Positive Solutions for Third-Order 𝑝-Laplacian Functional Dynamic Equations on Time Scales

Abstract

We study the following third-order 𝑝-Laplacian functional dynamic equation on time scales: Φ_𝑝(𝑢^Δ∇(𝑡))^∇+𝑎(𝑡)𝑓(𝑢(𝑡),𝑢(𝜇(𝑡)))=0, 𝑡∈(0,𝑇)_𝐓,  𝑢(𝑡)=𝜑(𝑡),  𝑡∈[−𝑟,0]_𝐓,  𝑢^Δ(0)=𝑢^Δ∇(𝑇)=0, and 𝑢(𝑇)+𝐵₀(𝑢^Δ(𝜂))=0. By applying the Five-Functional Fixed Point Theorem, the existence criteria of three positive solutions are established.

1. Introduction

Recently, much attention has been paid to the existence of positive solutions for the boundary value problems with 𝑝-Laplacian operator on time scales; for example, see [1–22] and the references therein. But, to the best of our knowledge, there is not much concerning 𝑝-Laplacian functional dynamic equations on time scales [6, 12–14, 19, 21, 22], especially for the third-order 𝑝-Laplacian functional dynamic equations on time scales [14, 22].

In [14], Song and Gao were concerned with the existence of positive solutions for the 𝑝-Laplacian functional dynamic equation on time scales: Φ_𝑝𝑢^Δ∇(𝑡)^∇+𝑎(𝑡)𝑓𝑢(𝑡),𝑢𝜇(𝑡)=0,𝑡∈(0,𝑇)_𝐓,𝑢(𝑡)=𝜑(𝑡),𝑡∈[−𝑟,0]_𝐓,𝑢^Δ(0)=𝑢^Δ∇(𝑇)=0,𝑢(𝑇)+𝐵₀𝑢^Δ𝜂=0, (1) where 𝜂∈(0,𝜌(𝑇))_𝐓 and Φ_𝑝(𝑠) is 𝑝-Laplacian operator; that is, Φ_𝑝(𝑠)=|𝑠|^𝑝−2𝑠, 𝑝>1, (Φ_𝑝)⁻¹=Φ_𝑞, 1/𝑝+1/𝑞=1, and(C₁)𝑓∶(ℝ⁺)²→ℝ⁺ is continuous;(C₂)𝑎∶𝐓→ℝ⁺ is left dense continuous (i.e., 𝑎∈𝐶_𝐥𝐝(𝐓,ℝ⁺)) and does not vanish identically on any closed subinterval of [0,𝑇], where 𝐶_𝐥𝐝(𝐓,ℝ⁺) denotes the set of all left dense continuous functions from 𝐓 to ℝ⁺;(C₃)𝜑∶[−𝑟,0]_𝐓→ℝ⁺ is continuous and 𝑟>0;(C₄)𝜇∶[0,𝑇]_𝐓→[−𝑟,𝑇]_𝐓 is continuous, 𝜇(𝑡)≤0 for all 𝑡;(C₅)𝐵₀∶ℝ→ℝ is continuous and satisfies the condition that there are 𝐴≥𝐵≥0 such that 𝐵𝑣≤𝐵₀(𝑣)≤𝐴𝑣,∀ℝ. (2)The existence of two positive solutions to problem (1) was obtained by using a double fixed point theorem due to Avery et al. [23] in a cone.

In [22], Wang and Guan considered the existence of positive solutions to problem (1) by applying the well-known Leggett-Williams Fixed Point Theorem.

Motivated by [14, 22], we will show that problem (1) has at least three positive solutions by means of the Five-Functional Fixed Point Theorem [24] (which is a generalization of the Leggett-Williams Fixed Point Theorem [25]). It is worth noting that the Five-Functional Fixed Point Theorem is used extensively in yielding three solutions for BVPs of differential equations, difference equations, and/or dynamic equations on time scales; see [6, 26, 27] and references therein.

Throughout this work we assume knowledge of time scales and time-scale notation, first introduced by Hilger [28]. For more on time scales, please see the texts by Bohner and Peterson [29, 30].

In the remainder of this section, we state the following theorem, which is crucial to our proof.

Let 𝛾, 𝛽, 𝜃 be nonnegative, continuous, and convex functionals on 𝑃 and let 𝛼, 𝜓 be nonnegative, continuous, and concave functionals on 𝑃. Then, for nonnegative real numbers ℎ, 𝑎, 𝑏, 𝑑, and 𝑐, we define the convex sets 𝑃=(𝑥),𝑃=(𝑥)(𝑥),𝑄=(𝑥)(𝑥),𝑃=(𝑥)(𝑥)(𝑥),𝑄=(𝑥)(𝑥)(𝑥).𝛾,𝑐𝑥∈𝑃∶𝛾<𝑐𝛾,𝛼,𝑎,𝑐𝑥∈𝑃∶𝑎≤𝛼,𝛾≤𝑐𝛾,𝛽,𝑑,𝑐𝑥∈𝑃∶𝛽≤𝑑,𝛾≤𝑐𝛾,𝜃,𝛼,𝑎,𝑏,𝑐𝑥∈𝑃∶𝑎≤𝛼,𝜃≤𝑏,𝛾≤𝑐𝛾,𝛽,𝜓,ℎ,𝑑,𝑐𝑥∈𝑃∶ℎ≤𝜓,𝛽≤𝑑,𝛾≤𝑐 (3)

Theorem 1 (see [24]). Let 𝑃 be a cone in a real Banach space 𝐸. Suppose there exist positive numbers 𝑐 and 𝑀; nonnegative, continuous, and concave functionals 𝛼 and 𝜓 on 𝑃; and nonnegative, continuous, and convex functionals 𝛾, 𝛽, and 𝜃 on 𝑃, with 𝛼(𝑥)≤𝛽(𝑥),‖𝑥‖≤𝑀𝛾(𝑥) (4) for all 𝑥∈𝑃(𝛾,𝑐). Suppose 𝐹∶𝑃(𝛾,𝑐)⟶𝑃(𝛾,𝑐) (5) is completely continuous and there exist nonnegative numbers ℎ, 𝑎, 𝑘, 𝑏, with 0<𝑎<𝑏 such that(i){𝑥∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑘,𝑐)∶𝛼(𝑥)>𝑏}≠Ø and 𝛼(𝐹𝑥)>𝑏 for 𝑥∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑘,𝑐);(ii){𝑥∈𝑄(𝛾,𝛽,𝜓,ℎ,𝑎,𝑐)∶𝛽(𝑥)<𝑎}≠Ø and 𝛽(𝐹𝑥)<𝑎 for 𝑥∈𝑄(𝛾,𝛽,𝜓,ℎ,𝑎,𝑐);(iii)𝛼(𝐹𝑥)>𝑏 for 𝑥∈𝑃(𝛾,𝛼,𝑏,𝑐) with 𝜃(𝐹𝑥)>𝑘;(iv)𝛽(𝐹𝑥)<𝑎 for 𝑥∈𝑄(𝛾,𝛽,𝑎,𝑐) with 𝜓(𝐹𝑥)<ℎ.
Then 𝐹 has at least three fixed points 𝑥₁,𝑥₂,𝑥₃∈𝑃(𝛾,𝑐) such that 𝛽𝑥₁<𝑎,𝑏<𝛼𝑥₂,𝑎<𝛽𝑥₃𝑤𝑖𝑡ℎ𝛼𝑥₃<𝑏. (6)

2. Existence of Three Positive Solutions

We note that 𝑢(𝑡) is a solution of BVP (1) if and only if 𝑢(𝑡)=^𝑇0(𝑇−𝑠)Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠−𝐵₀^𝜂₀Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟+−𝑎,𝑢)∇𝑟∇𝑠_𝑡0(𝑡−𝑠)Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)[]−𝑎,𝑢∇𝑟∇𝑠,𝑡∈0,𝑇_𝐓,𝜑(𝑡),𝑡∈[−𝑟,0]_𝐓. (7)

Let 𝐸=𝐶_𝐥𝐝([0,𝑇]_𝐓,ℝ) be endowed with ‖𝑢‖=sup_{𝑡∈[0,𝑇]𝐓}|𝑢(𝑡)|, so 𝐸 is a Banach space. Define cone 𝑃⊂𝐸 by 𝑃=𝑢∈𝐸∶𝑢isconcaveandnonnegativevaluedon[0,𝑇]_𝐓,𝑢^Δ(0)=0. (8)

For each 𝑢∈𝐸, extend 𝑢(𝑡) to [−𝑟,𝑇]_𝐓 with 𝑢(𝑡)=𝜑(𝑡) for 𝑡∈[−𝑟,0]_𝐓.

Define 𝐹∶𝑃→𝐸 by (𝑡)=𝐹𝑢)(_𝑇0(𝑇−𝑠)Φ_𝑞×^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠−𝐵₀^𝜂₀Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)+−𝑎,𝑢∇𝑟∇𝑠_𝑡0(𝑡−𝑠)Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)[]−𝑎,𝑢∇𝑟∇𝑠,𝑡∈0,𝑇_𝐓. (9) We seek a point, 𝑢₁, of 𝐹 in the cone 𝑃. Define 𝑢(𝑡)=𝑢₁(𝑡),𝑡∈[0,𝑇]_𝐓,𝜑(𝑡),𝑡∈[−𝑟,0]_𝐓. (10)

Then 𝑢(𝑡) denotes a positive solution of BVP (1).

We have the following results.

Lemma 2. Let 𝑢∈𝑃, and then(1)𝐹∶𝑃→𝑃 is completely continuous;(2)𝑢(𝑡)≥((𝑇−𝑡)/𝑇)‖𝑢‖ for 𝑡∈[0,𝑇]_𝐓;(3)𝑢(𝑡) is decreasing [0,𝑇]_𝐓;(4)(𝑇−𝜍)𝑢(𝜏)≤(𝑇−𝜏)𝑢(𝜍) for 0<𝜏<𝜍<𝑇 and 𝜏,𝜍∈𝐓.

Proof. (1)–(3) are Lemma 3.1 of [14]. It is easy to conclude that (4) is satisfied by the concavity of 𝑢.
Let 𝑙∈𝐓 be fixed such that 0<𝑙<𝜂<𝑇, and set 𝑌₁=𝑡∈[0,𝑇]_𝐓(𝑡);𝑌∶𝜇<0₂=𝑡∈[0,𝑇]_𝐓(𝑡);𝑌∶𝜇≥0₃=𝑌₁∩[0,𝑙]_𝐓. (11)
Throughout this paper, we assume 𝑌₃≠Ø and ∫_𝑌3𝑎(𝑟)∇𝑟>0.
We define the nonnegative, continuous, and concave functionals 𝛼, 𝜓 and the nonnegative, continuous, and convex functionals 𝛽, 𝜃, 𝛾 on the cone 𝑃, respectively, as 𝛾(𝑢)=𝜃(𝑢)=max_{𝑡∈𝜂,𝑇𝐓}𝑢(𝑡)𝜂,𝛼(𝑢)=𝑢=min_{[]𝑡∈0,𝑙𝐓}𝑢(𝑡)(𝑙),𝛽(𝑢)=𝑢=max_{[]𝑡∈𝑙,𝑇𝐓}𝑢(𝑡)(𝑙),𝜓(𝑢)=𝑢=min_{𝑡∈0,𝜂𝐓}𝑢(𝑡)=𝑢𝜂. (12)
We observe that 𝛼(𝑢)=𝛽(𝑢) for each 𝑢∈𝑃.
In addition, by Lemma 2, we have 𝛾(𝑢)=𝑢(𝜂)≥((𝑇−𝜂)/𝑇)‖𝑢‖. Hence ‖𝑢‖≤(𝑇/(𝑇−𝜂))𝛾(𝑢) for all 𝑢∈𝑃.
For convenience, we define 𝜇=𝑇𝑇+𝜂+𝐵Φ_𝑞^𝑇₀𝑎(𝑟)∇𝑟,𝛿=𝐴_𝑌3Φ_𝑞^𝑠₀𝑎(𝑟)()Φ∇𝑟∇𝑠,𝜆=𝑇𝑇+𝑙+𝐵_𝑞^𝑇₀𝑎(𝑟)∇𝑟. (13)

We now state growth conditions on 𝑓 so that BVP (1) has at least three positive solutions.

Theorem 3. Let 0<𝑎<((𝑇−𝑙)/𝑇)𝑏<((𝑇−𝜂)(𝑇−𝑙)/𝑇²)𝑐, 𝜇𝑏<𝛿𝑐, and suppose that 𝑓 satisfies the following conditions: (H₁)𝑓(𝑢,𝜑(𝑠))<Φ_𝑝(𝑐/𝜇), if 0≤𝑢≤(𝑇/(𝑇−𝜂))𝑐, uniformly in 𝑠∈[−𝑟,0]_𝐓, and 𝑓(𝑢₁,𝑢₂)<Φ_𝑝(𝑐/𝜇), if 0≤𝑢_𝑖≤(𝑇/(𝑇−𝜂))𝑐, 𝑖=1,2;(H₂)𝑓(𝑢,𝜑(𝑠))>Φ_𝑝(𝑏/𝛿), if 𝑏≤𝑢≤(𝑇/(𝑇−𝜂))²𝑏, uniformly in 𝑠∈[−𝑟,0]_𝐓;(H₃)𝑓(𝑢,𝜑(𝑠))<Φ_𝑝(𝑎/𝜆), if 0≤𝑢≤(𝑇/(𝑇−𝑙))𝑎, uniformly in 𝑠∈[−𝑟,0]_𝐓, and 𝑓(𝑢₁,𝑢₂)<Φ_𝑝(𝑎/𝜆), if 0≤𝑢_𝑖≤(𝑇/(𝑇−𝑙))𝑎, 𝑖=1,2.
Then BVP (1) has at least three positive solutions of the form 𝑢(𝑡)=𝑢_𝑖(𝑡),𝑡∈[0,𝑇]_𝐓𝜑(𝑡)[],𝑖=1,2,3,,𝑡∈−𝑟,0_𝐓, (14) where max_{𝑡∈[𝑙,𝑇]𝐓}𝑢₁(𝑡)<𝑎, min_{𝑡∈[0,𝑙]𝐓}𝑢₂(𝑡)>𝑏, and 𝑎<max_{𝑡∈[𝑙,𝑇]𝐓}𝑢₃(𝑡) with min_{𝑡∈[0,𝑙]𝐓}𝑢₃(𝑡)<𝑏.

Proof. Let 𝑢∈𝑃(𝛾,𝑐), and then 𝛾(𝑢)=max_{𝑡∈[𝜂,𝑇]𝐓}𝑢(𝑡)=𝑢(𝜂)≤𝑐, and consequently, 0≤𝑢(𝑡)≤𝑐 for 𝑡∈[𝜂,𝑇]_𝐓. Since 𝑢(𝜂)≥((𝑇−𝜂)/𝑇)𝑢(0), so ‖𝑢‖=𝑢(0)≤(𝑇/(𝑇−𝜂))𝑢(𝜂)≤(𝑇/(𝑇−𝜂))𝑐, and this implies (𝑡)≤𝑇0≤𝑢𝑇−𝜂𝑐,for𝑡∈[0,𝑇]_𝐓. (15)
From (H₁), we have 𝛾()=()𝜂=𝐹𝑢𝐹𝑢_𝑇0(𝑇−𝑠)Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠−𝐵₀^𝜂₀Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)+−𝑎,𝑢∇𝑟∇𝑠_𝜂0𝜂−𝑠Φ_𝑞^𝑠₀−𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠≤^𝑇₀𝑇Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠+𝐵^𝑇₀Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟+,𝑢)∇𝑟∇𝑠_𝜂0𝑇Φ_𝑞^𝑇₀𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)Φ,𝑢∇𝑟∇𝑠=𝑇𝑇+𝜂+𝐵_𝑞𝑌1𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟+,𝜑)∇𝑟_𝑌2𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)Φ,𝑢∇𝑟<𝑇𝑇+𝜂+𝐵_𝑞^𝑇₀𝑎(𝑟)𝑐∇𝑟𝜇=𝑐. (16)
Therefore 𝐹𝑢∈𝑃(𝛾,𝑐). (17)
We now turn to property (i) of Theorem 1. Choosing 𝑢≡(𝑇/(𝑇−𝜂))𝑏, 𝑘=(𝑇/(𝑇−𝜂))𝑏, it follows that 𝛼(𝑢)(𝑙)=𝑇=𝑢𝜃(𝑢)𝜂=𝑇𝑇−𝜂𝑏>𝑏,=𝑢𝛾(𝑢)𝜂=𝑇𝑇−𝜂𝑏=𝑘,=𝑢𝑇−𝜂𝑏<𝑐, (18) which shows that {𝑢∈𝑃(𝛾,𝜃,𝛼,𝑏,𝑘,𝑐)∶𝛼(𝑢)>𝑏}≠Ø, and, for 𝑢∈𝑃(𝛾,𝜃,𝛼,𝑏,(𝑇/(𝑇−𝜂))𝑏,𝑐), we have (𝑡)≤𝑇𝑏≤𝑢𝑇−𝜂²𝑏,for𝑡∈[0,𝑙]_𝐓. (19)
From (H₂), we have 𝛼()=(𝑙)=𝐹𝑢𝐹𝑢)(_𝑇0(𝑇−𝑠)Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠−𝐵₀^𝜂₀Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)+−𝑎,𝑢∇𝑟∇𝑠_𝑙0(𝑡−𝑠)Φ_𝑞^𝑠₀−𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠≥−𝐵₀^𝜂₀Φ_𝑞^𝑠₀−𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠≥𝐴^𝜂₀Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠≥𝐴^𝑙₀Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠≥𝐴_𝑌3Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝜑𝜇(𝑟)∇𝑟∇𝑠>𝐴_𝑌3Φ_𝑞^𝑠₀𝑎(𝑟)𝑏∇𝑟∇𝑠𝛿=𝑏. (20)
We conclude that (i) of Theorem 1 is satisfied.
We next address (ii) of Theorem 1. If we take 𝑢≡((𝑇−𝜂)/𝑇)𝑎, ℎ=((𝑇−𝜂)/𝑇)𝑎, then 𝛾(𝑢)=𝑢𝜂=𝑇−𝜂𝑇𝜓(𝑢)𝜂=𝑎<𝑐,=𝑢𝑇−𝜂𝑇𝛽(𝑢)(𝑙)=𝑎=ℎ,=𝑢𝑇−𝜂𝑇𝑎<𝑎. (21)
From this we know that {𝑢∈𝑄(𝛾,𝛽,𝜓,ℎ,𝑎,𝑐)∶𝛽(𝑢)<𝑎}≠Ø. If 𝑢∈𝑄(𝛾,𝛽,𝜓,((𝑇−𝜂)/𝑇)𝑎,𝑎,𝑐), then (𝑡)≤𝑇0≤𝑢𝑇−𝑙𝑎,for𝑡∈[0,𝑇]_𝐓. (22)
From (H₃), we have 𝛽()=(𝑙)=𝐹𝑢𝐹𝑢)(_𝑇0(𝑇−𝑠)Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠−𝐵₀^𝜂₀Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)+−𝑎,𝑢∇𝑟∇𝑠_𝑙0(𝑡−𝑠)Φ_𝑞^𝑠₀(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)≤−𝑎,𝑢∇𝑟∇𝑠_𝑇0𝑇Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠+𝐵^𝑇₀Φ_𝑞^𝑠₀𝑎(𝑟)𝑓𝑢(𝑟),𝑢𝜇(𝑟)∇𝑟∇𝑠+^𝑙₀𝑇Φ_𝑞^𝑇₀𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)()Φ,𝑢∇𝑟∇𝑠=𝑇𝑇+𝑙+𝐵_𝑞𝑌1𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)+,𝜑∇𝑟_𝑌2𝑎(𝑟)𝑓𝑢(𝑟)𝜇(𝑟)()Φ,𝑢∇𝑟<𝑇𝑇+𝑙+𝐵_𝑞^𝑇₀𝑎(𝑟)𝑎∇𝑟𝜆=𝑎. (23)
Now we show that (iii) of Theorem 1 is satisfied. If 𝑢∈𝑃(𝛾,𝛼,𝑏,𝑐) and 𝜃(𝐹𝑢)=𝐹𝑢(𝜂)>(𝑇/(𝑇−𝜂))𝑏, then 𝛼(𝐹𝑢)≥(𝐹𝑢)(𝑙)=𝑇−𝑙𝑇𝐹𝑢(𝑙)≥𝑇−𝑙𝑇𝜂>𝐹𝑢𝑇−𝑙𝑇−𝜂𝑏>𝑏. (24)
Finally, if 𝑢∈𝑄(𝛾,𝛽,𝑎,𝑐) and 𝜓(𝐹𝑢)=𝐹𝑢(𝜂)<((𝑇−𝜂)/𝑇)𝑎, then from (4) of Lemma 2 we have 𝛽()(𝑙)≤𝑇𝐹𝑢=𝐹𝑢(𝑙)≤𝑇𝑇−𝑙𝐹𝑢𝜂𝑇−𝜂𝐹𝑢<𝑎, (25) which shows that condition (iv) of Theorem 1 is fulfilled.
Thus, all the conditions of Theorem 1 are satisfied. Hence, 𝐹 has at least three fixed points 𝑢₁, 𝑢₂, 𝑢₃ satisfying 𝛽𝑢₁<𝑎,𝑏<𝛼𝑢₂,𝑎<𝛽𝑢₃with𝛼𝑢₃<𝑏. (26)
Let 𝑢(𝑡)=𝑢_𝑖(𝑡),𝑡∈[0,𝑇]_𝐓𝜑(𝑡)[],𝑖=1,2,3,,𝑡∈−𝑟,0_𝐓, (27) which are three positive solutions of BVP (1).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees and the editors for their helpful comments and suggestions. Research was supported by the Postdoctoral Fund in China (Grant no. 2013M531717), the Excellent Young Teacher Training Program of Lanzhou University of Technology (Grant no. Q200907), and the Natural Science Foundation of Gansu Province of China (Grant no. 1310RJYA080).