Oscillation Theorems for Second-Order Nonlinear Neutral Delay Differential Equations

Abstract

We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature.

1. Introduction

In this paper, we study the oscillatory behavior of a class of second-order nonlinear neutral delay differential equations 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+𝑞(𝑡)𝑓(𝑥(𝜎(𝑡)))=0, (1) where 𝑡∈𝐼∶=[𝑡₀,∞),  𝑡₀>0, 𝑧(𝑡)∶=𝑥(𝑡)+𝑝(𝑡)𝑥(𝜏(𝑡)), and 𝛼>0 is a constant. We assume that the following conditions hold:(A₁)𝑟, 𝑝, 𝑞∈𝐶(𝐼,ℝ),  𝑟(𝑡)>0,  𝑝(𝑡)≥0,  𝑞(𝑡)≥0, and 𝑞(𝑡) is not identically zero for large 𝑡;(A₂)𝑓∈𝐶(ℝ,ℝ),  𝑢𝑓(𝑢)>0, for all 𝑢≠0, and there exists a positive constant 𝑘 such that 𝑓(𝑢)|𝑢|^𝛼−1𝑢≥𝑘,∀𝑢≠0; (2)(A₃)𝜎∈𝐶(𝐼,ℝ), 𝜎(𝑡)≤𝑡, and lim_𝑡→∞𝜎(𝑡)=∞;(A₄)𝜏∈𝐶¹(𝐼,ℝ), 𝜏(𝑡)≤𝑡, and lim_𝑡→∞𝜏(𝑡)=∞;(A₅)𝜏^′(𝑡)≥𝜏₀>0 and 𝜏∘𝜎=𝜎∘𝜏.

By a solution of (1) we mean a function 𝑥∈𝐶([𝑇_𝑥,∞),ℝ),  𝑇_𝑥≥𝑡₀, such that 𝑟|𝑧^′|^𝛼−1𝑧^′∈𝐶¹([𝑇_𝑥,∞),ℝ) and 𝑥 satisfies (1) on [𝑇_𝑥,∞). We consider only those solutions of (1) which satisfy sup{|𝑥(𝑡)|∶𝑡≥𝑇}>0, for all 𝑇≥𝑇_𝑥, and assume that (1) possesses such solutions. As customary, we say that a solution of (1) is oscillatory if it has arbitrarily large zeros on the interval [𝑇_𝑥,∞); otherwise, it is called nonoscillatory. Equation (1) is termed oscillatory if all its solutions are oscillatory.

An increasing interest in oscillation of solutions to functional differential equations during the last few decades has been stimulated by applications arising in engineering and natural sciences; see Hale [1]. This resulted in publication of several monographs [2–5] and numerous research articles [6–20]; see also the references cited there. Prior to presenting our oscillation criteria, we briefly comment on a number of closely related results for (1) and its particular cases which motivated the present study. In the sequel, the following notation is frequently used: 𝑓₊(𝑡)∶=max0,𝑓(𝑡),𝑄(𝑡)∶=min𝑞(𝑡),𝑞(𝜏(𝑡)). (3)

Grace and Lalli [10] studied a second-order nonlinear neutral delay differential equation 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏)^′′+𝑞(𝑡)𝑓(𝑥(𝑡−𝜎))=0 (4) under the assumptions that (𝑡)𝑓(𝑢)0≤𝑝<1,𝑢≥𝑘>0,∀𝑢≠0,_∞𝑡0𝑟⁻¹(𝑡)𝑑𝑡=∞. (5) They proved that (4) is oscillatory if there exists a function 𝜌∈𝐶¹(𝐼,(0,∞)) such that ^∞_𝑡0𝜌(𝑡)𝑞(𝑡)()−𝜌1−𝑝𝑡−𝜎^′(𝑡)²𝑟(𝑡−𝜎)4𝑘𝜌(𝑡)𝑑𝑡=∞. (6) Hasanbulli and Rogovchenko [11] obtained several oscillation criteria for a nonlinear neutral differential equation 𝑟(𝑡)𝑥(𝑡)+𝑝(𝑡)𝑥(𝑡−𝜏)^′′+𝑞(𝑡)𝑓(𝑥(𝑡),𝑥(𝜎(𝑡)))=0 (7) in the case where 0≤𝑝(𝑡)<1. Ye and Xu [18, Theorem 2.1] proved the following result for (1).

Theorem 1. Suppose that 0≤𝑝(𝑡)<1, 𝜎∈𝐶¹(𝐼,ℝ), and 𝜎^′(𝑡)>0. Assume also that conditions (𝐴₁)--(𝐴₄) are satisfied and ^∞_𝑡0𝑟^−1/𝛼(𝑡)𝑑𝑡=∞. (8) If there exists a function 𝜌∈𝐶¹(𝐼,(0,∞)) such that ^∞_𝑡0𝜌(𝑡)𝑞(𝑡)1−𝑝(𝜎(𝑡))^𝛼−1𝑘(𝛼+1)^𝛼+1𝜌^′₊(𝑡)^𝛼+1𝑟(𝜎(𝑡))𝜌(𝑡)𝜎^′(𝑡)^𝛼𝑑𝑡=∞, (9) then (1) is oscillatory.

In a special case 𝑓(𝑢)∶=|𝑢|^𝛼−1𝑢, (1) reduces to a quasilinear neutral differential equation 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′+𝑞(𝑡)|𝑥(𝜎(𝑡))|^𝛼−1𝑥(𝜎(𝑡))=0. (10) Equation (10) was studied by Sun et al. [17] and Zhong et al. [20] who established the following results.

Theorem 2 (see [17, Theorem 3.4]). Suppose that 𝛼≥1, 𝜎(𝑡)≥𝜏(𝑡), and 0≤𝑝(𝑡)≤𝑝₀<∞. Assume also that conditions (𝐴₁), (𝐴₃)--(𝐴₅), and (8) are satisfied. If there exists a function 𝜌∈𝐶¹(𝐼,(0,∞)) such that ^∞_𝑡0𝜌(𝑡)𝑄(𝑡)2^𝛼−1−1+𝑝^𝛼₀/𝜏₀(𝛼+1)^𝛼+1𝜌^′₊(𝑡)^𝛼+1𝑟(𝜏(𝑡))𝜏₀𝜌(𝑡)^𝛼𝑑𝑡=∞, (11) then (10) is oscillatory.

Theorem 3 (see [20, Theorem 3.1]). Assume that 𝑟^′(𝑡)≥0, 𝜏(𝑡)=𝑡−𝜏≤𝑡, 0≤𝑝(𝑡)=𝑝₀<∞, 𝑝₀≠1, 𝜎∈𝐶¹(𝐼,ℝ), and 𝜎^′(𝑡)>0. Suppose also that conditions (𝐴₁), (𝐴₃), and (8) are satisfied. If there exist an 𝜀∈(0,1) and a function 𝜌∈𝐶¹(𝐼,(0,∞)) such that ^∞_𝑡0(1−𝜀)^𝛼1+𝑝₀(1+𝜀)^𝛼𝜌(𝑡)𝑞(𝑡)−1(𝛼+1)^𝛼+1𝜌^′₊(𝑡)^𝛼+1𝑟(𝜎(𝑡))𝜌(𝑡)𝜎^′(𝑡)^𝛼𝑑𝑡=∞, (12) then (10) is oscillatory.

The purpose of this note is to refine Theorems 1–3 in some cases. In what follows, all functional inequalities are assumed to hold for all 𝑡 large enough. Without loss of generality, we can deal only with positive solutions of (1).

2. Main Results

For a more compact presentation of conditions in our results, we use the notation 𝑅(𝑙,𝑡)∶=_𝑙^𝜎(𝑡)𝑟^−1/𝛼(𝑠)𝑑𝑠^𝑡_𝑙𝑟^−1/𝛼(𝑠)𝑑𝑠⁻¹. (13)

Theorem 4. Let 0<𝛼≤1 and 0≤𝑝(𝑡)≤𝑝₀<∞. Assume also that conditions (𝐴₁)--(𝐴₅) and (8) are satisfied. If there exists a function 𝜌∈𝐶¹(𝐼,(0,∞)) such that ^∞_𝑡∗∗𝜌(𝑡)𝑄(𝑡)𝑅^𝛼𝑡_∗−𝜌,𝑡_′+(𝑡)^𝛼+1𝑘(𝛼+1)^𝛼+1𝜌^𝛼(𝑡)𝑟(𝑡)+𝑝^𝛼₀𝑟(𝜏(𝑡))𝜏₀^𝛼+1𝑑𝑡=∞, (14) for all sufficiently large 𝑡_∗ and for some 𝑡_∗∗≥𝑡_∗≥𝑡₀, then (1) is oscillatory.

Proof. Let 𝑥(𝑡) be a nonoscillatory solution of (1); we assume that it is eventually positive. Then there exists a 𝑡₁≥𝑡₀ such that 𝑥(𝑡)>0, 𝑥(𝜏(𝑡))>0, and 𝑥(𝜎(𝑡))>0, for all 𝑡≥𝑡₁. It follows from (1) that 𝑟(𝑡)𝑧^′(𝑡)^𝛼−1𝑧^′(𝑡)^′≤−𝑘𝑞(𝑡)𝑥^𝛼(𝜎(𝑡))≤0,∀𝑡≥𝑡₁. (15) Using condition (8), we conclude that there exists a 𝑡₂≥𝑡₁ such that 𝑧^′(𝑡)>0, for all 𝑡≥𝑡₂. Hence, for all 𝑡≥𝑡₂, inequality (15) reduces to 𝑟(𝑡)𝑧^′(𝑡)^𝛼′≤−𝑘𝑞(𝑡)𝑥^𝛼(𝜎(𝑡))≤0, (16) and there exists a 𝑡₃≥𝑡₂ such that, for all 𝑡≥𝑡₃, 𝑝^𝛼₀𝑟(𝜏(𝑡))𝑧^′(𝜏(𝑡))^𝛼′𝜏^′(𝑡)≤−𝑘𝑝^𝛼₀𝑞(𝜏(𝑡))𝑥^𝛼(𝜎(𝜏(𝑡))). (17) Using the assumption 𝜏^′(𝑡)≥𝜏₀>0, we have, for all 𝑡≥𝑡₃, 𝑝^𝛼₀𝜏₀𝑟(𝜏(𝑡))𝑧^′(𝜏(𝑡))^𝛼′≤−𝑘𝑝^𝛼₀𝑞(𝜏(𝑡))𝑥^𝛼(𝜎(𝜏(𝑡))). (18) Combining inequalities (16) and (18), using the condition 𝜏  ∘  𝜎 = 𝜎  ∘  𝜏 and an auxiliary result due to Baculíková and Džurina [7, Lemma 2], we conclude that 𝑟(𝑡)𝑧^′(𝑡)^𝛼′+𝑝^𝛼₀𝜏₀𝑟(𝜏(𝑡))𝑧^′(𝜏(𝑡))^𝛼′≤−𝑘𝑞(𝑡)𝑥^𝛼(𝜎(𝑡))+𝑝^𝛼₀𝑞(𝜏(𝑡))𝑥^𝛼(𝜏(𝜎(𝑡𝑞(𝑡)(𝜏(𝑡𝑥)))≤−𝑘min,𝑞))^𝛼(𝜎(𝑡))+𝑝^𝛼₀𝑥^𝛼(𝜏(𝜎(𝑡(𝑡)𝑧)))≤−𝑘𝑄^𝛼(𝜎(𝑡)), (19) for all 𝑡≥𝑡₃. Define a new function 𝜔(𝑡) by 𝜔(𝑡)(𝑡)𝑟(𝑡𝑧∶=𝜌)^′(𝑡)^𝛼𝑧^𝛼(𝑡). (20) Then 𝜔(𝑡)>0, for all 𝑡≥𝑡₃. Differentiation of (20) yields 𝜔^′(𝑡)=𝜌^′(𝑡)𝑟(𝑡)𝑧^′(𝑡)^𝛼𝑧^𝛼(𝑡)(𝑡)𝑟(𝑡)𝑧+𝜌^′(𝑡)^𝛼′𝑧^𝛼(𝑡)(𝑡)𝑟(𝑡)𝑧−𝛼𝜌^′(𝑡)^𝛼+1𝑧^𝛼+1(𝑡)(𝑡)𝑧≤𝜌𝑟(𝑡)^′(𝑡)^𝛼′𝑧^𝛼(𝑡)+𝜌^′₊(𝑡)𝜌(𝑡)𝜔(𝑡)−𝛼𝜌(𝑡)𝑟(𝑡)^1/𝛼𝜔^{(𝛼+1)/𝛼}(𝑡). (21) Let 𝜌𝐴∶=^′+(𝑡)𝜌(𝑡)𝛼,𝐵∶=𝜌(𝑡)𝑟(𝑡)^1/𝛼,𝑢∶=𝜔(𝑡). (22) Using the inequality 𝐴𝑢−𝐵𝑢^{(𝛼+1)/𝛼}≤𝛼^𝛼(𝛼+1)^𝛼+1𝐴^𝛼+1𝐵^𝛼,𝐵>0, (23) we deduce from (21) that 𝜔^′(𝑡)(𝑡)𝑧≤𝜌𝑟(𝑡)^′(𝑡)^𝛼′𝑧^𝛼(𝑡)+1(𝛼+1)^𝛼+1𝜌^′₊(𝑡)^𝛼+1𝑟(𝑡)𝜌^𝛼(𝑡). (24) Define another function 𝜈(𝑡) by 𝜈(𝑡)(𝑡)𝑟(𝜏(𝑡𝑧∶=𝜌))^′(𝜏(𝑡))^𝛼𝑧^𝛼(𝜏(𝑡.)) (25) Observe that 𝜈(𝑡)>0, for all 𝑡≥𝑡₃. Differentiation of (25) yields 𝜈^′(𝑡)=𝜌^′(𝑡)𝑟(𝜏(𝑡))𝑧^′(𝜏(𝑡))^𝛼𝑧^𝛼(𝜏(𝑡(𝑡)(𝑡))𝑧))+𝜌𝑟(𝜏^′(𝜏(𝑡))^𝛼′𝑧^𝛼(𝜏(𝑡(𝑡)𝑟(𝜏(𝑡𝑧))−𝛼𝜌))^′(𝜏(𝑡))^𝛼𝑧^′(𝜏(𝑡))𝜏^′(𝑡)𝑧^𝛼+1(𝜏(𝑡(𝑡)𝑟(𝜏(𝑡𝑧))≤𝜌))^′(𝜏(𝑡))^𝛼′𝑧^𝛼(𝜏(𝑡+𝜌))_′+(𝑡)𝜌(𝑡)𝜈(𝑡)−𝛼𝜏^′(𝑡)𝜌(𝑡)𝑟(𝜏(𝑡))^1/𝛼𝜈^{(𝛼+1)/𝛼}(𝑡). (26) Let 𝜌𝐴∶=^′+(𝑡)𝜌(𝑡),𝐵∶=𝛼𝜏^′(𝑡)𝜌(𝑡)𝑟(𝜏(𝑡))^1/𝛼,𝑢∶=𝜈(𝑡). (27) Using the inequalities (23) and (26) along with the fact that 𝑧^′(𝑡)>0, we have 𝜈^′(𝑡)(𝑡)(𝑡))𝑧≤𝜌𝑟(𝜏^′(𝜏(𝑡))^𝛼′𝑧^𝛼(𝑡)+1(𝛼+1)^𝛼+1𝜌^′₊(𝑡)^𝛼+1𝑟(𝜏(𝑡))𝜌(𝑡)𝜏^′(𝑡)^𝛼. (28) Combining (24) and (28) and using the inequality (19), we obtain 𝜔^′(𝑡)+𝑝^𝛼₀𝜏₀𝜈^′(𝑡)(𝑡)𝑧≤𝜌𝑟(𝑡)^′(𝑡)^𝛼′+𝑝^𝛼₀/𝜏₀𝑟(𝜏(𝑡))𝑧^′(𝜏(𝑡))^𝛼′𝑧^𝛼(𝑡)+𝜌^′₊(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜌^𝛼(𝑡)𝑟(𝑡)+𝑝^𝛼₀𝑟(𝜏(𝑡))𝜏₀^𝛼+1(𝑡)𝑄(𝑡)𝑧≤−𝑘𝜌^𝛼(𝜎(𝑡))𝑧^𝛼(𝑡)+𝜌^′₊(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜌^𝛼(𝑡)𝑟(𝑡)+𝑝^𝛼₀𝑟(𝜏(𝑡))𝜏₀^𝛼+1. (29) Since (𝑟(𝑡)(𝑧^′(𝑡))^𝛼)^′≤0, we have 𝑧(𝑡)≥𝑟^1/𝛼(𝑡)𝑧^′(𝑡)^𝑡_𝑡2𝑟^−1/𝛼(𝑠)𝑑𝑠, (30) and thus 𝑧(𝑡)∫^𝑡_𝑡2𝑟^−1/𝛼(𝑠)𝑑𝑠^′≤0. (31) Consequently, 𝑧^𝛼(𝜎(𝑡))𝑧^𝛼(𝑡)≥𝑅^𝛼𝑡₂,𝑡. (32) Substitution of (32) in (29) yields 𝜔^′(𝑡)+𝑝^𝛼₀𝜏₀𝜈^′(𝑡)≤−𝑘𝜌(𝑡)𝑄(𝑡)𝑅^𝛼𝑡₂+𝜌,𝑡_′+(𝑡)^𝛼+1(𝛼+1)^𝛼+1𝜌^𝛼(𝑡)𝑟(𝑡)+𝑝^𝛼₀𝑟(𝜏(𝑡))𝜏₀^𝛼+1. (33) Integrating (33) from 𝑡₃ to 𝑡, we have ^𝑡_𝑡3𝜌(𝑠)𝑄(𝑠)𝑅^𝛼𝑡₂−𝜌,𝑠_′+(𝑠)^𝛼+1𝑘(𝛼+1)^𝛼+1𝜌^𝛼(𝑠)𝑟(𝑠)+𝑝^𝛼₀𝑟(𝜏(𝑠))𝜏₀^𝛼+1≤𝜔𝑡𝑑𝑠₃𝑘+𝑝^𝛼₀𝑘𝜏₀𝜈𝑡₃. (34) Passing in (34) to the limit as 𝑡→∞, we obtain contradiction with condition (14). Therefore, (1) is oscillatory.

Proceeding as in the proof of Theorem 4 and using another result by Baculíková and Džurina [7, Lemma 1], we obtain the following oscillation criterion for (1), for 𝛼≥1.

Theorem 5. Assume that 𝛼≥1 and 0≤𝑝(𝑡)≤𝑝₀<∞. Let conditions (𝐴₁)--(𝐴₅) and (8) be satisfied. If there exists a function 𝜌∈𝐶¹(𝐼,(0,∞)) such that ^∞_𝑡∗∗2^1−𝛼𝜌(𝑡)𝑄(𝑡)𝑅^𝛼𝑡_∗−𝜌,𝑡_′+(𝑡)^𝛼+1𝑘(𝛼+1)^𝛼+1𝜌^𝛼(𝑡)𝑟(𝑡)+𝑝^𝛼₀𝑟(𝜏(𝑡))𝜏₀^𝛼+1𝑑𝑡=∞, (35) for all sufficiently large 𝑡_∗ and for some 𝑡_∗∗≥𝑡_∗≥𝑡₀, then (1) is oscillatory.

3. Examples and Discussion

Example 1. For 𝑡≥1, consider a second-order neutral differential equation 1𝑥(𝑡)+3𝑥(𝑡−2)^′′+𝛾𝑡²𝑥(𝑡)=0, (36) where 𝛾>0 is a constant. We have 𝛼=1,  𝑟(𝑡)=1,  𝑝(𝑡)=𝑝₀=1/3,  𝜏(𝑡)=𝑡−2,  𝑞(𝑡)=𝛾/𝑡²,  𝜎(𝑡)=𝑡,  𝑓(𝑢)=𝑢, and 𝑘=1. Choose 𝜌(𝑡)=𝑡 and denote the left hand side of (14) by 𝜓(𝑡_∗∗). Then 𝜓𝑡_∗∗=1𝛾−3^∞_𝑡∗∗𝑑𝑡𝑡1=∞,providedthat𝛾>3. (37) Hence, (36) is oscillatory by Theorem 4 for any 𝛾>1/3. On the other hand, an application of Theorem 1 yields oscillation of (36) for 𝛾>3/8, whereas Theorem 3 implies that (36) is oscillatory if 𝛾>(4+𝜀)/(3(4−4𝜀)), for some 𝜀∈(0,1). Therefore, we observe that our Theorem 4 improves Theorems 1 and 3.

Example 2. For 𝑡≥1, consider a second-order neutral differential equation 1𝑥(𝑡)+3𝑥𝑡3^′′+𝛾𝑡²𝑥(𝑡)=0, (38) where 𝛾>0 is a constant. We have 𝛼=1,  𝑟(𝑡)=1,  𝑝(𝑡)=𝑝₀=1/3,  𝜏(𝑡)=𝑡/3,  𝑞(𝑡)=𝛾/𝑡²,  𝜎(𝑡)=𝑡,  𝑓(𝑢)=𝑢, and 𝑘=1. Let 𝜌(𝑡)=𝑡 and let 𝜓 be defined as in Example 1. Then 𝜓𝑡_∗∗=𝛾−1^∞_𝑡∗∗𝑑𝑡𝑡=∞, (39) provided that 𝛾>1. Therefore, by Theorem 4, (38) is oscillatory for any 𝛾>1, whereas an application of Theorem 2 yields oscillation of (38) for all 𝛾>3/2. Hence, Theorem 4 improves Theorem 2.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research of the first author is supported by the AMEP of Linyi University, China.