A New Oscillation Criterion for Forced Second-Order Quasilinear Differential Equations

Abstract

By using the generalized variational principle and Riccati technique, a new oscillation criterion is established for second-order quasilinear differential equation with an oscillatory forcing term, which improves and generalizes some of new results in the literature.

1. Introduction

In this paper, we are concerned with oscillation for the second order forced quasilinear differential equation𝑝(𝑡)𝑦^′(𝑡)^𝛼−1𝑦^′(𝑡)^′+𝑞(𝑡)𝑦(𝑡)^𝛽−1𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀, (1.1) where 𝑝,𝑞,𝑒∈𝐶([𝑡₀,∞),ℝ) with 𝑝(𝑡)>0 and 0<𝛼≤𝛽 being constants.

By a solution of (1.1), we mean a function 𝑦(𝑡)∈𝐶¹([𝑇_𝑦,∞),ℝ), where 𝑇_𝑦≥𝑡₀ depends on the particular solution, which has the property 𝑝(𝑡)|𝑦′(𝑡)|^𝛼−1𝑦′(𝑡)∈𝐶¹[𝑇_𝑦,∞) and satisfies (1.1). We restrict our attention to the nontrivial solutions 𝑦(𝑡) of (1.1) only, that is, to solutions 𝑦(𝑡) such that sup{|𝑦(𝑡)|∶𝑡≥𝑇}>0 for all 𝑇≥𝑇_𝑦. A nontrivial solution of (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise, it is said to be nonoscillatory. Equation (1.1) is said to be oscillatory if all its nontrivial solutions are oscillatory.

We note that when 0<𝛼=𝛽, (1.1) turns into the half-linear differential equation𝑝(𝑡)𝑦^′(𝑡)^𝛼−1𝑦^′(𝑡)^′+𝑞(𝑡)𝑦(𝑡)^𝛼−1𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀, (1.2) while 0<𝛼<𝛽, (1.1) is a super-half-linear differential equation.

Oscillation for (1.2) and its special case (without a forcing term)𝑝(𝑡)𝑦^′(𝑡)^𝛼−1𝑦^′(𝑡)^′+𝑞(𝑡)𝑦(𝑡)^𝛼−1𝑦(𝑡)=0,𝑡≥𝑡₀, (1.3) are widely studied by many authors (see [1–6] and references cited therein) since the foundation work of Elbert [7]. Manojlović [8], Ayanlar and Tiryaki [9] obtained some Kamenev type oscillation criteria for (1.3) by using integral averaging techniques; a more detailed description of oscillation criteria can be found in the book in [10].

The half-linear differential equation (1.3) has similar properties to linear differential equation; for example, Sturm's comparison theorem and separation theorem (see Elbert [7], Jaros and Kusano [5], Li and Yeh [11] for details) are still true for (1.3). In particular, the Riccati type equation (related to (1.3) by the substitution 𝑤=𝑝|𝑦^′|^𝛼−1𝑦′/|𝑦|^𝛼−1𝑦)𝑤^′+𝑞(𝑡)+𝛼𝑝^−1/𝛼(𝑡)|𝑤|^{(𝛼+1)/𝛼}=0 (1.4) and the (𝛼+1)-degree functional𝐼𝑦∶=^𝑏_𝑎𝑞(𝑡)𝑦^𝛼+1−𝑝(𝑡)𝑦^′𝛼+1𝑑𝑡 (1.5) play the same role as the classical Riccati equation and quadratic functional in the linear oscillation theory. (The proof of the relationship between (1.3), (1.4), and (1.5) is based on the recently established Picone's identity, see [5].)

When 𝛼=1, (1.2) is reduced to the forced linear differential equation𝑝(𝑡)𝑦^′(𝑡)^′+𝑞(𝑡)𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀. (1.6)

In paper [12], based on the oscillatory behavior of the forcing term, Wong proved the following result for (1.6).

Theorem 1.1. Suppose that, for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑒(𝑡)≤0 for 𝑡∈[𝑠₁,𝑡₁] and 𝑒(𝑡)≥0 for 𝑡∈[𝑠₂,𝑡₂]. Let 𝐷(𝑠_𝑖,𝑡_𝑖)={𝑢∈𝐶¹[𝑠_𝑖,𝑡_𝑖]∶𝑢(𝑡)≠0,𝑢(𝑠_𝑖)=𝑢(𝑡_𝑖)=0} for 𝑖=1,2. If there exists 𝑢∈𝐷(𝑠_𝑖,𝑡_𝑖) such that 𝑄_𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝑞(𝑡)𝑢²(𝑡)−𝑝(𝑡)𝑢^′(𝑡)²𝑑𝑡>0,𝑖=1,2, (1.7) then (1.6) is oscillatory.

This result involves the “oscillatory interval” of 𝑒(𝑡) and Leighton's variational principle for (1.6). Recently, using a similar method, Wong's result was extended to (1.2) by Li and Cheng [1] with a positive and nondecreasing function 𝜙∈𝐶¹([𝑡₀,∞),ℝ).

Theorem 1.2. Suppose that, for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑒(𝑡)≤0 for 𝑡∈[𝑠₁,𝑡₁] and 𝑒(𝑡)≥0 for 𝑡∈[𝑠₂,𝑡₂]. Let 𝐷(𝑠_𝑖,𝑡_𝑖)={𝑢∈𝐶¹[𝑠_𝑖,𝑡_𝑖]∶𝑢(𝑡)≢0,𝑢(𝑠_𝑖)=𝑢(𝑡_𝑖)=0} for 𝑖=1,2. If there exist 𝐻∈𝐷(𝑠_𝑖,𝑡_𝑖) and a positive, nondecreasing function 𝜙∈𝐶¹([𝑡₀,∞),ℝ⁺) such that ^𝑡𝑖_𝑠𝑖𝐻²(𝑡)𝜙(𝑡)𝑞(𝑡)𝑑𝑡>𝐾^𝑡𝑖_𝑠𝑖𝑝(𝑡)𝜙(𝑡)|𝐻(𝑡)|^𝛼−12𝐻^′(𝑡+|𝐻(𝑡𝜙))|^′(𝑡)𝜙(𝑡)^𝛼+1𝑑𝑡 (1.8) for 𝑖=1,2, where 𝐾=(1/(𝛼+1))^𝛼+1, then (1.2) is oscillatory.

However, inequality (1.8) has no relation to the (𝛼+1)-degree functional (1.5), and Theorem 1.2 cannot be applied to oscillation when 𝛼>1, since |𝐻(𝑡)|^𝛼−1 is the denominator of the fraction in the right-side integral of (1.8), and 𝐻(𝑠_𝑖)=𝐻(𝑡_𝑖)=0.

Zheng and Meng [4] improved the paper [1] and gave an oscillation criterion without any restriction on the signs of 𝑞, 𝑒, and 𝜙^′, which is closely related to the (𝛼+1)-degree functional (1.5). Here we list the main result of Zheng and Meng (Jaros et al. [6] obtained a similar result using the generalized Pocone's formulae with 𝜙(𝑡)≡1).

Theorem 1.3. Suppose that, for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑒(𝑡)≤0 for 𝑡∈[𝑠₁,𝑡₁] and 𝑒(𝑡)≥0 for 𝑡∈[𝑠₂,𝑡₂]. Let 𝐷(𝑠_𝑖,𝑡_𝑖)={𝑢∈𝐶¹[𝑠_𝑖,𝑡_𝑖]∶𝑢(𝑡)≢0,𝑢(𝑠_𝑖)=𝑢(𝑡_𝑖)=0} for 𝑖=1,2. If there exist 𝐻∈𝐷(𝑠_𝑖,𝑡_𝑖) and a positive function 𝜙∈𝐶¹([𝑡₀,∞),ℝ⁺) such that 𝑄^𝜙_𝑖(𝐻)∶=^𝑡𝑖_𝑠𝑖𝜙𝑄_𝑒𝐻^𝛼+1−𝑝𝐻^′+𝐻𝜙^′(𝛼+1)𝜙^𝛼+1(𝑡)𝑑𝑡>0 (1.9) for 𝑖=1,2, then (1.1) is oscillatory, where 𝑄_𝑒(𝑡) is the same as (2.4).

Meanwhile, among the oscillation criteria, Komkov [13] gave a generalized Leighton's variational principle, which also can be applied to oscillation for linear equation.

Theorem 1.4. Suppose that there exist a 𝐶¹ function 𝑢(𝑡) defined on [𝑠₁,𝑡₁] and a function 𝐺(𝑢) such that 𝐺(𝑢(𝑡)) is not constant on [𝑠₁,𝑡₁], 𝐺(𝑢(𝑠₁))=𝐺(𝑢(𝑡₁))=0, 𝑔(𝑢)=𝐺^′(𝑢) is continuous, ^𝑡1_𝑠1𝑞(𝑡)𝐺(𝑢(𝑡))−𝑝(𝑡)𝑢^′(𝑡)²𝑑𝑡>0, (1.10) and (𝑔(𝑢(𝑡)))²≤4𝐺(𝑢(𝑡)) for 𝑡∈[𝑠₁,𝑡₁]. Then every solution of the equation (𝑝(𝑡)𝑦′(𝑡))′+𝑞(t)𝑦(𝑡)=0 must vanish on [𝑠₁,𝑡₁].

As far as we know, there is no oscillation criterion which relates to the generalized Leighton's variational principle for (1.1). The question then arises as to whether a more general result can be established to this equation. In this paper, we give an affirmative answer for this question and give a new oscillation criterion for the quasilinear differential equation (1.1); this oscillation criterion is closely related to the generalized Leighton's variational principle, which generalizes and improves the results mention above.

2. Main Results

Firstly, we give an inequality, which is a transformation of Young's inequality.

Lemma 2.1 (see [14]). Supposing that 𝑋 and 𝑌 are nonnegative, then 𝛾𝑋𝑌^𝛾−1−𝑋^𝛾≤𝛾−1𝑌^𝛾,𝛾>1, (2.1) where equality holds if and only if 𝑋=𝑌.

Now, we will give our main results.

Theorem 2.2. Assume that, for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑒(𝑡𝑠)≤0,𝑡∈₁,𝑡₁,≥0,𝑡∈𝑠₂,𝑡₂. (2.2) Let 𝑢∈𝐶¹[𝑠_𝑖,𝑡_𝑖], and nonnegative functions 𝐺₁,𝐺₂ satisfying 𝐺_𝑖(𝑢(𝑠_𝑖))=𝐺_𝑖(𝑢(𝑡_𝑖))=0, 𝑔_𝑖(𝑢)=𝐺^′_𝑖(𝑢) are continuous and (𝑔_𝑖(𝑢(𝑡)))^𝛼+1≤(𝛼+1)^𝛼+1𝐺^𝛼_𝑖(𝑢(𝑡)) for 𝑡∈[𝑠_𝑖,𝑡_𝑖], 𝑖=1,2. If there exists a positive function 𝜙∈𝐶¹([𝑡₀,∞),ℝ) such that 𝑄^𝜙_𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝜙𝑄_𝑒𝐺_𝑖(𝑢)−𝑝𝑢^′+𝐺_𝑖^1/(𝛼+1)(𝑢)𝜙^′(𝛼+1)𝜙^𝛼+1(𝑡)𝑑𝑡>0 (2.3) for 𝑖=1,2, then (1.1) is oscillatory, where 𝑄_𝑒(𝑡)=𝛼^−𝛼/𝛽𝛽𝛽−𝛼^{(𝛼−𝛽)/𝛽}𝑞(𝑡)^𝛼/𝛽|𝑒(𝑡)|^{(𝛽−𝛼)/𝛽} (2.4) with the convention that 0⁰=1.

Proof. Suppose on the contrary that there is a nontrivial nonoscillatory solution. We assume that 𝑦(𝑡)>0 on [𝑇₀,∞) for some 𝑇₀≥𝑡₀. Set 𝑤(𝑡)(𝑡)𝑝(𝑡𝑦=𝜙)^′(𝑡)^𝛼−1𝑦^′(𝑡)𝑦(𝑡)^𝛼−1𝑦(𝑡),𝑡≥𝑇₀. (2.5) Then for all 𝑡≥𝑇₀, we obtain the Riccati type equation 𝑤^′(𝑡)=𝜙^′(𝑡)𝜙(𝑡)𝑤(𝑡)−𝜙(𝑡)𝑞(𝑡)𝑦^𝛽−𝛼−𝑒(𝑡)𝑦(𝑡)^𝛼−1𝑦(𝑡)|𝑤(𝑡−𝛼)|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼. (2.6) By the assumption, we can choose 𝑡₁>𝑠₁≥𝑇₀ so that 𝑒(𝑡)≤0 on the interval 𝐼₁=[𝑠₁,𝑡₁]. For given 𝑡∈𝐼₁, setting 𝐹(𝑥)=𝑞(𝑡)𝑥^𝛽−𝛼−𝑒(𝑡)/𝑥^𝛼, we have 𝐹^′(𝑥^∗)=0, 𝐹^′′(𝑥^∗)>0, where 𝑥^∗=[−𝛼𝑒(𝑡)/(𝛽−𝛼)𝑞(𝑡)]^1/𝛽. So 𝐹(𝑥) obtains its minimum on 𝑥^∗, and 𝐹(𝑥)≥𝐹𝑥^∗=𝑄_𝑒(𝑡). (2.7) So (2.6) and (2.7) imply that 𝑤(𝑡) satisfies 𝜙(𝑡)𝑄_𝑒(𝑡)≤−𝑤^′(𝑡)+𝜙^′(𝑡)𝜙(𝑡)𝑤(𝑡)|𝑤(𝑡−𝛼)|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼. (2.8) Multiplying 𝐺₁(𝑢(𝑡)) through (2.8) and integrating (2.8) from 𝑠₁ to 𝑡₁, using the fact that 𝐺₁(𝑢(𝑠₁))=𝐺₁(𝑢(𝑡₁))=0, we obtain ^𝑡1_𝑠1𝜙(𝑡)𝑄_𝑒(𝑡)𝐺₁(𝑢(𝑡≤))𝑑𝑡_𝑡1𝑠1𝐺₁(𝑢(𝑡))−𝑤^′(𝑡)+𝜙^′(𝑡)𝜙(𝑡)𝑤(𝑡)|𝑤(𝑡−𝛼)|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼𝑑𝑡=−𝐺₁(𝑢(𝑡))𝑤(𝑡)|^𝑡1_𝑠1+^𝑡1_𝑠1𝑔₁(𝑢(𝑡))𝑢^′(𝑡)𝑤(𝑡)+𝑑𝑡_𝑡1𝑠1𝐺₁(𝑢(𝑡𝜙))^′(𝑡)𝜙(𝑡)𝑤(𝑡)|𝑤(𝑡−𝛼)|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼=𝑑𝑡_𝑡1𝑠1𝑔₁(𝑢(𝑡))𝑢^′(𝑡)+𝐺₁(𝑢(𝑡𝜙))^′(𝑡)𝜙(𝑡)𝑤(𝑡)𝑑𝑡−𝛼^𝑡1_𝑠1𝐺₁(𝑢(𝑡|𝑤(𝑡)))|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼≤𝑑𝑡_𝑡1𝑠1𝑔₁(𝑢(𝑡))𝑢^′(𝑡)+𝐺₁(𝑢(𝑡𝜙))^′(𝑡)𝜙(𝑡)|𝑤(𝑡)|𝑑𝑡−𝛼^𝑡1_𝑠1𝐺₁(𝑢(𝑡|𝑤(𝑡)))|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼≤()𝑑𝑡𝛼+1^𝑡1_𝑠1𝐺₁^{𝛼/(𝛼+1)}(𝑢(𝑡))𝑢^′(𝑡)+𝐺₁(𝑢(𝑡𝜙))^′(𝑡)()𝜙(𝑡)|𝑤(𝑡𝛼+1)|𝑑𝑡−𝛼^𝑡1_𝑠1𝐺₁(𝑢(𝑡|𝑤(𝑡)))|^{(𝛼+1)/𝛼}𝑝(𝑡)𝜙(𝑡)^1/𝛼𝑑𝑡. (2.9) Let 𝛼𝑋=𝑝(𝑡)𝜙(𝑡)^{1/𝛼𝛼/(𝛼+1)}𝐺₁^{𝛼/(𝛼+1)}|𝑤(𝑡1)|,𝛾=1+𝛼,𝑌=𝛼𝜙(𝑡)𝑝(𝑡)^{𝛼/(𝛼+1)}𝑢^′(𝑡+𝐺)^11/(𝛼+1)𝜙^′(𝑡)(𝛼+1)𝜙(𝑡)^𝛼. (2.10) By Lemma 2.1 and (1.2), we have ^𝑡1_𝑠1𝜙(𝑡)𝑄_𝑒(𝑡)𝐺₁(𝑢(𝑡))𝑑𝑡≤^𝑡1_𝑠1𝜙(𝑡)𝑝(𝑡)|𝑢^′(𝑡)𝐺|+^11/(𝛼+1)(𝑢(𝑡))𝜙^′(𝑡)(𝛼+1)𝜙(𝑡)^𝛼+1𝑑𝑡, (2.11) which contradicts (2.3) with 𝑖=1.
When 𝑦(𝑡) is a negative solution for 𝑡≥𝑇₀>𝑡₀, we define the Riccati transformation as (2.5), and we obtain (2.8) also. In this case, we choose 𝑡₂>𝑠₂≥𝑇₀ so that 𝑒(𝑡)≥0 on the interval 𝐼₂=[𝑠₂,𝑡₂]. For given 𝑡∈𝐼₂, setting 𝐹(𝑥)=𝑞(𝑡)𝑥^𝛽−𝛼−(𝑒(𝑡)/𝑥^𝛼), we have 𝐹(𝑥)≥𝐹(𝑥^∗)=𝑄_𝑒(𝑡). We use 𝑢∈𝐶¹[𝑠₂,𝑡₂] and 𝐺₂ on [𝑠₂,𝑡₂] to reach a similar contradiction. The proof is complete.

Corollary 2.3. If 𝜙(𝑡)≡1 in Theorem 2.2 and (2.3) is replaced by 𝑄_𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝑄_𝑒(𝑡)𝐺_𝑖(𝑢(𝑡))−𝑝(𝑡)𝑢^′(𝑡)^𝛼+1𝑑𝑡>0 (2.12) for 𝑖=1,2, then (1.1) is oscillatory.

Remark 2.4. Corollary 2.3 is closely related to the generalized variational formulae similar to (1.10), so Theorem 2.2, Corollary 2.3 are generalizations of Theorems 1.1 and 1.3 (if we choose 𝐺₁(𝑢)=𝐺₂(𝑢)=𝑢^𝛼+1 in Corollary 2.3, then we obtain Corollary  2.3 of paper [4]), and when 0<𝛼=𝛽, we have 𝑄_𝑒(𝑡)=𝑞(𝑡), so Corollary 2.3 improves the main result of paper [1], since the positive constant 𝛼 in Theorem 2.2, Corollary 2.3 can be selected as any number lying in (0,∞).

Remark 2.5. Hypothesis (2.2) in Theorem 2.2, Corollary 2.3 can be replaced by the following condition 𝑒(𝑡𝑠)≥0,𝑡∈₁,𝑡₁,≤0,𝑡∈𝑠₂,𝑡₂. (2.13) The conclusion is still true for this case.

Now we give an example to illustrate the efficiency of our result.

Example 2.6. Consider the following forced quasilinear differential equation: 𝛾𝑡^−𝜆/3𝑦^′(𝑡)^′+𝑞(𝑡)𝑦(𝑡)²𝑦(𝑡)=−sin³𝑡,𝑡≥2𝜋, (2.14) where 𝛾,𝜆>0 are constants, 𝑞(𝑡)=𝑡^−𝜆exp(3sin𝑡) for 𝑡∈[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑞(𝑡)=𝑡^−𝜆exp(−3sin𝑡) for 𝑡∈[(2𝑛+1)𝜋,(2𝑛+2)𝜋), 𝑛>0 is an integer. Here 𝛼=1, 𝛽=3 in Theorem 2.2. Since 𝛼<𝛽, Theorems 1.1 and 1.2 cannot be applied to this case. However, we can obtain oscillation for (2.14) using Theorem 2.2. In fact, we can easily verify that 𝑄_𝑒(𝑡)=(3/2)√³2𝑡^−𝜆/3exp(sin𝑡)sin²𝑡 for 𝑡∈[2𝑛𝜋,(2𝑛+1)𝜋) and 𝑄_𝑒(𝑡)=(3/2)√³2𝑡^−𝜆/3exp(−sin𝑡)sin²𝑡 for 𝑡∈[(2𝑛+1)𝜋,(2𝑛+2)𝜋). For any 𝑇≥1, we choose 𝑛 sufficiently large so that 𝑛𝜋=2𝑘𝜋≥𝑇 and 𝑠₁=2𝑘𝜋 and 𝑡₁=(2𝑘+1)𝜋; we select 𝑢(𝑡)=sin𝑡≥0, 𝐺₁(𝑢)=𝑢²exp(−𝑢) (we note that (𝐺^′₁(𝑢))²≤4𝐺₁(𝑢) for 𝑢≥0), 𝜙(𝑡)=𝑡^𝜆/3, then we have ^𝑡1_𝑠1𝜙(𝑡)𝑄_𝑒(𝑡)𝐺₁(𝑢(𝑡3))𝑑𝑡=2³√2^{(2𝑘+1)𝜋}_2𝑘𝜋sin⁴𝑡𝑑𝑡=98³√2,_𝑡1𝑠1𝜙(𝑡)𝑝(𝑡)𝑢^′(𝑡+𝐺)^11/(𝛼+1)(𝑢(𝑡))𝜙^′(𝑡)(𝛼+1)𝜙(𝑡)^𝛼+1𝑑𝑡=𝛾^{(2𝑘+1)𝜋}_2𝑘𝜋|+𝜆||()|cos𝑡sin𝑡exp3sin𝑡/22𝑡²𝑑𝑡<𝛾^{(2𝑘+1)𝜋}_2𝑘𝜋1+𝜆𝑒^3/22²𝑑𝑡=𝛾1+𝜆𝑒^3/22²𝜋. (2.15) So we have 𝑄^𝜙₁(𝑢)>0 provided 0<𝛾<9√³2/2(2+𝜆𝑒^3/2)²𝜋. Similarly, for 𝑠₂=(2𝑘+1)𝜋 and 𝑡₂=(2𝑘+2)𝜋, we select 𝑢(𝑡)=sin𝑡≤0, 𝐺₂(𝑢)=𝑢²exp(𝑢) (we note that (𝐺^′₂(𝑢))²≤4𝐺₂(𝑢) for 𝑢≤0), and we can show that the integral inequality 𝑄^𝜙₂(𝑢)>0 for 0<𝛾<9√³2/2(2+𝜆𝑒^3/2)²𝜋. So (2.14) is oscillatory for 0<𝛾<9√³2/2(2+𝜆𝑒^3/2)²𝜋 by Theorem 2.2. We note further that Theorem 1.3 cannot be applied to (2.14) with 𝐺(𝑢)=𝑢².

Acknowledgments

Thanks are due to the referees for giving the author some useful comments which improve the main result. This paper was partially supported by the NSF of China (Grant 10801089), NSF of Shandong (Grant ZR2009AM011, ZR2009AQ010), and NSF of Jining University (Grant 2010QNKJ09).