New Oscillation Criteria for Second-Order Delay Differential Equations with Mixed Nonlinearities

Abstract

We establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the form (𝑝(𝑡)𝑥′(𝑡))′+∑^𝑛𝑖=1𝑝_𝑖(𝑡)𝑥(𝑡−𝜏_𝑖)+∑^𝑛𝑖=1𝑞_𝑖(𝑡)|𝑥(𝑡−𝜏_𝑖)|^𝛼𝑖sgn𝑥(𝑡−𝜏_𝑖)=𝑒(𝑡),𝑡≥0, where 𝑝(𝑡), 𝑝_𝑖(𝑡), 𝑞_𝑖(𝑡), and 𝑒(𝑡) are continuous functions defined on [0,∞), and 𝑝(𝑡)>0, 𝑝^(𝑡)≥0, and 𝛼₁>⋯>𝛼_𝑚>1>𝛼_𝑚+1>⋯>𝛼_𝑛>0. No restriction is imposed on the potentials 𝑝_𝑖(𝑡), 𝑞_𝑖(𝑡), and 𝑒(𝑡) to be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided.

1. Introduction

We consider the second-order delay differential equations containing mixed nonlinearities of the form 𝑝(𝑡)𝑥^(𝑡)^+^𝑛_𝑖=1𝑝_𝑖(𝑡)𝑥𝑡−𝜏_𝑖+^𝑛_𝑖=1𝑞_𝑖(𝑡)||𝑥(𝑡−𝜏_𝑖)||^𝛼𝑖sgn𝑥𝑡−𝜏_𝑖=𝑒(𝑡),𝑡≥0.(1.1)

In what follows we assume that 𝜏_𝑖≥0, 𝑝∈𝐶¹[0,∞), 𝑝(𝑡)>0,𝑝^(𝑡)≥0,𝑝_𝑖,𝑞_𝑖,𝑒∈𝐶[0,∞), 𝛼₁>⋯>𝛼_𝑚>1>𝛼_𝑚+1>⋯>𝛼_𝑛>0(𝑛>𝑚≥1), and ^∞01𝑝(𝑡)𝑑𝑡=∞.(1.2)

As usual, a solution 𝑥(𝑡) of (1.1) is called oscillatory if it is defined on some ray [𝑇,∞) with 𝑇≥0 and has arbitrary large zeros, otherwise, it is called nonoscillatory. Equation (1.1) is called oscillatory if all of its extendible solutions are oscillatory.

Recently, Mustafa [1] has studied the oscillatory solutions of certain forced Emden-Fowler like equations 𝑥^(𝑡)+𝑎(𝑡)||𝑥(𝑡)||^𝜆sgn𝑥(𝑡)=𝑒(𝑡),𝑡≥𝑡₀≥1.(1.3) Sun and Wong [2], as well as Sun and Meng [3] have established oscillation criteria for the second-order equation 𝑝(𝑡)𝑥^(𝑡)^+𝑞(𝑡)𝑥(𝑡)+^𝑛_𝑖=1𝑞_𝑖(𝑡)||𝑥(𝑡)||^𝛼𝑖sgn𝑥(𝑡)=𝑒(𝑡),𝑡≥0.(1.4) Later in [4], Li and Chen have extended (1.4) to the delay differential equation 𝑝(𝑡)𝑥^(𝑡)^+𝑞(𝑡)𝑥(𝑡−𝜏)+^𝑛_𝑖=1𝑞_𝑖(𝑡)||𝑥(𝑡−𝜏)||^𝛼𝑖sgn𝑥(𝑡−𝜏)=𝑒(𝑡),𝑡≥0.(1.5) As it is indicated in [2, 3], further research on the oscillation of equations of mixed type is necessary as such equations arise in mathematical modeling, for example, in the growth of bacteria population with competitive species. In this paper, we will continue in the direction to study the oscillatory properties of (1.1). We will employ the method in study of Kong in [5] and the arithmetic-geometric mean inequality (see [6]) to establish the interval oscillation criteria for the unforced (1.1) and forced (1.1), which extend and improve the known results. Our results are generalizations of the main results in [3, 4]. We also give an example to illustrate the sharpness of our main results.

2. Main Results

We need the following lemma proved in [2, 3] for our subsequent discussion.

Lemma 2.1. For any given 𝑛-tuple {𝛼₁,𝛼₂,…,𝛼_𝑛} satisfying 𝛼₁>𝛼₂>⋯>𝛼_𝑚>1>𝛼_𝑚+1>⋯>𝛼_𝑛>0, there corresponds an n-tuple {𝜂₁,𝜂₂,…,𝜂_𝑛} such that ^𝑛_𝑖=1𝛼_𝑖𝜂_𝑖=1,(a) which also satisfies either ^𝑛_𝑖=1𝜂_𝑖<1,0<𝜂_𝑖<1,(b) or ^𝑛_𝑖=1𝜂_𝑖=1,0<𝜂_𝑖<1.(c)

For a given set of exponents {𝛼_𝑖} satisfying 𝛼₁>𝛼₂>⋯>𝛼_𝑚>1>𝛼_𝑚+1>⋯>𝛼_𝑛>0, Lemma 2.1 ensures the existence of an 𝑛-tuple {𝜂₁,𝜂₂,…,𝜂_𝑛} such that either (a) and (b) hold or (a) and (c) hold. When 𝑛=2 and 𝛼₁>1>𝛼₂>0, in the first case, we have 𝜂₁=1−𝛼²1−𝜂₀𝛼₁−𝛼₂,𝜂₂=𝛼¹1−𝜂₀−1𝛼₁−𝛼₂,(2.1) where 𝜂₀ can be any positive number satisfying 0<𝜂₀<(𝛼₁−1)/𝛼₁. This will ensure that 0<𝜂₁, 𝜂₂<1, and conditions (a) and (b) are satisfied. In the second case, we simply solve (a) and (c) and obtain 𝜂₁=1−𝛼²𝛼₁−𝛼₂,𝜂₂=𝛼¹−1𝛼₁−𝛼₂.(2.2)

Following Philos [7], we say that a continuous function 𝐻(𝑡,𝑠) belongs to a function class 𝒟_𝑎,𝑏, denoted by 𝐻∈𝒟_𝑎,𝑏, if 𝐻(𝑏,𝑠)>0,𝐻(𝑠,𝑎)>0 for 𝑏>𝑠>𝑎, and 𝐻(𝑡,𝑠) has continuous partial derivatives 𝜕𝐻(𝑡,𝑠)/𝜕𝑡 and 𝜕𝐻(𝑡,𝑠)/𝜕𝑠 in [𝑎,𝑏]×[𝑎,𝑏]. Set ℎ₁(𝑡,𝑠)=𝜕𝐻(𝑡,𝑠)/𝜕𝑡2√𝐻(𝑡,𝑠),ℎ²(𝑡,𝑠)=−𝜕𝐻(𝑡,𝑠)/𝜕𝑠2√𝐻(𝑡,𝑠).(2.3)

Based on Lemma 2.1, we have the following interval criterion for oscillation of (1.1).

Theorem 2.2. If, for any 𝑇≥0, there exist 𝑎₁, 𝑏₁, 𝑐₁, 𝑎₂,𝑏₂ and 𝑐₂ such that 𝑇≤𝑎₁<𝑐₁<𝑏₁≤𝑎₂<𝑐₂<𝑏₂,𝑝𝑞_𝑖(𝑡)≥0,𝑡∈𝑎₁−𝜏_𝑖,𝑏₁∪𝑎₂−𝜏_𝑖,𝑏₂,𝑖=1,2,…,𝑛,_𝑖𝑒(𝑡)(𝑡≤)≥0,0,𝑡∈𝑡∈𝑎₁𝑎₁−𝜏_𝑖,𝑏₁∪𝑎₂−𝜏_𝑖,𝑏₂,𝑖=1,2,…,𝑛,−𝜏_𝑖,𝑏₁,𝑒(𝑡)≥0,𝑡∈𝑎₂−𝜏_𝑖,𝑏₂,(2.4) and there exist 𝐻_𝑗∈𝒟_{𝑎𝑗,𝑏𝑗} such that 1𝐻_𝑗𝑐_𝑗,𝑎_𝑗^𝑐𝑗𝑎_𝑗𝑄^𝑗+1(𝑠)𝐻_𝑗𝑠,𝑎_𝑗−𝑝(𝑠)ℎ^2𝑗1𝑠,𝑎_𝑗𝑑𝑠𝐻_𝑗𝑏_𝑗,𝑐_𝑗^𝑏𝑗𝑐_𝑗𝑄^𝑗(𝑠)𝐻_𝑗𝑏_𝑗,𝑠−𝑝(𝑠)ℎ^2𝑗2𝑏_𝑗,𝑠𝑑𝑠>0,(2.5) for 𝑗=1,2, where ℎ_𝑗1, ℎ_𝑗2 are defined as in (2.3), 𝜂₁,𝜂₂,…,𝜂_𝑛 are positive constants satisfying (a) and (b) in Lemma 2.1, 𝜂₀=1−∑^𝑛𝑖=1𝜂_𝑖, and 𝑄_𝑗(𝑡)=^𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎^𝑗𝑡−𝑎_𝑗+𝜏_𝑖+𝜂⁰⁻¹||𝑒(𝑡)||^𝜂0𝑛_𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑡−𝑡𝑎−_𝑗𝑎+^𝑗𝜏_𝑖^{𝛼𝑖𝜂𝑖},(2.6) then (1.1) is oscillatory.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that 𝑥(𝑡)>0 for all 𝑡≥𝑡₁−𝜏≥0, where 𝑡₁ depends on the solution 𝑥(𝑡) and 𝜏=max{𝜏_𝑖}, 𝑖=1,…,𝑛. When 𝑥(𝑡) is eventually negative, the proof follows the same argument by using the interval [𝑎₂,𝑏₂] instead of [𝑎₁,𝑏₁]. Choose 𝑎₁,𝑏₁≥𝑡₁ such that 𝑝_𝑖(𝑡),𝑞_𝑖(𝑡)≥0, 𝑒(𝑡)≤0 for 𝑡∈[𝑎₁−𝜏_𝑖,𝑏₁], and 𝑖=1,2,…,𝑛.
From (1.1), we have that 𝑥^(𝑡)≥0 for 𝑡∈[𝑎₁−𝜏_𝑖,𝑏₁]. If not, there exists 𝑡₂∈[𝑎₁−𝜏_𝑖,𝑏₁] such that 𝑥^(𝑡₂)<0. Because 𝑝(𝑡)𝑥^(𝑡)^≤0,(2.7) we have 𝑝(𝑡)𝑥^(𝑡)≤𝑝(𝑡₂)𝑥^(𝑡₂). Integrating from 𝑡₂ to 𝑡, we obtain 𝑥(𝑡)≤𝑥𝑡₂+𝑝𝑡₂𝑥^𝑡₂^𝜏𝑡₂1𝑝(𝑠)𝑑𝑠.(2.8) Noting the assumption (1.2), we have 𝑥(𝑡)≤0 for sufficient large 𝑡. This is a contradiction with 𝑥(𝑡)>0. From (2.7) and the conditions 𝑝(𝑡)>0, 𝑝^(𝑡)≥0, we obtain 𝑥^(𝑡)≤0 for 𝑡∈[𝑎₁−𝜏_𝑖,𝑏₁].
Employing the convexity of 𝑥(𝑡), we obtain 𝑥𝑡−𝜏_𝑖𝑥(𝑡)≥𝑡−𝑡𝑎−₁𝑎+¹𝜏_𝑖,𝑡∈𝑎₁,𝑏₁.(2.9)
Define 𝜔(𝑡)=−𝑝(𝑡)𝑥^(𝑡)𝑥(𝑡).(2.10)
Recall the arithmetic-geometric mean inequality ^𝑛_𝑖=0𝜂_𝑖𝑢_𝑖≥^𝑛_𝑖=0𝑢^𝜂𝑖_𝑖,𝑢_𝑖≥0,(2.11) where 𝜂₀=1−∑^𝑛𝑖=1𝜂_𝑖 and 𝜂_𝑖>0, 𝑖=1,2,…,𝑛, are chosen according to the given 𝛼₁,𝛼₂,…,𝛼_𝑛 as in Lemma 2.1 satisfying (a) and (b). Let 𝑢₀(𝑡)=𝜂⁰⁻¹||𝑒(𝑡)||,𝑢_𝑖(𝑡)=𝜂^𝑖−1𝑞_𝑖(𝑡)𝑥𝑡−𝜏_𝑖^𝛼𝑖.(2.12) We have 𝜔^(𝑡)=−(𝑝(𝑡)𝑥(𝑡))^𝑥(𝑡)+𝜔²(𝑡)=∑𝑝(𝑡)_𝑛𝑖=1𝑝_𝑖(𝑡)𝑥𝑡−𝜏_𝑖+∑^𝑛𝑖=1𝑞_𝑖(𝑡)𝑥𝑡−𝜏_𝑖^𝛼𝑖−𝑒(𝑡)𝑥(𝑡)+𝜔²(𝑡)≥𝑝(𝑡)_𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎¹𝑡−𝑎₁+𝜏_𝑖+𝜂⁰⁻¹||𝑒(𝑡)||^𝜂0∏^𝑛𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑥^{𝛼𝑖𝜂𝑖}𝑡−𝜏_𝑖𝑥(𝑡)+𝜔²(𝑡)=𝑝(𝑡)_𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎¹𝑡−𝑎₁+𝜏_𝑖+𝜂⁰⁻¹||𝑒(𝑡)||^𝜂0∏^𝑛𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑥^{𝛼𝑖𝜂𝑖}𝑡−𝜏_𝑖∏^𝑛𝑖=1𝑥^{𝛼𝑖𝜂𝑖}(𝑡)+𝜔²(𝑡)≥𝑝(𝑡)_𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎¹𝑡−𝑎₁+𝜏_𝑖+𝜂⁰⁻¹||𝑒(𝑡)||^𝜂0𝑛_𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑡−𝑡𝑎−₁𝑎+¹𝜏_𝑖^{𝛼𝑖𝜂𝑖}+𝜔²(𝑡)=𝑄₁𝑝(𝑡)(𝑡)+𝜔²(𝑡)𝑝(𝑡).(2.13)
Multiplying both sides of (2.13) by 𝐻₁(𝑏₁,𝑡)∈𝒟_𝑎1,𝑏1 and integrating by parts, we find that −𝜔𝑐₁𝐻₁𝑏₁,𝑐₁≥^𝑏1_𝑐1𝑄₁(𝑠)𝐻₁𝑏₁,𝑠−𝑝(𝑠)ℎ²¹²𝑏₁,𝑠𝑑𝑠.(2.14) That is, −𝜔𝑐₁≥1𝐻₁𝑏₁,𝑐₁^𝑏1𝑐₁𝑄₁(𝑠)𝐻₁𝑏₁,𝑠−𝑝(𝑠)ℎ²¹²𝑏₁,𝑠𝑑𝑠.(2.15)
On the other hand, multiplying both sides of (2.13) by 𝐻₁(𝑡,𝑎₁)∈𝒟_𝑎1,𝑏1 and integrating by parts, we can easily obtain 𝜔𝑐₁≥1𝐻₁𝑐₁,𝑎₁^𝑐1𝑎₁𝑄₁(𝑠)𝐻₁𝑠,𝑎₁−𝑝(𝑠)ℎ²¹¹𝑠,𝑎₁𝑑𝑠.(2.16)
Equations (2.15) and (2.16) yield 1𝐻₁𝑐₁,𝑎₁^𝑐1𝑎₁+1𝑄₁(𝑠)𝐻₁𝑠,𝑎₁−𝑝(𝑠)ℎ²¹¹𝑠,𝑎₁𝑑𝑠𝐻₁𝑏₁,𝑐₁^𝑏1𝑐₁𝑄₁(𝑠)𝐻₁𝑏₁,𝑠−𝑝(𝑠)ℎ²¹²𝑏₁,𝑠𝑑𝑠≤0,(2.17) which contradicts (2.5) for 𝑗=1. The proof of Theorem 2.2 is complete.

Remark 2.3. When 𝜏₁=⋯=𝜏_𝑛=0, Σ^𝑛𝑖=1𝑝_𝑖(𝑡)=𝑞(𝑡), the conditions 𝑞(𝑡)≥0 for 𝑡∈[𝑎₁,𝑏₁]∪[𝑎₂,𝑏₂], 𝑝^(𝑡)≥0 and (1.2) can be removed. Therefore, Theorem 2.2 reduces to Theorem 1 in [3].

Remark 2.4. When 𝜏₁=⋯=𝜏_𝑛=𝜏, Σ^𝑛_i=1𝑝_𝑖(𝑡)=𝑞(𝑡), Theorem 2.2 reduces to Theorem 1 in [4] for which the conditions 𝑞(𝑡)≥0 for 𝑡∈[𝑎₁−𝜏,𝑏₁]∪[𝑎₂−𝜏,𝑏₂],𝑝^(𝑡)≥0 and (1.2) are needed. There are some mistakes in the proof of Theorem 1 in [4].

The following theorem gives an oscillation criterion for the unforced (1.1).

Theorem 2.5. If, for any 𝑇≥0, there exist 𝑎, 𝑏, and 𝑐 such that 𝑇≤𝑎<𝑐<𝑏, 𝑝_𝑖(𝑡)≥0, and 𝑞_𝑖(𝑡)≥0 for 𝑡∈[𝑎−𝜏_𝑖,𝑏],𝑖=1,2,…,𝑛, and there exists 𝐻∈𝒟_𝑎,𝑏, such that 1𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝑠,𝑎)𝑄(𝑠)−𝑝(𝑠)ℎ²¹(𝑠,𝑎)𝑑𝑠+1𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝑏,𝑠)𝑄(𝑠)−𝑝(𝑠)ℎ²²(𝑏,𝑠)𝑑𝑠>0,(2.18) where 𝑄(𝑡)=^𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎𝑡−𝑎+𝜏_𝑖+^𝑛_𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑡−𝑡𝑎−+𝑎𝜏_𝑖^{𝛼𝑖𝜂𝑖},(2.19)𝜂₁,𝜂₂,…,𝜂_𝑛 are positive constants satisfying (a) and (c) in Lemma 2.1, and ℎ₁, ℎ₂ are defined as in (2.3), then the unforced (1.1) is oscillatory.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that 𝑥(𝑡)>0 for all 𝑡≥𝑡₁−𝜏≥0, where 𝑡₁ depends on the solution 𝑥(𝑡) and 𝜏=max{𝜏_𝑖}, 𝑖=1,…,𝑛. Similar to the proof in Theorem 2.2, we can obtain 𝑥𝑡−𝜏_𝑖𝑥(𝑡)≥𝑡−𝑡𝑎−+𝑎𝜏_𝑖,𝑡∈[𝑎,𝑏].(2.20) Define 𝜔(𝑡)=−𝑝(𝑡)𝑥^(𝑡)𝑥(𝑡).(2.21) Recall the arithmetic-geometric mean inequality ^𝑛_𝑖=1𝜂_𝑖𝑢_𝑖≥^𝑛_𝑖=1𝑢^𝜂𝑖_𝑖,𝑢_𝑖≥0,(2.22) where 𝜂_𝑖>0, 𝑖=1,2,…,𝑛, are chosen according to the given 𝛼₁,𝛼₂,…,𝛼_𝑛 as in Lemma 2.1 satisfying (a) and (c). Let 𝑢_𝑖=𝜂^𝑖−1𝑞_𝑖(𝑡)𝑥𝑡−𝜏_𝑖^𝛼𝑖.(2.23) We can obtain 𝜔^(𝑡)=−(𝑝(𝑡)𝑥(𝑡))^𝑥(𝑡)+𝜔²(𝑡)=∑𝑝(𝑡)_𝑛𝑖=1𝑝_𝑖(𝑡)𝑥𝑡−𝜏_𝑖+∑^𝑛𝑖=1𝑞_𝑖(𝑡)𝑥𝑡−𝜏_𝑖^𝛼𝑖𝑥(𝑡)+𝜔²(𝑡)≥𝑝(𝑡)_𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎𝑡−𝑎+𝜏_𝑖+∏^𝑛𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑥^{𝛼𝑖𝜂𝑖}𝑡−𝜏_𝑖𝑥(𝑡)+𝜔²(𝑡)=𝑝(𝑡)_𝑛_𝑖=1𝑝_𝑖(𝑡)𝑡−𝑎𝑡−𝑎+𝜏_𝑖+∏^𝑛𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑥^{𝛼𝑖𝜂𝑖}𝑡−𝜏_𝑖∏^𝑛𝑖=1𝑥^{𝛼𝑖𝜂𝑖}(𝑡)+𝜔²(𝑡)≥𝑝(𝑡)_𝑛_𝑖=1𝑝_𝑖(t)𝑡−𝑎𝑡−𝑎+𝜏_𝑖+^𝑛_𝑖=1𝜂^𝑖−1𝑞_𝑖(𝑡)^𝜂𝑖𝑡−𝑡𝑎−+𝑎𝜏_𝑖^{𝛼𝑖𝜂𝑖}+𝜔²(𝑡)=𝑄(𝑡)+𝜔₂𝑝(𝑡)(𝑡)𝑝(𝑡).(2.24)
Multiplying both sides of (2.24) by 𝐻(𝑏,𝑡)∈𝒟_𝑎,𝑏 and integrating by parts, we obtain ^𝑏𝑐𝐻(𝑏,𝑡)𝜔^(𝑡)𝑑𝑡≥^𝑏𝑐𝐻(𝑏,𝑡)𝑄(𝑡)𝑑𝑡+^𝑏𝑐𝐻(𝑏,𝑡)𝜔²(𝑡)−𝐻(𝑏,𝑐)𝜔(𝑐)≥_𝑏𝑐𝑝(𝑡)𝑑𝑡,𝐻(𝑏,𝑡)𝑄(𝑡)𝑑𝑡+^𝑏𝑐𝐻(𝑏,𝑡)𝜔²(𝑡)𝑝(𝑡)−2𝜔(𝑡)ℎ²=(𝑏,𝑡)√𝐻(𝑏,𝑡)𝑑𝑡_𝑏𝑐𝐻(𝑏,𝑡)𝑄(𝑡)−𝑝(𝑡)ℎ²²(𝑏,𝑡)𝑑𝑡+^𝑏𝑐𝐻(𝑏,𝑡)𝑝(𝑡)𝜔(𝑡)−√≥𝑝(𝑡)ℎ₂(𝑏,𝑡)²𝑑𝑡_𝑏𝑐𝐻(𝑏,𝑡)𝑄(𝑡)−𝑝(𝑡)ℎ²²(𝑏,𝑡)𝑑𝑡.(2.25) It follows that −𝜔(𝑐)≥1𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝑏,𝑡)𝑄(𝑡)−𝑝(𝑡)ℎ²²(𝑏,𝑡)𝑑𝑡.(2.26) On the other hand, multiplying both sides of (2.24) by 𝐻(𝑡,𝑎)∈𝒟_𝑎,𝑏 and integrating by parts, we have 𝜔(𝑐)≥1𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝑡,𝑎)𝑄(𝑡)−𝑝(𝑡)ℎ²¹(𝑡,𝑎)𝑑𝑡.(2.27) Equations (2.26) and (2.27) yield 1𝐻(𝑐,𝑎)^𝑐𝑎𝐻(𝑡,𝑎)𝑄(𝑡)−𝑝(𝑡)ℎ²¹(𝑡,𝑎)𝑑𝑡+1𝐻(𝑏,𝑐)^𝑏𝑐𝐻(𝑏,𝑡)𝑄(𝑡)−𝑝(𝑡)ℎ²²(𝑏,𝑡)𝑑𝑡<0,(2.28) which contradicts (2.24). The proof of Theorem 2.5 is complete.

Remark 2.6. When 𝜏₁=⋯=𝜏_𝑛=0, Σ^𝑛𝑖=1𝑝_𝑖(𝑡)=𝑞(𝑡), the conditions 𝑞(𝑡)≥0 for 𝑡∈[𝑎,𝑏], 𝑝^(𝑡)≥0 and (1.2) can be removed. Therefore, Theorem 2.5 reduces to Theorem 2 in [3].

Remark 2.7. When 𝜏₁=⋯=𝜏_𝑛=𝜏, Σ^𝑛𝑖=1𝑝_𝑖(𝑡)=𝑞(𝑡), Theorem 2.5 reduces to Theorem 2 in [4] for which the conditions 𝑞(𝑡)≥0 for 𝑡∈[𝑎−𝜏,𝑏], 𝑝^(𝑡)≥0 and (1.2) are needed.

3. Example

In this section, we provide an example to illustrate our results.

Consider the following equation: 𝑥^(𝑡)+𝑘sin𝑡|||𝑥𝑡−𝜋8|||^𝛼1sgn𝑥𝑡−𝜋8+𝑙cos𝑡|||𝑥𝑡−𝜋4|||^𝛼2sgn𝑥𝑡−𝜋4=−𝑚cos2𝑡,𝑡≥0,(3.1) where 𝑘, 𝑙, 𝑚 are positive constants, 𝛼₁>1, and 0<𝛼₂<1. Here 𝑝(𝑡)=1,𝑝𝜏₁(𝑡)=𝑝₂(𝑡)=0,𝑞₁(𝑡)=𝑘sin𝑡,𝑞₂(𝑡)=𝑙cos𝑡,₁=𝜋8,𝜏²=𝜋4,𝑒(𝑡)=−𝑚cos2𝑡.(3.2)

According to the direct computation, we have 𝑄_𝑗(𝑡)=𝑘₀||cos2𝑡||^𝜂0(sin𝑡)^𝜂1(cos𝑡)^𝜂2𝑡−𝑡𝑎−_𝑗𝑎+^𝑗𝜏₁^𝛼1𝜂1𝑡−𝑡𝑎−_𝑗𝑎+^𝑗𝜏₂^𝛼2𝜂2,𝑗=1,2,(3.3) where 𝑘₀=(𝜂⁰⁻¹/𝑚)^𝜂0(𝜂¹⁻¹/𝑘)^𝜂1(𝜂²⁻¹/𝑙)^𝜂2,𝜂₀ can be any positive number satisfying 0<𝜂₀<(𝛼₁−1)/𝛼₁, and 𝜂₁, 𝜂₂ satisfy (2.1). For any 𝑇≥0, we can choose 𝑎₁=2𝑖𝜋,𝑎₂=𝑏₁=2𝑖𝜋+𝜋4,𝑏²=2𝑖𝜋+𝜋2,𝑐¹=2𝑖𝜋+𝜋8,𝑐²=2𝑖𝜋+3𝜋8,(3.4) for 𝑖=0,1,…, and 𝐻₁(𝑡,𝑠)=𝐻₂(𝑡,𝑠)=(𝑡−𝑠)². By simple computation, we obtain ℎ_𝑗1(𝑡,𝑠)=ℎ_𝑗2(𝑡,𝑠)=1, 𝑗=1,2. From Theorem 2.2, we have that (3.1) is oscillatory if ^{2𝑖𝜋+𝜋/82𝑖𝜋}𝑄₁(𝑠)(𝑠−2𝑖𝜋)²𝑑𝑠+^{2𝑖𝜋+𝜋/42𝑖𝜋+𝜋/8}𝑄₁(𝑠)2𝑖𝜋+𝜋4−𝑠²𝑑𝑠>𝜋_{2𝑖𝜋+3𝜋/82𝑖𝜋+𝜋/4}4,𝑄₂(𝑠)𝑠−2𝑖𝜋−𝜋4²𝑑𝑠+^{2𝑖𝜋+𝜋/22𝑖𝜋+3𝜋/8}𝑄₂(𝑠)2𝑖𝜋+𝜋2−𝑠²𝑑𝑠>𝜋4.(3.5) If 𝐻₁(𝑡,𝑠)=𝐻₂(𝑡,𝑠)=sin²(𝑡−𝑠), by simple computation, we obtain ℎ_𝑗1(𝑡,𝑠)=ℎ_𝑗2(𝑡,𝑠)=cos(𝑡−𝑠) for 𝑗=1,2. From Theorem 2.2, we have that (3.1) is oscillatory if ^{2𝑖𝜋+𝜋/82𝑖𝜋}𝑄₁(𝑠)(𝑠−2𝑖𝜋)²𝑑𝑠+^{2𝑖𝜋+𝜋/42𝑖𝜋+𝜋/8}𝑄₁(𝑠)2𝑖𝜋+𝜋4−𝑠²𝑑𝑠>𝜋16+√2_{2𝑖𝜋+3𝜋/82𝑖𝜋+𝜋/4}8,𝑄₂(𝑠)𝑠−2𝑖𝜋−𝜋4²𝑑𝑠+^{2𝑖𝜋+𝜋/22𝑖𝜋+3𝜋/8}𝑄₂(𝑠)2𝑖𝜋+𝜋2−𝑠²𝑑𝑠>𝜋16+√28.(3.6)

Acknowledgments

The authors would like to thank the referees for their valuable comments which have led to an improvement of the presentation of this paper. This project is supported by the National Natural Science Foundation of China (10771118) and STPF of University in Shandong Province of China (J09LA04).