Open Access

A new system of generalized quasi-variational-like inclusions with noncompact valued mappings

Journal of Inequalities and Applications20122012:41

https://doi.org/10.1186/1029-242X-2012-41

Received: 13 October 2011

Accepted: 23 February 2012

Published: 23 February 2012

Abstract

In this article, we introduce and study a new system of generalized quasi-variational-like inclusions with noncompact valued mappings. By using the η- proximal mapping technique, we prove the existence of solutions and the convergence of some new N-step iterative algorithms for this system of generalized quasi-variational-like inclusions. Our results extend and improve some known results in the literature.

Mathematics Subject Classification 2000: 49H09; 49J40; 49H10.

Keywords

system of generalized quasi-variational-like inclusionsη- proximal mappingmonotone operatoriterative algorithm

1 Introduction

Let H be a real Hilbert space, and CB(H) be the family of all nonempty bounded closed subsets of H. We will consider the following problem:

For i, j = 1, 2, . . . , N, let A ij : H → CB(H), η i : H × H → H, g i : H → H, T i : H × H × × H N H be nonlinear mappings, and let φ i : H → R {+} be real function.

Find x 1 * , x 2 * , , x N * H , u 11 * A 11 x 1 * , u 12 * A 12 x 2 * , , u 1 N * A 1 N x N * , , u N 1 * A N 1 x 1 * , u N 2 * A N 2 x 2 * , , u N N * A N N x N * such that
T i ( u i 1 * , u i 2 * , , u i N * ) , η i ( x , g i ( x i * ) ) φ i ( g i ( x i * ) ) - φ i ( x ) , x H , i = 1 , 2 , N .
(1.1)

Problem (1.1) is called the set-valued nonlinear generalized quasi-variational-like inclusions.

Various special cases of the problem (1.1) had been studied by many authors before. Here, we mention some of them as follows:
  1. (1)
    If N = 2, A 11 = A 12 = A, A 21 = A 22 = B, T 1 = T, T 1(A(·), B(·)) : H → CB(H), then the problem (1.1) reduces to find x* H, u* Ax*, v* Bx* such that
    T ( u * , v * ) , η ( x , g ( x * ) ) φ ( g ( x * ) ) - φ ( x ) , x H .
    (1.2)
     
Problem (1.2) was introduced and studied by Ding [1] in 2001.
  1. (2)
    If N = 2, A 11 = A 12 = A 21 = A 22 = I, g i = I (identical operator), η(x, y) = x-y, φ 1 = φ 2 = φ, T : H × H → H, T 1(A 11 x, A 12 y) = ρ 1 T (A 12 y, A 11 x) + A 11 x - A 12 y, T 2(A 21 x, A 22 y) = ρ 2 T (A 21 x, A 22 y) + A 22 y - A 21 x, then the problem (1.1) reduces to find x*, y* H such that
    ρ 1 T ( y * , x * ) + x * - y * , x - x * + φ ( x ) - φ ( x * ) 0 , x H , ρ 1 > 0 ; ρ 2 T ( x * , y * ) + y * - x * , x - y * + φ ( x ) - φ ( y * ) 0 , x H , ρ 2 > 0 .
    (1.3)
     
Problem (1.3) was studied by He and Gu [2] in 2009.
  1. (3)
    Let K H be a closed convex subset, φ(x) = I K (x), the problem (1.3) reduces to find x*, y* K such that
    ρ 1 T ( y * , x * ) + x * - y * , x - x * 0 , x K , ρ 1 > 0 ; ρ 2 T ( x * , y * ) + y * - x * , x - y * 0 , x K , ρ 2 > 0 .
    (1.4)
     
Problem (1.4) was inspected and studied by Chang [3], Verma [4, 5] and Huang [6].
  1. (4)
    If N = 2, A 11 = A 12 = A 21 = A 22 = I, g : HH, T 1(x, y) = ρ 1 Ty + g(x) - g(y), T 2(x, y) = ρ 2 Tx + g(y) - g(x), (ρ 1, ρ 2 > 0), η(x, y) = g(x) - g(y), then the problem (1.1) reduces to find x*, y* H such that
    { ρ 1 T y * + g ( x * ) g ( y * ) , g ( x ) g ( x * ) 0 , x H ; ρ 2 T x * + g ( y * ) g ( x * ) , g ( x ) g ( y * ) 0 , x H .
    (1.5)
     
Problem (1.5) was introduced and studied by Hajjafar and Verma [7].
  1. (5)
    If N = 3, η(x, y) = x - y, then the problem (1.1) reduces to find x 1 * , x 2 * , x 3 * H , u i 1 * A i 1 x 1 * , u i 2 * A i 2 x 2 * , u i 3 * A i 3 x 3 * ( i = 1 , 2 , 3 ) such that
    T 1 ( u 11 * , u 12 * , u 13 * ) , x - g 1 ( x 1 * ) φ 1 ( g 1 ( x 1 * ) ) - φ 1 ( x ) , x H ; T 2 ( u 21 * , u 22 * , u 23 * ) , x - g 2 ( x 2 * ) φ 2 ( g 2 ( x 2 * ) ) - φ 2 ( x ) , x H ; T 3 ( u 31 * , u 32 * , u 33 * ) , x - g 3 ( x 3 * ) φ 3 ( g 3 ( x 3 * ) ) - φ 3 ( x ) , x H .
    (1.6)
     

Problem (1.6) was studied by Kazmi et al. [8].

For more special cases, please refer to [19] and the references therein.

Remark 1.1. Yang [10] pointed out a fact for the problem (1.4) discussed in reference [5], namely, if the problem (1.4) has a solution (x*, y*), then x* = y*. Therefore, actually, the problem(1.4) is a single variational inequality:
T ( x * , x * ) , x - x * 0 , x K .

In this article, we study the problem (1.1). By using the η proximal mapping technique, we prove the existence of solutions and approximate the solutions by some new N - step iterative algorithms. Our results extend and improve some known results in the references [19].

2 Preliminaries

In this article, we need the following concepts and lemmas.

Definition 2.1[1] A mapping g : HH is said to be
  1. (i)
    ξ-strongly monotone if there exists a constant ξ > 0 such that
    g ( x ) - g ( y ) , x - y ξ x - y 2 , x , y H .
     
  2. (ii)
    ζ-Lipschitz continuous if there exists a constant ζ > 0 such that
    g ( x ) - g ( y ) ζ x - y , x , y H .
     
Definition 2.2[1] A mapping η : H × HH is said to be
  1. (i)
    σ - strongly monotone if there exists a constant σ > 0 such that
    x - y , η ( x , y ) σ x - y 2 , x , y H ;
     
  2. (ii)
    τ - Lipschitz continuous if there exists a constant τ > 0 such that
    η ( x , y ) τ x - y , x , y H .
     
Definition 2.3[1, 11] Let A : HCB(H) be a set-valued mapping, T : H × H × × H N H is said to be
  1. (i)
    α - (A, g)-strongly monotone in the i th argument if α > 0 such that
    T ( , u i , ) - T ( , v i , ) , g ( x ) - g ( y ) α x - y 2 , x , y H , u i A x , v i A y .
     
  2. (ii)
    (s 1, s 2, . . . , s N )-Lipschitz continuous if there exist constants s 1, s 2, . . . , s N > 0 such that for all x i , y i H, i = 1, 2, . . . , N,
    T ( x 1 , x 2 , , x N ) - T ( y 1 , y 2 , , y N ) s 1 x 1 - y 1 + s 2 x 2 - y 2 + + s N x N - y N .
     
  3. (iii)
    A set-valued A is said to be δ - H - Lipschitz continuous if there exists a constant δ > 0 such that
    H ( A x , A y ) δ x - y , x , y H ,
     

where H(·,·) is the Hausdorff metric on CB(H).

Definition 2.4[1] A functional f : H ×H → R{+} is said to be 0-diagonally quasi-concave (in short,0-DQCV) in x, if for any finite set {x1, . . . , x N } H and for any y = i = 1 n λ i x i with λ i 0 and i = 1 n λ i = 1 ,
min 1 i n f ( x i , y ) 0 .
Definition 2.5[1] Let η : H × H → H be a single-valued mapping. A proper functional φ : H → R{+} is said to be η- subdifferentiable at a point x H, if there exists a point f * H such that
f * , η ( y , x ) φ ( y ) - φ ( x ) , y H ,
where f * is called a η- subgradient of φ at x. The set of all η- subgradients of φ at x is denoted by η φ(x). We have
η φ ( x ) = { f * H , f * , η ( y , x ) φ ( y ) - φ ( x ) , y H . }
(2.1)
Definition 2.6[1] Let η, φ be according to Definition 2.5, if for each x H and ρ > 0,there exists a unique point u H such that
u - x , η ( y , u ) ρ φ ( u ) - ρ φ ( y ) , y H ,
(2.2)
then the mapping x u denoted by J φ ρ , is said to be η- proximal mapping of φ. By (2.1) and the definition of J φ ρ , we have x - u ρ∂ η φ(x), it follows that
J φ ρ ( x ) = ( I + ρ η φ ) - 1 ( x ) .
Lemma 2.1[1] Let η : H × H → H be continuous and σ- strongly monotone such that η(x, y) = (y, x) for all x, y H. And for any given x H, the function h(y, u) = 〈x - u, η(y, u)〉 is 0-DQCV in y. Let φ : H → R {+} be a lower semicontinuous η- subdifferentiable proper functional on H, then for any given ρ > 0 and x H there exists a unique u H such that
u - x , η ( y , u ) ρ φ ( u ) - ρ φ ( y ) , y H .

i.e., u = J φ ρ ( x ) .

Lemma 2.2 Let η : H × H → H be σ- strongly monotone and τ - Lipschitz continuous such that η(x, y) = (y, x). Let h(y, u), φ, ρ be according to Lemma 2.1, then the η- proximal mapping J φ ρ ( x ) of φ is τ σ - Lipschitz continuous.

3 Main results

Theorem 3.1 ( x 1 * , x 2 * , , x N * ; u i 1 * , u i 2 * , , u i N * , i = 1 , 2 , , N . ) is a solution of problem (1.1) if and only if ( x 1 * , x 2 * , , x N * ; u i 1 * , u i 2 * , , u i N * , i = 1 , 2 , , N . ) satisfies the following relation: For every i = 1, 2, . . . N,
g i ( x i * ) = J φ i ρ i ( g i ( x i * ) - ρ i T i ( u i 1 * , u i 2 * , , u i N * ) ) ,
(3.1)

where J φ i ρ i = ( I + ρ i η φ i ) - 1 , ρ i > 0 .

Proof. Assume the ( x 1 * , x 2 * , , x N * ; u 11 * , u 12 * , , u 1 N * , , u N 1 * , u N 2 * , , u N N * ) satisfies relation (3.1). Since J φ i ρ i = ( I + ρ i η i φ i ) - 1 , we have
g i ( x i * ) + ρ i η i φ i ( g i ( x i * ) ) g i ( x i * ) - ρ i T i ( u i 1 * , u i 2 * , , u i N * ) .
i.e.,
- T 1 ( u i 1 * , u i 2 * , , u i N * ) η i φ i ( g i ( x i * ) ) .
By the Definition 2.5 of η i - subdifferential, the above relation holds if and only if
- T i ( u i 1 * , u i 2 * , , u i N * ) , η i ( x , g i ( x i * ) ) φ i ( x ) - φ i ( g i ( x i * ) ) , x H ,

and hence

T i ( u i 1 * , u i 2 * , , u i N * ) , η i ( x , g i ( x i * ) ) φ i ( g i ( x i * ) ) - φ i ( x ) , x H , i = 1, 2, . . . , N. i.e., ( x 1 * , x 2 * , , x N * ; u i 1 * , u i 2 * , , u i N * , i = 1 , 2 , , N . ) is a solution of the problem (1.1). □

Now, we give iterative algorithms of problem (1.1).

Algorithm(I) For given x 1 0 , x 2 0 , , x N 0 H , u i 1 0 A i 1 x 1 0 , u i 2 0 A i 2 x 2 0 , , u i N 0 A i N x N 0 , let
x 1 1 = x 1 0 - g 1 ( x 1 0 ) + J φ 1 ρ 1 ( g 1 ( x 1 0 ) - ρ 1 T 1 ( u 11 0 , u 12 0 , , u 1 N 0 ) ) ; x 2 1 = x 2 0 - g 2 ( x 2 0 ) + J φ 2 ρ 2 ( g 2 ( x 2 0 ) - ρ 2 T 2 ( u 21 0 , u 22 0 , , u 2 N 0 ) ) ; x N 1 = x N 0 - g N ( x N 0 ) + J φ N ρ N ( g N ( x N 0 ) - ρ N T N ( u N 1 0 , u N 2 0 , , u N N 0 ) ) .
By Nadler [12], for i = 1, 2, . . . , N, j = 1, 2, . . . , N, there exists u i j 1 A i j x j 1 such that
u i j 0 - u i j 1 ( 1 + 1 ) H ( A i j x j 0 , A i j x j 1 ) , j = 1 , 2 , , N .
Let
x i 2 = x i 1 - g i ( x i 1 ) + J φ i ρ i ( g i ( x i 1 ) - ρ i T i ( u i 1 1 , u i 2 1 , , u i N 1 ) ) , i = 1 , 2 , , N .
By induction, we can define sequences { x i n } , { u i j n } satisfying
x i n + 1 = x i n - g i ( x i n ) + J φ i ρ i ( g i ( x i n ) - ρ i T i ( u i 1 n , u i 2 n , , u i N n ) ) ,
where for any i, j = 1, 2, . . . , N; n = 0, 1, 2, . . . ,
u i j n A i j n x j n , u i j n u i j n + 1 ( 1 + 1 n + 1 ) H ( A i j ( x j n ) , A i j ( x j n + 1 ) ) .
Theorem 3.2 Let H be a real Hilbert space. For i, j = 1, 2, . . . , N, let set-valued mapping A ij : H → CB(H) be δ ij - H - Lipschitz continuous. Let mapping η i : H × H → H be σ i - strongly monotone and τ i -Lipschitz continuous such that η i (x, y) = i (y, x) for all x, y H and for any given x H, the function h i (y, u) = 〈x - g i (u), η i (y, u)〉 is 0-DQCU in y. Let mapping g i : H → H be ξ i - strongly monotone and ζ i - Lipschitz continuous, and T i : H × H × × H N H be (s1, . . . , s N )- Lipschitz continuous and α i - (A ij , g i )- strongly monotone in the i th argument. Let φ i : H → R {+} be a lower semicontinuous η i - subdifferentiable proper functional. If there exist ρ1, . . . , ρ N > 0 such that for all i = 1, 2, . . . , N
1 - 2 ξ i + ζ i 2 1 2 + τ i σ i ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 2 + k = 1 , k i N τ k σ k ρ k s i δ k i < 1 ;
(3.2)

then the iterative sequences { x 1 n } , , { x N n } , { u 11 n } , , { u 1 N n } , , { u N 1 n } , , { u N N n } , generated by algorithm (I) converge strongly to x 1 * , , x N * , u 11 * , , u 1 N * , , u N 1 * , , u N N * , respectively, and ( x 1 * , x 2 * , , x N * , u 11 * , u 12 * , , u 1 N * , , u N 1 * , u N 2 * , , u N N * ) is a solution of the problem (1.1).

Proof. For i = 1, 2, . . . , N, by algorithm (I) and Lemma 2.2, we have
x i n + 1 - x i n = x i n - g i x i n + J φ i ρ i g i x i n - ρ i T i u i 1 n , u i 2 n , , u i N n - x i n - 1 + g i x i n - 1 - J φ i ρ i g i x i n - 1 - ρ i T i u i 1 n - 1 , u i 2 n - 1 , , u i N n - 1 x i n - x i n - 1 - g i x i n + g i x i n - 1 + J φ i ρ i g i x i n - ρ i T i u i 1 n , u i 2 n , , u i N n - J φ i ρ i g i x i n - 1 - ρ i T i u i 1 n - 1 , u i 2 n - 1 , , u i N n - 1 x i n - x i n - 1 - g i x i n + g i x i n - 1 + τ i σ i g i x i n - g i x i n - 1 - ρ i T i u i 1 n , , u i N n + ρ i T i u i 1 n - 1 , , u i N n - 1 .
(3.3)
Since g i is ξ i - strongly monotone and ζ i - Lipschitz continuous, we obtain
x i n - x i n - 1 - g i x i n - g i x i n - 1 1 - 2 ξ i + ζ i 2 x i n - x i n - 1 .
(3.4)
Notice that,
g i x i n - g i x i n - 1 - ρ i T i u i 1 n , , u i N n - T i u i 1 n - 1 , , u i N n - 1 g i x i n - g i x i n - 1 - ρ i T i u i 1 n , u i 2 n , , u i , i - 1 n , u i i n , u i , i + 1 n , , u i N n - T i u i 1 n , u i 2 n , , u i , i - 1 n , u i , i n - 1 , u i , i + 1 n , , u i N n + ρ i T i u i 1 n , u i 2 n , , u i , i - 1 n , u i , i n - 1 , u i , i + 1 n , , u i N n - T i u i 1 n - 1 , u i 2 n - 1 , , u i N n - 1 .
(3.5)
Since T i is (s1, . . . , s N )- Lipschitz continuous and α i - (A ij , g i )- strongly monotone in the i th argument, we get
g i x i n - g i x i n - 1 - ρ i T i u i 1 n , u i 2 n , , u i , i - 1 n , u i i n , u i , i + 1 n , , u i N n - T i u i 1 n , u i 2 n , , u i , i - 1 n , u i i n - 1 , u i , i + 1 n , , u i N n 2 = g i x i n - g i x i n - 1 2 - 2 ρ i g i x i n - g i x i n - 1 , T i u i 1 n , u i 2 n , , u i , i - 1 n , u i , i n , u i , i + 1 n , , u i , N n - T i u i 1 n , u i 2 n , , u i , i - 1 n , u i , i n - 1 , u i , i + 1 n , , u i N n + ρ i 2 T i u i 1 n , u i 2 n , , u i , i - 1 n , u i , i n , u i , i + 1 n , , u i N n - T i u i 1 n , u i 2 n , , u i , i - 1 n , u i , i n - 1 , u i , i + 1 n , , u i N n 2 ξ i 2 x i n - x i n - 1 2 - 2 ρ i α i x i n - x i n - 1 2 + ρ i 2 s i 2 u i i n - u i i n - 1 2 ξ i 2 - 2 ρ i α i x i n - x i n - 1 2 + ρ i 2 s i 2 1 + 1 n 2 H A i i x i n , A i i x i n - 1 2 ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 + 1 n 2 x i n - x i n - 1 2 .
(3.6)
Therefore,
g i x i n - g i x i n - 1 - ρ i T i u i 1 n , u i 2 n , , u i , i - 1 n , u i i n , u i , i + 1 n , , u i N n - T i u i 1 n , u i 2 n , , u i , i - 1 n , u i i n - 1 , u i , i + 1 n , , u N n ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 + 1 n 2 1 2 x i n - x i n - 1 .
(3.7)
And
ρ i T i u i 1 n , u i 2 n , , u i , i - 1 n , u i i n - 1 , u i , i + 1 n , , u i N n - T i u i 1 n - 1 , u i 2 n - 1 , , u i , i - 1 n - 1 , u i i n - 1 , u i , i + 1 n - 1 , , u i N n - 1 ρ i s 1 u i 1 n - u i 1 n - 1 + s 2 u i 2 n - u i 2 n - 1 + + s i - 1 u i , i - 1 n - u i , i - 1 n - 1 + s i + 1 u i , i + 1 n - u i , i + 1 n - 1 + + s N u i N n - u i N n - 1 ρ i s 1 1 + 1 n H A i 1 x 1 n , A i 1 x 1 n - 1 + s 2 1 + 1 n H A i 2 x 2 n , A i 2 x 2 n - 1 + + s i - 1 1 + 1 n H A i , i - 1 x i - 1 n , A i , i - 1 x i - 1 n - 1 + s i + 1 1 + 1 n H A i , i + 1 x i + 1 n , A i , i + 1 x i + 1 n - 1 + + s N 1 + 1 n H A i N x N n , A i N x N n - 1 ρ i 1 + 1 n s 1 δ i 1 x 1 n - x 1 n - 1 + s 2 δ i 2 x 2 n - x 2 n - 1 + + s i - 1 δ i , i - 1 x i - 1 n - x i - 1 n - 1 + s i + 1 δ i , i + 1 x i + 1 n - x i + 1 n - 1 + + s N δ i N x N n - x N n - 1 .
(3.8)
It follows from (3.3)-(3.8) that for every i = 1, 2, . . . ,N,
x i n + 1 - x i n 1 - 2 ξ i + ζ i 2 + τ i σ i ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 + 1 n 2 1 2 x i n - x i n - 1 + ρ i τ i σ i 1 + 1 n k = 1 , k i N s k δ i k x k n - x k n - 1 .
(3.9)
So,
x 1 n + 1 - x 1 n + x 2 n + 1 - x 2 n + + x N n + 1 - x N n i = 1 N 1 - 2 ξ i + ζ i 2 + τ i σ i ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 + 1 n 2 1 2 x i n - x i n - 1 + ρ i τ i σ i k = 1 , k i N s k δ i k 1 + 1 n x k n - x k n - 1 = i = 1 N 1 - 2 ξ i + ζ i 2 + τ i σ i ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 + 1 n 2 1 2 + k = 1 , k i N ρ k τ k σ k s i δ k i 1 + 1 n x i n - x i n - 1 θ n x 1 n - x 1 n - 1 + x 2 n - x 2 n - 1 + + x N n - x N n - 1 ,
(3.10)
where
θ n = max i = 1 , 2 , N 1 - 2 ξ i + ζ i 2 + τ i σ i ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 + 1 n 2 1 2 + k = 1 , k i N ρ k τ k σ k s i δ k i 1 + 1 n
Letting
θ = max i = 1 , 2 , N 1 - 2 ξ i + ζ i 2 + τ i σ i ξ i 2 - 2 ρ i α i + ρ i 2 s i 2 δ i i 2 1 2 + k = 1 , k i N ρ k τ k σ k s i δ k i ,

from (3.2) we have 0 < θ < 1, and hence { x 1 n } { x N n } are also Cauchy sequences. Thus there exist x 1 * , , x N * H such that x i n x i * ( n ) , i = 1, 2, . . . ,N.

Now we prove u i j n u i j * ( n ) , for i = 1, 2, . . . ,N, j = 1, 2, . . . ,N. By u i j n - u i j n - 1 1 + 1 n H A i j x j n , A i j x j n - 1 1 + 1 n δ i j x j n - x j n - 1 .

It follows that { u i j n } are also Cauchy sequence. Therefore, there exist u i j * H such that u i j n u i j * ( n ) .

Note that
d u i j * , A i j x j * u i j * - u i j n d u i j n , A i j x j * u i j * - u i j n + H A i j x j n , A i j x j * u i j * - u i j n + δ i j x j n - x j * 0 ( n ) .
We have d u i j * , A i j x j * = 0 . Since A i j x j * is closed, u i j * A i j x j * , for each i = 1,2, . . . ,N, j = 1, 2, . . . ,N. By
x i n + 1 = x i n - g i x i n + J φ i ρ i g i x i n - ρ i T i u i 1 n , u i 2 n , , u i N n , i = 1 , 2 , , N ,
and the continuity of g i , J φ i ρ i , T i , let n → ∞, we have that
0 = - g i x i * + J φ i ρ i g i x i * - ρ i T i u i 1 * , u i 2 * , , u i N * ,
and
u i 1 * A i 1 x 1 * , u i 2 * A i 2 x 2 * , , u i N * A i N x N * .

0By Theorem 3.1, ( x 1 * , x 2 * , , x N * , u i 1 * , u i 2 * , , u i N * , i = 1 , 2 , , N ) is a solution of the problem (1.1). This completes the proof.□

Remark 3.1 For a suitable choice of T i , A ij , η i , g i and φ i , Theorem 3.2 includes many known results of generalized quasi-variational-like inclusions as special cases (see [18]),where φ i is nonconvex and A ij is noncompact.

Declarations

Acknowledgements

The author thanks to the guidance and support of my supervisor Pro. Li-Wei Liu who taught at the department of mathematics in Nanchang university. The author thanks the anonymous referees for reading this article carefully, providing valuable suggestions and comments. The work was supported by the National Science Foundation of China (No.10561007) and Science and Technology Research Project of Education Department in Jiangxi province (GJJ10269).

Authors’ Affiliations

(1)
Department of Science, Nanchang Institute of Technology

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© Cao; licensee Springer. 2012

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