Reverse-Order Lower and Upper Functions for Periodic Problems of Second-Order Singular Difference Equations

Abstract

We present sufficient conditions ensuring the lower and upper functions on the reversed-order for the periodic difference equations. This enables us to obtain the existence of positive periodic solutions of the second-order difference equation Δ²𝑢(𝑡−1)=𝑔(𝑡)/𝑢^𝜇(𝑡)−ℎ(𝑡)/𝑢^𝜆(𝑡)+𝑓(𝑡),𝑡∈ℤ, where 𝑔,ℎ∶ℤ→[0,∞), and 𝑓∶ℤ→ℝ are 𝑇-periodic functions, 𝜆,𝜇>0.

1. Introduction

The theory of nonlinear difference equations has been widely used to study discrete models appearing in many fields such as computer science, economics, neural network, ecology, and cybernetics, see for example, [1]. Recently, there are many papers to study the existence of positive periodic solutions for second-order difference equations, see [2–7] and their references therein. However, there are few techniques for studying the existence of positive solutions of difference equations with singularity, and thus the results in the field are very rare, see [8–13]. The existence of positive periodic solutions for continuous case has been studied by Torres, see [14, 15].

Let ℤ denote the integer set, for 𝑎,𝑏∈ℤ with 𝑎<𝑏, [𝑎,𝑏]_ℤ∶={𝑎,𝑎+1,…,𝑏}.

In 2012, Lu and Ma are concerned with the existence of positive periodic solutions of the second-order difference equation as follows: Δ²𝑢()=𝑔(𝑡)𝑡−1𝑢^𝜇(𝑡)−ℎ(𝑡)𝑢^𝜆(𝑡)+𝑓(𝑡),𝑡∈ℤ, (1) where 𝑔, ℎ∶ℤ→[0,∞), and 𝑓∶ℤ→ℝ are 𝑇-periodic functions, 𝜆, 𝜇>0. Special cases of (1) are Δ²𝑢()=𝑔(𝑡)𝑡−1𝑢^𝜇(𝑡)−ℎ(𝑡)𝑢^𝜆(𝑡)Δ,𝑡∈ℤ,₂𝑢()ℎ(𝑡)𝑡−1=−𝑢^𝜆(𝑡)(𝑡)Δ+𝑓,𝑡∈ℤ,₂𝑢()=𝑔(𝑡)𝑡−1𝑢^𝜇(𝑡)+𝑓(𝑡),𝑡∈ℤ. (2)(3)(4) In the related literature, it is said that (3) has an attractive singularity, whereas (4) has a repulsive singularity. They use the well-order lower and upper functions of (1) to show the existence of positive 𝑇-periodic solution of (1), (2), and (3), respectively, see [13].

However, the well-order lower and upper solutions lose their effects to deal with case (4). In this paper, we are devoted to constructing lower and upper functions on the reversed-order for (1) and dealing with the problems (1), (2), and (4), respectively.

The structure of the paper is as follows. Section 2 contains the tools needed in the proofs. In Section 3 we state and prove the main results and develop some corollaries for the equation with a singularity of mixed type. To illustrate the results, some examples are given.

2. Auxiliary Results

Let 𝐸∶={𝑢∶ℤ⟶ℝ∣𝑢(𝑡)=𝑢(𝑡+𝑇)} (5) under the norm ‖𝑢‖=max_{𝑡∈[1,𝑇]ℤ}|𝑢(𝑡)|. Then (𝐸,‖⋅‖) is a Banach space.

The proofs of our results rely on the method of upper and lower functions. The following lemmas are classical and can be found, for example, in [3]. We introduce them in a form suitable for us.

Lemma 1. Let there exist positive functions 𝛼, 𝛽∈𝐸, such that Δ²𝛼()≥𝑔(𝑡)𝑡−1𝛼^𝜇(𝑡)−ℎ(𝑡)𝛼^𝜆(𝑡)(𝑡)Δ+𝑓,𝑡∈ℤ,₂𝛽()≤𝑔(𝑡)𝑡−1𝛽^𝜇(𝑡)−ℎ(𝑡)𝛽^𝜆(𝑡)+𝑓(𝑡),𝑡∈ℤ, (6)(7) and 𝛼(𝑡)≤𝛽(𝑡) for 𝑡∈[1,𝑇]_ℤ. Then there exists at least one positive 𝑇-periodic solution to (1).
A function 𝛼∈𝐸 (resp., 𝛽∈𝐸) verifying (6) (resp, (7)) is called lower (resp, upper) function (solution) of (1). When the order between the lower and the upper functions is the inverse, an additional hypothesis is needed.

Definition 2. A function 𝜑∈𝐸 and 𝜑≥0 is said to verify the property (𝑃) if the implication 𝑢∈𝐸Δ²𝑢(𝑡−1)+𝜑(𝑡)𝑢(𝑡)≥0,𝑡∈[1,𝑇]_ℤ⟹𝑢(𝑡)≥0,for𝑡∈[1,𝑇]_ℤ (8) holds.

Lemma 3. Let there exist positive functions 𝛼, 𝛽∈𝐸 satisfying (6), (7), and 𝛽(𝑡)≤𝛼(𝑡), 𝑡∈[1,𝑇]_ℤ. Moreover, there exists 𝜑∈𝐸 with the property (𝑃), such that 𝑔(𝑡)𝑢^𝜇(𝑡)−ℎ(𝑡)𝑢^𝜆(𝑡)−𝑔(𝑡)𝑣^𝜇(𝑡)−ℎ(𝑡)𝑣^𝜆(𝑡)(𝑡𝑣(𝑡)(𝑡,[]≤𝜑)[−𝑢)]𝑓𝑜𝑟𝑡∈1,𝑇_ℤ, (9) where 𝛽(𝑡)≤𝑢(𝑡)≤𝑣(𝑡)≤𝛼(𝑡) for 𝑡∈[0,𝑇+1]_ℤ. Then there exists at least one positive 𝑇-periodic solution to (1).

Proof. From the condition (9), it follows that 𝑔(𝑡)𝑢^𝜇(𝑡)−ℎ(𝑡)𝑢^𝜆(𝑡)(𝑡)𝑢(𝑡)≤𝑔(𝑡)+𝜑𝑣^𝜇(𝑡)−ℎ(𝑡)𝑣^𝜆(𝑡)+𝜑(𝑡)𝑣(𝑡). (10) That is, the nonlinearity is increasing.
Define the operator 𝑇∶𝐸→𝐸 as the unique solution of problem (1) as follows: 𝑇𝑢(𝑡)=^𝑡+𝑇_𝑠=𝑡+1𝐺(𝑔(𝑠)𝑡,𝑠)𝑢^𝜇(𝑠)−ℎ(𝑠)𝑢^𝜆(𝑠)+𝜑(𝑡)𝑢(𝑡)+𝑓(𝑠), (11) where 𝐺(𝑡,𝑠) is the Green’s function of Δ²𝑢(𝑡−1)+𝜑(𝑡)𝑢(𝑡)=0,𝑡∈ℤ,𝑢(𝑡)=𝑢(𝑡+𝑇). (12) As 𝜑 satisfies the property (𝑃), it follows that 𝐺(𝑡,𝑠)>0. Now we divide the proof into three steps.
Step  1. We show 𝑇(𝐾)⊂𝐾, (13) where 𝐾={𝑢∈𝐸∣𝛽≤𝑢≤𝛼} is a nonempty bounded closed subset in 𝐸.
In fact, for 𝑢∈𝐾, set 𝑤=𝑇𝑢(𝑡). From the definitions of 𝛼, 𝛽, and 𝐾, combining with (9), we have Δ²()(𝑡(𝑡)(𝑡)=().𝑤−𝛽𝑡−1+𝜑)𝑤−𝛽≥0,𝑤−𝛽𝑤−𝛽𝑡+𝑇 (14) Using property (𝑃), we get 𝛽≤𝑤.
Analogously, we can prove that 𝑤≤𝛼. Thus, (13) holds.
Step  2. Let 𝑢₁=𝑇𝜂₁, 𝑢₂=𝑇𝜂₂, where 𝜂₁, 𝜂₂∈𝐾 satisfy 𝛽≤𝜂₁≤𝜂₂≤𝛼. Then we claim that 𝑢₁≤𝑢₂. (15) In fact, let 𝜔=𝑢₂−𝑢₁, it follows from (10) and (11) that 𝜔(𝑡)=𝑇𝜂₂−𝑇𝜂₁=^𝑡+𝑇_𝑠=𝑡+1𝐺(𝑔(𝑠)𝑡,𝑠)𝜂^𝜇₂(𝑠)−ℎ(𝑠)𝜂^𝜆₂(𝑠)+𝜑(𝑡)𝜂₂(𝑡)−𝑔(𝑠)𝜂^𝜇₁(𝑠)−ℎ(𝑠)𝜂^𝜆₁(𝑠)+𝜑(𝑡)𝜂₁(𝑡)≥0. (16)
Step  3. The sequences {𝛼_𝑛} and {𝛽_𝑛} are obtained by recurrence: 𝛼₀=𝛼,𝛽₀𝛼=𝛽,_𝑛=𝑇𝛼_𝑛−1,𝛽_𝑛=𝑇𝛽_𝑛−1,𝑛=1,2,…. (17) From the results of Steps  1 and  2, it follows that 𝛽=𝛽₀≤𝛽₁≤⋯≤𝛽_𝑛≤⋯≤𝛼_𝑛≤⋯≤𝛼₀=𝛼. (18) Moreover, from the definition of 𝑇, we get Δ²𝛼_𝑛(𝑡−1)+𝜑(𝑡)𝛼_𝑛(𝑡)=𝑔(𝑡)𝛼^𝜇_𝑛−1(𝑡)−ℎ(𝑡)𝛼^𝜆_𝑛−1(𝑡)+𝜑(𝑡)𝛼_𝑛−1(𝑡)(𝑡)𝛼+𝑓,𝑡∈ℤ,_𝑛(𝑡)=𝛼_𝑛(),Δ𝑡+𝑇₂𝛽_𝑛(𝑡−1)+𝜑(𝑡)𝛽_𝑛(𝑡)=𝑔(𝑡)𝛽^𝜇_𝑛−1(𝑡)−ℎ(𝑡)𝛽^𝜆_𝑛−1(𝑡)+𝜑(𝑡)𝛽_𝑛−1(𝑡)(𝑡)𝛽+𝑓,𝑡∈ℤ,_𝑛(𝑡)=𝛽_𝑛(𝑡+𝑇). (19)
This together with (18), we can easily get that there exists 𝐶_𝛼 depending only on 𝛼 but not on 𝑡 and 𝑛, such that |𝛼_𝑛|≤𝐶_𝛼, so {𝛼_𝑛} is bounded in 𝐸. Similarly, {𝛽_𝑛} is bounded in 𝐸. Therefore, we can conclude that {𝛼_𝑛} and {𝛽_𝑛} converge uniformly to the extremal solutions 𝑢 of the problem (1). Subsequently, there exists at least one positive 𝑇-periodic solutions of (1).

Property (𝑃) is just an anti-maximum principle for the linear operator 𝐿𝑢∶=Δ²𝑢(𝑡−1)+𝜑(𝑡)𝑢(𝑡) with periodic boundary conditions, and it is equivalent to have a nonnegative Green function. Reference [7] provides sufficient conditions for 𝜑 to verify property (𝑃). In particular, we have the following lemma.

Lemma 4 (see [7]). Let us assume 𝜑∈𝐸, 𝜑≢0 and the following conditions holds: 0≤𝜑(𝑡)<4sin²𝜋[]2𝑇,𝑓𝑜𝑟𝑡∈1,𝑇_ℤ. (20) Then 𝜑 verifies the property (𝑃).

To finish this section, we give a technical bound on the amplitude of oscillation of a periodic function.

Lemma 5 ([13, Lemma  2.2]). Given 𝑣∈𝐸, then 𝑀_𝑣−𝑚_𝑣≤𝑇4^𝑇_𝑠=1Δ²𝑣(𝑠−1)₊, (21) where 𝑀_𝑣=max𝑣(𝑡)∶𝑡∈[1,𝑇]_ℤ,𝑚_𝑣=min𝑣(𝑡)∶𝑡∈[1,𝑇]_ℤ. (22) Moreover, (21) is fulfilled as an equality if and only if 𝑣 is a constant function.

3. The Main Results

For the sake of brevity we will use the following notation throughout the paper: 𝐺=^𝑇_𝑠=1𝑔(𝑠),𝐻=^𝑇_𝑠=1ℎ(𝑠),𝐹=^𝑇_𝑠=1𝑓(𝑠),𝐹₊=^𝑇_𝑠=1𝑓(𝑠)₊,𝐹₋=^𝑇_𝑠=1𝑓(𝑠)₋. (23) The following theorems are the main results of the paper.

Theorem 6. Let 𝐺>0, 𝐹<0, let functions 𝑤, 𝜎∈𝐸 be such that the equalities Δ²𝑤()(𝑡)(𝑡)Δ𝑡−1=𝐻𝑔−𝐺ℎ,𝑓𝑜𝑟𝑡∈ℤ,₂𝜎()=|𝐹|𝑡−1𝐺𝑔(𝑡)+𝑓(𝑡),𝑓𝑜𝑟𝑡∈ℤ (24)(25) are fulfilled and let there exist 𝑥₀∈(0,+∞) such that 𝑥₀𝑤(𝑡)−𝑚_𝑤+𝜎(𝑡)−𝑚_𝜎≤𝐺𝑥₀𝐺𝐻+|𝐹|^1/𝜇−1𝑥₀𝐺^1/𝜆,𝑡∈[1,𝑇]_ℤ, (26) where 𝑚_𝑤=min𝑤(𝑡)∶𝑡∈[1,𝑇]_ℤ,𝑚_𝜎=min𝜎(𝑡)∶𝑡∈[1,𝑇]_ℤ. (27) Moreover, define 𝛽(𝑡)=1𝑥₀𝐺^1/𝜆+𝑥₀𝑤(𝑡)−𝑚_𝑤+𝜎(𝑡)−𝑚_𝜎,𝑓𝑜𝑟𝑡∈ℤ (28) and assume that 𝜑(𝑡)=𝜇𝑔(𝑡)/𝛽^1+𝜇(𝑡) verifies the property (𝑃). Then problem (1) has at least one positive 𝑇-periodic solution.

Proof. Let 𝛽 be defined by (28). Then 𝛽∈𝐸 and in view of (24) and (25), we have Δ²𝛽()=𝑥𝑡−1₀|𝐹|𝐻+𝐺𝑔(𝑡)−𝑥₀𝐺ℎ(𝑡)+𝑓(𝑡),for𝑡∈ℤ. (29) Moreover, according to (26) and (27) 1𝑥₀𝐺^1/𝜆(𝑡)≤𝐺≤𝛽𝑥₀𝐺𝐻+|𝐹|^1/𝜇,for𝑡∈ℤ. (30) Now (29) and (30) imply Δ²𝛽()≤𝑔(𝑡)𝑡−1𝛽^𝜇(𝑡)−ℎ(𝑡)𝛽^𝜆(𝑡)+𝑓(𝑡),for𝑡∈ℤ. (31) Consequently, 𝛽 is an upper function to (1).
Further, we can choose 𝑥₁∈(0,𝑥₀) such that 𝑥₁𝑤(𝑡)−𝑚_𝑤+𝜎(𝑡)−𝑚_𝜎≤1𝑥₁𝐺^1/𝜆−𝐺𝑥₁𝐺𝐻+|𝐹|^1/𝜇,for𝑡∈[1,𝑇]_ℤ (32) and put 𝛼(𝑡)=𝐺𝑥₁𝐺𝐻+|𝐹|^1/𝜇+𝑥₁𝑤(𝑡)−𝑚_𝑤+𝜎(𝑡)−𝑚_𝜎,for𝑡∈ℤ. (33) Then 𝛼∈𝐸, and in view of (24) and (25) we have Δ²𝛼(𝑡−1)=𝑥₁|𝐹|𝐻+𝐺𝑔(𝑡)−𝑥₁𝐺ℎ(𝑡)+𝑓(𝑡),𝑡∈ℤ. (34) Moreover, according to (27) and (32) 𝐺𝑥₀𝐺𝐻+|𝐹|^1/𝜇(𝑡)≤1≤𝛼𝑥₁𝐺^1/𝜆,for𝑡∈ℤ. (35) Now (34) and (35) imply Δ²𝛼()≥𝑔(𝑡)𝑡−1𝛼^𝜇(𝑡)−ℎ(𝑡)𝛼^𝜆(𝑡)+𝑓(𝑡),𝑡∈ℤ. (36) Consequently, 𝛼 is a lower function to (1) and according to (30) and (35) we have 𝛽(𝑡)≤𝛼(𝑡),for𝑡∈[1,𝑇]_ℤ. (37)
Furthermore, note that the function 𝜓𝑦=𝜇𝛽^1+𝜇1𝑦+𝑦^𝜇 (38) is nondecreasing for 𝑦≥𝛽. Therefore we have 𝑔(𝑡)𝜇𝛽^1+𝜇(𝑡)𝑢(𝑡)+1𝑢^𝜇(𝑡)−ℎ(𝑡)𝑢^𝜆(𝑡)(𝑡)𝜇≤𝑔𝛽^1+𝜇(𝑡)𝑣(𝑡)+1𝑣^𝜇(𝑡)−ℎ(𝑡)𝑣^𝜆(𝑡),𝑡∈[1,𝑇]_ℤ, (39) whenever 𝛽(𝑡)≤𝑢(𝑡)≤𝑣(𝑡) for 𝑡∈[1,𝑇]_ℤ, hence, we get 𝑔(𝑡)𝑢^𝜇(𝑡)−ℎ(𝑡)𝑢^𝜆(𝑡)−𝑔(𝑡)𝑣^𝜇(𝑡)−ℎ(𝑡)𝑣^𝜆(𝑡)≤𝜇𝑔(𝑡)𝛽^1+𝜇(𝑡)(𝑣(𝑡)−𝑢(𝑡)),for𝑡∈[1,𝑇]_ℤ. (40) Thus, the assertion follows from Lemma 3.

Remark 7. Note that for every 𝑞∈𝐸 such that ∑^𝑇_𝑠=1𝑞(𝑠)=0, the periodic solution 𝑣 of the equation Δ²𝑣(𝑡−1)=𝑞(𝑡),for𝑡∈[1,𝑇]_ℤ,𝑣(0)=𝑣(𝑇),Δ𝑣(0)=Δ𝑣(𝑇) (41) is given by the Green formula: 𝑣(𝑡)1=𝑐−𝑇^𝑡−1_𝑠=1(𝑇−𝑡)𝑠𝑞(𝑠)+^𝑇_𝑠=𝑡𝑡(𝑇−𝑠)𝑞(𝑠), (42) where 𝑐∈ℝ. Therefore, the periodic functions 𝑤 and 𝜎 with properties (24) and (25) exist and, moreover, are unique up to a constant term, the value of which has no influence on the validity of the condition (26). A similar observation can be made in relation to the formulations of the theorems given below.

Theorem 8. Let 𝜇>𝜆, 𝐻>0, 𝐺>0, 𝐹=0, let functions 𝑤, 𝜎∈𝐸 be such that (24) and (25) are fulfilled and let there exist 𝑥₀∈(0,+∞) such that 𝑥₀𝑤(𝑡)−𝑚_𝑤+𝜎(𝑡)−𝑚_𝜎≤1𝑥₀𝐻^1/𝜇−1𝑥₀𝐺^1/𝜆,𝑡∈[1,𝑇]_ℤ, (43) where 𝑚_𝑤 and 𝑚_𝜎 are defined by (27). Moreover, assume that 𝜑(𝑡)=𝜇𝑔(𝑡)/𝛽^1+𝜇(𝑡) verifies the property (𝑃), where 𝛽 is given by (28). Then problem (1) has at least one positive 𝑇-periodic solution.

Proof. Note that the inequality 𝜇>𝜆 implies lim_𝑥→0+1𝑥𝐺^1/𝜆−1𝑥𝐻^1/𝜇=+∞. (44) Therefore, analogously to the proof of Theorem 6, one can show that there exist lower and upper functions 𝛼, 𝛽 satisfying (37). Consequently, the assertion follows from Lemma 3 with 𝜑(𝑡)=𝜇𝑔(𝑡)/𝛽^1+𝜇(𝑡).

Corollary 9. Let 𝜇>𝜆, 𝐻>0, 𝐺>0, and let 𝑤∈𝐸 be such that (24) is fulfilled. Let 𝑀_𝑤−𝑚_𝑤≤𝐺^{(1+𝜇)/(𝜇−𝜆)}𝐻^{(1+𝜆)/(𝜇−𝜆)}1+𝜇𝜆(1+𝜆)𝜇^{((1+𝜇)𝜆)/(𝜇−𝜆)}𝜇−𝜆()𝜇,1+𝜆 (45) where 𝑚_𝑤 is given by (27) and 𝑀_𝑤=max𝑤(𝑡)∶𝑡∈[1,𝑇]_ℤ. (46) Moreover, let us define 𝛽(𝑡)=1+𝜇𝜆(1+𝜆)𝜇^{𝜇/(𝜇−𝜆)}𝐺𝐻^{1/(𝜇−𝜆)}+(1+𝜆)𝜇1+𝜇𝜆^{𝜆𝜇/(𝜇−𝜆)}×𝐻^{𝜆/(𝜇−𝜆)}𝐺^{𝜇/(𝜇−𝜆)}𝑤(𝑡)−𝑚_𝑤,𝑓𝑜𝑟𝑡∈ℤ, (47) and assume that 𝜑(𝑡)=𝜇𝑔(𝑡)/𝛽^1+𝜇(𝑡) verifies the property (𝑃). Then problem (2) has at least one positive solution.

Proof. Put 𝑓≡0 and 𝑥₀=𝐻^{𝜆/(𝜇−𝜆)}𝐺^{𝜇/(𝜇−𝜆)}(1+𝜆)𝜇1+𝜇𝜆^{𝜆𝜇/(𝜇−𝜆)}. (48) Then the assertion follows from Theorem 8.

Corollary 10. Let 𝐺>0, 𝐹<0, let 𝜎∈𝐸 be such that (25) is fulfilled, and let 𝑀_𝜎−𝑚_𝜎^𝜇|𝐹|<𝐺, (49) where 𝑚_𝜎 is defined by (27) and 𝑀_𝜎=max𝜎(𝑡)∶𝑡∈[1,𝑇]_ℤ. (50) Moreover, if 𝜇|𝐹|^{(1+𝜇)/𝜇}𝑔(𝑡)𝐺^1/𝜇−|𝐹|^1/𝜇𝑀_𝜎−𝜎(𝑡)^1+𝜇<4sin²𝜋[]2𝑇,𝑡∈1,𝑇_ℤ (51) holds. Then problem (4) has at least one positive 𝑇-periodic solution.

Proof. Put 𝐻=0, 𝑥₀=|𝐹|^𝜆/𝜇𝐺𝐺^1/𝜇−|𝐹|^1/𝜇𝑀_𝜎−𝑚_𝜎^𝜆 (52) and define a function 𝛽 by (28). Let 𝛽(𝑡)=(𝐺/|𝐹|)^1/𝜇−𝑀_𝜎+𝜎(𝑡) for 𝑡∈[1,𝑇]_ℤ. Then (51) guaranties that 𝜑(𝑡)=𝜇𝑔(𝑡)/𝛽^1+𝜇(𝑡) satisfies the property (𝑃). Moreover, (51) yields (25). Therefore, the assertion follows from Theorem 6.

Corollary 11. Let 𝐺>0, 𝐹<0 and 𝑇4𝜇𝐺^1/(1+𝜇)|𝐹|^1/𝜇+𝑇4𝐹₋|𝐹|^1/𝜇≤𝐺^1/𝜇. (53) Then problem (4) has at least one positive 𝑇-periodic solution.

Proof. According to Lemma 5, 𝑀_𝜎−𝑚_𝜎≤𝑇4𝐹₋. (54) Then (53) implies (51). Consequently, the assertion follows from Corollary 10.

Example 12. Let us consider the boundary value problem: Δ²𝑢()=1𝑡−1𝑢−1[]4𝑇,𝑡∈1,𝑇_ℤ,𝑢(0)=𝑢(𝑡),Δ𝑢(0)=Δ𝑢(𝑇), (55) where 𝑇>3 is an integer.
Obviously, (1/𝑢)−(1/4)→+∞ as 𝑢→0. Let 𝑔≡1, 𝑓≡−1/4𝑇, 𝜇=1. Then 𝐺=𝑇, 𝐹=−1/4 and 𝑇4𝜇𝐺^1/(1+𝜇)|𝐹|^1/𝜇+𝑇4𝐹₋|𝐹|^1/𝜇=9𝑇64<𝑇=𝐺^1/𝜇. (56) Thus, from Corollary 11, problem (55) has at least one positive 𝑇-periodic solution.

Acknowledgments

The authors are very grateful to the anonymous referees for their valuable suggestions. This work was supported by the NSFC (no. 11061030, no. 11361054), Gansu provincial National Science Foundation of China (no. 1208RJZA258) and SRFDP (no. 20126203110004).