On Delay-Independent Criteria for Oscillation of Higher-Order Functional Differential Equations

Abstract

We investigate the oscillation of the following higher-order functional differential equation: 𝑥^(𝑛)(𝑡)+𝑞(𝑡)|𝑥(𝑡−𝜏)|^𝜆−1𝑥(𝑡−𝜏)=𝑒(𝑡), where 𝑞(𝑡) and 𝑒(𝑡) are continuous functions on [𝑡₀,∞), 1>𝜆>0 and 𝜏≠0 are constants. Unlike most of delay-dependent oscillation results in the literature, two delay-independent oscillation criteria for the equation are established in both the case 𝜏>0 and the case 𝜏<0 under the assumption that the potentials 𝑞(𝑡) and 𝑒(𝑡) change signs on [𝑡₀,∞).

1. Introduction

Consider the following 𝑛th-order forced functional differential equation of the form:𝑥^(𝑛)(𝑡)+𝑞(𝑡)||𝑥(𝑡−𝜏)||^𝜆−1𝑥(𝑡−𝜏)=𝑒(𝑡),(1.1) where 𝑛≥1 is an integer, 𝑞(𝑡), 𝑒(𝑡)∈𝐶[𝑡₀,∞), 𝜆>0, and 𝜏≠0 are constants.

We are here only concerned with the nonconstant solutions of (1.1) that are defined for all large 𝑡. The oscillatory behavior is considered in the usual sense, that is, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is called nonoscillatory.

The oscillatory behavior of (1.1) with 𝜏=0 has been studied by many authors. In early papers [1, 2], by assuming that 𝑒(𝑡)=ℎ^(𝑛)(𝑡), where ℎ(𝑡) is an oscillatory function satisfying lim_𝑡→∞ℎ(𝑡)=0, the author proved that the forced equation would remain oscillatory if the unforced equation is oscillatory. However, the potential 𝑞(𝑡) is usually assumed to be nonnegative in [1, 2].

When 𝑞(𝑡)<0, 𝜏=0, and 𝜆>1, Agarwal and Grace [3] studied the oscillation of (1.1) by using a method of general means without imposing the Kartsatos condition. Following this method, the oscillation of (1.1) with 𝜏=0 was studied in [4] for both the case 𝑞(𝑡)≥0 and 𝑞(𝑡)<0 on [𝑡₀,∞). When 𝑞(𝑡) changes its sign on [𝑡₀,∞), 𝜏=0, and 0<𝜆<1, oscillation criteria for (1.1) were given in [5]. Sun and Saker [6], Sun and Mingarelli [7], and Yang [8] studied the oscillation for a generalized form of (1.1) with 𝜏=0. When 𝜏>0, there have been many oscillation criteria for equations of the type (1.1). For example, see [9–14] and references cited therein. We see that all these oscillation criteria depend on time delay.

To the best of our knowledge, little has been known about the oscillatory behavior of (1.1) in the case of oscillatory potentials when 𝜏<0. Particularly, little has been known about the delay-independent criteria for oscillation of (1.1). Unlike most of papers devoted on delay-dependent oscillation criteria for functional differential equations, the main purpose of this paper is to establish two delay-independent oscillation criteria for (1.1) in both the case 𝜏>0 and the case 𝜏<0, where the potential 𝑒(𝑡) is not imposed on the Kartsatos condition, and the potential 𝑞(𝑡) may change its sign. Finally, two interesting examples are worked out to illustrate the main results.

2. Main Results

Theorem 2.1. Assume that 0<𝜆<1 and 𝜏>0. If limsup_𝑡→∞1𝑡−𝑡₀^𝑛liminf_𝑡→∞∫^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)+𝑄(𝑡,𝑠)𝑑𝑠=+∞,(2.1)1𝑡−𝑡₀^𝑛∫^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)−𝑄(𝑡,𝑠)𝑑𝑠=−∞,(2.2) where 𝑄(𝑡,𝑠)=(𝜆−1)𝜆^{𝜆/(1−𝜆)}(𝑛!)^{𝜆/(𝜆−1)}(𝑡−𝑠)^𝑛𝑞(𝑠)^1/(1−𝜆),(2.3)𝑞(𝑠)=max{−𝑞(𝑠),0}, then all solutions of (1.1) are oscillatory for any 𝜏>0.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1.1). Without loss of generality, we may assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀. When 𝑥(𝑡) is eventually negative, the proof follows the same argument. Multiplying (1.1) by (𝑡−𝑠)^𝑛 and integrating it from 𝑡₀ to 𝑡 yields ^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)𝑑𝑠=≥^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑥^(𝑛)(𝑠)𝑑𝑠+^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆(𝑠−𝜏)𝑑𝑠_𝑡𝑡0=−(_𝑛−1𝑡−𝑠)^𝑛𝑥^(𝑛)(𝑠)𝑑𝑠−^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆(𝑠−𝜏)𝑑𝑠_𝑖=0+𝑐_𝑛𝑐_𝑖𝑡−𝑡₀^𝑛−𝑖𝑥^{(𝑛−𝑖−1)}𝑡₀^{𝑡𝑡+𝜏}_0+𝜏𝑥(𝑠−𝜏)𝑑𝑠−^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆(𝑠−𝜏)𝑑𝑠,(2.4) where 𝑐₀=0, 𝑐_𝑖=𝑛(𝑛−1)⋯(𝑛−𝑖+1), and 𝑐_𝑛=𝑛!. Since 𝜏>0, ^𝑡𝑡₀𝑥(𝑠)𝑑𝑠=^{𝑡𝑡+𝜏}_0+𝜏𝑥(𝑠−𝜏)𝑑𝑠,(2.5) we have ^{𝑡𝑡+𝜏}_0+𝜏𝑥(𝑠−𝜏)𝑑𝑠≥^𝑡𝑡_0+𝜏𝑥(𝑠−𝜏)𝑑𝑠.(2.6) This together with (2.4) and (2.5) yield ^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)𝑑𝑠≥−^𝑛−1_𝑖=0𝑐_𝑖𝑡−𝑡₀^𝑛−𝑖𝑥^{(𝑛−𝑖−1)}𝑡₀−𝑐_𝑛^{𝑡0𝑡+𝜏}₀+𝑥(𝑠−𝜏)𝑑𝑠_𝑡𝑡0𝑐_𝑛𝑥(𝑠−𝜏)−(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆(𝑠−𝜏)𝑑𝑠.(2.7) For given 𝑡 and 𝑠(𝑡>𝑠), set 𝐹(𝑥)=𝑐_𝑛𝑥−(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆,𝑥>0,0<𝜆<1.(2.8) It is not difficult to see that 𝐹(𝑥) obtains its minimum at 𝑥=[𝑐_𝑛/(𝑡−𝑠)^𝑛𝑞(𝑠)]^1/(1−𝜆) and 𝐹_min=(𝜆−1)𝜆^{𝜆/(1−𝜆)}(𝑛!)^{𝜆/(𝜆−1)}(𝑡−𝑠)^𝑛𝑞(𝑠)^1/(1−𝜆)=𝑄(𝑡,𝑠).(2.9) It implies that 𝑐_𝑛𝑥(𝑠−𝜏)−(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆(𝑠−𝜏)≥𝑄(𝑡,𝑠).(2.10) Therefore, for any 𝜏>0, multiplying (2.7) by (𝑡−𝑡₀)^−𝑛, using (2.10), and taking liminf on both sides of (2.7), we get a contradiction with (2.1). This completes the proof of Theorem 2.1.

Theorem 2.2. Assume that 0<𝜆<1 and 𝜏<0. If (2.1) and (2.2) hold, then all solutions of (1.1) satisfying 𝑥(𝑡)=𝑂(𝑡^𝑛) are oscillatory for any 𝜏<0.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1.1) satisfying 𝑥(𝑡)=𝑂(𝑡^𝑛). Without loss of generality, we may assume that 𝑥(𝑡)>0 for 𝑡≥𝑡₀, and there exists a positive constant 𝑀>0 such that 𝑥(𝑡)≤𝑀𝑡^𝑛. Similar to the corresponding computation in Theorem 2.1 and noting that 𝜏<0, we have ^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)𝑑𝑠≥−^𝑛−1_𝑖=0𝑐_𝑖𝑡−𝑡₀^𝑛−𝑖𝑥^{(𝑛−𝑖−1)}𝑡₀+𝑐_𝑛^𝑡0_𝑡0+𝜏−𝑐_𝑛𝑥(𝑠−𝜏)𝑑𝑠^{𝑡𝑡+𝜏}𝑥(𝑠−𝜏)𝑑𝑠+^𝑡𝑡₀𝑐_𝑛𝑥(𝑠−𝜏)−(𝑡−𝑠)^𝑛𝑞(𝑠)𝑥^𝜆(𝑠−𝜏)𝑑𝑠.(2.11) Since 𝑥(𝑡)≤𝑀𝑡^𝑛, we get ^{𝑡𝑡+𝜏}𝑥(𝑠−𝜏)𝑑𝑠≤𝑀𝑛+1(𝑡−𝜏)^𝑛+1−𝑡^𝑛+1.(2.12) Then, for any 𝜏<0, multiplying (2.11) by (𝑡−𝑡₀)^−𝑛, using (2.10) and (2.12), and taking liminf on both sides of (2.11), we get a contradiction with (2.1). This completes the proof of Theorem 2.2.

The main results in this paper can also be extended to the case of time-varying delay. That is, we can consider the following equation: 𝑥^(𝑛)(𝑡)+𝑞(𝑡)||𝑥(𝜎(𝑡))||^𝜆−1𝑥(𝜎(𝑡))=𝑒(𝑡),(2.13) where 𝜎(𝑡) is continuously differentiable on [𝑡₀,∞), lim_𝑡→∞𝜎(𝑡)=∞, and 𝜎^(𝑡)>0 for 𝑡 sufficiently large. Without loss of generality, say 𝜎^(𝑡)>0 for 𝑡≥𝑡₀. Similar to the analysis as before, we have the following delay-independent and derivative-dependent oscillation criteria for (2.13).

Theorem 2.3. Assume that 0<𝜆<1 and 𝜎(𝑡)≤𝑡. If limsup_𝑡→∞1𝑡−𝑡₀^𝑛liminf_𝑡→∞∫^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)+𝑄(𝑡,𝑠)𝑑𝑠=+∞,1𝑡−𝑡₀^𝑛∫^𝑡𝑡₀(𝑡−𝑠)^𝑛𝑒(𝑠)−𝑄(𝑡,𝑠)𝑑𝑠=−∞,(2.14) where 𝑄(𝑡,𝑠)=(𝜆−1)𝜆^{𝜆/(1−𝜆)}𝑛!𝜎^(𝑠)^{𝜆/(𝜆−1)}(𝑡−𝑠)^𝑛𝑞(𝑠)^1/(1−𝜆),(2.15)𝑞(𝑠)=max{−𝑞(𝑠),0}, then all solutions of (2.13) are oscillatory.

Theorem 2.4. Assume that 0<𝜆<1 and 𝜎(𝑡)≥𝑡. If (2.1) and (2.2) hold, and there exists a continuous function 𝜙(𝑡)≥0 on [𝑡₀,∞) such that ∫^𝑡𝜎_−1(𝑡)𝜙(𝑠)𝑑𝑠=𝑂(𝑡^𝑛), where 𝜎⁻¹ is the inverse of 𝜎(𝑡), then all solutions of (2.13) satisfying 𝑥(𝑡)=𝑂(𝜙(𝑡)) are oscillatory.

3. Examples

In this section, we work out two examples to illustrate the main results.

Example 3.1. Consider the following equation: 𝑥^(𝑛)(𝑡)+𝑡^𝛼sin𝑡||𝑥(𝑡−𝜏)||^𝜆−1𝑥(𝑡−𝜏)=𝑡^𝛽cos𝑡,𝑡≥0,(3.1) where 𝜏≠0, 𝛼≥0, 𝛽>0, and 0<𝜆<1 are constants. Note that 𝑄(𝑡,𝑠)≥Λ(𝑡−𝑠)^{𝑛/(1−𝜆)}𝑠^{𝛼/(1−𝜆)},(3.2) where Λ=(𝜆−1)𝜆^{𝜆/(1−𝜆)}(𝑛!)^{𝜆/(𝜆−1)}<0. We have ^𝑡0𝑄(𝑡,𝑠)𝑑𝑠≥=ΛΛ𝐵^𝑡0(𝑡−𝑛𝑠)^{𝑛/(1−𝜆)}𝑠^{𝛼/(1−𝜆)}𝑑𝑠1−𝜆,1−𝛼𝜆𝑡^{((𝑛+𝛼)/(1−𝜆))+1},(3.3) where the Beta function 𝐵(𝑛/(1−𝜆),𝛼/(1−𝜆)) is a positive constant. On the other hand, ^𝑡0(𝑡−𝑠)^𝑛𝑠^𝛽cos𝑠𝑑𝑠=𝑡^𝑛+1+𝛽¹⁰(1−𝑢)^𝑛𝑢^𝛽cos𝑡𝑢𝑑𝑢=𝑡^𝑛+1+𝛽𝐼_𝑛,𝛽(𝑡),(3.4) where 𝐼_𝑛,𝛽(𝑡) has the asymptotic formula 𝐼_𝑛,𝛽(𝑡)=−Γ(𝑛+1)𝑡^−𝑛−1cos𝑡−(𝑛+1)𝜋2+𝑜𝑡^−𝑛−1,(3.5) as 𝑡→∞ [15, pages 49 and 50]. By Theorems 2.1 and 2.2, we have that if 𝛽>𝑛+𝛼1−𝜆+1,(3.6) then all solutions of (3.1) are oscillatory for any 𝜏>0, and all solutions of (3.1) satisfying 𝑥(𝑡)=𝑂(𝑡^𝑛) are oscillatory for any 𝜏<0.

Example 3.2. Consider the following equation: 𝑥^(𝑛)(𝑡)+𝑡^𝛼sin𝑡|||𝑥√𝑡|||^𝜆−1𝑥√𝑡=𝑡^𝛽cos𝑡,𝑡≥0,(3.7) where 𝛼, 𝛽, and 𝜆 are defined as in Example 3.1. Similar to the computation in Example 3.1, we have ^𝑡0𝑄(𝑡,𝑠)𝑑𝑠≥=ΛΛ𝐵^𝑡0(𝑡−𝑛𝑠)^{𝑛/(1−𝜆)}𝑠^{(𝛼+𝜆)/(1−𝜆)}𝑑𝑠1−𝜆,𝛼1−+𝜆𝜆𝑡^{(𝑛+𝛼+𝜆)/(1−𝜆)+1},(3.8) where Λ=(𝜆−1)𝜆^{𝜆/(1−𝜆)}(𝑛!/2)^{𝜆/(𝜆−1)}. Following the same argument in Example 3.1, we have that all solutions of (3.7) are oscillatory if 𝛽>(𝑛+𝛼+𝜆)/(1−𝜆)+1.

Acknowledgment

This paper was supported by the Natural Science Foundation of Shandong Province under Grant no. ZR2010AL002.