be μ-Lipschitz continuous, if there exists a constant μ > 0 such that
T x - T y μ x - y , x , y H .

Remark 2.2. As T = I, the identity operator is 1-strongly monotone and 1-Lipschitz continuous.

Remark 2.3. Obviously, whenever operator T is r-strongly monotone and μ-Lipschitz continuous, it follows that μr.

3 Projection methods

In this section, we establish the equivalence between the new system of general nonconvex variational inequalities (2.1) and the fixed point problem with the projection technique. This alternative formulation enable us to suggest and analyze an explicit projection method for solving system (2.1).

Lemma 3.1. x*, y* K r is a solution of the system of general nonconvex variational inequalities (2.1), if and only if
x * = P K r [ g ( y * ) - ρ T 1 ( y * , x * ) ] ,
(3.1a)
y * = P K r [ h ( x * ) - η T 2 ( x * , y * ) ] ,
(3.1b)

where P K r is the projection of H onto the uniformly prox-regular set K r .

Proof. Let x*, y* K r be a solution of (2.1). From (2.7), we have that the problem (2.1a) is equivalent to that
0 ρ T 1 ( y * , x * ) + x * - g ( y * ) + N K r P ( x * ) .
(3.2)
where N K r P ( x * ) is proximal normal cone of K r at x* in the sense of nonconvex analysis. Indeed, if ρT1(y*, x*) + x* - g(y*) = 0, because the vector zero always belongs to any normal cone, then (3.2) is valid. If ρT1(y*, x*) + x* - g(y*) ≠ 0, then for all x H : g(x) K r , it follows from (2.1a) that
- ( ρ T 1 ( y * , x * ) + x * - g ( y * ) ) , g ( x ) - x * 1 2 r g ( x ) - x * 2 .
By using Lemma 2.1, we obtain
- ( ρ T 1 ( y * , x * ) + x * - g ( y * ) ) N K r P ( x * )

and so (3.2) holds also. Consequently, the general nonconvex variational inequality (2.1a) is equivalent to (3.2), which is called variational inclusion associated with the problem (2.1a).

On the other hand, (3.2) can be written as
g ( y * ) - ρ T 1 ( y * , x * ) x * + N K r P ( x * ) = ( I + N K r P ) x * ,
where I is identity operator. Moreover, we have
x * = P K r [ g ( y * ) - ρ T 1 ( y * , x * ) ] ,

where we have used the well-known fact that P K r = ( I + ρ N K r P ) - 1 . In a similar way, we can obtain (3.1b). This proves our assertions. □

Algorithm 3.1. For arbitrarily chosen initial points x0, y0 K r , compute the sequences {x n } and {y n } in the following way:
x n + 1 = ( 1 - α n ) x n + α n P K r [ g ( y n ) - ρ T 1 ( y n , x n ) ] , ρ > 0 ,
(3.3a)
y n + 1 = ( 1 - β n ) x n + 1 + β n P K r [ h ( x n + 1 ) - η T 2 ( x n + 1 , y n ) ] , η > 0 ,
(3.3b)

where {α n }, {β n } are two sequences in [0,1].

If β n = 1, K r = K, then Algorithm 3.1 reduces to the following explicit projection method for solving the system of variational inequalities (2.3), which is mainly due to Noor and Noor [14]:

Algorithm 3.2. For arbitrarily chosen initial points x0, y0 K, compute the sequences {x n } and {y n } in the following way:
x n + 1 = ( 1 α n ) x n + α n P K [ g ( y n ) ρ T 1 ( y n , x n ) ] , ρ > 0 , y n + 1 = P K [ h ( x n + 1 ) η T 2 ( x n + 1 , y n ) ] , η > 0 ,

where {α n }, {β n } are two sequences in [0,1].

If g = h = I, K r = K, and T1, T2 : KK are two univariate nonlinear operators, then Algorithm 3.1 reduces to the following explicit projection method for solving the system of variational inequalities (2.4):

Algorithm 3.3. For arbitrarily chosen initial points x0, y0 K, compute the sequences {x n } and {y n } in the following way:
x n + 1 = ( 1 α n ) x n + α n P K [ y n ρ T 1 ( y n ) ] , ρ > 0 , y n + 1 = ( 1 β n ) x n + 1 + β n P K [ x n + 1 η T 2 ( x n + 1 ) ] , η > 0 ,

where {α n }, {β n } are two sequences in [0,1]. Algorithm 3.3 extends and improves the two-step projection methods of Verma [13].

If g = h, T1 = T2 = T is the univariate nonlinear operator, we again use the fixed point formulation (3.1) to suggest and analyze the following explicit projection method, known as Mann iteration:

Algorithm 3.4. For arbitrarily chosen initial points x0 K r , compute the sequence {x n } in the following way:
x n + 1 = ( 1 - α n ) x n + α n P K r [ g ( x n ) - ρ T x n ] , ρ > 0 ,

where {α n } is a sequence in [0,1].

Remark 3.1. Algorithm 3.4 includes the projection methods of Noor [10] as special cases.

4 Main results

We now consider the convergence analysis of Algorithm 3.1, and this is the main motivation of our next result. In a similar way, we consider the convergence criteria of other algorithms.

Theorem 4.1. Let P K r be a Lipschitz continuous operator with constant δ = r r - r . Let T i : K r × K r K r be r i -strongly monotone and μ i -Lipschitz continuous in the first variable, i = 1, 2, and g, h : K r K r be strongly monotone with constants r3, r4 and Lipschitz continuous with constants μ3, μ4, respectively. If there exist constants ρ, η > 0 such that
ρ - r 1 μ 1 2 < δ 2 r 1 2 - μ 1 2 [ δ 2 - ( 1 - δ k 1 ) 2 ] δ μ 1 2 , δ r 1 > μ 1 δ 2 - ( 1 - δ k 1 ) 2 , δ k 1 < 1 ,
(4.1)
η - r 2 μ 2 2 < δ 2 r 2 2 - μ 2 2 [ δ 2 - ( 1 - δ k 2 ) 2 ] δ μ 2 2 , δ r 2 > μ 2 δ 2 - ( 1 - δ k 2 ) 2 , δ k 2 < 1 ,
(4.2)
where
k 1 = 1 - 2 r 3 + μ 3 2 , k 2 = 1 - 2 r 4 + μ 4 2 ,

and α n , β n [0, 1], n = 0 α n = , n = 0 ( 1 - β n ) < , then the sequences {x n } and {y n } obtained from Algorithm 3.1 converges to a solution of the system of general nonconvex variational inequalities (2.1), respectively.

Proof. Let x*, y* K r be a solution of (2.1). From (3.1a) and (3.3a) and the Lipschitz continuous property of operator P K r , we can obtain
x n + 1 - x * = ( 1 - α n ) ( x n - x * ) + α n { P K r [ g ( y n ) - ρ T 1 ( y n , x n ) ] - P K r [ g ( y * ) - ρ T 1 ( y * , x * ) ] } ( 1 - α n ) x n - x * + α n P K r [ g ( y n ) - ρ T 1 ( y n , x n ) ] - P K r [ g ( y * ) - ρ T 1 ( y * , x * ) ] ( 1 - α n ) x n - x * + α n δ g ( y n ) - g ( y * ) - ρ [ T 1 ( y n , x n ) - T 1 ( y * , x * ) ] ( 1 - α n ) x n - x * + α n δ g ( y n ) - g ( y * ) - ( y n - y * ) + α n δ ( y n - y * ) - ρ [ T 1 ( y n , x n ) - T 1 ( y * , x * ) ] .
(4.3)
Since the operator T1 is r1-strongly monotone and μ1-Lipschitz continuous definition in the first variable, it follows that
y n y * ρ [ T 1 ( y n , x n ) T 1 ( y * , x * ) ] 2 = y n y * 2 2 ρ T 1 ( y n , x n ) T 1 ( y * , x * ) , y n y * + ρ 2 T 1 ( y n , x n ) T 1 ( y * , x * ) 2 y n y * 2 2 ρ r 1 y n y * 2 + ρ 2 T 1 ( y n , x n ) T 1 ( y * , x * ) 2 ( 1 2 ρ r 1 + ρ 2 μ 1 2 ) y n y * 2 ,
(4.4)
which implies that
y n - y * - ρ [ T 1 ( y n , x n ) - T 1 ( y * , x * ) ] 1 - 2 ρ r 1 + ρ 2 μ 1 2 y n - y * .
(4.5)
In a similar way, we have (note that μ3r3, from Remark 2.3)
g ( y n ) - g ( y * ) - ( y n - y * ) 1 - 2 r 3 + μ 3 2 y n - y * .
(4.6)
Consequently, from (4.3), (4.5), and (4.6), we have
x n + 1 - x * ( 1 - α n ) x n - x * + α n δ k 1 + 1 - 2 ρ r 1 + ρ 2 μ 1 2 y n - y * = ( 1 - α n ) x n - x * + α n θ 1 y n - y * ,
(4.7)

where θ 1 = δ ( k 1 + 1 - 2 ρ r 1 + ρ 2 μ 1 2 ) , k 1 = 1 - 2 r 3 + μ 3 2 . From (4.1), we obtain that θ1 (0, 1).

On the other hand, it follows from (3.1b) and (3.3b) that
y n + 1 - y * ( 1 - β n ) x n + 1 - y * + β n P K r [ h ( x n + 1 ) - η T 2 ( x n + 1 , y n ) ] - P K r [ h ( x * ) - η T 2 ( x * , y * ) ] ( 1 - β n ) x n + 1 - y * + β n δ h ( x n + 1 ) - h ( x * ) - η [ T 2 ( x n + 1 , y n ) - T 2 ( x * , y * ) ] ( 1 - β n ) x n + 1 - x * + ( 1 - β n ) x * - y * + β n δ h ( x n + 1 ) - h ( x * ) - ( x n + 1 - x * ) + β n δ x n + 1 - x * - η [ T 2 ( x n + 1 , y n ) - T 2 ( x * , y * ) ] .
(4.8)
Similarly, from the properties defined on T2 in the first variable, we have
x n + 1 x * η [ T 2 ( x n + 1 , y n ) T 2 ( x * , y * ) ] 2 = x n + 1 x * 2 2 η T 2 ( x n + 1 , y n ) T 2 ( x * , y * ) , x n + 1 x * + η 2 T 2 ( x n + 1 , y n ) T 2 ( x * , y * ) 2 x n + 1 x * 2 2 η r 2 x n + 1 x * 2 + η 2 T 2 ( x n + 1 , y n ) T 2 ( x * , y * ) 2 ( 1 2 η r 2 + η 2 μ 2 2 ) x n + 1 x * 2 ,
(4.9)
and
h ( x n + 1 ) - h ( x * ) - ( x n + 1 - x * ) 2 ( 1 - 2 r 4 + μ 4 2 ) x n + 1 - x * 2
(4.10)
Moreover, from (4.8)-(4.10), we have
y n + 1 - y * ( 1 - β n ) x n + 1 - x * + ( 1 - β n ) x * - y * + β n δ k 2 + 1 - 2 η r 2 + η 2 μ 2 2 x n + 1 - x * ( 1 - β n ) x n + 1 - x * + ( 1 - β n ) x * - y * + β n θ 2 x n + 1 - x * ,
(4.11)
where θ 2 = δ ( k 2 + 1 - 2 η r 2 + η 2 μ 2 2 ) , k 2 = 1 - 2 r 4 + μ 4 2 . From (4.2), we obtain that θ2 (0, 1). For all n ≥ 1, it follows from (4.11) that
y n - y * ( 1 - β n - 1 ) x n - x * + 1 - β n - 1 x * - y * + β n - 1 θ 2 x n - x * = 1 - 1 - θ 2 β n - 1 x n - x * + 1 - β n - 1 x * - y * .
(4.12)
Substituting (4.12) into (4.7), we have (note that θ1, θ2 (0, 1))
x n + 1 - x * ( 1 - α n ) x n - x * + α n θ 1 1 - 1 - θ 2 β n - 1 x n - x * + 1 - β n - 1 x * - y * 1 - α n x n - x * + α n θ 1 x n - x * + α n 1 - β n - 1 θ 1 x * - y * 1 - 1 - θ 1 α n x n - x * + α n 1 - β n - 1 x * - y * .
(4.13)
Since 1 - θ 1 > 0 , n = 0 α n = and n = 0 ( 1 - β n ) < , we apply Lemma 2.3 to get
lim n x n - x * = 0 .
(4.14)
Combining (4.12) and (4.14), we have
lim n y n - y * = 0 .
(4.15)

It follows that lim n x n = x*, lim n y n = y*, satisfying the general system of nonconvex variational inequalities (2.1). This completes the proof. □

Theorem 4.2. Let K be a nonempty and convex subset of Hilbert space H. Let T i : K × KK be r i -strongly monotone and μ i -Lipschitz continuous in the first variable, i = 1, 2, and g, h : KK be strongly monotone with constants r3, r4 and Lipschitz continuous with constants μ3, μ4, respectively. If there exist constants ρ, η > 0 such that
| ρ r 1 μ 1 2 | < r 1 2 μ 1 2 ( 2 k 1 k 1 2 ) μ 1 2 , r 1 > μ 1 2 k 1 k 1 2 , k 1 < 1 ,
(4.16)
| η r 2 μ 2 2 | < r 2 2 μ 2 2 ( 2 k 2 k 2 2 ) μ 2 2 , r 2 > μ 2 2 k 2 k 2 2 , k 2 < 1 ,
(4.17)
where
k 1 = 1 - 2 r 3 + μ 3 2 , k 2 = 1 - 2 r 4 + μ 4 2 ,

and α n [0, 1], n = 0 α n = , then the sequences {x n } and {y n } obtained from Algorithm 3.2 converges to a solution of the system of general variational inequalities (2.3), respectively.

Proof. If K r = K, β n = 1, Algorithm 3.1 reduces to Algorithm 3.2. Moreover, we can obtain r = ∞ and δ = 1 from Remark 2.1, and n = 0 ( 1 - β n ) = 0 . Then the conclusion follows immediately from Theorem 4.1. This completes the proof. □

Theorem 4.3. Let K be a nonempty and convex subset of Hilbert space H, and T i : KK be r i -strongly monotone and μ i -Lipschitz continuous, i = 1, 2. If there exist constants ρ, η such that
0 < ρ < 2 r 1 μ 1 2 , 0 < η < 2 r 2 μ 2 2 .
(4.18)

and α n , β n [0, 1], n = 0 α n = , n = 0 ( 1 - β n ) < , then the sequences {x n } and {y n } obtained from Algorithm 3.3 converges to a solution of the system of variational inequalities (2.4), respectively.

Proof. If g = h = I, K r = K, Algorithm 3.1 reduces to Algorithm 3.3. Moreover, we can obtain r = ∞ and δ = 1 from Remark 2.1, and k1 = k2 = 0 from Remark 2.2 (r3 = μ3 = r4 = μ4 = 1). Then the conclusion follows immediately from Theorem 4.1. This completes the proof. □

Declarations

Acknowledgements

This study was supported by the National Science Foundation of China (11001287), Natural Science Foundation Project of Chongqing (CSTC 2010BB9254) and Science and Technology Research Project of Chongqing Municipal Education Commission (KJ 110701).

Authors’ Affiliations

(1)
College of Mathematics and Statistics, Chongqing Technology and Business University
(2)
College of Computer Science and Information Engineering, Chongqing Technology and Business University

References

  1. Stampacchia G: Formes bilinearies coercivities sur les ensembles convexes. C R Acad Sci Paris 1964, 258: 4413–4416.MATHMathSciNetGoogle Scholar
  2. Noor MA: Some developments in general variational inequalities. Appl Math Comput 2004, 152: 199–277. 10.1016/S0096-3003(03)00558-7MATHMathSciNetView ArticleGoogle Scholar
  3. Bounkhel M, Tadj L, Hamdi A: Iterative schemes to solve nonconvex variational problems. J Inequal Pure Appl Math 2003, 4: 1–14.MathSciNetGoogle Scholar
  4. Noor MA: On implicit methods for nonconvex variational inequalities. J Optim Theory Appl 2010, 147: 411–417. 10.1007/s10957-010-9717-yMATHMathSciNetView ArticleGoogle Scholar
  5. Pang LP, Shen J, Song HS: A modified predictor-corrector algorithm for solving non-convex generalized variational inequalities. Comput Math Appl 2007, 54: 319–325. 10.1016/j.camwa.2006.07.010MATHMathSciNetView ArticleGoogle Scholar
  6. Verma RU: Generalized system for relaxed cocoercive variational inequalities and projection methods. J Optim Theory Appl 2004, 121(1):203–210.MATHMathSciNetView ArticleGoogle Scholar
  7. Noor MA: Some iterative methods for nonconvex variational inequalities. Comput Math Model 2010, 21(1):97–108. 10.1007/s10598-010-9057-7MATHMathSciNetView ArticleGoogle Scholar
  8. Clarke FH, Ledyaev YS, Wolenski PR: Nonsmooth Analysis and Control Theory. Springer, Berlin; 1998.Google Scholar
  9. Poliquin RA, Rockafellar RT, Thibault L: Local differentiability of distance functions. Trans Am Math Soc 2000, 352: 5231–5249. 10.1090/S0002-9947-00-02550-2MATHMathSciNetView ArticleGoogle Scholar
  10. Noor MA: Projection methods for nonconvex variational inequalities. Optim Lett 2009, 3: 411–418. 10.1007/s11590-009-0121-1MATHMathSciNetView ArticleGoogle Scholar
  11. Noor MA: Iterative methods for general nonconvex variational inequalities. Albanian J Math 2009, 3(1):117–127.MATHMathSciNetGoogle Scholar
  12. Huang Z, Noor MA: An explicit projection method for a system of nonlinear variational inequalities with different ( γ , r )-cocoercive mappings. Appl Math Comput 2007, 190: 356–361. 10.1016/j.amc.2007.01.032MATHMathSciNetView ArticleGoogle Scholar
  13. Verma RU: General Convergence analysis for two-step projection methods and applications to variational problems. Appl Math Lett 2005, 18: 1286–1292. 10.1016/j.aml.2005.02.026MATHMathSciNetView ArticleGoogle Scholar
  14. Noor MA, Noor KI: Projection algorithms for solving system of general variational inequalities. Nonlinear Anal 2009, 70: 2700–2706. 10.1016/j.na.2008.03.057MATHMathSciNetView ArticleGoogle Scholar
  15. Wen DJ: Projection methods for a generalized system of nonconvex variational inequalities with different nonlinear operators. Nonlinear Anal 2010, 73: 2292–2297. 10.1016/j.na.2010.06.010MATHMathSciNetView ArticleGoogle Scholar
  16. Ceng LC, Teboulle M, Yao JC: Weak convergence of an iterative method for pseudomonotone variational inequalities and fixed-point problems. J Optim Theory Appl 2010, 146: 19–31. 10.1007/s10957-010-9650-0MATHMathSciNetView ArticleGoogle Scholar
  17. Wen DJ: Strong convergence theorems for equilibrium problems and k -strict pseudo-contractions in Hilbert spaces. Abstract Appl Anal 2011., 2011: Article ID 276874, doi:10.1155/2011/276874Google Scholar
  18. Xu HK: Iterative algorithms for nonlinear operators. J Lond Math Soc 2002, 2: 240–256.View ArticleGoogle Scholar

Copyright

© Wen et al; licensee Springer. 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.