Generalized Variational Oscillation Principles for Second-Order Differential Equations with Mixed-Nonlinearities

Abstract

Using generalized variational principle and Riccati technique, new oscillation criteria are established for forced second-order differential equation with mixed nonlinearities, which improve and generalize some recent papers in the literature.

1. Introduction

In this paper, we consider the second-order forced differential equation with mixed nonlinearities: 𝑟(𝑡)||𝑦^(𝑡)||^𝛼−1𝑦^(𝑡)^+𝑝(𝑡)||𝑦(𝑡)||^𝛼−1𝑦(𝑡)+^𝑚_𝑗=1𝑞_𝑗(𝑡)||𝑦(𝑡)||^𝛽𝑗−1𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀,(1.1) where 𝑟,𝑝,𝑞_𝑗(1≤𝑗≤𝑚),𝑒∈𝐶([𝑡₀,∞),ℝ) with 𝑟(𝑡)>0 and 0<𝛼<𝛽₁<𝛽₂<⋯<𝛽_𝑚 are real numbers, 𝑝,𝑞_𝑗 (1≤𝑗≤𝑚), and 𝑒 might change signs.

In this paper, we are concerned with the nonhomogeneous equation (1.1). By a solution of (1.1), we mean that a function 𝑦∈𝐶¹[𝑇_𝑦,∞)(𝑇_𝑦≥𝑡₀, 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). 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 solutions are oscillatory.

When 𝑚=0, we have the following second-order half-linear differential equation without or with forcing term: 𝑟(𝑟𝑡()𝑡||)𝑦||_𝑦^(𝑡)||^𝛼−1𝑦^(𝑡)^+𝑞(𝑡)||𝑦(𝑡)||^𝛼−1𝑦(𝑡)=0,𝑡≥𝑡₀,(1.2)(𝑡)||^𝛼−1𝑦^(𝑡)^+𝑞(𝑡)||𝑦(𝑡)||^𝛼−1𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀.(1.3) There are a lot of papers involved oscillation (see [1–6]) for these equations since the foundation work of Elbert [2]. In paper [1], using Leighton’s variational principle (see [3]) for (1.3), the following result was obtained by Li and Cheng.

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 exist 𝐻∈𝐷(𝑠_𝑖,𝑡_𝑖) and a positive, nondecreasing function 𝜌∈𝐶¹([𝑡₀,∞),ℝ) such that ^𝑡𝑖_𝑠𝑖𝐻²(𝑡)𝜌(𝑡)𝑞(𝑡)𝑑𝑡>1𝛼+1^𝛼+1^𝑡𝑖_𝑠𝑖𝑟(𝑡)𝜌(𝑡)||𝐻(𝑡)||^𝛼−12||𝐻^(𝑡)||+||𝐻(𝑡)||𝜌′𝜌^𝛼+1𝑑𝑡(1.4) for 𝑖=1,2. Then, (1.3) is oscillatory.

Unfortunately, Theorem 1.1 cannot be applied to the case where 𝛼>1, since for 𝜌(𝑡)≡1, the term |𝐻(𝑡)|^𝛼−1 will appear as a denominator in (1.4) so that the requirement 𝐻(𝑠_𝑖)=𝐻(𝑡_𝑖)=0 will cause trouble. This certainly calls for investigation of oscillation criteria that can handle with such cases.

When 𝛼=1, (1.2) and (1.3) are reduced to the linear differential equation: 𝑟(𝑟𝑡()𝑡𝑦)_𝑦^(𝑡)^+𝑞(𝑡)𝑦(𝑡)=0,𝑡≥𝑡₀,(1.5)(𝑡)^+𝑞(𝑡)𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀.(1.6) In paper [7], Wong proved the following result for (1.6).

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 exists 𝑢∈𝐷(𝑠_𝑖,𝑡_𝑖) such that 𝑄_𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝑞(𝑡)𝑢²(𝑡)−𝑟(𝑡)𝑢^(𝑡)²𝑑𝑡>0,𝑖=1,2,(1.7) then (1.6) is oscillatory.

On the other hand, among the oscillation criteria, Komkov [8] gave a generalized Leighton’s variational principle, which also can be applied to oscillation for (1.5).

Theorem 1.3. 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_𝑠1𝑞(𝑡)𝐺(𝑢(𝑡))−𝑟(𝑡)𝑢^(𝑡)²𝑑𝑡>0,(1.8) and (𝑔(𝑢(𝑡)))²≤4𝐺(𝑢(𝑡)) for 𝑡∈[𝑠₁,𝑡₁]. Then, every solution of (1.5) must vanish on [𝑠₁,𝑡₁].

We note that when 𝐺(𝑢)≡𝑢², the left-hand side of (1.8) is the energy functional related to (1.5).

When 𝑝(𝑡)≡0, 𝑚=1, (1.1) turns into the quasilinear differential equation: 𝑟(𝑡)||𝑦^(𝑡)||^𝛼−1𝑦^(𝑡)^+𝑞(𝑡)||𝑦(𝑡)||^𝛽−1𝑦(𝑡)=𝑒(𝑡),𝑡≥𝑡₀,(1.9) where 𝑝,𝑞,𝑒∈𝐶([𝑡₀,∞),ℝ) with 𝑝(𝑡)>0 and 0<𝛼≤𝛽 being constants. In paper [9], using the generalized variational principle, Shao proved the following result for (1.9).

Theorem 1.4. Assume that for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑒(𝑡)≤0,𝑡∈𝑠¹≥0,𝑡∈𝑠₂,𝑡₁,,𝑡₂.(1.10) 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^𝛼𝑖(𝑢(𝑡)) for 𝑡∈[𝑠_𝑖,𝑡_𝑖], 𝑖=1,2. If there exists a positive function 𝜙∈𝐶¹([𝑡₀,∞),ℝ) such that 𝑄^𝜙𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝜙(𝑡)⎡⎢⎢⎣𝑄_𝑒(𝑡)𝐺_𝑖(𝑢(𝑡))−𝑟(𝑡)||𝑢^(𝑡)||+𝐺^{𝑖1/(𝛼+1)}(𝑢(𝑡))||𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼+1⎤⎥⎥⎦𝑑𝑡>0(1.11) for 𝑖=1,2. Then (1.9) is oscillatory, where 𝑄_𝑒(𝑡)=𝛼^−𝛼/𝛽𝛽(𝛽−𝛼)^{(𝛼−𝛽)/𝛽}[𝑞(𝑡)]^𝛼/𝛽||𝑒(𝑡)||^{(𝛽−𝛼)/𝛽},(1.12) with the convention that 0⁰=1.

Recently, using Riccati transformation, the following oscillation criteria were given for (1.1) by Zheng et al. [10].

Theorem 1.5. Assume that for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that 𝑞_𝑗(𝑡)≥0(1≤𝑗≤𝑚) for 𝑡∈[𝑠₁,𝑡₁]∪[𝑠₂,𝑡₂] and 𝑒(𝑡)≤0,𝑡∈𝑠¹≥0,𝑡∈𝑠₂,𝑡₁,,𝑡₂.(1.13) Let 𝐷(𝑠_𝑖,𝑡_𝑖)={𝑢∈𝐶¹[𝑠_𝑖,𝑡_𝑖]∶𝑢^𝛼+1(𝑡)>0,𝑡∈(𝑠_𝑖,𝑡_𝑖),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 ^𝑡𝑖_𝑠𝑖𝜙(𝑡)𝑝(𝑡)+^𝑚_𝑗=1𝑄_𝑗(𝑡)𝐻^𝛼+1(𝑡)−𝑟(𝑡)||𝐻^(𝑡)||+||𝐻(𝑡)𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼+1𝑑𝑡>0(1.14) for 𝑖=1,2. Then (1.1) is oscillatory, where 𝑄_𝑗(𝑡)=𝛼^{−𝛼/𝛽𝑗}𝛽_𝑗𝑚𝛽_𝑗−𝛼^{(𝛼−𝛽𝑗)/𝛽𝑗}𝑞_𝑗(𝑡)^{𝛼/𝛽𝑗}||𝑒(𝑡)||^{(𝛽𝑗−𝛼)/𝛽𝑗},1≤𝑗≤𝑚,(1.15) with the convention that 0⁰=1.

The purpose of this paper is to obtain new oscillation criteria for (1.1) based on generalized variational principles. Roughly, if the existence of a “positive” solution of a functional relation implies the “positivity” of an associated functional over a set of “admissible” functions, then we say that a variational oscillation principle is valid. For instance, in Theorem 1.1, 𝐻∈𝐷(𝑠_𝑖,𝑡_𝑖) is admissible, and the functional is ^𝑡𝑖_𝑠𝑖𝛼+11^𝛼+1𝑝(𝑡)𝜌(𝑡)||𝐻(𝑡)||^𝛼−12||𝐻^(𝑡)||+||𝐻(𝑡)||𝜌^(𝑡)𝜌(𝑡)^𝛼+1−𝐻²(𝑡)𝜌(𝑡)𝑞(𝑡)𝑑𝑡.(1.16) Our emphasis will be directed towards oscillation criteria that are closely related to the generalized energy functional (the generalization of (𝛼+1)-degree energy functional) for half-linear equations (see [4, 11–13] for more details on these functionals), which improve the results mentioned above. Examples will also be given to illustrate the effectiveness of our main results.

2. Main Results

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

Lemma 2.1 (see [14]). Suppose that 𝑋 and 𝑌 are nonnegative, then 𝛾𝑋𝑌^𝛾−1−𝑋^𝛾≤(𝛾−1)𝑌^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(𝑡)≤0,𝑡∈𝑠¹≥0,𝑡∈𝑠₂,𝑡₁,,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^𝛼𝑖(𝑢(𝑡)) for 𝑡∈[𝑠_𝑖,𝑡_𝑖],𝑖=1,2. If there exists a positive function 𝜙∈𝐶¹([𝑡₀,∞),ℝ) such that 𝑄^𝜙𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝜙(𝑡)𝐺_𝑖(𝑢(𝑡))𝑝(𝑡)+^𝑚_𝑗=1−𝑟(𝑡)||𝑢_𝑄_𝑗(𝑡)(2.3)(𝑡)||+𝐺^{𝑖1/(𝛼+1)}(𝑢(𝑡))||𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼+1⎤⎥⎥⎦𝑑𝑡>0(2.4) for 𝑖=1,2, where 𝑄_𝑗(𝑡) is defined as (1.15) with the convention that 0⁰=1. Then, (1.1) is oscillatory.

Proof. Suppose to the contrary that there is a nontrivial nonoscillatory solution 𝑦=𝑦(𝑡). We assume that 𝑦(𝑡)≠0 on [𝑇₀,∞) for some 𝑇₀≥𝑡₀. Set 𝑤(𝑡)=𝜙(𝑡)𝑟(𝑡)||𝑦^(𝑡)||^𝛼−1𝑦^(𝑡)||𝑦(𝑡)||^𝛼−1𝑦(𝑡),𝑡≥𝑇⁰.(2.5) Then differentiating (2.5) and making use of (1.1), it follows that for all 𝑡≥𝑇₀, 𝑤^(𝑡)=𝜙^(𝑡)𝜙(𝑡)𝑤(𝑡)−𝜙(𝑡)𝑝(𝑡)+||𝑦𝜙(𝑡)(||𝑡)𝑒^𝛼−1(𝑦𝑡)(𝑡)−𝛼(||𝑟𝑤(𝑡()𝑡)𝜙||(𝑡))^{(𝛼+1)/𝛼1/𝛼}−𝜙(𝑡)^𝑚_𝑗=1𝑞_𝑗(𝑡)||𝑦||^{𝛽𝑗−𝛼}.(2.6) By the assumptions, we can choose 𝑠_𝑖,𝑡_𝑖≥𝑇₀ for 𝑖=1,2 so that 𝑒(𝑡)≤0 on the interval 𝐼₁=[𝑠₁,𝑡₁], with 𝑠₁<𝑡₁ and 𝑦(𝑡)≥0, or 𝑒(𝑡)≥0 on the interval 𝐼₂=[𝑠₂,𝑡₂], with 𝑠₂<𝑡₂ and 𝑦(𝑡)≤0. For given 𝑡∈𝐼₁ or 𝑡∈𝐼₂, set 𝐹_𝑗(𝑥)=𝑞_𝑗(𝑡)𝑥^{𝛽𝑗−𝛼}−𝑒(𝑡)/𝑚𝑥^𝛼, 1≤𝑗≤𝑚, we have 𝐹^𝑗(𝑥^∗𝑗)=0, 𝐹^𝑗(𝑥^∗𝑗)>0, where 𝑥^∗𝑗=[−𝛼𝑒(𝑡)/𝑚(𝛽_𝑗−𝛼)𝑞_𝑗(𝑡)]^1/𝛽𝑗. So, 𝐹_𝑗(𝑥) obtains it minimum on 𝑥^∗𝑗 and 𝐹_𝑗(𝑥)≥𝐹_𝑗𝑥^∗𝑗=𝑄_𝑗(𝑡).(2.7) So on the interval 𝐼₁ or 𝐼₂, (2.6) and (2.2) imply that 𝑤(𝑡) satisfies 𝜙(𝑡)𝑝(𝑡)+^𝑚_𝑗=1𝑄_𝑗(𝑡)≤−𝑤^(𝑡)+𝜙^(𝑡)𝜙(𝑡)𝑤(𝑡)−𝛼(||𝑟𝑤(𝑡()𝑡)𝜙||(𝑡))^{(𝛼+1)/𝛼1/𝛼}.(2.8) Multiplying 𝐺_𝑖(𝑢(𝑡)) through (2.8) and integrating (2.8) from 𝑠_𝑖 to 𝑡_𝑖, using the fact that 𝐺_𝑖(𝑢(𝑠₁))=𝐺_𝑖(𝑢(𝑡₁))=0, we obtain ^𝑡𝑖_𝑠𝑖𝜙(𝑡)𝑝(𝑡)+^𝑚_𝑗=1≤𝑄_𝑗(𝑡)𝐺_𝑖(𝑢(𝑡))𝑑𝑡_{𝑡𝑖𝑠𝑖}𝐺_𝑖(𝑢(𝑡))−𝑤^(𝑡)+𝜙^(𝑡)=−𝐺_𝑖𝜙(𝑡)𝑤(𝑡)−𝛼(||𝑟𝑤(𝑡()𝑡)𝜙||(𝑡))^{(𝛼+1)/𝛼1/𝛼}𝑑𝑡+(𝑢(𝑡))𝑤(𝑡)||^𝑡𝑖_𝑠𝑖+^𝑡𝑖_𝑠𝑖𝑔_𝑖(𝑢(𝑡))𝑢^(𝑡)𝑤(𝑡)𝑑𝑡_{𝑡𝑖𝑠𝑖}𝐺_𝑖(𝑢(𝑡))𝜙^(𝑡)=𝜙(𝑡)𝑤(𝑡)−𝛼(||𝑟𝑤(𝑡()𝑡)𝜙||(𝑡))^{(𝛼+1)/𝛼1/𝛼}𝑑𝑡_{𝑡𝑖𝑠𝑖}𝑔^𝑖(𝑢(𝑡))𝑢^(𝑡)+𝐺_𝑖(𝑢(𝑡))𝜙^(𝑡)−𝛼_{𝑡𝑖𝑠𝑖}𝜙(𝑡)𝑤(𝑡)𝑑𝑡𝐺_𝑖(𝑢(𝑡))||𝑤(𝑡)||^{(𝛼+1)/𝛼}(𝑟(𝑡)𝜙(𝑡))^1/𝛼≤𝑑𝑡_{𝑡𝑖𝑠𝑖}||𝑔^𝑖(𝑢(𝑡))||||𝑢^(𝑡)||+𝐺_𝑖(𝑢(𝑡))||𝜙^(𝑡)||−𝛼_{𝑡𝑖𝑠𝑖}𝜙(𝑡)||𝑤(𝑡)||𝑑𝑡𝐺_𝑖(𝑢(𝑡))||𝑤(𝑡)||^{(𝛼+1)/𝛼}(𝑟(𝑡)𝜙(𝑡))^1/𝛼≤(𝛼+1)_{𝑡𝑖𝑠𝑖}𝑑𝑡𝐺^{𝑖𝛼/(𝛼+1)}(𝑢(𝑡))||𝑢^(𝑡)||+𝐺_𝑖(𝑢(𝑡))||𝜙^(𝑡)||−𝛼_{𝑡𝑖𝑠𝑖}(𝛼+1)𝜙(𝑡)||𝑤(𝑡)||𝑑𝑡𝐺_𝑖(𝑢(𝑡))||𝑤(𝑡)||^{(𝛼+1)/𝛼}(𝑟(𝑡)𝜙(𝑡))^1/𝛼𝑑𝑡.(2.9) Let 𝑋=𝛼(𝑟(𝑡)𝜙(𝑡))^1/𝛼^{𝛼/(𝛼+1)}𝐺^{𝑖𝛼/(𝛼+1)}||𝑤(𝑡)||,𝛾=1+1𝑌=(𝛼𝜙(𝑡)𝑟(𝑡))_{𝛼/(𝛼+1)}𝛼,||𝑢^(𝑡)||+𝐺^{𝑖1/(𝛼+1)}||𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼,(2.10) by Lemma 2.1 and (2.9), we have ^𝑡𝑖_𝑠𝑖𝜙(𝑡)𝑝(𝑡)+^𝑚_𝑗=1𝑄_𝑗(𝑡)𝐺_𝑖(𝑢(𝑡))𝑑𝑡≤^𝑡𝑖_𝑠𝑖𝜙(𝑡)𝑟(𝑡)||𝑢^(𝑡)||+𝐺^{𝑖1/(𝛼+1)}(𝑢(𝑡))||𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼+1𝑑𝑡,(2.11) which contradicts with (2.3). This completes the proof of Theorem 2.2.

Corollary 2.3. If 𝜙(𝑡)≡1 in Theorem 2.2, and (2.3) is replaced by 𝑄_𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝑝(𝑡)+^𝑚_𝑗=1𝑄_𝑗(𝑡)𝐺_𝑖(𝑢(𝑡))−𝑟(𝑡)||𝑢^(𝑡)||^𝛼+1𝑑𝑡>0,(2.12) for 𝑖=1,2. Then, (1.1) is oscillatory.

If we choose 𝐺₁(𝑢)=𝐺₂(𝑢)=𝑢^𝛼+1 in Corollary 2.3, then we have the following corollary.

Corollary 2.4. Suppose that for any 𝑇≥𝑡₀, there exist 𝑇≤𝑠₁<𝑡₁≤𝑠₂<𝑡₂ such that (2.2) is true. 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 𝑢∈𝐷(𝑠_𝑖,𝑡_𝑖) such that 𝑄_𝑖(𝑢)∶=^𝑡𝑖_𝑠𝑖𝑝(𝑡)+^𝑚_𝑗=1𝑄_𝑗(𝑡)||𝑢(𝑡)||^𝛼+1−𝑟(𝑡)||𝑢^(𝑡)||^𝛼+1𝑑𝑡>0,(2.13) for 𝑖=1,2. Then, (1.3) is oscillatory.

Remark 2.5. Corollary 2.4 is closely related to the (𝛼+1)-degree functional (1.8), so Theorem 2.2, Corollaries 2.3, and 2.4 are generalizations of Theorem 1.2, and improvement of Theorem 1.1 since the positive constant 𝛼 in Theorem 2.2 and Corollary 2.3 can be selected as any number lying in (0,∞). We note further that in most cases, oscillation criteria are obtained using the same auxiliary function on [𝑠₁,𝑡₁] and [𝑠₂,𝑡₂], we note that such functions can be selected differently.

Remark 2.6. If 𝐺(𝑢)≡𝑢^𝛼+1, then Theorem 2.2 reduces to Theorem 1.5, and if 𝑝(𝑡)≡0, 𝑗=1, Theorem 2.2 reduces to Theorem 1.4. So Theorem 2.2 and Corollary 2.3 are generalizations of the papers by Zheng et al. [10] and Shao [9].

Remark 2.7. The hypothesis (2.2) in Theorem 2.2 and Corollary 2.3 can be replaced by the following condition: 𝑒(𝑡)≥0,𝑡∈𝑠¹≤0,𝑡∈𝑠₂,𝑡₁,,𝑡₂.(2.14) The conclusion is still true for these cases.

Example 2.8. Consider the following forced mixed nonlinearities differential equation: 𝛾𝑡^−𝜆/3𝑦^(𝑡)^+𝑝(𝑡)𝑦(𝑡)+𝑞(𝑡)||𝑦(𝑡)||²𝑦(𝑡)=−sin³𝑡,𝑡≥2𝜋,(2.15) where 𝛾,𝜆>0 are constants, 𝑞(𝑡)=𝑡^−𝜆exp(3sin𝑡), 𝑝(𝑡)=𝑡^−𝜆/3exp(sin𝑡), for 𝑡∈[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑞(𝑡)=𝑡^−𝜆exp(−3sin𝑡), 𝑝(𝑡)=𝑡^−𝜆/3exp(−sin𝑡), for 𝑡∈[(2𝑛+1)𝜋,(2𝑛+2)𝜋), 𝑛>0 is an integer, Shao [9] obtain oscillation for (2.15) when 𝐾(𝑡)≡0. Using Theorem 2.2, we can easily verify that 𝑄₁(𝑡)=(3/2)√³2𝑡^−𝜆/3exp(sin𝑡)sin²𝑡 for 𝑡∈[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑄₁(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)𝑝(𝑡)+𝑄₁(𝑡)𝐺₁(𝑢(𝑡))𝑑𝑡=^𝜋0sin²𝑡𝑑𝑡+32³√2^𝜋0sin⁴𝑡𝑑𝑡=𝜋2+89³√2,_𝑡1𝑠1𝜙(𝑡)𝑝(𝑡)||𝑢^(𝑡)||+𝐺^11/(𝛼+1)(𝑢(𝑡))||𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼+1=𝛾_{(2𝑘+1)𝜋2𝑘𝜋}𝑑𝑡|cos𝑡|+𝜆||sin𝑡||ex2p𝑡(3sin𝑡/2)²<𝛾_{(2𝑘+1)𝜋2𝑘𝜋}𝑑𝑡1+𝜆𝑒2^3/2²𝑑𝑡=𝛾1+𝜆𝑒^3/22²𝜋.(2.16) So we have 𝑄^𝜙1(𝑢)>0 provided, 0<𝛾<(4𝜋+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), we can show that the integral inequality 𝑄^𝜙2(𝑢)>0 for 0<𝛾<(4𝜋+9√³2)/2(2+𝜆𝑒^3/2)²𝜋. So (2.15) is oscillatory for 0<𝛾<(4𝜋+9√³2)/2(2+𝜆𝑒^3/2)²𝜋 by Theorem 2.2.

Example 2.9. Consider the following forced mixed nonlinearities differential equation: 𝑡^−𝜆||𝑦^(𝑡)||^𝛼−1𝑦^(𝑡)^+𝑝(𝑡)||𝑦(𝑡)||^𝛼−1𝑦(𝑡)+𝑞(𝑡)𝑦³(𝑡)=−sin^1/3𝑡,(2.17) for 𝑡≥2𝜋, where 𝑝(𝑡)=𝐾𝑡^−𝜆exp(sin𝑡), 𝑞(𝑡)=𝑡^−9𝜆/5exp(9sin𝑡/5), for 𝑡∈[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑝(𝑡)=𝐾𝑡^−𝜆exp(−sin𝑡), 𝑞(𝑡)=𝑡^−9𝜆/5exp(−9sin𝑡/5), for 𝑡∈[(2𝑛+1)𝜋,(2𝑛+2)𝜋), 𝑛>0 is an integer, 𝐾,𝜆>0 are constants and 𝛼=5/3>1, 𝛽=3. Obviously, Theorem 1.1 cannot be applied to this case. However, we conclude that (2.17) is oscillatory for 𝐾>(3/4)(1+3𝜆𝑒/8)^8/3𝜋−9/5^5/94^4/9. Since the zeros of the forcing term −sin^1/3𝑡 are 𝑛𝜋, let 𝑢(𝑡)=sin𝑡 and 𝜙(𝑡)=𝑡^𝜆. Using Theorem 2.2, we can easily verify that 𝑄(𝑡)=(9/5^5/94^4/9)𝑡^−𝜆exp(sin𝑡)sin^4/27𝑡 for 𝑡∈[2𝑛𝜋,(2𝑛+1)𝜋), and 𝑄(𝑡)=(9/5^5/94^4/9)𝑡^−𝜆exp(−sin𝑡)sin^4/27𝑡 for 𝑡∈[(2𝑛+1)𝜋,(2𝑛+2)𝜋). For any 𝑇≥1, choose 𝑛 sufficiently large so that 𝑛𝜋=2𝑘𝜋≥𝑇 and 𝑠₁=2𝑘𝜋 and 𝑡₁=(2𝑘+1)𝜋. For 𝑡∈[𝑠₁,𝑡₁], we select 𝐺₁(𝑢)=𝑢^8/3exp(−𝑢) (we note that (𝐺^1(𝑢))^8/3≤(8/3)^8/3(𝐺₁(𝑢))^5/3 for 𝑢≥0). It is easy to verify the following estimations: ^𝑡1_𝑠1𝜙(𝑡=)(𝑝(𝑡)+𝑄(𝑡))𝐺₁(𝑢(𝑡))𝑑𝑡_{(2𝑘+1)𝜋2𝑘𝜋}sin^8/3𝑡𝐾+95^5/94^4/9>𝐾+9sin^4/27𝑡𝑑𝑡5^5/94^4/9^{(2𝑘+1)𝜋2𝑘𝜋}sin³𝑡𝑑𝑡=4_𝑡1𝑠13𝐾+5^5/994^4/9,𝜙(𝑡)𝑟(𝑡)||𝑢^(𝑡)||+𝐺^11/(𝛼+1)(𝑢(𝑡))||𝜙^(𝑡)||(𝛼+1)𝜙(𝑡)^𝛼+1=𝑑𝑡_{(2𝑘+1)𝜋2𝑘𝜋}|cos𝑡|+3𝜆𝑒^{−3sin𝑡/8}||sin𝑡||8𝑡^8/3<𝑑𝑡_{(2𝑘+1)𝜋2𝑘𝜋}1+38𝜆𝑒^8/3𝑑𝑡=1+38𝜆𝑒^8/3𝜋.(2.18) So we have 𝑄^𝜙1(𝑢)>0. Similarly, for 𝑠₂=(2𝑘+1)𝜋 and 𝑡₂=(2𝑘+2)𝜋, we select 𝑢(𝑡)=sin𝑡<0, 𝐺₂(𝑢)=𝑢^8/3exp(𝑢) (we note that (𝐺^2(𝑢))^8/3≤(8/3)^8/3(𝐺₂(𝑢))^5/3 for 𝑢≤0), we can show that the integral inequality 𝑄^𝜙2(𝑢)>0. So (2.17) is oscillatory for 𝐾>(3/4)(1+3𝜆𝑒/8)^8/3𝜋−9/5^5/94^4/9 by Theorem 2.2.

Acknowledgment

This research was partially supported by the NSF of China (Grants nos. 11171178 and 11271225) and Science and Technology Project of High Schools of Shandong Province (Grant no. J12LI52).