# Bounded Oscillation of a Forced Nonlinear Neutral Differential Equation

Authors: Zeqing Liu; Yuguang Xu; Shin Min Kang; Young Chel Kwun
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with thenth-order forced nonlinear neutral differential equation [x(t)-p(t)x(τ(t))](n)+∑i=1mqi(t)fi(x(σi1(t)),x(σi2(t)),…,x(σiki(t)))=g(t),t≥t0. Some necessary and sufficient conditions for the oscillation of bounded solutions and several sufficient conditions for the existence of uncountably many bounded positive and negative solutions of the above equation are established. The results obtained in this paper improve and extend essentially some known results in the literature. Five interesting examples that point out the importance of our results are also included.


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## Body

## 1.  Introduction

Consider the followingnth-order forced nonlinear neutral differential equation:(1.1)[x(t)-p(t)x(τ(t))](n)+∑i=1mqi(t)fi(x(σi1(t)),x(σi2(t)),…,x(σiki(t)))=g(t),t≥t0,
where t0∈ℝ and n,m,ki∈ℕ are constants for 1≤i≤m. In what follows, we assume that(A1)  p,g,τ,σij∈C([t0,+∞),ℝ) and qi∈C([t0,+∞),ℝ+) satisfy that
(1.2)lim⁡t→+∞τ(t)=lim⁡t→+∞σij(t)=+∞,1≤j≤ki,1≤i≤m,
and there exists 1≤i0≤m such that qi0 is positive eventually:(A2)  τ is strictly increasing and τ(t)<t in [t0,+∞);(A3)  fi∈C(ℝki,ℝ) satisfies that
(1.3)fi(u1,u2,…,uki)>0,∀(u1,u2,…,uki)∈(R+∖{0})ki,fi(u1,u2,…,uki)<0,∀(u1,u2,…,uki)∈(R-∖{0})ki
for 1≤i≤m.During the last decades, the oscillation criteria and the existence results of nonoscillatory solutions for various linear and nonlinear differential equations have been studied extensively, for example, see [1–28] and the references cited therein. In particular, Zhang and Yan [25] obtained some sufficient conditions for the oscillation of the first-order linear neutral delay differential equation with positive and negative coefficients:(1.4)[x(t)-p(t)x(t-τ)]′+q(t)x(t-σ)-r(t)x(t-δ)=0,t≥t0,
where p,q,r∈C([t0,+∞),ℝ+),τ>0, and σ≥δ≥0. Das and Misra [7] studied the nonhomogeneous neutral delay differential equation:(1.5)[x(t)-cx(t-τ)]′+q(t)f(x(t-σ))=g(t),t≥t0,
where q,g∈C([T,+∞),ℝ+∖{0}),σ>0,τ>0,c∈[0,1),f:ℝ→ℝ, tf(t)>0 for t≠0, f is nondecreasing, Lipschitzian, and satisfies ∫0k(1/f(t))dt<+∞ for every k>0, and they obtained a necessary and sufficient condition for the solutions of (1.5) to be oscillatory or tend to zero asymptotically. Parhi and Rath [18] extended Das and Misra’s result to the following forced first-order neutral differential equation with variable coefficients:(1.6)[x(t)-p(t)x(t-τ)]′+q(t)f(x(t-σ))=g(t),t≥0,
where p∈C(ℝ+,ℝ), and they got necessary and sufficient conditions which ensures every solution of (1.6) is oscillatory or tends to zero or to ±∞ as t→+∞. By using Banach’s fixed point theorem, Zhang et al. [24] proved the existence of a nonoscillatory solution for the first-order linear neutral delay differential equation:(1.7)[x(t)+p(t)x(t-τ)]′+∑i=1nfi(t)x(t-σi)=0,t≥t0,
where p∈C([t0,+∞),ℝ),τ>0,σi∈ℝ+, and fi∈C([t0,+∞),ℝ) for 1≤i≤m. Çakmak and Tiryaki [6] showed several sufficient conditions for the oscillation of the forced second-order nonlinear differential equations with delayed argument in the form:(1.8)x′′(t)+p(t)f(x(α(t)))=g(t),t≥t0≥0,
where p,α,g∈C([t0,+∞),ℝ), α(t)≤t,limt→+∞α(t)=+∞, and f∈C(ℝ,ℝ). Travis [20] investigated the oscillatory behavior of the second-order differential equation with functional argument:(1.9)x′′(t)+p(t)f(x(t),x(α(t)))=0,t≥t0,
where p,α∈C([t0,+∞),ℝ) and f∈C(ℝ2,ℝ) satisfies that f(s,t) has the same sign of s and t when they have the same sign. Lin [12] got some sufficient conditions for oscillation and nonoscillation of the second order nonlinear neutral differential equation:(1.10)[x(t)-p(t)x(t-τ)]′′+q(t)f(x(t-σ))=0,t≥0,
where p,q∈C(ℝ+,ℝ),p¯∈[0,1) with 0≤p(t)≤p¯ eventually, f∈C(ℝ,ℝ), f is nondecreasing and tf(t)>0 for t≠0. Kulenović and Hadžiomerspahić [9] deduced the existence of a nonoscillatory solution for the neutral delay differential equation of second order with positive and negative coefficients:(1.11)[x(t)+cx(t-τ)]′′+q1(t)x(t-σ1)-q2(t)x(t-σ2)=0,t≥t0,
where c≠±1, τ>0,σi∈ℝ+,qi∈C([t0,+∞),ℝ+), and ∫t0+∞qi(t)dt<+∞ for i∈{1,2}. Utilizing the fixed point theorems due to Banach, Schauder and Krasnoselskii, and Zhou and Zhang [27], and Zhou et al. [28] established some sufficient conditions for the existence of a nonoscillatory solution of the following higher-order neutral functional differential equations:(1.12)[x(t)+cx(t-τ)](n)+(-1)n+1[P(t)x(t-σ)-Q(t)x(t-δ)]=0,t≥t0,[x(t)+p(t)x(t-τ)](n)+∑i=1mqi(t)fi(x(t-σi))=g(t),t≥t0,
where c∈ℝ∖{±1},τ,σ,δ,σi∈ℝ+,P,Q∈C([t0,+∞),ℝ+), and p,g,fi∈C([t0,+∞),ℝ) for 1≤i≤m. Li et al. [11] investigated the existence of an unbounded positive solution, bounded oscillation, and nonoscillation criteria for the following even-order neutral delay differential equation with unstable type:(1.13)[x(t)-p(t)x(t-τ)](n)-q(t)|x(t-σ)|α-1x(t-σ)=0,t≥t0,
where τ>0,σ>0,α≥1, and p,q∈C([t0,+∞),ℝ+). Zhang and Yan [22] obtained some sufficient conditions for oscillation of all solutions of the even-order neutral differential equation with variable coefficients and delays:(1.14)[x(t)+p(t)x(τ(t))](n)+q(t)x(σ(t))=0,t≥t0,
where n is even, p,q,τ,σ∈C([t0,+∞),ℝ+),p(t)<1,τ(t)≤t and σ(t)≤t for t∈[t0,+∞), and limt→+∞τ(t)=limt→+∞σ(t)=+∞. Yilmaz and Zafer [21] discussed sufficient conditions for the existence of positive solutions and the oscillation of bounded solutions of the nth-order neutral type differential equations:(1.15)[x(t)+cx(τ(t))](n)+q(t)f(x(σ(t)))=0,t≥t0,[x(t)+p(t)x(τ(t))](n)+q(t)f(x(σ(t)))=g(t),t≥t0,
where c∈ℝ∖{±1}, τ,σ∈C([t0,+∞),ℝ+),p,q,g∈C([t0,+∞),ℝ), and f∈C(ℝ,ℝ). Bolat and Akin [4, 5] got sufficient criteria for oscillatory behaviour of solutions for the higher-order neutral type nonlinear forced differential equations with oscillating coefficients:(1.16)[x(t)+p(t)x(τ(t))](n)+∑i=1mqi(t)fi(x(σi(t)))=0,t≥t0,[x(t)+p(t)x(τ(t))](n)+∑i=1mqi(t)fi(x(σi(t)))=g(t),t≥t0,
where n∈ℕ∖{1},m∈ℕ,p,fi,g,τ,σi∈C([t0,+∞),ℝ), fi is nondecreasing and ufi(u)>0 for u≠0,σi∈C1([t0,+∞),ℝ),σi′(t)>0,σi(t)≤t for t∈[t0,+∞), limt→+∞τ(t)=limt→+∞σi(t)=+∞ for 1≤i≤m, and p and g are oscillating functions. Zhou and Yu [26] attempted to extend the result of Bolat and Akin [4] and established a necessary and sufficient condition for the oscillation of bounded solutions of the higher-order nonlinear neutral forced differential equation of the form:(1.17)[x(t)-p(t)x(τ(t))](n)+∑i=1mqi(t)fi(x(σi(t)))=g(t),t≥t0,
where n∈ℕ∖{1},m∈ℕ, and(C1)  p,qi,τ,g∈C([t0,+∞),ℝ) for i=1,2,…,m and limt→+∞τ(t)=+∞;(C2)  p and g are oscillating functions;(C3)  σi∈C([t0,+∞),ℝ),σi′(t)>0,σi(t)≤t and limt→+∞σi(t)=+∞ for i=1,2,…,m;(C4)  fi∈C(ℝ,ℝ) is nondecreasing function, ufi(u)>0 for u≠0 and i=1,2,…,m.That is, they claimed the following result.Theorem 1.1 (see [26, Theorem 2.1]).
Assume that(C5) there is an oscillating function r∈C([t0,+∞),ℝ) such that r(n)(t)=g(t) and limt→+∞r(t)=0;(C6)  p is an oscillating function and |p(t)|≤p0<1/2;(C7)  qi(t)≥0, i=1,2,…,m.Then, every bounded solution of (1.17) either oscillates or tends to zero if and only if
(1.18)∫t0+∞sn-1qi(s)ds=+∞,i=1,2,…,m.We, unfortunately, point out that the necessary part in Theorem1.1 is false, see Remark 4.2 and Example 4.7 below. It is clear that (1.1) includes (1.4)–(1.17) as special cases. To the best of our knowledge, there is no literature referred to the oscillation and existence of uncountably many bounded nonoscillatory solutions of (1.1). The aim of this paper is to establish the bounded oscillation and the existence of uncountably many bounded positive and negative solutions for (1.1) without the monotonicity of the nonlinear term fi. Our results extend and improve substantially some known results in [4, 5, 9, 10, 20, 24, 26–28] and correct Theorem  2.1 in [26].The paper is organized as follows. In Section2, a few notation and lemmas are introduced and proved, respectively. In Section 3, by employing Krasnoselskii’s fixed point theorem and some techniques, the existence of uncountably many bounded positive and negative solutions for (1.1) are given, and some necessary and sufficient conditions for all bounded solutions of (1.1) to be oscillatory or tend to zero as t→+∞ are provided. In Section 4, a number of examples which clarify advantages of our results are constructed.


## 2.  Preliminaries

It is assumed throughout this paper thatℝ=(-∞,+∞),ℝ+=[0,+∞),ℝ-=(-∞,0] and(2.1)β=min⁡{t0,inf⁡{τ(t),σij(t):t∈[t0,+∞),1≤j≤ik,1≤i≤m}}.
By a solution of (1.1), we mean a function x∈C([β,+∞),ℝ) for some T≥t0+β, such that x(t)-p(t)x(τ(t)) is n times continuously differentiable in [T,+∞) and such that (1.1) is satisfied for t≥T. As is customary, a solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros. Otherwise, it is nonoscillatory, that is, if it is eventually positive or eventually negative. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.LetBC([β,+∞),ℝ) stand for the Banach space of all bounded continuous functions in [β,+∞) with the norm ∥x∥=supt≥β|x(t)| for each x∈BC([β,+∞),ℝ) and(2.2)A(N,M)={x∈BC([β,+∞),R):N≤x(t)≤M,t≥β}forM,N∈RwithM>N.
It is easy to see that A(N,M) is a bounded closed and convex subset of the Banach space BC([β,+∞),ℝ).Lemma 2.1.
Letn∈ℕ and x∈Cn([t0,+∞),ℝ) be bounded. If  x(n)(t)≤0 eventually, then(a)
limt→+∞x(t) exists and limt→+∞x(i)(t)=0 for 1≤i≤n-1; furthermore, there exists θ=0 for n odd and θ=1 for n even such that(b)
(-1)θ+ix(i)(t)≥0 eventually for 1≤i≤n;(c)
(-1)θ+ix(i) is nonincreasing eventually for 0≤i≤n-1.Proof.
Now, we consider two possible cases below.
Case 1.
Assume thatn=1. Let θ=0. Note that x′(t)≤0 eventually. It follows that there exists a constant t1>t0 satisfying x′(t)≤0, for all t≥t1, which yields that x is nonincreasing in [t1,+∞). Since x is bounded in [t0,+∞), it follows that limt→+∞x(t) exists.
Case 2.
Assume thatn≥2. Notice that θ+n is odd. It follows that (-1)θ+nx(n)(t)≥0 eventually, which implies that there exists a constant t1>t0 satisfying
(2.3)(-1)θ+nx(n)(t)≥0,∀t≥t1,
which means that
(2.4)(-1)θ+n-1x(n-1)(t)is nonincreasing in[t1,+∞).
Suppose that there exists a constantt2≥t1 satisfying (-1)θ+n-1x(n-1)(t2)<0, which together with (2.4) gives that
(2.5)(-1)θ+n-1x(n-1)(t)≤(-1)θ+n-1x(n-1)(t2)<0,∀t≥t2,
which guarantees that (-1)θ+n-2x(n-2)(t) is increasing in [t2,+∞) and
(2.6)(-1)θ+n-1x(n-2)(t)-(-1)θ+n-1x(n-2)(t2)=∫t2t(-1)θ+n-1x(n-1)(s)ds≤(-1)θ+n-1x(n-1)(t2)(t-t2)⟶-∞ast⟶+∞,
that is,
(2.7)lim⁡t→+∞x(n-2)(t)=-∞,
which means that
(2.8)lim⁡t→+∞x(n-3)(t)=lim⁡t→+∞x(n-4)(t)=⋅⋅⋅=lim⁡t→+∞x′(t)=lim⁡t→+∞x(t)=-∞,
which contradicts the boundedness of x. Consequently, we have
(2.9)(-1)θ+n-1x(n-1)(t)≥0,∀t≥t1.
Combining (2.4) and (2.9), we conclude easily that there exists a constant L≥0 with
(2.10)lim⁡t→+∞(-1)θ+n-1x(n-1)(t)=L.
Next, we claim thatL=0. Otherwise, there exists a constant b>t1 satisfying
(2.11)(-1)θ+n-1x(n-1)(t)≥L2>0,∀t≥b,
which yields that
(2.12)(-1)θ+n-1x(n-2)(t)-(-1)θ+n-1x(n-2)(b)=∫bt(-1)θ+n-1x(n-1)(s)ds≥L(t-b)2⟶+∞ast⟶+∞,
which gives that
(2.13)lim⁡t→+∞x(n-2)(t)=+∞,
which means that
(2.14)lim⁡t→+∞x(n-3)(t)=lim⁡t→+∞x(n-4)(t)=⋅⋅⋅=lim⁡t→+∞x′(t)=lim⁡t→+∞x(t)=+∞,
which contradicts the boundedness of x in [t0,+∞). Hence, L=0, that is,
(2.15)lim⁡t→+∞x(n-1)(t)=0.
Repeating the proof of (2.3)–(2.15), we deduce similarly that
(2.16)(-1)θ+jx(j)is nonincreasing and nonnegative in[t1,+∞),lim⁡t→+∞x(j)(t)=0,1≤j≤n-1,
which together with the boundedness of x implies that (-1)θx is nonincreasing in [t1,+∞) and limt→+∞x(t) exists.
Thus, (2.3) and (2.16) yield (a)–(c). This completes the proof.Lemma 2.2.
Letx,p,τ,r,y∈C([t0,+∞),ℝ) satisfy (A2) and
(2.17)y(t)=x(t)-p(t)x(τ(t))-r(t),∀t≥t0;(2.18)xis bounded andlim⁡t→+∞τ(t)=+∞;(2.19)lim⁡t→+∞y(t)=lim⁡t→+∞r(t)=0,|p(t)|≥p0>1eventually,
where p0 is a fixed constant. Then, limt→+∞x(t)=0.Proof.
Sinceτ is a strictly increasing continuous function, τ(t)<t in [t0,+∞) and limt→+∞τ(t)=+∞, it follows that the inverse function τ-1 of τ is also strictly increasing continuous, τ-1(t)>t in [τ(t0),+∞) and limj→∞τ-j(t)=+∞, where τ-j=τ-(j-1)(τ-1) for all j∈ℕ. Equation (2.18) implies that there exists a constant B>0 with
(2.20)|x(t)|≤B,∀t≥t0.
Using (2.18) and (2.19), we deduce that, for any ε>0, there exist sufficiently large numbers T>1+|t0| and K∈ℕ satisfying
(2.21)Bp0K<ε4,max⁡{|y(t)|,|r(t)|}<ε(p0-1)4,|p(t)|≥p0,∀t≥T.
In view of (2.17), (2.20), and (2.21), we infer that for all t≥T(2.22)|x(t)|=|x(τ-1(t))-y(τ-1(t))-r(τ-1(t))||p(τ-1(t))|≤|x(τ-1(t))|+|y(τ-1(t))|+|r(τ-1(t))||p(τ-1(t))|<1p0|x(τ-1(t))|+ε(p0-1)2p0≤1p0[1p0|x(τ-2(t))|+ε(p0-1)2p0]+ε(p0-1)2p0=1p02|x(τ-2(t))|+ε(p0-1)2p0(1+1p0)≤⋅⋅⋅≤1p0K|x(τ-K(t))|+ε(p0-1)2p0(1+1p0+⋅⋅⋅+1p0K-1)≤Bp0K+ε(p0-1)2p0⋅11-1/p0<ε,
which gives that limt→+∞x(t)=0. This completes the proof.Lemma 2.3.
Letx,p,τ,r, and y be in C([t0,+∞),ℝ) satisfying (A2), (2.17), (2.18), and
(2.23)lim⁡t→+∞|y(t)|=d>0,lim⁡t→+∞r(t)=0;(2.24)p1≥|p(t)|≥p0>1eventually,p02>p0+p1,
where d,p0, and p1 are constants. Then, there exists L>0 such that |x(t)|≥L eventually.Proof.
Obviously, (2.20) holds. It follows from (2.18), (2.23), and (2.24) that for ε=d[p0(p0-1)-p1]/(p0(p0-1)+p1)>0, there exist K∈ℕ and T>1+|t0| satisfying
(2.25)Bp0K<ε4p1,d-ε4<|y(t)|<d+ε4,|r(t)|<ε4p0,p1≥|p(t)|≥p0,∀t≥T.
Put L=d[p0(p0-1)-p1]/2p1p0(p0-1). In light of (2.17), we conclude that for each t≥T(2.26)x(t)=x(τ-1(t))p(τ-1(t))-y(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))=1p(τ-1(t))[x(τ-2(t))p(τ-2(t))-y(τ-2(t))p(τ-2(t))-r(τ-2(t))p(τ-2(t))]-y(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))=x(τ-2(t))Πi=12p(τ-i(t))-∑j=12y(τ-j(t))Πi=1jp(τ-i(t))-∑j=12r(τ-j(t))Πi=1jp(τ-i(t))=⋅⋅⋅=x(τ-K(t))Πi=1Kp(τ-i(t))-∑j=1Ky(τ-j(t))Πi=1jp(τ-i(t))-∑j=1Kr(τ-j(t))Πi=1jp(τ-i(t)),
which together with (2.20) and (2.25) yields that for any t≥T(2.27)|x(t)|≥|y(τ-1(t))||p(τ-1(t))|-|x(τ-K(t))|Πi=1K|p(τ-i(t))|-∑j=2K|y(τ-j(t))|Πi=1j|p(τ-i(t))|-∑j=1K|r(τ-j(t))|Πi=1j|p(τ-i(t))|≥d-ε/4p1-Bp0K-(d+ε4)∑j=2K1p0j-ε4p0∑j=1K1p0j≥d-ε/4p1-ε4p1-(d+ε4)⋅1/p021-1/p0-ε4p0⋅1/p01-1/p0=d-ε/2p1-d+ε/2p0(p0-1)=d[p0(p0-1)-p1]-(ε/2)[p0(p0-1)+p1]p1p0(p0-1)=L.
This completes the proof.Similar to the proof of Lemma  3.2 in [26], we have the following two lemmas.Lemma 2.4.
Letx,p,τ,r, and y be in C([t0,+∞),ℝ) satisfying (A2), (2.17), (2.18), and
(2.28)lim⁡t→+∞y(t)=lim⁡t→+∞r(t)=0;(2.29)|p(t)|≤p0<12eventually,
where p0 is a constant. Then, limt→+∞x(t)=0.Lemma 2.5.
Letx, p, τ, r, and y be in C([t0,+∞),ℝ) satisfying (A2), (2.17), (2.18), (2.23), and (2.29). Then, there exists L>0 such that |x(t)|≥L eventually.Lemma 2.6 (Krasnoselskii’s fixed point theorem).
LetX be a Banach space, let Y be a nonempty bounded closed convex subset of X, and let f, g be mappings of Y into X such that fx+gy∈Y for every pair x,y∈Y. If f is a contraction mapping and g is completely continuous, then the mapping f+g has a fixed point in Y.


## 3.  Main Results

First, we use the Krasnoselskii’s fixed point theorem to show the existence and multiplicity of bounded positive and negative solutions of (1.1).Theorem 3.1.
Let(A1),(A2), and (A3) hold. Assume that there exist p0,p1∈ℝ+∖{0},r0,r1∈ℝ+, and r∈Cn([t0,+∞),ℝ) satisfying
(3.1)p1≥p(t)≥p0>1eventually,p02>p0+p1;(3.2)r(n)(t)=g(t),-r0≤r(t)≤r1eventually;(3.3)∫t0+∞sn-1∑i=1mqi(s)ds<+∞.
Then, the following hold:(a)
for arbitrarily positive constantsM and N with
(3.4)(p0-1)M>(p1-1)N+p1r1p0+r0,
equation (1.1) has uncountably many bounded positive solutions x∈A(N,M) with
(3.5)N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤M;(b)
for arbitrarily positive constantsM and N with
(3.6)(p0-1)N>(p1-1)M+p1r0p0+r1,
equation (1.1) has uncountably many bounded negative solutions x∈A(-N,-M) with
(3.7)-N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤-M.Proof.
It follows from (3.1) and (3.2) that there exists an enough large constant T0 with τ-1(T0)>1+|t0|+|β| satisfying
(3.8)p0≤p(t)≤p1,r(n)(t)=g(t),-r0≤r(t)≤r1,∀t≥T0.
(a) Assume thatM and N are arbitrary positive constants satisfying (3.4). Let D∈((p1-1)N+(p1r1/p0),(p0-1)M-r0). First of all, we prove that there exist two mappings FD,GD:A(N,M)→BC([β,+∞),ℝ) and a constant TD>τ-1(T0) such that FD+GD has a fixed point x∈A(N,M), which is also a bounded positive solution of  (1.1) with N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤M. Put
(3.9)B=max⁡{|fi(u1,u2,…,uki)|:uj∈[N,M],1≤j≤ki,1≤i≤m}.
In light of (3.3), (3.9), and (A2), we infer that there exists a sufficiently large number TD>τ-1(T0) satisfying
(3.10)Bp0(n-1)!∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds<min⁡{M-D+M+r0p0,D+Np1-r1p0-N}.
Define two mappings FD,GD:A(N,M)→C([β,+∞),ℝ) by
(3.11)(FDx)(t)={Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t)),t≥TD(FDx)(TD),β≤t<TD,(3.12)(GDx)(t)={(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1×∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds,t≥TD,(GDx)(TD),β≤t<TD,
for each x∈A(N,M). In view of (3.1), (3.8), and (3.10)–(3.12), we conclude that for any x,u∈A(N,M) and t≥TD(3.13)|(FDx)(t)-(FDu)(t)|=|x(τ-1(t))-u(τ-1(t))p(τ-1(t))|≤1p0‖x-u‖,(FDx)(t)+(GDu)(t)=Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≤Dp0+Mp0+r0p0+Bp0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds<D+M+r0p0+min⁡{M-D+M+r0p0,D+Np1-r1p0-N}≤M,(FDx)(t)+(GDu)(t)=Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≥Dp1+Np1-r1p0-Bp0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds>D+Np1-r1p0-min⁡{M-D+M+r0p0,D+Np1-r1p0-N}≥N,|(GDu)(t)|=|(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds|≤Bp0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds<min⁡{M-D+M+r0p0,D+Np1-r1p0-N}<M,
which ensures that
(3.14)‖FDx-FDu‖=sup⁡t≥TD|(FDx)(t)-(FDu)(t)|≤1p0‖x-u‖,∀x,u∈A(N,M),(3.15)FDx+GDu∈A(N,M),∀x,u∈A(N,M),(3.16)‖GDu‖≤M,∀u∈A(N,M).
It follows from (3.11), (3.12), (3.15), and (3.16) that FD and GD map A(N,M) into BC([β,+∞),ℝ), respectively.
Now, we show thatGD is continuous in A(N,M). Let {xl}l∈ℕ⊂A(N,M) and x∈A(N,M) with liml→∞xl=x, given ε>0. It follows from the uniform continuity of fi in [N,M]ki for 1≤i≤m and liml→∞xl=x that there exist δ>0 and K∈ℕ satisfying
(3.17)|fi(ui1,ui2,…,uiki)-fi(vi1,vi2,…,viki)|<ε1+(1/p0(n-1)!)∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds,∀uij,vij∈[N,M],|uij-vij|<δ,1≤j≤ki,1≤i≤m,‖xl-x‖<δ,∀l≥K.
In view of (3.8), (3.12), (3.17), we arrive at
(3.18)∥GDxl-GDx∥=sup⁡t≥TD|(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)[fi(xl(σi1(s)),xl(σi2(s)),…,xl(σiki(s)))-fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))]ds∑i=1m|≤sup⁡t≥TD1p0(n-1)!×∫τ-1(t)+∞sn-1∑i=1mqi(s)|fi(xl(σi1(s)),xl(σi2(s)),…,xl(σiki(s)))-fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))|ds≤1p0(n-1)!∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds⋅ε1+1/p0(n-1)!∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds<ε,∀l≥K,
which means that GD is continuous in A(N,M).
Next, we show thatGD(A(N,M)) is equicontinuous in [β,+∞). Let ε>0. Taking into account (3.3) and (A2), we know that there exists T*>TD satisfying
(3.19)1p0(n-1)!∫τ-1(T*)+∞sn-1∑i=1mqi(s)ds<ε4.
Put
(3.20)B1=max⁡{sn-1∑i=1mqi(s):τ-1(TD)≤s≤τ-1(T*)}.
It follows from the uniform continuity of pτ-1 and τ-1 in [TD,T*] that there exists δ>0 satisfying
(3.21)|p(τ-1(t1))-p(τ-1(t2))|<εp02(n-1)!4[1+B∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds],∀t1,t2∈[TD,T*]with|t1-t2|<δ;|τ-1(t1)-τ-1(t2)|<εp0(n-1)!4B[1+B1+(n-1)∫τ-1(TD)+∞un-1∑i=1mqi(s)ds],∀t1,t2∈[TD,T*]with|t1-t2|<δ.
Let x∈A(N,M) and t1,t2∈[β,+∞) with |t1-t2|<δ. We consider three possible cases.
Case 1.
Lett1,t2∈[T*,+∞). In view of (3.8), (3.9), (3.12), and (3.19), we conclude that
(3.22)|(GDx)(t1)-(GDx)(t2)|=1(n-1)!|1p(τ-1(t1))×∫τ-1(t1)+∞(s-τ-1(t1))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds-1p(τ-1(t2))×∫τ-1(t2)+∞(s-τ-1(t2))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds1p(τ-1(t1))|≤Bp0(n-1)![∫τ-1(t1)+∞sn-1∑i=1mqi(s)ds+∫τ-1(t2)+∞sn-1∑i=1mqi(s)ds]<ε2.
Case 2.
Lett1,t2∈[TD,T*]. In terms of (3.8), (3.9), (3.12), (3.21), we arrive at
(3.23)|(GDx)(t1)-(GDx)(t2)|=1(n-1)!|1p(τ-1(t1))×∫τ-1(t1)+∞(s-τ-1(t1))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds-1p(τ-1(t2))×∫τ-1(t2)+∞(s-τ-1(t2))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds|≤1(n-1)!{|1p(τ-1(t1))-1p(τ-1(t2))|×∫τ-1(t1)+∞(s-τ-1(t1))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds+1p(τ-1(t2))×[|∫τ-1(t1)τ-1(t2)(s-τ-1(t1))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds|+∫τ-1(t2)+∞|(s-τ-1(t1))n-1-(s-τ-1(t2))n-1|×∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds]}≤B(n-1)!{|p(τ-1(t1))-p(τ-1(t2))|p(τ-1(t1))p(τ-1(t2))∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds+1p0×[|∫τ-1(t1)τ-1(t2)sn-1∑i=1mqi(s)ds|+∫τ-1(t2)+∞(n-1)smax⁡{n-2,0}|τ-1(t1)-τ-1(t2)|∑i=1mqi(s)ds]}≤Bp02(n-1)!|p(τ-1(t1))-p(τ-1(t2))|∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds+Bp0(n-1)![B1+(n-1)∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds]|τ-1(t1)-τ-1(t2)|<ε2.
Case 3.
Lett1,t2∈[β,TD]. By (3.12), we have
(3.24)|(GDx)(t1)-(GDx)(t2)|=|(GDx)(TD)-(GDx)(TD)|=0<ε.
Thus, GD(A(N,M)) is equicontinuous in [β,+∞). Consequently, GD(A(N,M)) is relatively compact by (3.16) and the continuity of GD. By means of (3.14), (3.15), and Lemma 2.6, we infer that FD+GD possesses a fixed point x∈A(N,M), that is,
(3.25)x(t)=Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds,∀t≥TD,
which gives that
(3.26)x(t)-p(t)x(τ(t))=-D+r(t)+(-1)n-1(n-1)!×∫t+∞(s-t)n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds,∀t≥τ-1(TD),[x(t)-p(t)x(τ(t))](n)=g(t)-∑i=1mqi(t)fi(x(σi1(t)),x(σi2(t)),…,x(σiki(t))),∀t≥τ-1(TD),
which mean that x∈A(N,M) is a bounded positive solution of (1.1) with
(3.27)N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤M.
LetD1 and D2 be two arbitrarily different numbers in ((p1-1)N+(p1r1/p0),(p0-1)M-r0). Similarly, we conclude that for each l∈{1,2} there exist two mappings FDj,GDj:A(N,M)→BC([β,+∞),ℝ) and a sufficiently large number TDl>τ-1(T0) satisfying (3.8)–(3.12), where D,TD,FD, and GD are replaced by Dl,TDl,FDl, and GDl, respectively, and FDl+GDl has a fixed point xl∈A(N,M), which is also a bounded positive solution with N≤liminft→+∞xl(t)≤limsupt→+∞xl(t)≤M, that is,
(3.28)xl(t)=Dlp(τ-1(t))+xl(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(xl(σi1(s)),xl(σi2(s)),…,xl(σiki(s)))ds,∀t≥TDl.
It follows from (3.3) that there exists T3>max{TD1,TD2} satisfying
(3.29)Bp0(n-1)!∫τ-1(T3)+∞sn-1∑i=1mqi(s)ds<|D1-D2|4p1.
Combining (3.8), (3.28), and (3.29), we conclude easily that
(3.30)|x1(t)-x2(t)|=|D1-D2p(τ-1(t))+x1(τ-1(t))-x2(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1×∑i=1mqi(s)[fi(x1(σi1(s)),x1(σi2(s)),…,x1(σiki(s)))-fi(x2(σi1(s)),x2(σi2(s)),…,x2(σiki(s)))]dsD1-D2p(τ-1(t))|≥|D1-D2|p(τ-1(t))-|x1(τ-1(t))-x2(τ-1(t))|p(τ-1(t))-2Bp(τ-1(t))(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds≥|D1-D2|p1-‖x1-x2‖p0-2Bp0(n-1)!∫τ-1(T3)+∞sn-1∑i=1mqi(s)ds>|D1-D2|p1-‖x1-x2‖p0-|D1-D2|2p1=|D1-D2|2p1-‖x1-x2‖p0,∀t≥T3,
which guarantees that
(3.31)‖x1-x2‖≥p0|D1-D2|2p1(1+p0)>0,
that is, x1≠x2. Hence, (1.1) has uncountably many bounded positive solutions x∈A(N,M) with N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤M.
(b) Assume thatM and N are arbitrary positive constants satisfying (3.6) and put
(3.32)B2=max⁡{|fi(u1,u2,…,uki)|:uj∈[-N,-M],1≤j≤ki,1≤i≤m}.
Let D∈((1-p0)N+r1,(1-p1)M-(p1r0/p0)). It follows from (3.3), (3.8), (3.32), and (A2) that there exists TD>τ-1(T0) satisfying
(3.33)B2p0(n-1)!∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds<min⁡{-M+M-Dp1-r0p0,N+D-N-r1p0}.
Let the mappings FD,GD:A(-N,-M)→C([β,+∞),ℝ) be defined by (3.11) and (3.12), respectively.
Using (3.1), (3.8), (3.11), (3.12), and (3.33), we deduce that for any x,u∈A(-N,-M) and t≥TD(3.34)(FDx)(t)+(GDu)(t)=Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≤Dp1-Mp1+r0p0+B2p0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds<D-Mp1+r0p0+min⁡{-M+M-Dp1-r0p0,N+D-N-r1p0}≤-M,(FDx)(t)+(GDu)(t)=Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≥Dp0-Np0-r1p0-B2p0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds>D-N-r1p0-min⁡{-M+M-Dp1-r0p0,N+D-N-r1p0}≥-N,
which give that
(3.35)FDx+GDu∈A(-N,-M),∀x,u∈A(-N,-M).
The rest of the proof is similar to the proof of (a) and is omitted. This completes the proof.Theorem 3.2.
Let(A1),(A2), and (A3), hold. Assume that there exist  p0,p1∈ℝ+∖{0},r0,r1∈ℝ+, and r∈Cn([t0,+∞),ℝ) satisfying  (3.2), (3.3), and
(3.36)p1≥-p(t)≥p0>1eventually.
Then, the following hold:(a)
for arbitrarily positive constantsM and N with
(3.37)N<M,(p02-p1)M>(p1-p0p1)p0N+p0r1+p1r0,
equation (1.1) has uncountably many bounded positive solutions x∈A(N,M) with
(3.38)N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤M;(b)
for arbitrarily positive constantsM and N with
(3.39)M<N,(p02-p1)N>(p1-p0p1)p0M+p1r1+p0r0,
equation (1.1) has uncountably many bounded negative solutions x∈A(-N,-M) with
(3.40)-N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤-M.Proof.
It follows from (3.2) and (3.36) that there exists a constant T0 with τ(T0)>1+|t0|+|β| satisfying
(3.41)p0≤-p(t)≤p1,r(n)(t)=g(t),-r0≤r(t)≤r1,∀t≥T0.
(a) Assume thatM and N are arbitrary positive constants satisfying (3.37). Let D∈(p1((M+r0)/p0+N),p0(N/p1+M)-r1) and B be defined by (3.9). In light of (3.3), (3.9), and (A2), there exists a sufficiently large number TD>τ-1(T0) satisfying
(3.42)Bp0(n-1)!∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds<min⁡{M-D+r1p0+Np1,Dp1-M+r0p0-N}.
Define two mappings FD,GD:A(N,M)→C([β,+∞),ℝ) by (3.12) and
(3.43)(FDx)(t)={-Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t)),t≥TD(FDx)(TD),β≤t<TD
for each x∈A(N,M). In view of (3.12), (3.36), and (3.41)–(3.43), we conclude that for any x,u∈A(N,M) and t≥TD(3.44)(FDx)(t)+(GDu)(t)=-Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≤Dp0-Np1+r1p0+Bp0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds<Dp0-Np1+r1p0+min⁡{M-D+r1p0+Np1,Dp1-M+r0p0-N}≤M,(FDx)(t)+(GDu)(t)=-Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≥Dp1-Mp0-r0p0-Bp0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds>Dp1-M+r0p0-min⁡{M-D+r1p0+Np1,Dp1-M+r0p0-N}≥N,
which imply (3.15). The rest of the proof is similar to that of Theorem 3.1 and is omitted.
(b) Assume thatM and N are arbitrary positive constants satisfying (3.39). Let D∈(-p0(N+(M/p1))M+r0,-Mp1-(p1/p0)(N+r1)) and B2 be defined by (3.32). Note that (3.3), (3.32), and (A2) yield that there exists a sufficiently large number TD>τ-1(T0) satisfying
(3.45)B2p0(n-1)!∫τ-1(TD)+∞sn-1∑i=1mqi(s)ds<min⁡{-M-Dp1-N+r1p0,N+D-r0p0+Mp1}.
Let the mappings FD,GD:A(-N,-M)→C([β,+∞),ℝ) be defined by (3.12) and (3.43), respectively.
Using (3.12), (3.36), (3.41), and (3.45), we infer that for any x,u∈A(N,M) and t≥TD(3.46)(FDx)(t)+(GDu)(t)=-Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≤Dp1+Np0+r1p0+B2p0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds<Dp1+Np0+r1p0+min⁡{-M-Dp1-N+r1p0,N+D-r0p0+Mp1}≤-M,(FDx)(t)+(GDu)(t)=-Dp(τ-1(t))+x(τ-1(t))p(τ-1(t))-r(τ-1(t))p(τ-1(t))+(-1)np(τ-1(t))(n-1)!×∫τ-1(t)+∞(s-τ-1(t))n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≥Dp0+Mp1-r0p0-B2p0(n-1)!∫τ-1(t)+∞sn-1∑i=1mqi(s)ds>Dp0+Mp1-r0p0-min⁡{-M-Dp1-N+r1p0,N+D-r0p0+Mp1}≥-N,
which give (3.15). The rest of the proof is similar to the proof of Theorem 3.1 and is omitted. This completes the proof.Theorem 3.3.
Let(A1) and (A3) hold. Assume that there exist  p0,p1∈ℝ+∖{0},  r0,r1∈ℝ+, and r∈Cn([t0,+∞),ℝ) satisfying (3.2), (3.3), and
(3.47)-p0≤p(t)≤p1eventually,p0+p1<1.
Then, the following hold:
(a) for arbitrarily positive constantsM and N with
(3.48)r0+r1+N<(1-p0-p1)M,
equation (1.1) has uncountably many bounded positive solutions x∈A(N,M) with
(3.49)N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤M;
for arbitrarily positive constants M and N with
(3.50)r0+r1+M<(1-p0-p1)N,
equation (1.1) has uncountably many bounded negative solutions x∈A(-N,-M) with
(3.51)-N≤liminf⁡t→+∞x(t)≤limsup⁡t→+∞x(t)≤-M.Proof.
It follows from (3.2) and (3.47) that there exists a constant T0>1+|t0|+|β| satisfying
(3.52)-p0≤p(t)≤p1,r(n)(t)=g(t),-r0≤r(t)≤r1,∀t≥T0.
(a) Assume thatM and N are arbitrary positive constants satisfying (3.48). Let D∈(p0M+r0+N,(1-p1)M1-r1) and B be defined by (3.9). In light of (3.3), (3.9), and (A2), we infer that there exists a sufficiently large number TD>max{T0,τ(T0)} satisfying
(3.53)Bp0(n-1)!∫TD+∞sn-1∑i=1mqi(s)ds<min⁡{M-D-p1M-r1,D-p0M-r0-N}.
Define two mappings FD,GD:A(N,M)→C([β,+∞),ℝ) by
(3.54)(FDx)(t)={D+p(t)x(τ(t))+r(t),t≥TD,(FDx)(TD),β≤t<TD,(3.55)(GDx)(t){(-1)n-1(n-1)!×∫t+∞(s-t)n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds,t≥TD(GDx)(TD),β≤t<TD,
for each x∈A(N,M). In view of (3.47) and (3.52)–(3.55), we conclude that for any x,u∈A(N,M) and t≥TD(3.56)|(FDx)(t)-(FDu)(t)|≤|p(t)(x(τ(t))-u(τ(t)))|≤(p0+p1)‖x-u‖,(FDx)(t)+(GDu)(t)=D+p(t)x(τ(t))+r(t)+(-1)n-1(n-1)!×∫t+∞(s-t)n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≤D+p1M+r1+B(n-1)!∫t+∞sn-1∑i=1mqi(s)ds<D+p1M+r1+min⁡{M-D-p1M-r1,D-p0M-r0-N}≤M,(FDx)(t)+(GDu)(t)=D+p(t)x(τ(t))+r(t)+(-1)n-1(n-1)!×∫t+∞(s-t)n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≥D-p0M-r0-B(n-1)!∫t+∞sn-1∑i=1mqi(s)ds>D-p0M-r0-min⁡{M-D-p1M-r1,D-p0M-r0-N}≥N,
which yield (3.15). The rest of the proof is similar to that of Theorem 3.1 and is omitted.
(b) Assume thatM and N are arbitrary positive constants satisfying (3.50). Let D∈(r0-(1-p1)N-M-N,p0-r1) and B2 be defined by (3.32). In light of (3.3), (3.32), and (A2), we infer that there exists a sufficiently large number TD>max{T0,τ(T0)} satisfying
(3.57)B2p0(n-1)!∫TD+∞sn-1∑i=1mqi(s)ds<min⁡{-M-D-p0N-r1,D+N(1-p1)-r0}.
Define two mappings FD,GD:A(-N,-M)→C([β,+∞),ℝ) by (3.54) and (3.55). In view of (3.47), (3.52), (3.54), (3.55), and (3.57), we conclude that (3.56) holds and
(3.58)(FDx)(t)+(GDu)(t)=D+p(t)x(τ(t))+r(t)+(-1)n-1(n-1)!×∫t+∞(s-t)n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≤D+p0N+r1+B(n-1)!∫t+∞sn-1∑i=1mqi(s)ds<D+p0N+r1+min⁡{-M-D-p0N-r1,D+N(1-p1)-r0}≤-M,∀x,u∈A(N,M),t≥TD,(FDx)(t)+(GDu)(t)=D+p(t)x(τ(t))+r(t)+(-1)n-1(n-1)!×∫t+∞(s-t)n-1∑i=1mqi(s)fi(u(σi1(s)),u(σi2(s)),…,u(σiki(s)))ds≥D-p1N-r0-B(n-1)!∫t+∞sn-1∑i=1mqi(s)ds>D-p1N-r0-min⁡{-M-D-p0N-r1,D+N(1-p1)-r0}≥-N,∀x,u∈A(N,M),t≥TD.
Thus, (3.15) follows from (3.58). The rest of the proof is similar to that of Theorem 3.1 and is omitted. This completes the proof.Second, we provide necessary and sufficient conditions for the oscillation of bounded solutions of (1.1).Theorem 3.4.
Let(A1),(A2), and (A3) hold. Assume that there exist  p0,p1∈ℝ+∖{0} and r∈Cn([t0,+∞),ℝ) satisfying (2.24) and
(3.59)lim⁡t→+∞r(t)=0,r(n)(t)=g(t)eventually.
Then, each bounded solution of (1.1) either oscillates or tends to 0 as t→+∞ if and only if
(3.60)∫t0+∞sn-1∑i=1mqi(s)ds=+∞.Proof.
Sufficiency.
Suppose, without loss of generality, that (1.1) possesses a bounded eventually positive solution x with limsupt→+∞x(t)>0, which together with (A1),(A3), (2.17), (2.24), and (3.60), yields that there exist constants M>0 and T>1+|t0|+|β| satisfying
(3.61)0<x(t)≤M,∀t≥T;(3.62)y(n)(t)=-∑i=1mqi(t)fi(x(σi1(t)),x(σi2(t)),…,x(σiki(t)))<0,∀t≥T.
Obviously (2.17), (2.24), (3.59), and the boundedness of x imply that y is bounded. It follows from (2.17), (3.62), Lemmas 2.1 and 2.2 that there exists a constant L satisfying
(3.63)lim⁡t→+∞y(t)=L≠0,lim⁡t→+∞y(i)(t)=0,1≤i≤n-1.
Thus, (A1), (3.61), (3.63), and Lemma 2.3 imply that there exist constants N and T1≥T0≥T satisfying
(3.64)inf⁡{σij(t):t≥T1,1≤j≤ki,1≤i≤m}≥T0,0<N≤x(t),|y(t)-L|<1,∀t≥T1.
Put
(3.65)B3=min⁡{fi(u1,u2,…,uki):uj∈[N,M],1≤j≤ki,1≤i≤m}.
Clearly, (A3) guarantees that B3>0. Integrating (3.62) from t to +∞, by (3.63) and (3.64), we have
(3.66)y(n-1)(t)=(-1)2∫t+∞∑i=1mqi(u1)fi(x(σi1(u1)),x(σi2(u1)),…,x(σiki(u1)))du1,∀t≥T1,
repeating this procedure, we obtain that
(3.67)y(n-2)(t)=(-1)3∫t+∞du2∫u2+∞∑i=1mqi(u1)fi(x(σi1(u1)),x(σi2(u1)),…,x(σiki(u1)))du1,∀t≥T1,⋯y′(t)=(-1)n∫t+∞dun-1∫un-1+∞dun-2⋯×∫u2+∞∑i=1mqi(u1)fi(x(σi1(u1)),x(σi2(u1)),…,x(σiki(u1)))du1,∀t≥T1,L-y(t)=lim⁡u→+∞y(u)-y(t)=(-1)n∫t+∞dun∫un+∞dun-1⋯×∫u2+∞∑i=1mqi(u1)fi(x(σi1(u1)),x(σi2(u1)),…,x(σiki(u1)))du1=(-1)n(n-1)!∫t+∞(s-t)n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds,∀t≥T1,
which together with (3.64) and (A3) means that
(3.68)1>|L-y(t)|=|(-1)n(n-1)!∫t+∞(s-t)n-1∑i=1mqi(s)fi(x(σi1(s)),x(σi2(s)),…,x(σiki(s)))ds|≥B3(n-1)!∫t+∞(s-t)n-1∑i=1mqi(s)ds,∀t≥T1,
which gives that
(3.69)∫T1+∞sn-1∑i=1mqi(s)ds<+∞,
which contradicts (3.60).
Necessity.
Suppose that (3.60) does not hold. Observe that limt→+∞r(t)=0 implies that there exist two positive constants r0 and r1 satisfying
(3.70)-r0≤r(t)≤r1eventually.
It follows from Theorem 3.1 or Theorem 3.2 that, for any positive constants M and N satisfying (3.4) or (3.37), (1.1) possesses uncountably many bounded positive solutions x∈A(N,M) with M≥limsupt→+∞x(t)≥liminft→+∞x(t)≥N. This is a contradiction. This completes the proof.As in the proof of Theorem3.4, by means of Lemmas 2.1, 2.4, and 2.5, we haveTheorem 3.5.
Let(A1) and (A3) hold. Assume that there exist  p0∈ℝ+∖{0} and r∈Cn([t0,+∞),ℝ) satisfying (2.29) and (3.59). Then, each bounded solution of (1.1) either oscillates or tends to 0 as t→+∞ if and only if (3.60) holds.


## 4.  Remarks and Examples

Now, we compare the results in Section3 with some known results in the literature. In order to illustrate the advantage and applications of our results, five nontrivial examples are constructed.Remark 4.1.
Theorems3.1–3.3 extend and improve the Theorem in [9], Theorem  8.4.2 in [10], Theorem  1 in [21], Theorems  1–3 in [24], Theorem  2.2 in [26], and Theorems  1–4 in [27, 28].Remark 4.2.
The sufficient part of Theorem3.5 is a generalization of Theorem  3.1 in [4, 5]. Theorem 3.5 corrects and perfects Theorem  2.1 in [26].The examples below show that our results extend indeed the corresponding results in [4, 5, 9, 10, 21, 24, 26–28]. Notice that none of the known results can be applied to these examples.Example 4.3.
Consider thenth-order forced nonlinear neutral differential equation:
(4.1)[x(t)-3+4tn1+tnx(t)](n)+(1+3+2t)[x5(3t+sin⁡t)+x3(t-1/t)](1+tn+3)[1+|x8(3t2)-2x21(t-t-1)|]+tx(3t-ln⁡t)x4(t2-t)x6(t-2)+5tx(t(1+1/t)t)(1+3t3n+1)[1+|x3(4t-cos⁡3t)-4x4(t-1)|]=12sin⁡(t+nπ2),t≥2,
where t0=2,m=2, and n∈ℕ. Put k1=4,k2=6,β=0,r0=r1=1/2,p0=3,p1=4,
(4.2)p(t)=3+4tn1+tn,q1(t)=1+3+2t1+tn+3,q2(t)=t1+3t3n+1,g(t)=12sin⁡(t+nπ2),r(t)=12sin⁡t,τ(t)=t,σ11(t)=3t+sin⁡t,σ12(t)=t-1t,σ13(t)=3t2,σ14(t)=t-t-1,σ21(t)=3t-ln⁡t,σ22(t)=t2-t,σ23(t)=t-2,σ24(t)=t(1+1t)t,σ25(t)=4t-cos⁡3t,σ26(t)=t-1,f1(u,v,w,z)=u5+v31+|w8-2z21|,f2(u,v,w,z,y,s)=uv4w6+5z1+|y3-4s4|,∀(t,u,v,w,z,y,s)∈[t0,+∞)×R6.
Clearly (A1), (A2), (A3), and (3.1)–(3.3) hold.
LetM and N be arbitrarily positive constants satisfying M>(3/2)N+7/12. It is easy to verify that (3.4) holds. It follows from Theorem 3.1 that (4.1) has uncountably many bounded positive solutions x∈A(N,M) with N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤M.
LetM and N be arbitrarily positive constants satisfying N>3M/2+7/12. It is easy to verify that (3.6) holds. It follows from Theorem 3.1 that (4.1) has uncountably many bounded negative solutions x∈A(-N,-M) with -N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤-M.Example 4.4.
Consider thenth-order forced nonlinear neutral differential equation:
(4.3)[x(t)+8+10t52+t5x(t2+1-1)](n)+(t2+3t3)x7(3t2)x(t-1)x3(tln⁡t)(2+sin⁡3(t2)+tn+5)[1+x2(t-1)x4(tln⁡t)]+(3+t2)[x5(t2+1)+7x3(t4-2)+x9(t+t)x8(t-4)](t+1+tn+3)[1+(x5(t+t)-4x4(t-4)-3)6]=(-1)nn!tn+1+n!(1-t)n+1,t≥3,
where t0=3,m=2, and n∈ℕ. Put k1=3,k2=4,β=-1,r0=1/2,r1=0,p0=4,p1=10,
(4.4)p(t)=-8+10t52+t5,q1(t)=t2+3t32+sin⁡3(t2)+tn+5,q2(t)=3+t2t+1+tn+3,g(t)=(-1)nn!tn+1+n!(1-t)n+1,r(t)=1t(1-t),τ(t)=t2+1-1,σ11(t)=3t2,σ12(t)=t-1,σ13(t)=tln⁡t,σ21(t)=t2+1,σ22(t)=t4-2,σ23(t)=t+t,σ24(t)=t-4,f1(u,v,w)=u7vw31+v2w4,f2(u,v,w,z)=u5+7v3+w9z81+(w5-4z4-3)6,∀(t,u,v,w,z)∈[t0,+∞)×R4.
Clearly (A1), (A2), (A3), (3.2), (3.3), and (3.36) hold.
LetM and N be arbitrarily positive constants satisfying M>(32/5)N+5/6. It is easy to verify that (3.37) holds. It follows from Theorem 3.2 that (4.3) has uncountably many bounded positive solutions x∈A(N,M) with N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤M.
LetM and N be arbitrarily positive constants satisfying N>(32/5)M+1/3. It is easy to verify that (3.39) holds. It follows from Theorem 3.2 that (4.3) has uncountably many bounded negative solutions x∈A(-N,-M) with -N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤-M.Example 4.5.
Consider thenth-order forced nonlinear neutral differential equation:
(4.5)[x(t)-2sin⁡(t2-t)5+sin⁡(t2-t)x((t-5)2)](n)+(3t-1+t5)x3(t-4)(t2+1+tn+6)[2+cos⁡5x(t+1-3)]+(1-t+1ln⁡2t+t4)x9(2t+sin⁡(t2+1))(1-2t3+3t4+tn+5)ln⁡[2+x2(t21+2t)]=(-1)ncos⁡(t+nπ2),t≥1,
where t0=1,m=2, and n∈ℕ. Put k1=k2=2,β=-4,r0=r1=1,p0=1/2,p1=1/3,
(4.6)p(t)=2sin⁡(t2-t)5+sin⁡(t2-t),q1(t)=3t-1+t5t2+1+tn+6,q2(t)=1-t+1ln⁡2t+t41-2t3+3t4+tn+5,g(t)=(-1)ncos⁡(t+nπ2),r(t)=(-1)ncos⁡t,τ(t)=(t-4)2σ11(t)=t-5,σ12(t)=t+1-3,σ21(t)=2t+sin⁡(t2+1),σ22(t)=t21+2t,f1(u,v)=u32+cos⁡5v,f2(u,v)=u9ln⁡(2+v2),∀(t,u,v)∈[t0,+∞)×R2.
Clearly (A1), (A3), (3.2), (3.3), and (3.47) hold.
LetM and N be arbitrarily positive constants satisfying M>6N+12. It is easy to verify that (3.48) holds. It follows from Theorem 3.3 that  (4.5) has uncountably many bounded positive solutions x∈A(N,M) with N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤M.
LetM and N be arbitrarily positive constants satisfying N>6M+12. It is easy to verify that (3.50) holds. It follows from Theorem 3.3 that  (4.5) has uncountably many bounded negative solutions x∈A(-N,-M) with -N≤liminft→+∞x(t)≤limsupt→+∞x(t)≤-M.Example 4.6.
Consider thenth-order forced nonlinear neutral differential equation:
(4.7)[x(t)-(-1)n(5+9ln⁡2t)1+ln⁡2tx(t-1)](n)+(t8+9t5+3)[2x3(tln⁡2t)+5x7(t-16)]+t2(2+sin⁡(t3-5t))x9(t-sin⁡tt)[x4(t-cos⁡t)+4x6(1+t+t2+t31+t+t2)]2+[(t+1)2-t]x5(tarctan(t3+1)/(1+t+1))ln⁡(1+x6(t+1)/(1+x2(t-2)))[2tn+3+tsin⁡(3t5-1)][1+x2(t2-t)x4(t2+t)]=(-1)nn!(ln⁡t-∑i=1n(1/i))tn+1,t≥4,
where t0=4,m=3, and n∈ℕ. Put k1=2,k2=3,k3=5,β=-12,p0=5,p1=9,
(4.8)p(t)=(-1)n(5+9ln⁡2t)1+ln⁡2t,q1(t)=t8+9t5+3,q2(t)=t2(2+sin⁡(t3-5t)),q3=(t+1)2-t2tn+3+tsin⁡(3t5-1),g(t)=(-1)nn!(ln⁡t-∑i=1n1/i)tn+1,r(t)=ln⁡tt,τ(t)=t-1,σ11(t)=tln⁡2t,σ12(t)=t-16,σ21(t)=t-sin⁡tt,σ22(t)=t-cos⁡t,σ23(t)=1+t+t2+t31+t+t2,σ31(t)=tarctan(t3+1)1+t+1,σ32(t)=t+1,σ33(t)=t-2,σ34(t)=t2-t,σ35(t)=t2+t,f1(u,v)=2u3+5v7,f2(u,v,w)=u9(v4+4w6)2,f3(u,v,w,y,z)=u5ln⁡(1+v6/(1+w2))1+y2z4,∀(t,u,v,w,y,z)∈[t0,+∞)×R5.
Clearly (A1), (A2), (A3), (2.24), (3.59), and (3.60) hold. It follows from Theorem 3.4 that each bounded solution of (4.7) either oscillates or tends to 0 as t→+∞.Example 4.7.
Consider thenth-order forced nonlinear neutral differential equation:
(4.9)[x(t)-(-1)ncos⁡3(3t-1)4+cos⁡3(3t-1)x(t-sin⁡t)](n)+(t3+2t2-t+1)x5(t-2-1)1+x2(t-2-1)+t2-1[x3(t-1/t)+5x7(t-1/t)]ln⁡(2+x6(t-1/t))t2n+1+2tnln⁡3(1+t2)+1=2nsin⁡(2t+nπ/4)e2t,t≥6,
where t0=6,m=2, and n∈ℕ. Put k1=k2=1,β=1,p0=1/3,
(4.10)p(t)=(-1)ncos⁡3(3t-1)4+cos⁡3(3t-1),q1(t)=t3+2t2-t+1,q2(t)=t2-1t2n+1+2tnln⁡3(1+t2)+1,g(t)=2nsin⁡(2t+nπ/4)e2t,r(t)=sin⁡(2t)e2t,τ(t)=t-sin⁡t,σ1(t)=t-2-1σ2(t)=t-1t,f1(u)=u51+u2,f2(u)=u3+5u7ln⁡(2+u6),∀(t,u)∈[t0,+∞)×R.
Clearly (A1), (A3), (2.29), (3.59), and (3.60) hold. It follows from Theorem 3.5 that each bounded solution of (4.9) either oscillates or tends to 0 as t→+∞.Next, we prove that the necessary part of Theorem  2.1 in [26] does not hold by means of (4.9). It is easy to verify that the conditions of Theorem  2.1 in [26] are fulfilled. Suppose that the necessary part of Theorem  2.1 in [26] is true. Because each bounded solution of (4.9) either oscillates or tends to 0 as t→+∞, it follows that the necessary part of Theorem  2.1 in [26] gives that(4.11)∫t0+∞sn-1qi(s)ds=+∞,i∈{1,2},
which yields that(4.12)+∞=∫t0+∞sn-1q2(s)ds=∫t0+∞sn-1s2-1s2n+1+2snln⁡3(1+s2)+1ds≤∫t0+∞1sn+1ds<+∞,
which is a contradiction.



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*Source: 546302-2012-05-17.xml*