Some Properties of Third-Order Differential Equations with Mixed Arguments

Abstract

We offer a new comparison the principle for deducing properties of third-order differential equations with mixed arguments, (𝑟(𝑡)[𝑥^′(𝑡)]^𝛾)^′′+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))+𝑝(𝑡)ℎ(𝑥(𝜎(𝑡)))=0, from those of the corresponding differential equations, without deviating arguments. The presented technique permits to extend immediately the results known for an equation without deviating arguments to a more general equation with advanced and delay arguments.

1. Introduction

We consider third-order differential equations with mixed arguments 𝑟(𝑡)𝑥^′(𝑡)^𝛾′′+𝑞(𝑡)𝑓(𝑥(𝜏(𝑡)))+𝑝(𝑡)ℎ(𝑥(𝜎(𝑡)))=0, (𝐸) where we assume that 𝑟,𝑞,𝜏,𝑝,𝜎∈𝐶([𝑡₀,∞)), 𝑓,ℎ∈𝐶((−∞,∞)), and (𝐻₁)𝛾 is the ratio of two positive odd integers, (𝐻₂)𝑟(𝑡)>0, 𝑞(𝑡)>0, and 𝑝(𝑡)>0, (𝐻₃)𝜎(𝑡)≥𝑡, 𝜏(𝑡)≤𝑡, lim_𝑡→∞𝜏(𝑡)=∞, and 𝜏^′(𝑡)>0, (𝐻₄)𝑥𝑓(𝑥)>0, 𝑓^′(𝑥)≥0 for 𝑥≠0, and −𝑓(−𝑥𝑦)≥𝑓(𝑥𝑦)≥𝑓(𝑥)𝑓(𝑦) for 𝑥𝑦>0, (𝐻₅)𝑥ℎ(𝑥)>0, ℎ^′(𝑥)≥0 for 𝑥≠0, and −ℎ(−𝑥𝑦)≥ℎ(𝑥𝑦)≥ℎ(𝑥)ℎ(𝑦) for 𝑥𝑦>0.

In this paper, we study the canonical case of (𝐸), that is, 𝑅(𝑡)=^𝑡_𝑡0𝑟^−1/𝛾(𝑠)d𝑠→∞as𝑡→∞. (1)

There is a permanent interest in studying the properties of third-order differential equations. Various techniques have been presented for such equations (see [1–21]). Comparison theorems, especially, are a very strong and effective tool in the oscillation theory. Mahfoud [18] has presented a very useful comparison technique for studying the properties of delay differential equations through those of differential equations without delay. However, the corresponding result for advanced differential equation is still missing. In this paper, we fill up this gap in the oscillation theory; and moreover, we present such a comparison result that works also for the differential equation with mixed arguments.

2. The Main Results

The following result is crucial for our intended comparison theorem.

Lemma 1. Assume that 𝑧(𝑡)>0, 𝑧^′(𝑡)>0, and (𝑟(𝑡)[𝑧^′(𝑡)]^𝛾)^′>0, eventually. Then, for arbitrary 𝑘∈(0,1), 𝑧[𝜎(𝑡𝑅(𝜎(𝑡)]≥𝑘))𝑅(𝑡)𝑧(𝑡), (2) eventually.

Proof. It follows from the monotonicity of 𝑤(𝑡)=𝑟(𝑡)[𝑧^′(𝑡)]^𝛾 that 𝑧[𝜎(𝑡)]−𝑧(𝑡)=_𝑡^𝜎(𝑡)𝑧^′(𝑠)=d𝑠_{𝑡𝜎(𝑡)}𝑤^1/𝛾(𝑠)𝑟^−1/𝛾(𝑠)d𝑠≥𝑤^1/𝛾(𝑡)(𝑅(𝜎(𝑡))−𝑅(𝑡)), (3) that is, 𝑧[𝜎(𝑡)]𝑧(𝑡)𝑤≥1+^1/𝛾(𝑡)𝑧(𝑡)[𝑅(𝜎(𝑡))−𝑅(𝑡)]. (4) On the other hand, since 𝑧(𝑡)→∞ as 𝑡→∞, then for any 𝑘∈(0,1) there exists a 𝑡₁ large enough, such that 𝑘𝑧(𝑡)≤𝑧(𝑡)−𝑧𝑡₁=^𝑡_𝑡1𝑤^1/𝛾(𝑠)𝑟^−1/𝛾(𝑠)d𝑠≤𝑤^1/𝛾(𝑡)^𝑡_𝑡1𝑟^−1/𝛾(𝑠)d𝑠=𝑤^1/𝛾(𝑡)𝑅(𝑡)−𝑅𝑡₁≤𝑤^1/𝛾(𝑡)𝑅(𝑡) (5) or equivalently 𝑤^1/𝛾(𝑡)𝑧(𝑡)≥𝑘𝑅(𝑡). (6) Using (6) in (4), we obtain that 𝑧[𝜎(𝑡)]𝑧(𝑡)𝑘≥1+𝑅(𝑡)[𝑅(𝜎(𝑡(𝑡𝑅(𝜎(𝑡))−𝑅)]≥𝑘))𝑅(𝑡). (7)

We introduce the structure of nonoscillatory solutions.

Lemma 2. Every nonoscillatory solution 𝑥(𝑡) of (𝐸) satisfies one of the folowing: 𝑥(𝑡)𝑥^′(𝑡)<0,𝑥(𝑡)𝑟(𝑡)𝑥^′(𝑡)^𝛾′𝑥(𝑡𝑟(𝑡𝑥>0,))^′(𝑡)^𝛾′′<0,𝑥(𝑡)𝑥^′(𝑡)>0,𝑥(𝑡)𝑟(𝑡)𝑥^′(𝑡)^𝛾′𝑥(𝑡𝑟(𝑡𝑥>0,))^′(𝑡)^𝛾′′<0, 𝑁₀𝑁₂ eventually.

This structure follows from canonical case of (𝐸), and the proof can be omitted. It is well known that for a particular case of (𝐸), namely, for the equation 𝑥^′′′(𝑡)+𝑝(𝑡)𝑥(𝑡)=0, (8) there always exists a solution satisfying 𝑁₀, and this fact leads to the following definition, which is due to Kiguradze and Chanturia [15].

Definition 3. One can say that (𝐸) enjoys property (A) if all its nonoscillatory solutions 𝑥(𝑡) satisfy 𝑁₀.

We are prepared to present the main result.

Theorem 4. Assume that for some 𝑘∈(0,1) the differential inequality 𝑟(𝑡)𝑧^′(𝑡)^𝛾′′+𝑞𝜏⁻¹(𝑡)𝜏^′𝜏⁻¹(𝑡)𝑓(𝑧(𝑡(𝑡)ℎ𝑘𝑅(𝜎(𝑡))+𝑝))𝑅(𝑡)ℎ(𝑧(𝑡))sgn𝑧(𝑡)≤0 (9) enjoys property (A), then so does (𝐸).

Proof. Assume the contrary, that is, we admit that (𝐸) possesses a positive solution 𝑥(𝑡) satisfying 𝑁₂. Thus, it follows from Lemma 1 that, for every 𝑘∈(0,1), 𝑟(𝑡)𝑥^′(𝑡)^𝛾′′(𝑡)𝑓(𝑥(𝜏(𝑡(𝑡)ℎ𝑘𝑅(𝜎(𝑡+𝑞)))+𝑝))𝑅(𝑡)ℎ(𝑥(𝑡))≤0. (10) Integrating from 𝑡 to ∞, we obtain that 𝑟(𝑡)𝑥^′(𝑡)^𝛾′≥^∞_𝑡𝑞(𝑠)𝑓(𝑥(𝜏(𝑠+)))d𝑠_∞𝑡𝑝(𝑠)ℎ𝑘𝑅(𝜎(𝑠))𝑅(𝑠)ℎ(𝑥(𝑠=))d𝑠_{∞𝜏(𝑡)}𝑞𝜏⁻¹(𝑠)𝜏^′𝜏⁻¹(𝑠)𝑓(𝑥(𝑠+))d𝑠_∞𝑡𝑝(𝑠)ℎ𝑘𝑅(𝜎(𝑠))𝑅(𝑠)ℎ(𝑥(𝑠≥))d𝑠_∞𝑡𝑞𝜏⁻¹(𝑠)𝜏^′𝜏⁻¹(𝑠)𝑓(𝑥(𝑠(𝑠)ℎ𝑘𝑅(𝜎(𝑠))+𝑝))𝑅(𝑠)ℎ(𝑥(𝑠))d𝑠. (11) Integrating twice from 𝑡₁ to 𝑡, we are led to 𝑥(𝑡)≥^𝑡_𝑡1𝑟^−1/𝛾(𝑣)×^𝑣_𝑡1^∞_𝑢𝑞𝜏⁻¹(𝑠)𝜏^′𝜏⁻¹(𝑠)𝑓(𝑥(𝑠(𝑠)ℎ𝑘𝑅(𝜎(𝑠))+𝑝))𝑅(𝑠)ℎ(𝑥(𝑠))d𝑠d𝑢^1/𝛾d𝑣. (12) Let us denote the right-hand side by 𝑧(𝑡). Then, 𝑥(𝑡)≥𝑧(𝑡), 𝑧(𝑡) satisfies 𝑁₂, and 0=𝑟(𝑡)𝑧^′(𝑡)^𝛾′′+𝑞𝜏⁻¹(𝑡)𝜏^′𝜏⁻¹(𝑡)𝑓(𝑥(𝑡(𝑡)ℎ𝑘𝑅(𝜎(𝑡))+𝑝))𝑅(𝑡)ℎ(𝑥(𝑡≥𝑟(𝑡𝑧)))^′(𝑡)^𝛾′′+𝑞𝜏⁻¹(𝑡)𝜏^′𝜏⁻¹(𝑡)𝑓(𝑥(𝑡(𝑡)ℎ𝑘𝑅(𝜎(𝑡))+𝑝))𝑅(𝑡)ℎ(𝑧(𝑡)), (13) which contradicts with the assumptions of the theorem, and we conclude that (𝐸) has property (A).

Theorem 4 reduces the examination of properties of differential equations with mixed arguments to that of simpler equations without deviating arguments and permits to extend immediately the criteria known for property (A) of equations without deviating arguments to more general equations with both advanced and delay arguments. We provide some applications of our main result.

For our further references, we set that 𝑄₁(𝑡)=𝑞𝜏⁻¹(𝑠)𝜏^′𝜏⁻¹(𝑠)(𝑠)𝑘𝑅(𝜎(𝑠+𝑝))𝑅(𝑡)^𝛾,𝑄(𝑡)=^∞_𝑡𝑄₁(𝑠)d𝑠, (14) where 𝑘∈(0,1).

Theorem 5. If, for some 𝑘∈(0,1), liminf_𝑡→∞1𝑄(𝑡)^∞_𝑡𝑠𝑟(𝑠)^1/𝛾𝑄^1+1/𝛾(𝑠)1d𝑠>𝛾+1^1+1/𝛾, (15) then 𝑟(𝑡)𝑥^′(𝑡)^𝛾′′+𝑞(𝑡)𝑥^𝛾(𝜏(𝑡))+𝑝(𝑡)𝑥^𝛾(𝜎(𝑡))=0 𝐸₁ enjoys property (A).

Proof. By Theorem 4, it is sufficient to show that the differential inequality 𝑟(𝑡)𝑥^′(𝑡)^𝛾′′+𝑄₁(𝑡)𝑥^𝛾(𝑡)sgn𝑥(𝑡)≤0 𝐸₂ has property (A) for some 𝑘∈(0,1). Assuming the contrary, we admit that 𝐸₂ does not property (A), that is, it possesses an eventually positive solution 𝑥(𝑡) satisfying 𝑁₂. We define have the following: 𝑤(𝑡)=𝑟(𝑡)𝑥^′(𝑡)^𝛾′𝑥^𝛾(𝑡)>0. (16) Differentiating 𝑤(𝑡), one gets in view of 𝐸₂ that 𝑤^′(𝑡)=𝑟(𝑡)𝑥^′(𝑡)^𝛾′′𝑥^𝛾(𝑡)𝑟(𝑡𝑥−𝛾)^′(𝑡)^𝛾′𝑥^𝛾(𝑡)𝑥^′(𝑡)𝑥(𝑡)≤𝑄₁(𝑡)(𝑡)𝑥−𝛾𝑤^′(𝑡)𝑥(𝑡). (17) On the other hand, using the monotonicity of [𝑟(𝑡)[𝑥^′(𝑡)]^𝛾]^′, we have 𝑟(𝑡)𝑥^′(𝑡)^𝛾≥^𝑡_𝑡0𝑟(𝑠)𝑥^′(𝑠)^𝛾′≥𝑟(𝑡)𝑥^′(𝑡)^𝛾′𝑡−𝑡₀≥ℓ𝑡𝑟(𝑡)𝑥^′(𝑡)^𝛾′, (18) eventually; let us say that for 𝑡≥𝑡₁ or equivalently 𝑥^′(𝑡)≥ℓ𝑡𝑟(𝑡)𝑥^′(𝑡)^{𝛾′1/𝛾}𝑟^−1/𝛾(𝑡), (19) where ℓ∈(0,1) will be specified latter. Setting the last inequality into (17), we obtain 𝑤^′(𝑡)≤−𝑄₁(𝑡)−𝛾ℓ^1/𝛾𝑤^1+1/𝛾(𝑡)𝑡^1/𝛾𝑟^1/𝛾(𝑡). (20) Integrating the last inequality from 𝑡 to ∞, we have 𝑤(𝑡)≥𝑄(𝑡)+𝛾ℓ^1/𝛾∞_𝑡𝑤^1+1/𝛾(𝑠)𝑠𝑟(𝑠)^1/𝛾d𝑠 (21) or 𝑤(𝑡)𝑄(𝑡)≥1+𝛾ℓ^1/𝛾𝑄(𝑡)^∞_𝑡𝑠𝑟(𝑠)^1/𝛾𝑄^1+1/𝛾(𝑠𝑤(𝑠))𝑄(𝑠)^1+1/𝛾d𝑠, (22) eventually; let us say that 𝑡≥𝑡₁. Since 𝑤(𝑡)>𝑄(𝑡), then inf_𝑡≥𝑡1𝑤(𝑡)𝑄(𝑡)=𝜆≥1. (23) Thus, 𝑤(𝑡)𝑄(𝑡)≥1+𝛾ℓ^1/𝛾𝜆^1+1/𝛾𝑄(𝑡)^∞_𝑡𝑠𝑟(𝑠)^1/𝛾𝑄^1+1/𝛾(𝑠)d𝑠. (24) From (15), we see that there exist some ℓ∈(0,1) and some positive 𝜂, such that ℓ^1/𝛾𝑄(𝑡)^∞_𝑡𝑠𝑟(𝑠)^1/𝛾𝑄^1+1/𝛾(𝑠)d𝑠>𝜂>𝛾+1^{−(𝛾+1)/𝛾}. (25) Combining (24) with (25), we have 𝑤(𝑡)𝑄(𝑡)≥1+𝛾𝜆^1+1/𝛾𝜂. (26) Therefore, 𝜆≥1+𝛾𝜆^1+1/𝛾𝜂>1+𝛾𝜆^1+1/𝛾𝛾+1^{−(𝛾+1)/𝛾} (27) or equivalently 10>+𝛾𝛾+1𝜆𝛾+1𝛾+1^1+1/𝛾−𝜆.𝛾+1 (28) This contradicts with the fact that the function 𝑔(𝛼)=1+𝛾𝛾+1𝛼𝛾+1^1+1/𝛾−𝛼 (29) is positive for all 𝛼>0, and we conclude that 𝑥(𝑡) cannot satisfy (𝑁₂); consequently, 𝐸₁ has property (A).

In the following criteria, we utilize existing results known for differential equations without deviating arguments to provide new criteria for differential equations with mixed arguments.

Corollary 6. Assume that for some 𝑘∈(0,1) the differential equation 𝑥^′′′(𝑡)+𝑞𝜏⁻¹(𝑡)𝜏^′𝜏⁻¹(𝑡)(𝑡)𝜎(𝑡)+𝑘𝑝𝑡𝑥(𝑡)=0 𝐸₃ enjoys property (A), then so does the equation with mixed arguments 𝑥^′′′(𝑡)+𝑞(𝑡)𝑥(𝜏(𝑡))+𝑝(𝑡)𝑥(𝜎(𝑡))=0. 𝐸₄

Proof. By Theorem 4,  𝐸₄ enjoys property (A), provided that, for some 𝑘∈(0,1), differential inequality 𝑥^′′′(𝑡)+𝑞𝜏⁻¹(𝑡)𝜏^′𝜏⁻¹(𝑡)(𝑡)𝜎(𝑡)+𝑘𝑝𝑡𝑥(𝑡)sgn𝑥(𝑡)≤0 (30) has property (A). But by Corollary 1 in [16], this is equivalent to property (A) of the corresponding differential equation 𝐸₃.

Corollary 7. Assume that at least one of the following conditions: liminf_𝑡→∞𝑡^2∞_𝑡𝑞𝜏⁻¹(𝑠)𝜏^′𝜏⁻¹(𝑠)(𝑠)𝜎(𝑠)+𝑝𝑠1d𝑠>3√3,liminf_𝑡→∞𝑡^∞_𝑡𝑠𝑞𝜏⁻¹(𝑠)𝜏^′𝜏⁻¹(𝑠)(𝑠)𝜎(𝑠)2+𝑝d𝑠>3√3 (31) holds, then 𝐸₃ has property (A).

Proof. It is known (see, e.g., [12, 15]) that liminf_𝑡→∞𝑡^2∞_𝑡𝑃(𝑠)1d𝑠>3√3 (32) guarantees property (A) of 𝑥^′′′(𝑡)+𝑃(𝑡)𝑥(𝑡)=0. (33) The rest follows from Corollary 6, where constant 𝑘∈(0,1) is eliminated due to the sharp inequality of the used criterion.

We support our results by several illustrative examples.

3. Examples

Example 8. Consider the third-order nonlinear differential equation 𝑡⁻¹𝑥^′(𝑡)^3′′+𝑎𝑡⁶𝑥³()+𝑏𝛿𝑡𝑡⁶𝑥³(𝜆𝑡)=0, 𝐸_𝑥1 where 𝑎,𝑏>0, 0<𝛿≤1, and 𝜆≥1. Simple computation shows that here (15) reduces to 𝛿⁵𝑎+𝑘𝑏𝜆⁴>54⁴, (34) for some 𝑘∈(0,1), which is equivalent to 𝛿⁵𝑎+𝑏𝜆⁴>54⁴ (35) and this by Theorem 4 guarantees that 𝐸_𝑥1 enjoys property (A).

Example 9. Consider the third-order differential equation with mixed arguments 𝑥^′′′(𝑡)+𝑎𝑡³𝑥()+𝑏𝛿𝑡𝑡³𝑥(𝜆𝑡)=0, 𝐸_𝑥2 where 𝑎,𝑏>0, 0<𝛿≤1, and 𝜆≥1. Both conditions of Corollary 7 reduce to 𝛿²2𝑎+𝑏𝜆>3√3, (36) which guarantees property (A) of (𝐸_𝑥2).

Our results take into account the value of advanced argument, which can be seen in the following example.

Example 10. Consider the third-order nonlinear differential equation 𝑥^′′′(𝑡)+𝑎𝑡³𝑥()+𝑏𝛿𝑡𝑡⁴𝑥𝑡²=0, 𝐸_𝑥3 where 𝑎,𝑏>0, 0<𝛿<1. Both conditions of Corollary 7 reduce to 𝛿²2𝑎+𝑏>3√3, (37) which yields property (A) of 𝐸_𝑥3.

4. Conclusion and Discussion

In the paper, we presented new comparison theorems for studying the properties of third-order differential equations with mixed arguments through those of the corresponding differential equations without deviating arguments. The presented technique permits to extend immediately the results known for an equation without deviating arguments to a more general equation with advanced and delay arguments. The results obtained have been supported by several illustrative examples.

Acknowledgment

This work was supported by S.G.A. KEGA Grant no. 020TUKE-4/2012.