Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

Abstract

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^(𝑛−1)]^+∑^𝑚𝑖=1𝑄_𝑖(𝑡)𝑓_𝑖(𝑥(𝑡−𝜎_𝑖))=0,𝑡≥𝑡₀, where 𝑚≥1,𝑛≥2 are integers, 𝜏>0,𝜎_𝑖≥0,𝑟,𝑃,𝑄_𝑖∈𝐶([𝑡₀,∞),𝑅),𝑓_𝑖∈𝐶(𝑅,𝑅)(𝑖=1,2,…,𝑚), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general 𝑄_𝑖(𝑡)(𝑖=1,2,…,𝑚) which means that we allow oscillatory 𝑄_𝑖(𝑡)(𝑖=1,2,…,𝑚). In particular, our results improve essentially and extend some known results in the recent references.

1. Introduction

Consider the higher-order nonlinear neutral differential equation 𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^(𝑛−1)^+^𝑚_𝑖=1𝑄_𝑖(𝑡)𝑓_𝑖𝑥𝑡−𝜎_𝑖=0,𝑡≥𝑡₀.(1.1) With respect to (1.1), throughout, we shall assume the following:(i)𝑚≥1,𝑛≥2 are integers, 𝜏>0,𝜎_𝑖≥0,(ii)𝑟,𝑃,𝑄_𝑖∈𝐶([𝑡₀,∞),𝑅),𝑟(𝑡)>0,𝑓_𝑖∈𝐶(𝑅,𝑅),𝑖=1,2,…,𝑚.

Let 𝜌=max_{1≤𝑖≤𝑚}{𝜏,𝜎_𝑖}. By a solution of (1.1), we mean a function 𝑥(𝑡)∈𝐶([𝑡₁−𝜌,∞),𝑅) for some 𝑡₁≥𝑡₀ which has the property that 𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏)∈𝐶^𝑛−1([𝑡₁,∞),𝑅) and 𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^(𝑛−1)∈𝐶¹([𝑡₁,∞),𝑅) and satisfies (1.1) on [𝑡₁,∞).

A nontrivial solution of (1.1) is called oscillatory if it has arbitrarily large zeros, and, otherwise, it is nonoscillatory.

The existence of nonoscillatory solutions of higher-order nonlinear neutral differential equations received much less attention, which is due mainly to the technical difficulties arising in its analysis.

In 1998, Kulenovic and Hadziomerspahic [1] investigated the existence of nonoscillatory solutions of second-order nonlinear neutral differential equation (𝑥(𝑡)+𝑐𝑥(𝑡−𝜏))^+𝑄₁(𝑡)𝑥𝑡−𝜎₁−𝑄₂(𝑡)𝑥𝑡−𝜎₂=0,𝑡≥𝑡₀,(E0) where 𝑐 is a constant.

In 2006, Zhang and Wang [2] investigated the second neutral delay differential equation with positive and negative coefficients: 𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^^+𝑄₁(𝑡)𝑓𝑥𝑡−𝜎₁−𝑄₂(𝑡)𝑔𝑥𝑡−𝜎₂=0,𝑡≥𝑡₀,(𝐸) where 𝜏>0,𝜎_𝑖≥0,𝑄₁,𝑄₂∈𝐶([𝑡₀,∞),𝑅⁺),𝑓,𝑔∈𝐶(𝑅,𝑅),𝑥𝑓(𝑥)>0,𝑥𝑔(𝑥)>0,(𝑥≠0). By using Banach contraction mapping principle, they proved the following theorem which extends the results in [1].

Theorem A. ([2, Theorem  2.3]). Assume that(𝐻₁)𝑓 and 𝑔 satisfy local Lipschitz condition and 𝑥𝑓(𝑥)>0,𝑥𝑔(𝑥)>0,for𝑥≠0; (𝐻₂)𝑄_𝑖(𝑡)≥0,𝑖=1,2,𝑎𝑄₁(𝑡)−𝑄₂(𝑡) is eventually nonnegative for every 𝑎>0;(𝐻₃)∫^∞𝑡₀∫^𝑡𝑡₀(𝑄_𝑖(𝑡)/𝑟(𝑠))𝑑𝑠𝑑𝑡<∞,𝑖=1,2 hold if one of the following two conditions is satisfied:(𝐻₄)𝑃(𝑡)>1 eventually, and 0<𝑃₂≤𝑃₁<𝑃²²<+∞,(𝐻₅)𝑃(𝑡)<−1 eventually, and −∞<𝑃₂≤𝑃₁<−1,where 𝑃₁=limsup_𝑡→∞𝑃(𝑡), 𝑃₂=liminf_𝑡→∞𝑃(𝑡), then (1.1) has a nonoscillatory solution.

In 2007, Zhou [3] studies the existence of nonoscillatory solution of the following second-order nonlinear differential equation. 𝑟(𝑡)(𝑥(𝑡)+𝑃(𝑡)𝑥(𝑡−𝜏))^^+^𝑚_𝑖=1𝑄_𝑖(𝑡)𝑓_𝑖𝑥𝑡−𝜎_𝑖=0,𝑡≥𝑡₀,(E') where 𝑓_𝑖∈𝐶(𝑅,𝑅)(𝑖=1,2,…,𝑚). By using Krasnoselskii’s fixed point theorem, they proved the following theorem.

Theorem B. ([3, Theorem  1]). Assume that there exist nonnegative constants 𝑐₁ and 𝑐₂ such that 𝑐₁+𝑐₂<1, −𝑐₂≤𝑃(𝑡)≤𝑐₁. Further, assume that ^∞𝑡₀^𝑡𝑡₀||𝑄_𝑖(𝑡)||𝑟(𝑠)𝑑𝑠𝑑𝑡<∞,𝑖=1,2,…,𝑚.(1.2) Then (1.1) has a bounded nonoscillatory solution.

In this paper, by using Krasnoselskii’s fixed point theorem and some new techniques, we obtain some sufficient conditions for the existence of a nonoscillatory solution of (1.1) for general 𝑄_𝑖(𝑡)(𝑖=1,2,…,𝑚) which means that we allow oscillatory 𝑄_𝑖(𝑡)(𝑖=1,2,…,𝑚). Meanwhile, we extend the main results of [2, 3].

2. Main Result

The following fixed point theorem will be used to prove the main results in this section.

Lemma 2.1 (see [3, Krasnoselskii’s fixed point theorem]). Let 𝑋 be a Banach space, let Ω be a bounded closed convex subset of 𝑋, and let 𝑆₁, 𝑆₂ be maps of Ω into 𝑋 such that 𝑆₁𝑥+𝑆₂𝑦∈Ω for every pair 𝑥,𝑦∈Ω. If 𝑆₁ is a contraction and 𝑆₂ is completely continuous, then the equation 𝑆₁𝑥+𝑆₂𝑥=𝑥(2.1) has a solution in Ω.

Theorem 2.2. Assume that there exist nonnegative constants 𝑐₁ and 𝑐₂ such that 𝑐₁+𝑐₂<1, −1<−𝑐₂≤𝑃(𝑡)≤𝑐₁<1. Further, assume that ^∞𝑡₀^𝑡𝑡₀𝑠^𝑛−2||𝑄_𝑖(𝑡)||𝑟(𝑠)𝑑𝑠𝑑𝑡<∞,𝑖=1,2,…,𝑚.(2.2) Then (1.1) has a bounded nonoscillatory solution.

Proof. By interchanging the order of integral, we note that (2.2) is equivalent to ^∞𝑡₀𝑠^𝑛−2^∞𝑠||𝑄_𝑖(𝑡)||𝑟(𝑠)𝑑𝑠𝑑𝑡<∞,𝑖=1,2,…,𝑚.(2.3) By (2.3), we choose 𝑇>𝑡₀ sufficiently large such that 1(𝑛−2)!^∞𝑇𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠<1−𝑐¹−𝑐₂4,(2.4) where 𝑀=max_{((1−𝑐1−𝑐2)/2)≤𝑥≤1}{|𝑓_𝑖(𝑥)|∶1≤𝑖≤𝑚}.
Let 𝐶([𝑡₀,∞),𝑅) be the set of all continuous functions with the norm ‖𝑥‖=sup_𝑡≥𝑡0|𝑥(𝑡)|<∞. Then 𝐶([𝑡₀,∞),𝑅) is a Banach space. We define a bounded, closed, and convex subset Ω of 𝐶([𝑡₀,∞),𝑅) as follows: Ω=𝑥=𝑥(𝑡)∈𝐶𝑡₀,∞,𝑅∶1−𝑐¹−𝑐₂2≤𝑥(𝑡)≤1,𝑡≥𝑡⁰.(2.5)
Define two maps 𝑆₁ and 𝑆₂∶Ω→𝐶([𝑡₀,∞),𝑅) as follows: 𝑆₁𝑥(𝑡)=3+𝑐¹−3𝑐₂𝑆4−𝑃(𝑡)𝑥(𝑡−𝜏),𝑡≥𝑇,₁𝑆𝑥(𝑇),𝑡₀≤𝑡≤𝑇,₂𝑥(𝑡)=⎧⎪⎨⎪⎩((𝑛−−1)2)^𝑛−1!^∞𝑡(𝑠−𝑡)^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1𝑆𝑄_𝑖(𝑢)𝑓_𝑖𝑥𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠,𝑡≥𝑇,₂𝑥(𝑇),𝑡₀≤𝑡≤𝑇.(2.6)
(i) We shall show that for any 𝑥,𝑦∈Ω, 𝑆₁𝑥+𝑆₂𝑦∈Ω.
In fact, 𝑥,𝑦∈Ω, and 𝑡≥𝑇, we get 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)≤3+𝑐¹−3𝑐₂+14−𝑃(𝑡)𝑥(𝑡−𝜏)(𝑛−2)!^∞𝑡(𝑠−𝑡)^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1≤3+𝑐₁||𝑄_𝑖(𝑢)𝑓_𝑖𝑦𝑢−𝜎_𝑖||𝑑𝑢𝑑𝑠−3𝑐₂4+𝑐²+1(𝑛−2)!^∞𝑇𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1≤3+𝑐₁||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠−3𝑐₂4+𝑐²+1−𝑐¹−𝑐₂4=1.(2.7)
Furthermore, we have 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)≥3+𝑐¹−3𝑐₂−14−𝑃(𝑡)𝑥(𝑡−𝜏)(𝑛−2)!^∞𝑡𝑠^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1≥3+𝑐₁||𝑄_𝑖(𝑢)𝑓_𝑖𝑦𝑢−𝜎_𝑖||𝑑𝑢𝑑𝑠−3𝑐₂4−𝑐¹−1(𝑛−2)!^∞𝑇𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1≥3+𝑐₁||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠−3𝑐₂4−𝑐¹−1−𝑐¹−𝑐₂4=1−𝑐¹−𝑐₂2.(2.8) Hence, 1−𝑐₁−𝑐₂2≤𝑆¹𝑥(𝑡)+𝑆₂𝑦(𝑡)≤1,for𝑡≥𝑡₀.(2.9) Thus, we have proved that 𝑆₁𝑥+𝑆₂𝑦∈Ω for any 𝑥,𝑦∈Ω.
(ii) We shall show that 𝑆₁ is a contraction mapping on Ω.
In fact, for 𝑥,𝑦∈Ω and 𝑡≥𝑇, we have ||𝑆₁𝑥(𝑡)−𝑆₁𝑦(𝑡)||≤||𝑃(𝑡)||||𝑥(𝑡−𝜏)−𝑦(𝑡−𝜏)||≤𝑐₀‖𝑥−𝑦‖,(2.10) where 𝑐₀=max{𝑐₁,𝑐₂}. This implies that ‖‖𝑆₁𝑥−𝑆₁𝑦‖‖≤𝑐₀‖𝑥−𝑦‖.(2.11) Since 0<𝑐₀<1, we conclude that 𝑆₁ is a contraction mapping on Ω.
(iii) We now show that 𝑆₂ is completely continuous.
First, we will show that 𝑆₂ is continuous. Let 𝑥_𝑘=𝑥_𝑘(𝑡)∈Ω be such that 𝑥_𝑘(𝑡)→𝑥(𝑡) as 𝑘→∞. Because Ω is closed, 𝑥=𝑥(𝑡)∈Ω. For 𝑡≥𝑇, we have ||𝑆≤₂𝑥_𝑘(1𝑡)−𝑆₂𝑥(𝑡)||(𝑛−2)!^∞𝑡𝑠^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1≤1||𝑄_𝑖(𝑢)||||𝑓_𝑖𝑥_𝑘𝑢−𝜎_𝑖−𝑓_𝑖𝑥𝑢−𝜎_𝑖||𝑑𝑢𝑑𝑠(𝑛−2)!^∞𝑇𝑠^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1||𝑄_𝑖(𝑢)||||𝑓_𝑖𝑥_𝑘𝑢−𝜎_𝑖−𝑓_𝑖𝑥𝑢−𝜎_𝑖||𝑑𝑢𝑑𝑠.(2.12) Since |𝑓_𝑖(𝑥_𝑘(𝑡−𝜎_𝑖))−𝑓_𝑖(𝑥(𝑡−𝜎_𝑖))|→0 as 𝑘→∞ for 𝑖=1,2,…,𝑚, by applying the Lebesgue dominated convergence theorem, we conclude that lim_𝑘→∞‖(𝑆₂𝑥_𝑘)(𝑡)−(𝑆₂𝑥)(𝑡)‖=0. This means that 𝑆₂ is continuous.
Next, we show that 𝑆₂Ω is relatively compact. It suffices to show that the family of functions {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞). The uniform boundedness is obvious. For the equicontinuity, according to Levitan's result [4], we only need to show that, for any given 𝜀>0, [𝑇,∞) can be decomposed into finite subintervals in such a way that on each subinterval all functions of the family have change of amplitude less than 𝜀. By (2.3), for any 𝜀>0, take 𝑇^∗≥𝑇 large enough so that 1(𝑛−2)!^∞𝑇∗𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠<𝜀2.(2.13) Then, for 𝑥∈Ω, 𝑡₂≥𝑡₁≥𝑇^∗, ||𝑆₂𝑥𝑡₂−𝑆₂𝑥𝑡₁||≤1(𝑛−2)!^∞𝑡₂𝑠^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1+1||𝑄_𝑖(𝑢)||||𝑓_𝑖𝑥𝑢−𝜎_𝑖||𝑑𝑢𝑑𝑠(𝑛−2)!^∞𝑡₁𝑠^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1≤1||𝑄_𝑖(𝑢)||||𝑓_𝑖𝑥𝑢−𝜎_𝑖||𝑑𝑢𝑑𝑠(𝑛−2)!^∞𝑡₂𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1+1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠(𝑛−2)!^∞𝑡₁𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1<𝜀||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠2+2𝜀=𝜀.(2.14) For 𝑥∈Ω, 𝑇≤𝑡₁<𝑡₂≤𝑇^∗+1, ||𝑆₂≤𝑥𝑡₂1−𝑆₂𝑥𝑡₁||(𝑛−2)!|||||^𝑡2_𝑡1𝑠−𝑡₁^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1+1𝑄_𝑖(𝑢)𝑓_𝑖𝑥𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠|||||(𝑛−2)!|||||^∞𝑡₂𝑠−𝑡²^𝑛−2−𝑠−𝑡₁^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1≤1𝑄_𝑖(𝑢)𝑓_𝑖𝑥𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠|||||(𝑛−2)!^𝑡2_𝑡1𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1+1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠(𝑛−3)!𝑡²−𝑡₁^∞𝑡₂(𝑠−𝜉)^𝑛−3^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1≤1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠(𝑛−2)!^𝑡2_𝑡1𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1+1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠(𝑛−3)!𝑡²−𝑡₁^∞𝑇𝑠^𝑛−2^∞𝑠𝑀𝑟(𝑠)^𝑚_𝑖=1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠,(2.15) where 𝑡₁<𝜉<𝑡₂.
Then there exists 𝛿>0 such that ||𝑆₂𝑥𝑡₂−𝑆₂𝑥𝑡₁||<𝜀,if0<𝑡₂−𝑡₁<𝛿.(2.16) For any 𝑥∈Ω, 𝑡₀≤𝑡₁<𝑡₂≤𝑇, it is easy to see that ||𝑆₂𝑥𝑡₂−𝑆₂𝑥𝑡₁||=0<𝜀.(2.17) Therefore, {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞), and hence 𝑆₂Ω is relatively compact. By Lemma 2.1, there is 𝑥₀∈Ω such that 𝑆₁𝑥₀+𝑆₂𝑥₀=𝑥₀. It is easy to see that 𝑥₀(𝑡) is a nonoscillatory solution of (1.1). The proof is complete.

Theorem 2.3. Assume that −∞<𝑐₁≤𝑃(𝑡)≤𝑐₂<−1 and (2.2) holds. Then (1.1) has a bounded nonoscillatory solution.

Proof. We choose positive constants 𝑀₁,𝑀₂,𝛼 such that −𝑐₁𝑀₁<𝛼<(−𝑐₂−1)𝑀₂. 𝑐=min{(𝛼+𝑀₁𝑐₁)𝑐₂/𝑐₁,((−𝑐₂−1)𝑀₂)−𝛼}. Choosing 𝑇>𝑡₀ sufficiently large such that 1(𝑛−2)!^∞𝑇𝑠^𝑛−2^∞𝑠𝑀^𝑟(𝑠)^𝑚_𝑖=1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠<𝑐,(2.18) where 𝑀^=max_{𝑀1≤𝑥≤𝑀2}{|𝑓_𝑖(𝑥)|∶1≤𝑖≤𝑚}.
Let 𝐶([𝑡₀,∞),𝑅) be the set as in the proof of Theorem 2.2. We define a bounded, closed, and convex subset Ω of 𝐶([𝑡₀,∞),𝑅) as follows: Ω=𝑥=𝑥(𝑡)∈𝐶𝑡₀,∞,𝑅∶𝑀₁≤𝑥(𝑡)≤𝑀₂,𝑡≥𝑡₀.(2.19)
Define two maps 𝑆₁ and 𝑆₂∶Ω→𝐶([𝑡₀,∞),𝑅) as follows: 𝑆₁𝑥(𝑡)=⎧⎪⎨⎪⎩−𝛼𝑃𝑆(𝑡+𝜏)−𝑃𝑥((𝑡𝑡++𝜏𝜏)),𝑡≥𝑇,₁𝑆𝑥(𝑇),𝑡₀≤𝑡≤𝑇,₂𝑥(𝑡)=⎧⎪⎨⎪⎩((𝑛−−1)2)^𝑛−1!𝑃(𝑡1+𝜏)^∞𝑡+𝜏(𝑠−𝑡−𝜏)^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1𝑆𝑄_𝑖(𝑢)𝑓_𝑖𝑥𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠,𝑡≥𝑇,₂𝑥(𝑇),𝑡₀≤𝑡≤(2𝑇.2.0)
(i) We shall show that for any 𝑥,𝑦∈Ω, 𝑆₁𝑥+𝑆₂𝑦∈Ω.
In fact, for every 𝑥,𝑦∈Ω, and 𝑡≥𝑇, we get 𝑆₁𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)≥−𝑐₁𝛼+𝑐𝑐₂≥𝑀₁,𝑥(𝑡)+𝑆₂𝑦(𝑡)≤−𝛼𝑐₂−𝑀²𝑐₂−𝑐𝑐₂≤𝑀₂.(2.21)
Thus, we have proved that 𝑆₁𝑥+𝑆₂𝑦∈Ω. Since −∞<𝑐₁≤𝑃(𝑡)≤𝑐₂<−1, we get that 𝑆₁ is a contraction mapping. We also can prove that {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞), and hence 𝑆₂Ω is relatively compact. So by Lemma 2.1, there is 𝑥₀∈Ω such that 𝑆₁𝑥₀+𝑆₂𝑥₀=𝑥₀. That is, 𝑥₀(𝑡)=−𝛼𝑃(𝑡+𝜏)−𝑥⁰(𝑡+𝜏)×𝑃(𝑡+𝜏)+((𝑛−−1)2)^𝑛−1!𝑃(𝑡1+𝜏)_∞𝑡+𝜏(𝑠−𝑡−𝜏)^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1𝑄_𝑖(𝑢)𝑓_𝑖𝑥₀𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠.(2.22) It is easy to see that 𝑥₀(𝑡) is a bounded nonoscillatory solution of (1.1).
The proof is complete.

Theorem 2.4. Assume that 1<𝑐₁≤𝑃(𝑡)≤𝑐₂<+∞ and (2.2) holds. Then (1.1) has a bounded nonoscillatory solution.

Proof. We choose positive constants 𝑀₃,𝑀₄,𝛼 such that 𝑀₄+𝑐₂𝑀₃<𝛼<𝑐₁𝑀₄. 𝑐=min{𝛼−𝑀₄−𝑐₂𝑀₃,𝑐₁𝑀₄−𝛼}. Choosing 𝑇>𝑡₀ sufficiently large such that 1(𝑛−2)!^∞𝑇𝑠^𝑛−2^∞𝑠𝑀^𝑟(𝑠)^𝑚_𝑖=1||𝑄_𝑖(𝑢)||𝑑𝑢𝑑𝑠<𝑐,(2.23) where 𝑀^=max_{𝑀3≤𝑥≤𝑀4}{|𝑓_𝑖(𝑥)|∶1≤𝑖≤𝑚}.
Let 𝐶([𝑡₀,∞),𝑅) be the set as in the proof of Theorem 2.2. We define a bounded, closed, and convex subset Ω of 𝐶([𝑡₀,∞),𝑅) as follows: Ω=𝑥=𝑥(𝑡)∈𝐶𝑡₀,∞,𝑅∶𝑀₃≤𝑥(𝑡)≤𝑀₄,𝑡≥𝑡₀.(2.24)
Define two maps 𝑆₁ and 𝑆₂∶Ω→𝐶([𝑡₀,∞),𝑅) as follows: 𝑆₁𝑥(𝑡)=⎧⎪⎨⎪⎩𝛼𝑃𝑆(𝑡+𝜏)−𝑃𝑥((𝑡𝑡++𝜏𝜏)),𝑡≥𝑇,₁𝑆𝑥(𝑇),𝑡₀≤𝑡≤𝑇,₂𝑥(𝑡)=⎧⎪⎨⎪⎩((𝑛−−1)2)^𝑛−1!𝑃(𝑡1+𝜏)^∞𝑡+𝜏(𝑠−𝑡−𝜏)^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1𝑆𝑄_𝑖(𝑢)𝑓_𝑖𝑥𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠,𝑡≥𝑇,₂𝑥(𝑇),𝑡₀≤𝑡≤(2𝑇.2.5)
(i) We shall show that for any 𝑥,𝑦∈Ω, 𝑆₁𝑥+𝑆₂𝑦∈Ω.
In fact, for every 𝑥,𝑦∈Ω and 𝑡≥𝑇, we get 𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)≥𝑐1₂𝛼−𝑀₄−𝑐≥𝑀₃,𝑆₁𝑥(𝑡)+𝑆₂𝑦(𝑡)≤𝑐𝛼₁+𝑐𝑐₁≤𝑀₄.(2.26)
Thus, we have proved that 𝑆₁𝑥+𝑆₂𝑦∈Ω. Since 1<𝑐₁≤𝑃(𝑡)≤𝑐₂<+∞, we get 𝑆₁ is a contraction mapping. We also can prove that {𝑆₂𝑥∶𝑥∈Ω} is uniformly bounded and equicontinuous on [𝑡₀,∞), and, hence, 𝑆₂Ω is relatively compact. So by Lemma 2.1, there is 𝑥₀∈Ω such that 𝑆₁𝑥₀+𝑆₂𝑥₀=𝑥₀. That is, 𝑥₀(𝑡)=𝛼𝑃(𝑡+𝜏)−𝑥⁰(𝑡+𝜏)×𝑃(𝑡+𝜏)+((𝑛−−1)2)^𝑛−1!𝑃(𝑡1+𝜏)_∞𝑡+𝜏(𝑠−𝑡−𝜏)^𝑛−2^∞𝑠1𝑟(𝑠)^𝑚_𝑖=1𝑄_𝑖(𝑢)𝑓_𝑖𝑥₀𝑢−𝜎_𝑖𝑑𝑢𝑑𝑠.(2.27) It is easy to see that 𝑥₀(𝑡) is a bounded nonoscillatory solution of (1.1).
The proof is complete.

Remark 2.5. If we let 𝑛=2 in Theorem 2.2, we get the Theorem 1 in [3]. In the case where 𝑛=2, 𝑟(𝑡)≡1, Theorem 2.2 improves essentially Theorem 2.2 in [5].

Remark 2.6. The conditions of Theorem 2.4 relaxing the hypotheses (𝐻₄) of Theorem 3 in [2].

Remark 2.7. Theorems 2.3 and 2.4 improve essentially Theorem 3 in [2], we allow that 𝑄_𝑖(𝑡)(𝑖=1,2,…,𝑚) are oscillatory.

Acknowledgments

This research was supported by Natural Science Foundations of Shandong Province of China (ZR2009AM011 and ZR2009AQ010) and Doctor of Ministry of Education (20103705110003).