Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 485203, 9 pages
http://dx.doi.org/10.5402/2011/485203
Research Article

Existence for Nonoscillatory Solutions of Higher-Order Nonlinear Differential Equations

1Department of Basic Courses, Qingdao Technological University (Linyi), Feixian 273400, Shandong, China
2Department of Mathematics, Qufu Normal University, Qufu 273165, Shandong, China

Received 8 August 2011; Accepted 22 September 2011

Academic Editor: Z.Β Dosla

Copyright Β© 2011 Yazhou Tian and Fanwei Meng. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The existence of nonoscillatory solutions of the higher-order nonlinear differential equation [π‘Ÿ(𝑑)(π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ))(π‘›βˆ’1)]ξ…ž+βˆ‘π‘šπ‘–=1𝑄𝑖(𝑑)𝑓𝑖(π‘₯(π‘‘βˆ’πœŽπ‘–))=0,𝑑β‰₯𝑑0, where π‘šβ‰₯1,𝑛β‰₯2 are integers, 𝜏>0,πœŽπ‘–β‰₯0,π‘Ÿ,𝑃,π‘„π‘–βˆˆπΆ([𝑑0,∞),𝑅),π‘“π‘–βˆˆπΆ(𝑅,𝑅)(𝑖=1,2,…,π‘š), is studied. Some new sufficient conditions for the existence of a nonoscillatory solution of above equation are obtained for general 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š) which means that we allow oscillatory 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š). 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 ξ€Ίπ‘Ÿ(𝑑)(π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ))(π‘›βˆ’1)ξ€»ξ…ž+π‘šξ“π‘–=1𝑄𝑖(𝑑)𝑓𝑖π‘₯ξ€·π‘‘βˆ’πœŽπ‘–ξ€Έξ€Έ=0,𝑑β‰₯𝑑0.(1.1) With respect to (1.1), throughout, we shall assume the following:(i)π‘šβ‰₯1,𝑛β‰₯2 are integers, 𝜏>0,πœŽπ‘–β‰₯0,(ii)π‘Ÿ,𝑃,π‘„π‘–βˆˆπΆ([𝑑0,∞),𝑅),π‘Ÿ(𝑑)>0,π‘“π‘–βˆˆπΆ(𝑅,𝑅),𝑖=1,2,…,π‘š.

Let 𝜌=max1β‰€π‘–β‰€π‘š{𝜏,πœŽπ‘–}. By a solution of (1.1), we mean a function π‘₯(𝑑)∈𝐢([𝑑1βˆ’πœŒ,∞),𝑅) for some 𝑑1β‰₯𝑑0 which has the property that π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ)βˆˆπΆπ‘›βˆ’1([𝑑1,∞),𝑅) and π‘Ÿ(𝑑)(π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ))(π‘›βˆ’1)∈𝐢1([𝑑1,∞),𝑅) and satisfies (1.1) on [𝑑1,∞).

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 (π‘₯(𝑑)+𝑐π‘₯(π‘‘βˆ’πœ))ξ…žξ…ž+𝑄1ξ€·(𝑑)π‘₯π‘‘βˆ’πœŽ1ξ€Έβˆ’π‘„2ξ€·(𝑑)π‘₯π‘‘βˆ’πœŽ2ξ€Έ=0,𝑑β‰₯𝑑0,(E0) where 𝑐 is a constant.

In 2006, Zhang and Wang [2] investigated the second neutral delay differential equation with positive and negative coefficients: ξ€Ίπ‘Ÿ(𝑑)(π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ))ξ…žξ€»ξ…ž+𝑄1ξ€·π‘₯ξ€·(𝑑)π‘“π‘‘βˆ’πœŽ1ξ€Έξ€Έβˆ’π‘„2ξ€·π‘₯ξ€·(𝑑)π‘”π‘‘βˆ’πœŽ2ξ€Έξ€Έ=0,𝑑β‰₯𝑑0,(𝐸) where 𝜏>0,πœŽπ‘–β‰₯0,𝑄1,𝑄2∈𝐢([𝑑0,∞),𝑅+),𝑓,π‘”βˆˆπΆ(𝑅,𝑅),π‘₯𝑓(π‘₯)>0,π‘₯𝑔(π‘₯)>0,(π‘₯β‰ 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 π‘₯𝑓(π‘₯)>0,π‘₯𝑔(π‘₯)>0,forπ‘₯β‰ 0; (𝐻2)𝑄𝑖(𝑑)β‰₯0,𝑖=1,2,π‘Žπ‘„1(𝑑)βˆ’π‘„2(𝑑) is eventually nonnegative for every π‘Ž>0;(𝐻3)βˆ«βˆžπ‘‘0βˆ«π‘‘π‘‘0(𝑄𝑖(𝑑)/π‘Ÿ(𝑠))𝑑𝑠𝑑𝑑<∞,𝑖=1,2 hold if one of the following two conditions is satisfied:(𝐻4)𝑃(𝑑)>1 eventually, and 0<𝑃2≀𝑃1<𝑃22<+∞,(𝐻5)𝑃(𝑑)<βˆ’1 eventually, and βˆ’βˆž<𝑃2≀𝑃1<βˆ’1,where 𝑃1=limsupπ‘‘β†’βˆžπ‘ƒ(𝑑), 𝑃2=liminfπ‘‘β†’βˆžπ‘ƒ(𝑑), 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. ξ€Ίπ‘Ÿ(𝑑)(π‘₯(𝑑)+𝑃(𝑑)π‘₯(π‘‘βˆ’πœ))ξ…žξ€»ξ…ž+π‘šξ“π‘–=1𝑄𝑖(𝑑)𝑓𝑖π‘₯ξ€·π‘‘βˆ’πœŽπ‘–ξ€Έξ€Έ=0,𝑑β‰₯𝑑0,(E') where π‘“π‘–βˆˆπΆ(𝑅,𝑅)(𝑖=1,2,…,π‘š). 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, βˆ’π‘2≀𝑃(𝑑)≀𝑐1. Further, assume that ξ€œβˆžπ‘‘0ξ€œπ‘‘π‘‘0||𝑄𝑖||(𝑑)π‘Ÿ(𝑠)𝑑𝑠𝑑𝑑<∞,𝑖=1,2,…,π‘š.(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 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š) which means that we allow oscillatory 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š). 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 Ξ© be a bounded closed convex subset of 𝑋, and let 𝑆1, 𝑆2 be maps of Ξ© into 𝑋 such that 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ© for every pair π‘₯,π‘¦βˆˆΞ©. If 𝑆1 is a contraction and 𝑆2 is completely continuous, then the equation 𝑆1π‘₯+𝑆2π‘₯=π‘₯(2.1) has a solution in Ξ©.

Theorem 2.2. Assume that there exist nonnegative constants 𝑐1 and 𝑐2 such that 𝑐1+𝑐2<1, βˆ’1<βˆ’π‘2≀𝑃(𝑑)≀𝑐1<1. Further, assume that ξ€œβˆžπ‘‘0ξ€œπ‘‘π‘‘0π‘ π‘›βˆ’2||𝑄𝑖||(𝑑)π‘Ÿ(𝑠)𝑑𝑠𝑑𝑑<∞,𝑖=1,2,…,π‘š.(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 ξ€œβˆžπ‘‘0π‘ π‘›βˆ’2ξ€œβˆžπ‘ ||𝑄𝑖||(𝑑)π‘Ÿ(𝑠)𝑑𝑠𝑑𝑑<∞,𝑖=1,2,…,π‘š.(2.3) By (2.3), we choose 𝑇>𝑑0 sufficiently large such that 1ξ€œ(π‘›βˆ’2)!βˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||(𝑒)𝑑𝑒𝑑𝑠<1βˆ’π‘1βˆ’π‘24,(2.4) where 𝑀=max((1βˆ’π‘1βˆ’π‘2)/2)≀π‘₯≀1{|𝑓𝑖(π‘₯)|∢1β‰€π‘–β‰€π‘š}.
Let 𝐢([𝑑0,∞),𝑅) be the set of all continuous functions with the norm β€–π‘₯β€–=sup𝑑β‰₯𝑑0|π‘₯(𝑑)|<∞. Then 𝐢([𝑑0,∞),𝑅) is a Banach space. We define a bounded, closed, and convex subset Ξ© of 𝐢([𝑑0,∞),𝑅) as follows: 𝑑Ω=π‘₯=π‘₯(𝑑)βˆˆπΆξ€·ξ€Ί0ξ€Έξ€ΈβˆΆ,∞,𝑅1βˆ’π‘1βˆ’π‘22≀π‘₯(𝑑)≀1,𝑑β‰₯𝑑0ξ‚Ό.(2.5)
Define two maps 𝑆1 and 𝑆2βˆΆΞ©β†’πΆ([𝑑0,∞),𝑅) as follows: 𝑆1π‘₯ξ€Έξƒ―(𝑑)=3+𝑐1βˆ’3𝑐24ξ€·π‘†βˆ’π‘ƒ(𝑑)π‘₯(π‘‘βˆ’πœ),𝑑β‰₯𝑇,1π‘₯𝑑(𝑇),0𝑆≀𝑑≀𝑇,2π‘₯ξ€Έ(⎧βŽͺ⎨βŽͺβŽ©π‘‘)=(βˆ’1)π‘›βˆ’1(ξ€œπ‘›βˆ’2)!βˆžπ‘‘(π‘ βˆ’π‘‘)π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ𝑆𝑑𝑒𝑑𝑠,𝑑β‰₯𝑇,2π‘₯𝑑(𝑇),0≀𝑑≀𝑇.(2.6)
(i) We shall show that for any π‘₯,π‘¦βˆˆΞ©, 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ©.
In fact, π‘₯,π‘¦βˆˆΞ©, and 𝑑β‰₯𝑇, we get 𝑆1π‘₯𝑆(𝑑)+2𝑦(𝑑)≀3+𝑐1βˆ’3𝑐24+1βˆ’π‘ƒ(𝑑)π‘₯(π‘‘βˆ’πœ)ξ€œ(π‘›βˆ’2)!βˆžπ‘‘(π‘ βˆ’π‘‘)π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖(𝑒)π‘“π‘–ξ€·π‘¦ξ€·π‘’βˆ’πœŽπ‘–||ξƒͺ≀𝑑𝑒𝑑𝑠3+𝑐1βˆ’3𝑐24+𝑐2+1ξ€œ(π‘›βˆ’2)!βˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||≀(𝑒)𝑑𝑒𝑑𝑠3+𝑐1βˆ’3𝑐24+𝑐2+1βˆ’π‘1βˆ’π‘24=1.(2.7)
Furthermore, we have 𝑆1π‘₯𝑆(𝑑)+2𝑦(𝑑)β‰₯3+𝑐1βˆ’3𝑐24βˆ’1βˆ’π‘ƒ(𝑑)π‘₯(π‘‘βˆ’πœ)ξ€œ(π‘›βˆ’2)!βˆžπ‘‘π‘ π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖(𝑒)π‘“π‘–ξ€·π‘¦ξ€·π‘’βˆ’πœŽπ‘–||ξƒͺβ‰₯𝑑𝑒𝑑𝑠3+𝑐1βˆ’3𝑐24βˆ’π‘1βˆ’1ξ€œ(π‘›βˆ’2)!βˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||β‰₯(𝑒)𝑑𝑒𝑑𝑠3+𝑐1βˆ’3𝑐24βˆ’π‘1βˆ’1βˆ’π‘1βˆ’π‘24=1βˆ’π‘1βˆ’π‘22.(2.8) Hence, 1βˆ’π‘1βˆ’π‘22≀𝑆1π‘₯𝑆(𝑑)+2𝑦(𝑑)≀1,for𝑑β‰₯𝑑0.(2.9) Thus, we have proved that 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ© for any π‘₯,π‘¦βˆˆΞ©.
(ii) We shall show that 𝑆1 is a contraction mapping on Ξ©.
In fact, for π‘₯,π‘¦βˆˆΞ© and 𝑑β‰₯𝑇, we have ||𝑆1π‘₯𝑆(𝑑)βˆ’1𝑦||≀||𝑃||||π‘₯||(𝑑)(𝑑)(π‘‘βˆ’πœ)βˆ’π‘¦(π‘‘βˆ’πœ)≀𝑐0β€–π‘₯βˆ’π‘¦β€–,(2.10) where 𝑐0=max{𝑐1,𝑐2}. This implies that ‖‖𝑆1π‘₯βˆ’π‘†1𝑦‖‖≀𝑐0β€–π‘₯βˆ’π‘¦β€–.(2.11) Since 0<𝑐0<1, we conclude that 𝑆1 is a contraction mapping on Ξ©.
(iii) We now show that 𝑆2 is completely continuous.
First, we will show that 𝑆2 is continuous. Let π‘₯π‘˜=π‘₯π‘˜(𝑑)∈Ω be such that π‘₯π‘˜(𝑑)β†’π‘₯(𝑑) as π‘˜β†’βˆž. Because Ξ© is closed, π‘₯=π‘₯(𝑑)∈Ω. For 𝑑β‰₯𝑇, we have ||𝑆2π‘₯π‘˜ξ€Έξ€·π‘†(𝑑)βˆ’2π‘₯ξ€Έ||≀1(𝑑)ξ€œ(π‘›βˆ’2)!βˆžπ‘‘π‘ π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||||𝑓(𝑒)𝑖π‘₯π‘˜ξ€·π‘’βˆ’πœŽπ‘–ξ€Έξ€Έβˆ’π‘“π‘–ξ€·π‘₯ξ€·π‘’βˆ’πœŽπ‘–||ξƒͺ≀1ξ€Έξ€Έπ‘‘π‘’π‘‘π‘ ξ€œ(π‘›βˆ’2)!βˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ 1π‘Ÿξƒ©(𝑠)π‘šξ“π‘–=1||𝑄𝑖||||𝑓(𝑒)𝑖π‘₯π‘˜ξ€·π‘’βˆ’πœŽπ‘–ξ€Έξ€Έβˆ’π‘“π‘–ξ€·π‘₯ξ€·π‘’βˆ’πœŽπ‘–||ξƒͺ𝑑𝑒𝑑𝑠.(2.12) Since |𝑓𝑖(π‘₯π‘˜(π‘‘βˆ’πœŽπ‘–))βˆ’π‘“π‘–(π‘₯(π‘‘βˆ’πœŽπ‘–))|β†’0 as π‘˜β†’βˆž for 𝑖=1,2,…,π‘š, by applying the Lebesgue dominated convergence theorem, we conclude that limπ‘˜β†’βˆžβ€–(𝑆2π‘₯π‘˜)(𝑑)βˆ’(𝑆2π‘₯)(𝑑)β€–=0. This means that 𝑆2 is continuous.
Next, we show that 𝑆2Ξ© is relatively compact. It suffices to show that the family of functions {𝑆2π‘₯∢π‘₯∈Ω} is uniformly bounded and equicontinuous on [𝑑0,∞). The uniform boundedness is obvious. For the equicontinuity, according to Levitan's result [4], we only need to show that, for any given πœ€>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 πœ€. By (2.3), for any πœ€>0, take π‘‡βˆ—β‰₯𝑇 large enough so that 1ξ€œ(π‘›βˆ’2)!βˆžπ‘‡βˆ—π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||πœ€(𝑒)𝑑𝑒𝑑𝑠<2.(2.13) Then, for π‘₯∈Ω, 𝑑2β‰₯𝑑1β‰₯π‘‡βˆ—, ||𝑆2π‘₯𝑑2ξ€Έβˆ’ξ€·π‘†2π‘₯𝑑1ξ€Έ||≀1ξ€œ(π‘›βˆ’2)!βˆžπ‘‘2π‘ π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||||𝑓(𝑒)𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–||ξƒͺ+1ξ€Έξ€Έπ‘‘π‘’π‘‘π‘ ξ€œ(π‘›βˆ’2)!βˆžπ‘‘1π‘ π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||||𝑓(𝑒)𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–||ξƒͺ≀1ξ€Έξ€Έπ‘‘π‘’π‘‘π‘ ξ€œ(π‘›βˆ’2)!βˆžπ‘‘2π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||+1(𝑒)π‘‘π‘’π‘‘π‘ ξ€œ(π‘›βˆ’2)!βˆžπ‘‘1π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||<πœ€(𝑒)𝑑𝑒𝑑𝑠2+πœ€2=πœ€.(2.14) For π‘₯∈Ω, 𝑇≀𝑑1<𝑑2β‰€π‘‡βˆ—+1, ||𝑆2π‘₯𝑑2ξ€Έβˆ’ξ€·π‘†2π‘₯𝑑1ξ€Έ||≀1|||||ξ€œ(π‘›βˆ’2)!𝑑2𝑑1ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’2ξ€œβˆžπ‘ 1π‘Ÿξƒ©(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ|||||+1𝑑𝑒𝑑𝑠|||||ξ€œ(π‘›βˆ’2)!βˆžπ‘‘2ξ‚ƒξ€·π‘ βˆ’π‘‘2ξ€Έπ‘›βˆ’2βˆ’ξ€·π‘ βˆ’π‘‘1ξ€Έπ‘›βˆ’2ξ‚„ξ€œβˆžπ‘ 1π‘Ÿξƒ©(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ|||||≀1ξ€Έξ€Έπ‘‘π‘’π‘‘π‘ ξ€œ(π‘›βˆ’2)!𝑑2𝑑1π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||+1(𝑒)𝑑𝑒𝑑𝑠(ξ€·π‘‘π‘›βˆ’3)!2βˆ’π‘‘1ξ€Έξ€œβˆžπ‘‘2(π‘ βˆ’πœ‰)π‘›βˆ’3ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||≀1(𝑒)π‘‘π‘’π‘‘π‘ ξ€œ(π‘›βˆ’2)!𝑑2𝑑1π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||+1(𝑒)𝑑𝑒𝑑𝑠𝑑(π‘›βˆ’3)!2βˆ’π‘‘1ξ€Έξ€œβˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€π‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||(𝑒)𝑑𝑒𝑑𝑠,(2.15) where 𝑑1<πœ‰<𝑑2.
Then there exists 𝛿>0 such that ||𝑆2π‘₯𝑑2ξ€Έβˆ’ξ€·π‘†2π‘₯𝑑1ξ€Έ||<πœ€,if0<𝑑2βˆ’π‘‘1<𝛿.(2.16) For any π‘₯∈Ω, 𝑑0≀𝑑1<𝑑2≀𝑇, it is easy to see that ||𝑆2π‘₯𝑑2ξ€Έβˆ’ξ€·π‘†2π‘₯𝑑1ξ€Έ||=0<πœ€.(2.17) Therefore, {𝑆2π‘₯∢π‘₯∈Ω} is uniformly bounded and equicontinuous on [𝑑0,∞), and hence 𝑆2Ξ© is relatively compact. By Lemma 2.1, there is π‘₯0∈Ω such that 𝑆1π‘₯0+𝑆2π‘₯0=π‘₯0. It is easy to see that π‘₯0(𝑑) is a nonoscillatory solution of (1.1). The proof is complete.

Theorem 2.3. Assume that βˆ’βˆž<𝑐1≀𝑃(𝑑)≀𝑐2<βˆ’1 and (2.2) holds. Then (1.1) has a bounded nonoscillatory solution.

Proof. We choose positive constants 𝑀1,𝑀2,𝛼 such that βˆ’π‘1𝑀1<𝛼<(βˆ’π‘2βˆ’1)𝑀2. 𝑐=min{(𝛼+𝑀1𝑐1)𝑐2/𝑐1,((βˆ’π‘2βˆ’1)𝑀2)βˆ’π›Ό}. Choosing 𝑇>𝑑0 sufficiently large such that 1ξ€œ(π‘›βˆ’2)!βˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€ξ…žπ‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||(𝑒)𝑑𝑒𝑑𝑠<𝑐,(2.18) where π‘€ξ…ž=max𝑀1≀π‘₯≀𝑀2{|𝑓𝑖(π‘₯)|∢1β‰€π‘–β‰€π‘š}.
Let 𝐢([𝑑0,∞),𝑅) be the set as in the proof of Theorem 2.2. We define a bounded, closed, and convex subset Ξ© of 𝐢([𝑑0,∞),𝑅) as follows: 𝑑Ω=π‘₯=π‘₯(𝑑)βˆˆπΆξ€·ξ€Ί0ξ€Έξ€Έ,∞,π‘…βˆΆπ‘€1≀π‘₯(𝑑)≀𝑀2,𝑑β‰₯𝑑0ξ€Ύ.(2.19)
Define two maps 𝑆1 and 𝑆2βˆΆΞ©β†’πΆ([𝑑0,∞),𝑅) as follows: 𝑆1π‘₯ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩(𝑑)=βˆ’π›Όβˆ’π‘ƒ(𝑑+𝜏)π‘₯(𝑑+𝜏)𝑆𝑃(𝑑+𝜏),𝑑β‰₯𝑇,1π‘₯ξ€Έ(𝑇),𝑑0𝑆≀𝑑≀𝑇,2π‘₯ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩(𝑑)=(βˆ’1)π‘›βˆ’11(π‘›βˆ’2)!ξ€œπ‘ƒ(𝑑+𝜏)βˆžπ‘‘+𝜏(π‘ βˆ’π‘‘βˆ’πœ)π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ𝑆𝑑𝑒𝑑𝑠,𝑑β‰₯𝑇,2π‘₯ξ€Έ(𝑇),𝑑0≀𝑑≀𝑇.(2.20)
(i) We shall show that for any π‘₯,π‘¦βˆˆΞ©, 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ©.
In fact, for every π‘₯,π‘¦βˆˆΞ©, and 𝑑β‰₯𝑇, we get 𝑆1π‘₯𝑆(𝑑)+2𝑦(𝑑)β‰₯βˆ’π›Όπ‘1+𝑐𝑐2β‰₯𝑀1,𝑆1π‘₯𝑆(𝑑)+2𝑦(𝑑)β‰€βˆ’π›Όπ‘2βˆ’π‘€2𝑐2βˆ’π‘π‘2≀𝑀2.(2.21)
Thus, we have proved that 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ©. Since βˆ’βˆž<𝑐1≀𝑃(𝑑)≀𝑐2<βˆ’1, we get that 𝑆1 is a contraction mapping. We also can prove that {𝑆2π‘₯∢π‘₯∈Ω} is uniformly bounded and equicontinuous on [𝑑0,∞), and hence 𝑆2Ξ© is relatively compact. So by Lemma 2.1, there is π‘₯0∈Ω such that 𝑆1π‘₯0+𝑆2π‘₯0=π‘₯0. That is, π‘₯0(𝑑)=βˆ’π›Όπ‘ƒβˆ’π‘₯(𝑑+𝜏)0(𝑑+𝜏)𝑃+(𝑑+𝜏)(βˆ’1)π‘›βˆ’11(π‘›βˆ’2)!π‘ƒΓ—ξ€œ(𝑑+𝜏)βˆžπ‘‘+𝜏(π‘ βˆ’π‘‘βˆ’πœ)π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯0ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ𝑑𝑒𝑑𝑠.(2.22) It is easy to see that π‘₯0(𝑑) is a bounded nonoscillatory solution of (1.1).
The proof is complete.

Theorem 2.4. Assume that 1<𝑐1≀𝑃(𝑑)≀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{π›Όβˆ’π‘€4βˆ’π‘2𝑀3,𝑐1𝑀4βˆ’π›Ό}. Choosing 𝑇>𝑑0 sufficiently large such that 1ξ€œ(π‘›βˆ’2)!βˆžπ‘‡π‘ π‘›βˆ’2ξ€œβˆžπ‘ π‘€ξ…žξ…žπ‘Ÿ(𝑠)π‘šξ“π‘–=1||𝑄𝑖||(𝑒)𝑑𝑒𝑑𝑠<𝑐,(2.23) where π‘€ξ…žξ…ž=max𝑀3≀π‘₯≀𝑀4{|𝑓𝑖(π‘₯)|∢1β‰€π‘–β‰€π‘š}.
Let 𝐢([𝑑0,∞),𝑅) be the set as in the proof of Theorem 2.2. We define a bounded, closed, and convex subset Ξ© of 𝐢([𝑑0,∞),𝑅) as follows: 𝑑Ω=π‘₯=π‘₯(𝑑)βˆˆπΆξ€·ξ€Ί0ξ€Έξ€Έ,∞,π‘…βˆΆπ‘€3≀π‘₯(𝑑)≀𝑀4,𝑑β‰₯𝑑0ξ€Ύ.(2.24)
Define two maps 𝑆1 and 𝑆2βˆΆΞ©β†’πΆ([𝑑0,∞),𝑅) as follows: 𝑆1π‘₯ξ€ΈβŽ§βŽͺ⎨βŽͺβŽ©π›Ό(𝑑)=βˆ’π‘ƒ(𝑑+𝜏)π‘₯(𝑑+𝜏)𝑆𝑃(𝑑+𝜏),𝑑β‰₯𝑇,1π‘₯ξ€Έ(𝑇),𝑑0𝑆≀𝑑≀𝑇,2π‘₯ξ€ΈβŽ§βŽͺ⎨βŽͺ⎩(𝑑)=(βˆ’1)π‘›βˆ’11(π‘›βˆ’2)!ξ€œπ‘ƒ(𝑑+𝜏)βˆžπ‘‘+𝜏(π‘ βˆ’π‘‘βˆ’πœ)π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ𝑆𝑑𝑒𝑑𝑠,𝑑β‰₯𝑇,2π‘₯ξ€Έ(𝑇),𝑑0≀𝑑≀𝑇.(2.25)
(i) We shall show that for any π‘₯,π‘¦βˆˆΞ©, 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ©.
In fact, for every π‘₯,π‘¦βˆˆΞ© and 𝑑β‰₯𝑇, we get 𝑆1π‘₯𝑆(𝑑)+2𝑦1(𝑑)β‰₯𝑐2ξ€·π›Όβˆ’π‘€4ξ€Έβˆ’π‘β‰₯𝑀3,𝑆1π‘₯𝑆(𝑑)+2𝑦𝛼(𝑑)≀𝑐1+𝑐𝑐1≀𝑀4.(2.26)
Thus, we have proved that 𝑆1π‘₯+𝑆2π‘¦βˆˆΞ©. Since 1<𝑐1≀𝑃(𝑑)≀𝑐2<+∞, we get 𝑆1 is a contraction mapping. We also can prove that {𝑆2π‘₯∢π‘₯∈Ω} is uniformly bounded and equicontinuous on [𝑑0,∞), and, hence, 𝑆2Ξ© is relatively compact. So by Lemma 2.1, there is π‘₯0∈Ω such that 𝑆1π‘₯0+𝑆2π‘₯0=π‘₯0. That is, π‘₯0𝛼(𝑑)=π‘ƒβˆ’π‘₯(𝑑+𝜏)0(𝑑+𝜏)𝑃+(𝑑+𝜏)(βˆ’1)π‘›βˆ’11(π‘›βˆ’2)!π‘ƒΓ—ξ€œ(𝑑+𝜏)βˆžπ‘‘+𝜏(π‘ βˆ’π‘‘βˆ’πœ)π‘›βˆ’2ξ€œβˆžπ‘ 1ξƒ©π‘Ÿ(𝑠)π‘šξ“π‘–=1𝑄𝑖(𝑒)𝑓𝑖π‘₯0ξ€·π‘’βˆ’πœŽπ‘–ξƒͺ𝑑𝑒𝑑𝑠.(2.27) It is easy to see that π‘₯0(𝑑) 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, π‘Ÿ(𝑑)≑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 𝑄𝑖(𝑑)(𝑖=1,2,…,π‘š) 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).

References

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