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Strong convergence theorems by Halpern-Mann iterations for multi-valued relatively nonexpansive mappings in Banach spaces with applications

Journal of Inequalities and Applications20122012:73

https://doi.org/10.1186/1029-242X-2012-73

©  Zhu et al; licensee Springer. 2012

Received: 29 January 2012

Accepted: 29 March 2012

Published: 29 March 2012

Abstract

In this article, an iterative sequence for relatively nonexpansive multi-valued mapping by modifying Halpern and Mann's iterations is introduced, and then some strong convergence theorems are proved. At the end of the article some applications are given also.

AMS Subject Classification: 47H09; 47H10; 49J25.

Keywords

multi-valued mappingrelatively nonexpansivefixed pointiterative sequence

1 Introduction

Throughout this article, we denote by ℕ and ℝ the sets of positive integers and real numbers, respectively. Let D be a nonempty closed subset of a real Banach space E. A single-valued mapping T : D → D is called nonexpansive if ∥Tx - Ty∥ ≤ ∥x - y∥ for all x, y ∈ D. Let N(D) and CB(D) denote the family of nonempty subsets and nonempty closed bounded subsets of D, respectively. The Hausdorff metric on CB(D) is defined by
H A 1 , A 2 = max sup x ∈ A 1 d ( x , A 2 ) , sup y ∈ A 2 d ( y , A 1 ) ,

for A1,A2 ∈ CB(D), where d(x, A1) = inf{∥x - y∥, y ∈ A1}. The multi-valued mapping T : D → CB(D) is called nonexpansive if H(T(x),T(y)) ≤ ∥x - y∥ for all x, y ∈ D. An element p ∈ D is called a fixed point of T : D → N(D) if p ∈ T(p). The set of fixed points of T is represented by F(T).

Let E be a real Banach space with dual E*. We denote by J the normalized duality mapping from E to 2E*defined by
J ( x ) = x * ∈ E * : x , x * = x 2 = x * 2 , x ∈ E .

where 〈·,·〉 denotes the generalized duality pairing.

A Banach space E is said to be strictly convex if x + y 2 < 1 for all x, y ∈ U = {z ∈ E : ∥z∥ = 1} with x ≠ y. E is said to be uniformly convex if, for each ϵ ∈ (0, 2], there exists δ > 0 such that x + y 2 < 1 - δ for all x, y ∈ U with ∥x - y∥ ≥ ϵ. E is said to be smooth if the limit
lim t → 0 x + t y - x t

exists for all x, y ∈ U. E is said to be uniformly smooth if the above limit exists uniformly in x, y ∈ U.

Remark 1.1. The following basic properties for Banach space E and for the normalized duality mapping J can be found in Cioranescu [1].
  1. (i)

    If E is an arbitrary Banach space, then J is monotone and bounded;

     
  2. (ii)

    If E is a strictly convex Banach space, then J is strictly monotone;

     
  3. (iii)

    If E is a a smooth Banach space, then J is single-valued, and hemi-continuous, i.e., J is continuous from the strong topology of E to the weak star topology of E*;

     
  4. (iv)

    If E is a uniformly smooth Banach space, then J is uniformly continuous on each bounded subset of E;

     
  5. (v)

    If E is a reflexive and strictly convex Banach space with a strictly convex dual E* and J*: E* → E is the normalized duality mapping in E*, then J-1 = J*, J J* = IE*, and J* J = I E ;

     
  6. (vi)

    If E is a smooth, strictly convex, and reflexive Banach space, then the normalized duality mapping J is single-valued, one-to-one and onto;

     
  7. (vii)

    A Banach space E is uniformly smooth if and only if E* is uniformly convex. If E is uniformly smooth, then it is smooth and reflexive.

     
Next we assume that E is a smooth, strictly convex, and reflexive Banach space and C is a nonempty closed convex subset of E. In the sequel, we always use ϕ : E × E → ℝ+ to denote the Lyapunov functional defined by
ϕ ( x , y ) = x 2 - 2 x , J y + y 2 , ∀ x , y ∈ E .
(1.2)
It is obvious from the definition of ϕ that
x - y 2 ≤ ϕ ( x , y ) ≤ x - y 2 , ∀ x , y ∈ E .
(1.3)
ϕ x , J - 1 λ J y + ( 1 - λ ) J z ≤ λ ϕ ( x , y ) + ( 1 - λ ) ϕ ( x , z ) ,
(1.4)

for all λ ∈ [0,1] and x,y,z ∈ E.

Following Alber [2], the generalized projection Π C : E → C is defined by
∏ C ( x ) = arg inf y ∈ C ϕ ( y , x ) , ∀ x ∈ E .

Let D be a nonempty subset of a smooth Banach space. A mapping T : D → E is relatively expansive [3–5], if the following properties are satisfied:

(R1) F ( T ) ≠ ∅ ;

(R2) ϕ(p,Tx) ≤ ϕ(p,x) for all p ∈ F(T) and x ∈ D;

(R3) I - T is demi-closed at zero, that is, whenever a sequence {x n } in D converges weakly to p and {x n - Tx n } converges strongly to 0, it follows that p ∈ F(T).

If T satisfies (R1) and (R2), then T is called quasi-ϕ-nonexpansive [6].

Recently, Nilsrakoo and Saejung [7] introduced the following iterative sequence for finding a fixed point of relatively nonexpansive mapping T : D → E. Given x1 ∈ D,
x n + 1 = ∏ D J - 1 α n J u + ( 1 - α n ) β n J x n + ( 1 - β n ) J T x n

where D is nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, Π D is the generalized projection of E onto D and {α n } and {β n } are two sequences in [0,1].

They proved strong convergence theorems in uniformly convex and uniformly smooth Banach space E.

Iterative methods for approximating fixed points of multi-valued mappings in Banach spaces have been studied by some authors, see for instance [8–15].

Let D be a nonempty closed convex subset of a smooth Banach space E. We define a relatively nonexpansive multi-valued mapping as follows.

Definition 1.2. A multi-valued mapping T : D → N(D) is called relatively nonexpansive, if the following conditions are satisfied:

(S1) F ( T ) ≠ ∅

(S2) ϕ(p,z) ≤ ϕ(p, x), ∀x ∈ D, z ∈ T(x), p ∈ F(T);

(S3) I - T is demi-closed at zero, that is, whenever a sequence {x n } in D which weakly to p and limn→∞d(x n , T(x n )) = 0, it follows that p ∈ F(T).

If T satisfies (S1) and (S2), then multi-valued mapping T is called quasi-ϕ-nonexpansive.

In this article, inspired by Nilsrakoo and Saejung [7], we introduce the following iterative sequence for finding a fixed point of relatively nonexpansive multi-valued mapping T : D → N(D). Given u ∈ E,x i ∈ D,
x n + 1 = ∏ D J - 1 α n J u + ( 1 - α n ) β n J x n + ( 1 - β n ) J w n

where w n ∈ Tx n for all n ∈ ℕ, D is a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E, Π D is the generalized projection of E onto D and {α n }, {β n } are sequences in [0,1]. We proved the strong convergence theorems in uniformly convex and uniformly smooth Banach space E.

2 Preliminaries

In the sequel, we denote the strong convergence and weak convergence of the sequence {x n } by x n → x and x n ⇀ x, respectively.

First, we recall some conclusions.

Lemma 2.1 [16, 17]. Let E be a smooth, strictly convex, and reflexive Banach space and C be a nonempty closed convex subset of E. Then the following conclusions hold:
  1. (a)

    ϕ(x, Π C y) + ϕ(Π C y, y) ≤ ϕ(x, y) for all x ∈ C and y ∈ E;

     
  2. (b)
    If x ∈ E and z ∈ C, then
    z = ∏ C x ⇔ z - y , J x - J z ≥ 0 , ∀ y ∈ C ;
     
  3. (c)

    For x, y ∈ E, ϕ(x, y) = 0 if and only x = y.

     

Remark 2.2. If E is a real Hilbert space H, then ϕ(x, y) = ∥x - y∥2 and Π C is the metric projection P C of H onto C.

Lemma 2.3 [18]. Let E be a uniformly convex Banach space, r > 0 be a positive number and B r (0) be a closed ball of E. Then, for any given sequence x i i = 1 ∞ ⊂ B r ( 0 ) and for any given sequence λ i i = 1 ∞ of positive numbers with ∑ i = 1 ∞ λ i = 1 , then there exists a continuous, strictly increasing, and convex function g : [0, 2r) → [0, ∞) with g(0) = 0 such that for any positive integers i, j with i < j,
∑ n = 1 ∞ λ n x n 2 ≤ ∑ n = 1 ∞ λ n x n 2 - λ i λ j g x i - x j
(2.1)

In what follows, we need the following lemmas for proof of our main results.

Lemma 2.4 [17]. Let E be a uniformly convex and smooth Banach space and let {x n } and {y n } be two sequences of E such that {x n } or {y n } is bounded. If limn→∞ϕ(x n , y n ) = 0. Then limn→∞∥x n -y n ∥ = 0.

Let E be a reflexive, strictly convex, and smooth Banach space. The duality mapping J* from E* onto E** = E coincides with the inverse of the duality mapping J from E onto E*, that is, J* = J-1. We make use the following mapping V : E × E* → ℝ studied in Alber [19]:
V ( x , x * ) = x 2 - 2 x , x * + x * 2
(2.2)

for all x ∈ E and x* ∈ E*. Obviously, V(x, x*) = ϕ(x, J-1(x*)) for all x ∈ E and x* ∈ E*. We know the following lemma.

Lemma 2.5 [20]. Let E be a reflexive, strictly convex, and smooth Banach space, and let V as in (2.2). Then
V ( x , x * ) + 2 J - 1 ( x * ) - x , y * ≤ V ( x , x * + y * ) ,
(2.3)

for all x ∈ E and x*,y* ∈ E*.

Lemma 2.6 [21]. Assume that {α n } is a sequence of nonnegative real numbers such that
α n + 1 ≤ ( 1 - γ n ) α n + γ n δ n ,
where {γ n } is a sequence in (0,1) and {δ n } is a sequence such that
  1. (a)

    lim n → ∞ γ n = 0 , ∑ n = 1 ∞ γ n = ∞ ;

     
  2. (b)

    lim supn→∞≤ 0.

     

Then limn→∞α n = 0.

Lemma 2.7 [22]. Let {α n } be a sequence of real numbers such that there exists a subsequence {n i } of {n} such that α n i < α n i + 1 for all i ∈ ℕ. Then there exists a nondecreasing sequence {m k } ⊂ ℕ such that m k → ∞ and the following properties are satisfied for all (sufficiently large) numbers k ∈ ℕ:
α m k ≤ α m k + 1 a n d α k ≤ α m k + 1 .

In fact, m k = max{j ≤ k : α j < αj+1}.

3 Main results

Lemma 3.1 Let E be a strictly convex and smooth Banach space, and D a nonempty closed subset of E. Suppose T : D → N(D) is a quasi-ϕ-nonexpansive multi-valued mapping. Then F(T) is closed and convex.

Proof. First, we show F(T) is closed. Let {x n } be a sequence in F(T) such that x n → x*. Since T is quasi-ϕ-nonexpansive, we have
ϕ ( x n , z ) ≤ ϕ ( x n , x * )
for all z ∈ T(x*) and for all n ∈ ℕ. Therefore,
ϕ ( x * , z ) = lim n → ∞ ϕ ( x n , z ) ≤ lim n → ∞ ϕ ( x n , x * ) = ϕ ( x * , x * ) = 0 .
By Lemma 2.1(c), we obtain x* = z. Hence, T(x*) = {x*}. So, we have x* ∈ F(T). Next, we show F(T) is convex. Let x, y ∈ F(T) and t ∈ (0,1), put p = tx + (1 - t)y. We show p ∈ F(T). Let w ∈ F(p), we have
ϕ ( p , w ) = p 2 - 2 p , J w + w 2 = p 2 - 2 t x + ( 1 - t ) y , J w + w 2 = p 2 - 2 t x , J w - 2 ( 1 - t ) y , J w + w 2 = p 2 + t ϕ ( x , w ) + ( 1 - t ) ϕ ( y , p ) - t x 2 - t ( 1 - t ) p 2 = p 2 - 2 t x + ( 1 - t ) y , J p + p 2 = p 2 - 2 p , J p + p 2 = 0 .

By Lemma 2.1(c), we obtain p = w. Hence, T(p) = {p}. So, we have p ∈ F(T). Therefore, F(T) is convex.

Lemma 3.2. Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E and T : D → N(D) be a relatively nonexpansive multi-valued mapping. If {x n } is a bounded sequence such that limn→∞d(x n ,Tx n ) and x* = ΠF(T)x, then
lim n → ∞ sup x n - x * , J x - J x * ≤ 0 .
Proof. From (S3) of the mapping T, we choose a subsequence x n i of {x n } such that x n i ⇀ y ∈ F ( T ) and
lim n → ∞ sup x n - x * , J x - J x * = lim i → ∞ x n i - x * , J x - J x * .
By Lemma 2.1(b), we immediately obtain that
lim n → ∞ sup x n - x * , J x - J x * = y - x * , J x - J x * ≤ 0 .
Lemma 3.3. Let D be a nonempty closed convex subset of a reflexive, strictly convex, and smooth Banach space E and T : D → N(D) be a relatively nonexpansive multi-valued mapping. Let {x n } be a sequence in D defined as follows: u ∈ E, x1 ∈ D and
x n + 1 = ∏ D J - 1 α n J u + ( 1 - α n ) β n J x n + ( 1 - β n ) J w n ,
(3.1)

where w n ∈ Tx n for all n ∈ ℕ, {α n }, {β n } are sequences in [0,1]. Then {x n } is bounded.

Proof. Let p ∈ F(T) and y n = J - 1 β n J x n + ( 1 - β n ) J w n for all n ∈ ℕ. Then
x n + 1 ≡ ∏ D J - 1 α n J u + ( 1 - α n ) J y n
for all n ∈ ℕ. By using (1.4), we have
ϕ ( p , y n ) = ϕ p , J - 1 β n J x n + ( 1 - β n ) J w n ≤ β n ϕ ( p , x n ) + ( 1 - β n ) ϕ ( p , w n ) ≤ β n ϕ ( p , x n ) + ( 1 - β n ) ϕ ( p , x n ) = ϕ ( p , x n )
and
ϕ ( p , x n + 1 ) = ϕ p , ∏ D J - 1 α n J u + ( 1 - α n ) J y n ≤ ϕ p , J - 1 α n J u + ( 1 - α n ) J y n ≤ α n ϕ ( p , u ) + ( 1 - α n ) ϕ ( p , y n ) ≤ α n ϕ ( p , u ) + ( 1 - α n ) ϕ ( p , x n ) ≤ max ϕ ( p , u ) , ϕ ( p , x n ) ≤ ⋯ ≤ max ϕ ( p , u ) , ϕ ( p , x 1 ) .

This implies that {x n } is bounded.

Theorem 3.4 Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : D → N(D) be a relatively nonexpansive multivalued mapping. Let {α n } and {β n } be sequences in (0,1) satisfying

(C1) limn→∞, α n = 0;

(C2) ∑ n = 1 ∞ α n = ∞ ;

(C3) lim infn→∞β n (1- β n ) > 0.

Then {x n } defined by (3.1) converges strongly to ΠF(T)u, where ΠF(T)is the generalized projection from E onto F(T).

Proof. By Lemma 3.1, F(T) is closed and convex. So, we can define the generalized projection ΠF(T)onto F(T). Putting u* = ∏F(T)u, by Lemma 3.3 we know that {x n } is bounded and hence, {w n } is bounded. Let g : [0,2r] → [0,∞) be a function satisfying the properties of Lemma 2.3, where r = sup{∥u∥, ∥x n ∥, ∥w n ∥ : n ∈ ℕ}. Put
y n ≡ J - 1 β n J u + ( 1 - β n ) J w n
Then
ϕ ( u * , y n ) = ϕ u * , J - 1 β n J x n + ( 1 - β n ) J w n = u * 2 - 2 u * , β n J x n + ( 1 - β n ) J w n + β n J x n + ( 1 - β n ) J w n 2 ≤ u * 2 - 2 β n u * , J x n - 2 ( 1 - β n ) u * , J w n + β n x n 2 + ( 1 - β n ) w n 2 - β n ( 1 - β n ) g J x n - J w n = ϕ u * , x n - β n 1 - β n g J x n - J w n
(3.2)
and
ϕ u * , x n + 1 = ϕ u * , ∏ D J - 1 α n J u + ( 1 - α n ) J y n ≤ ϕ u * , J - 1 α n J u + ( 1 - α n ) J y n ≤ α n ϕ ( u * , u ) + ( 1 - α n ) ϕ ( u * , y n ) ≤ α n ϕ ( u * , u ) + ( 1 - α n ) ϕ ( u * , x n ) - β n ( 1 - β n ) g J x n - J w n .
(3.3)
for all n ∈ ℕ. Put
M = sup ϕ ( u * , u ) - ϕ ( u * , x n ) + β n ( 1 - β n ) g | | J x n - J w n : n ∈ ℕ
It follows from (3.3) that
β n ( 1 - β n ) g J x n - J w n ≤ ϕ ( u * , x n ) - ϕ u * , x n + 1 + α n M .
(3.4)
Let z n ≡ J - 1 α n J u + ( 1 - α n ) J y n . Then x n + 1 = ∏ C z n for all n ∈ ℕ. It follows from (2.3) and (3.2) that
ϕ u * , x n + 1 ≤ ϕ u * , J - 1 α n J u + ( 1 - α n ) J y n = V u * , α n J u + ( 1 - α n ) J y n ≤ V u * , α n J u + ( 1 - α n ) J y n - α n ( J u - J u * ) - 2 J - 1 α n J u + ( 1 - α n ) J y n - u * , - α n J u - J u * = V u * , α n J u * + ( 1 - α n ) J y n + 2 α n z n - u * , J u - J u * = ϕ u * , J - 1 α n J u * + ( 1 - α n ) J y n + 2 α n z n - u * , J u - J u * = u * 2 - 2 u * , α n J u * + ( 1 - α n ) J y n + α n J u * + ( 1 - α n ) J y n 2 + 2 α n z n - u * , J u - J u * ≤ u * 2 - 2 α n u * , J u * - 2 ( 1 - α n ) u * , J y n + α n u * 2 + ( 1 - α n ) y n 2 + 2 α n z n - u * , J u - J u * = α n ϕ ( u , u * ) + ( 1 - α n ) ϕ ( u * , y n ) + 2 α n z n - u * , J u - J u * ≤ ( 1 - α n ) ϕ ( u * , x n ) + 2 α n z n - u * , J u - J u * .
(3.5)

The rest of proof will be divided into two parts:

Case (1). Suppose that there exists n0 ∈ ℕ such that ϕ ( u * , x n ) n = n 0 ∞ is nonincreasing. In this situation, {ϕ(u*, x n )} is then convergent. Then limn→∞(ϕ(u*, x n ) - ϕ(u*, xn+1)) = 0. This together with (C1), (C3), and (3.4), we obtain
lim n → ∞ g J x n - J w n = 0 .
Therefore,
lim n → ∞ J x n - J w n = 0 .
Since J-1 is uniformly norm-to-norm continuous on every bounded subset of E, we have
lim n → ∞ x n - w n = 0 .
(3.6)
Since d x n , T x n ≤ x n - w n , we obtain
lim n → ∞ d ( x n , T x n ) = 0
(3.7)
Then,
ϕ ( w n , y n ) = ϕ w n , J - 1 β n J x n + ( 1 - β n ) J w n ≤ β n ϕ ( w n , x n ) + ( 1 - β n ) ϕ ( w n , w n ) = β n ϕ ( w n , x n ) → 0 .
(3.8)
and
ϕ ( y n , z n ) ≤ α n ϕ ( y n , u ) + ( 1 - α n ) ϕ ( y n , y n ) = α n ϕ ( y n , u ) → 0 .
(3.9)
From (3.8), (3.9) and Lemma 2.3, we have
lim n → ∞ w n - y n = 0
and
lim n → ∞ y n - z n = 0
This together with (3.6) gives
lim n → ∞ x n - z n = 0
(3.10)
From (3.7), (3.10) and invoking Lemma 3.2, we have
lim n → ∞ z n - u * , J u - J u * = lim n → ∞ x n - u * , J u - J u * ≤ 0

Hence the conclusion follows by Lemma 2.5.

Case (2). Suppose that there exists a subsequence {n i } of {n} such that
ϕ ( u * , x n i ) < ϕ ( u * , x n i + 1 )
for all i ∈ ℕ. Then, by Lemma 2.7, there exists a nondecreasing sequence {m k } ⊂ ℕ, m k → ∞ such that
ϕ u * , x m k ≤ ϕ u * , x m k + 1 a n d ϕ u * , x k ≤ ϕ u * , x m k + 1
for all k ∈ ℕ. This together with (3.4) gives
β m k ( 1 - β m k ) g J x m k - J w m k ≤ ϕ u * , x m k - ϕ u * , x m k + 1 + α m k M ≤ α m k M
for all k ∈ N. Then, by conditions (C1) and (C3)
lim k → ∞ g J x m k - J w m k = 0
By the same argument as Case (1), we get
lim k → ∞ sup z m k - u * , J u - J u * ≤ 0 .
(3.11)
From (3.5), we have
ϕ u * , x m k + 1 ≤ 1 - α m k ϕ u * , x m k + 2 α m k z m k - u * , J u - J u *
(3.12)
Since ϕ u * , x m k ≤ ϕ u * , x m k + 1 , we have
α m k ϕ u * , x m k ≤ ϕ u * , x m k - ϕ u * , x m k + 1 + 2 α m k z m k - u * , J u - J u * ≤ 2 α m k z m k - u * , J u - J u *
In particular, since α m k > 0 , we get
ϕ u * , x m k ≤ 2 z m k - u * , J u - J u *
It follows from (3.11) that lim k → ∞ ϕ u * , x m k = 0 . This together with (3.12) gives
lim k → ∞ ϕ u * , x m k + 1 = 0

But ϕ u * , x k ≤ ϕ u * , x m k + 1 for all k ∈ ℕ. We conclude that x k → u*.

This implies that limn→∞x n = u* and the proof is finished.

Letting β n = β gives the following result.

Corollary 3.5. Let D be a nonempty closed convex subset of a uniformly convex and uniformly smooth Banach space E and T : D → N(D) be a relatively nonexpansive multivalued mapping. Let {x n } be a sequence in D defined as follows: u ∈ E,x1 ∈ D and
x n + 1 = ∏ D J - 1 α n J u + ( 1 - α n ) β J x n + ( 1 - β ) J w n ,

where w n ∈ Tx n for all n ∈ ℕ, {α n } is a sequence in [0,1] satisfying condition (C1) and (C2), and β ∈ (0,1). Then {x n } converges strongly to ∏F(T)u.

4 Application to zero point problem of maximal monotone mappings

Let E be a smooth, strictly convex, and reflexive Banach space. An operator A : E → 2E*is said to be monotone, if 〈x - y, x* - y*〉 ≥ 0 whenever x, y ∈ E, x* ∈ Ax, y* ∈ Ay. We denote the zero point set {x ∈ E : 0 ∈ Ax} of A by A-10. A monotone operator A is said to be maximal, if its graph G(A) := {(x, y) : y ∈ Ax} is not properly contained in the graph of any other monotone operator. If A is maximal monotone, then A-10 is closed and convex. Let A be a maximal monotone operator, then for each r > 0 and x ∈ E, there exists a unique x r ∈ D(A) such that J(x) ∈ J(x r ) + rA(x r ) (see, for example, [19]). We define the resolvent of A by J r x = x r . In other words J r = (J + rA)-1 J, ∀r > 0. We know that J r is a single-valued relatively nonexpansive mapping and A-10 = F(J r ),∀r > 0, where F(J r ) is the set of fixed points of J r .

We have the following

Theorem 4.1 Let E, {α n }, and {β n } be the same as in Theorem 3.4. Let A : E → 2E*be a maximal monotone operator and J r = (J + rA)-1J for all r > 0 such that A - 1 0 ≠ ∅ . Let {x n } be the sequence generated by
x n + 1 = J - 1 α n J x 1 + ( 1 - α n ) β n J x n + ( 1 - β n ) J J r n x n ,

then {x n } converges strongly to Π A -10x1.

Proof. In Theorem 3.4 taking D = E, T = J r , r > 0, then T : E → E is a single-valued relatively nonexpansive mapping and A-10 = F(T) = F(J r ),∀r > 0 is a nonempty closed convex subset of E. Therefore all the conditions in Theorem 3.4 are satisfied. The conclusion of Theorem 4.1 can be obtained from Theorem 3.4 immediately.

Declarations

Acknowledgements

This study was supported by Scientific Research Fund of Sichuan Provincial Education Department (11ZB146) and Yunnan University of Finance and Economics.

Authors’ Affiliations

(1)
Department of Mathematics, Yibin University, Yibin, P. R. China
(2)
College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, China

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© Zhu 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.

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