# Positive Solutions for a Higher-Order Nonlinear Neutral Delay Differential Equation

Authors: Zeqing Liu; Ming Jia; Jeong Sheok Ume; Shin Min Kang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper deals with the higher-order nonlinear neutral delay differential equation(dn/dtn)[x(t)+∑i=1mpi(t)x(Ti(t))]+(dn−1/dtn−1)f(t,x(α1(t)),…,x(αk(t)))+h(t,x(β1(t)),…,x(βk(t)))=g(t), t≥to, where n,m,k∈ℕ,
pi,τi,βj,g∈C([to,+∞),ℝ),
αj∈Cn−1([to,+∞),ℝ),
f∈Cn−1([to,+∞)×ℝk,ℝ),
h∈C([to,+∞)×ℝk,ℝ), and limt→+∞τi(t)=limt→+∞αj(t)=limt→+∞βj(t)=+∞,
i∈{1,2,…,m},j∈{1,2,…,k}. By making use of the Leray-Schauder nonlinear alterative theorem, we establish the existence of uncountably many bounded positive solutions for the above equation. Our results improve and generalize some corresponding results in the field. Three examples are given which illustrate the advantages of the results presented in this paper.


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## Body

## 1. Introduction and Preliminaries

This paper is concerned with the higher-order nonlinear neutral delay differential equation:(1.1)dndtn[x(t)+∑i=1mpi(t)x(τi(t))]+dn-1dtn-1f(t,x(α1(t)),…,x(αk(t)))+h(t,x(β1(t)),…,x(βk(t)))=g(t),t≥t0,
where n,m,k∈ℕ,pi,τi,βj,g∈C([t0,+∞),ℝ),αj∈Cn-1([t0,+∞),ℝ),f∈Cn-1([t0,+∞)×ℝk,ℝ),h∈C([t0,+∞)×ℝk,ℝ), and (1.2)lim⁡t⟶+∞⁡τi(t)=lim⁡t⟶+∞⁡αj(t)=lim⁡t⟶+∞⁡βj(t)=+∞,i∈{1,2,…,m},j∈{1,2,…,k}.Theory of neutral delay differential equations has undergone a rapid development in the last over thirty years. We refer the readers to [1–8] and the references therein for a wealth of reference materials on the subject. The authors [1–8] and others discussed the oscillation, nonoscillation, and existence of a nonoscillatoy solution for some special cases of (1.1) under various conditions. By using the Banach fixed point theorem, Zhang et al. [4] and Kulenović and Hadžiomerspahić [1] studied, respectively, the existence of a nonoscillatory solution for the first-order neutral delay differential equation: (1.3)ddt[x(t)+p(t)x(t-τ)]+P(t)x(t-σ)-Q(t)x(t-δ)=0,t≥t0,
where τ>0, σ,δ∈ℝ+, P,Q∈C([t0,+∞),ℝ+), and p∈C([t0,+∞),ℝ), and the second-order neutral delay differential equation with positive and negative coefficients: (1.4)d2dt2[x(t)+px(t-τ)]+P(t)x(t-σ)-Q(t)x(t-δ)=0,t≥t0,
where p∈ℝ∖{±1}, σ,δ∈ℝ+ and P,Q∈C([t0,+∞),ℝ+). Zhang et al. [6] considered the second-order nonlinear neutral differential equation with positive and negative terms: (1.5)d2dt2[x(t)-px(τ(t))]+f1(t,x(σ1(t)))-f2(t,x(σ2(t)))=0,t≥t0
and its corresponding equation with forced term: (1.6)d2dt2[x(t)-px(τ(t))]+f1(t,x(σ1(t)))-f2(t,x(σ2(t)))=g(t),t≥t0,
where t≥t0, p,τ,σi∈C([t0,∞),ℝ), fi∈C([t0,∞)×ℝ,ℝ), and limt→+∞τ(t)=limt→+∞σi(t)=+∞ for i∈{1,2}. Lin [2] investigated sufficient conditions of oscillation and nonoscillation for the second-order nonlinear neutral differential equation: (1.7)d2dt2[x(t)-p(t)x(t-τ)]+q(t)f(x(t-σ))=0,t≥0,
where τ>0,σ>0, p,q∈C(ℝ+,ℝ+), f∈C(ℝ,ℝ) with xf(x)>0 for all x≠0. Liu and Huang [3] used the coincidence degree theory to establish the existence and uniqueness of T-periodic solutions for the second-order neutral functional differential equation of the form (1.8)d2dt2[x(t)+Bx(t-δ)]+Cdx(t)dt+g(x(t-τ(t)))=p(t),t≥0,
where τ,p,g:ℝ→ℝ are continuous functions, B, δ, C are constants, τ and p are T-periodic, C≠0,|B|≠1, and T>0. Zhou and Zhang [8] extended the results in [1] to the higher-order neutral functional differential equation with positive and negative coefficients: (1.9)dndtn[x(t)+px(t-τ)]+(-1)n+1[P(t)x(t-σ)-Q(t)x(t-δ)]=0,t≥t0,
where p∈ℝ∖{±1}, τ,σ,δ∈ℝ+ and P,Q∈C([t0,+∞),ℝ+). Zhou et al. [7] used the Krasnoselskii fixed point theorem and the Schauder fixed point theorem to prove the existence results of a nonoscillatory solution for the forced higher-order nonlinear neutral functional differential equation: (1.10)dndtn[x(t)+p(t)x(t-τ)]+∑i=1mqi(t)f(x(t-σi))=g(t),t≥t0,
where τ,σi∈ℝ+, p,qi,g∈C([t0,+∞),ℝ) for i∈{1,2,…,m} and f∈C(ℝ,ℝ). Zhang et al. [5] obtained some sufficient conditions for the oscillation of all solutions of the even order nonlinear neutral differential equations with variable coefficients: (1.11)dndtn[x(t)+p(t)x(τ(t))]+q(t)f(x(σ(t)))=0,t≥t0,
where n is an even number, p,q,σ,τ∈C([t0,+∞),ℝ+) with 0≤p(t)<1, for all t≥t0, limt→+∞τ(t)=limt→+∞,σi(t)=+∞ and f∈C([t0,+∞),ℝ).The purpose of this paper is to investigate the solvability of (1.1). By constructing appropriate mappings and using the Laray-Schauder nonlinear alternative theorem, we establish a few sufficient conditions which ensure the existence of uncountably many bounded positive solutions for (1.1). Our results improve and generalize some corresponding results in [1, 2, 4, 6–8]. Three examples are given to illustrate the advantages of the results presented in this paper.Throughout this paper, we assume thatℝ, ℝ+, and ℕ denote the sets of all real numbers, nonnegative numbers, and positive integers, respectively, and (1.12)ν=inf⁡{τi(t),αj(t),βj(t):t∈[t0,+∞),i∈{1,2,…,m},j∈{1,2,…,k}}.
Let CB([ν,+∞),ℝ) stand for the Banach space of all continuous and bounded functions in [ν,+∞) with norm ∥x∥=supt≥ν|x(t)| for all x∈CB([ν,+∞),ℝ) and (1.13)E(N)={x∈CB([ν,+∞),R):x(t)≥Nfort≥ν},U(M)={x∈E(N):‖x‖<M},
where M,N∈ℝ+ with M>N>0. Clearly, E(N) is a nonempty closed convex subset of CB([ν,+∞),ℝ) and U(M) is an open subset of E(N).By a solution of (1.1), we mean a function x∈C([ν,+∞),ℝ) with some T≥t0+|ν| such that x(t)+∑i=1mpi(t)x(τi(t)) is n times continuously differentiable in [T,+∞) and f(t,x(α1(t)),…,x(αk(t))) is n-1 times continuously differentiable in [T,+∞) and (1.1) holds for t≥T.Lemma 1.1 (the Leray-Schauder nonlinear alterative theorem [9]).
LetE be a closed convex subset of a Banach space X and let U be an open subset of E with p⋆∈U. Also, G:U¯→E is a continuous, condensing mapping with G(U¯) bounded, where U¯ denotes the closure of U Then,
(A1)
G has a fixed point in U¯, or(A2)
there arex∈∂U and λ∈(0,1) with x=(1-λ)p⋆+λGx.


## 2. Main Results

Now, we apply the Leray-Schauder nonlinear alterative theorem to investigate the existence of uncountably many bounded positive solutions of (1.1) under certain conditions.Theorem 2.1.
Assume that there exist constantsM,N,p0,t1 and functions F,H∈C([t0,+∞),ℝ+) satisfying
(2.1)|f(t,u1,…,uk)|≤F(t),∀(t,u1,…,uk)∈[t0,+∞)×[N,M]k,(2.2)|h(t,v1,…,vk)|≤H(t),∀(t,v1,…,vk)∈[t0,+∞)×[N,M]k,(2.3)max⁡⁡{∫t0+∞F(s)ds,∫t0+∞sn-1max⁡⁡{|g(s)|,H(s)}ds}<+∞,(2.4)0<N<(1-2p0)M,∑i=1m|pi(t)|≤p0<12,∀t≥t1≥t0.
Then, (1.1) has uncountably many bounded positive solutions in U(M)¯.Proof.
LetL∈(p0M+N,(1-p0)M). It follows from (2.3) and (2.4) that there exists a constant T>1+|t0|+|t1|+|ν| satisfying
(2.5)∫T+∞F(s)ds+∫T+∞sn-1[|g(s)|+H(s)]ds<min⁡⁡{L-p0M-N,(1-p0)M-L,M-N2}.
Choose ϵ0∈(0,min{L-p0M-N,(1-p0)M-L,(M-N/2)}) with
(2.6)∫T+∞F(s)ds+∫T+∞sn-1[|g(s)|+H(s)]ds<min⁡⁡{L-p0M-N,(1-p0)M-L,M-N2}-ϵ0.
Put p⋆=M-ϵ0. Clearly, p⋆∈U(M). Define two mappings AL,BL:U(M)¯→CB([ν,+∞),ℝ) by
(2.7)(ALx)(t)={L-∑i=1mpi(t)x(τi(t))+(-1)n(n-1)!∫t+∞(s-t)n-1g(s)ds,t≥T,(ALx)(T),ν≤t<T,(2.8)(BLx)(t)={∫t+∞f(s,x(α1(s)),…,x(αk(s)))ds+(-1)n-1(n-1)!∫t+∞(s-t)n-1h(s,x(β1(s)),…,x(βk(s)))ds,t≥T,(BLx)(T),ν≤t<T
for all x∈U(M)¯. It is clear to see that ALx and BLx are continuous for each x∈U(M)¯. Let DL=AL+BL. In view of (2.1), (2.2), and (2.4)–(2.8), we get that
(2.9)(ALx)(t)+(BLx)(t)=L-∑i=1mpi(t)x(τi(t))+(-1)n(n-1)!∫t+∞(s-t)n-1g(s)ds+∫t+∞f(s,x(α1(s)),…,x(αk(s)))ds+(-1)n-1(n-1)!∫t+∞(s-t)n-1h(s,x(β1(s)),…,x(βk(s)))ds≥L-p0M-∫T+∞F(s)ds-∫T+∞sn-1[|g(s)|+H(s)]ds≥L-p0M-min⁡⁡{L-p0M-N,(1-p0)M-L,M-N2}+ε0>N,∀(t,x)∈[T,+∞)×U(M)¯,
which gives that DL:U(M)¯→E(N).
Now, we show thatBL:U(M)¯→CB([ν,+∞),ℝ) is continuous and compact. Let {xm}m∈ℕ⊆U(M)¯ be an arbitrary sequence and x∈C([ν,+∞),ℝ) with
(2.10)‖xm-x‖⟶asm⟶∞.
Since U(M)¯ is closed, it follows that x∈U(M)¯. For any (s,m)∈[T,+∞)×ℕ, put
(2.11)Fm(s)=|f(s,xm(α1(s)),…,xm(αk(s)))-f(s,x(α1(s)),…,x(αk(s)))|,Hm(s)=|h(s,xm(β1(s)),…,xm(βk(s)))-h(s,x(β1(s)),…,x(βk(s)))|.
It follows from (2.1), (2.2), and (2.11) that
(2.12)|Fm(s)|≤2F(s),|Hm(s)|≤2H(s),∀(s,m)∈[T,+∞)×N,
which together with (2.8)–(2.11), the continuity of f,h,αj,βj for j∈{1,2,…,k}, and the Lebesgue dominated convergence theorem yields that
(2.13)|(BLxm)(t)-(BLx)(t)|≤∫t+∞|f(s,xm(α1(s)),…,xm(αk(s)))-f(s,x(α1(s)),…,x(αk(s)))|ds+1(n-1)!×∫t+∞(s-t)n-1|h(s,xm(β1(s)),…,xm(βk(s)))-h(s,x(β1(s)),…,x(βk(s)))|ds≤∫T+∞Fm(s)ds+1(n-1)!∫T+∞sn-1Hm(s)ds,∀t≥T,limsup⁡m→∞‖BLxm-BLx‖≤limsup⁡m→∞(∫T+∞Fm(s)ds+1(n-1)!∫T+∞sn-1Hm(s)ds)=0,
which means that BL is continuous in U(M)¯. It follow from (2.1), (2.2), (2.6), and (2.8) that
(2.14)‖BLx‖=sup⁡t≥ν⁡|(BLx)(t)|≤∫T+∞F(s)ds+1(n-1)!∫T+∞sn-1H(s)ds<min⁡⁡{L-p0M-N,(1-p0)M-L,M-N2}-ϵ0<M,∀x∈U(M)¯,
which yields that BL(U(M)¯) is uniformly bounded in [ν,+∞).
Letε be an arbitrary positive number. Equation (2.3) ensures that there exists T*>T satisfying
(2.15)∫T*+∞F(s)ds+∫T*+∞sn-1H(s)ds<ε2.
Set
(2.16)δ=ε1+4[Q+M+Q(T*-T)n-1],Q=max⁡⁡{F(t),H(t):t∈[T,T*]}.
For any x∈U(M)¯ and t1,t2∈[ν,+∞) with |t1-t2|<δ, we consider the following three cases.Case 1 (T*≤t1<t2).
In view of (2.1), (2.2), (2.8), and (2.15), we deduce that
(2.17)|(BLx)(t2)-(BLx)(t1)|=|∫t2+∞f(s,x(α1(s)),…,x(αk(s)))ds+(-1)n-1(n-1)!∫t2+∞(s-t2)n-1h(s,x(β1(s)),…,x(βk(s)))ds-∫t1+∞f(s,x(α1(s)),…,x(αk(s)))ds-(-1)n-1(n-1)!∫t1+∞(s-t1)n-1h(s,x(β1(s)),…,x(βk(s)))ds|≤2∫T*+∞F(s)ds+2(n-1)!∫T*+∞sn-1H(s)ds<ε.Case 2 (T≤t1<t2≤T*).
Suppose thatn=1. It follows from (2.1), (2.2), (2.6), and (2.8) that
(2.18)|(BLx)(t2)-(BLx)(t1)|=|∫t2+∞f(s,x(α1(s)),…,x(αk(s)))ds+∫t2+∞h(s,x(β1(s)),…,x(βk(s)))ds-∫t1+∞f(s,x(α1(s)),…,x(αk(s)))ds-∫t1+∞h(s,x(β1(s)),…,x(βk(s)))ds|≤∫t1t2(F(s)+H(s))ds≤2Q|t1-t2|<ε.Suppose that n∈N∖{1}. It follows from the mean value theorem that, for each s∈(t2,+∞), there exists ζ∈(s-t2,s-t1) satisfying
(2.19)|(s-t2)n-1-(s-t1)n-1|=(n-1)ζn-2|t1-t2|≤(n-1)sn-1|t1-t2|,
which together with (2.1), (2.2), (2.6), and (2.8) yields that
(2.20)|(BLx)(t2)-(BLx)(t1)|=|∫t2+∞f(s,x(α1(s)),…,x(αk(s)))ds+(-1)n-1(n-1)!∫t2+∞(s-t2)n-1h(s,x(β1(s)),…,x(βk(s)))ds-∫t1+∞f(s,x(α1(s)),…,x(αk(s)))ds-(-1)n-1(n-1)!∫t1+∞(s-t1)n-1h(s,x(β1(s)),…,x(βk(s)))ds|≤∫t1t2F(s)ds+1(n-1)!(∫t2+∞|(s-t2)n-1-(s-t1)n-1|H(s)ds+∫t1t2(s-t1)n-1H(s)ds)≤Q|t1-t2|+(n-1)|t1-t2|(n-1)!∫t2+∞sn-1H(s)ds+(T*-T)n-1Q|t1-t2|≤[Q+M+(T*-T)n-1Q]|t1-t2|<ε.Case 3 (ν≤t1<t2≤T).
Equation (2.8) gives that
(2.21)|(BLx)(t2)-(BLx)(t1)|=|(BLx)(T)-(BLx)(T)|=0.
Thus, BL(U(M)¯) is equicontinuous in [ν,+∞). Hence, BL(U(M)¯) is a relatively compact subset of C([ν,+∞),ℝ). That is, BL is a compact mapping.
Note that for anyx,y∈U(M)¯ and t≥T(2.22)|(ALx)(t)-(ALy)(t)|=|L-∑i=1mpi(t)x(τi(t))+(-1)n(n-1)!∫t+∞(s-t)n-1g(s)ds-L+∑i=1mpi(t)y(τi(t))-(-1)n(n-1)!∫t+∞(s-t)n-1g(s)ds|≤∑i=1m|pi(t)|‖x-y‖≤p0‖x-y‖,which implies that
(2.23)‖ALx-ALy‖≤p0‖x-y‖,∀x,y∈U(M)¯,
which together with (2.4) gives that AL is a contraction mapping. It follows that DL:U(M)¯→E(N) is a continuous and condensing mapping. Let x0(t)=N for all t∈[ν,+∞). Notice that x0∈U(M)¯. Thus, (2.4)–(2.7), (2.14), and (2.23) yield that
(2.24)‖DLx‖≤‖ALx‖+‖BLx‖≤‖ALx-ALx0‖+‖ALx0‖+M≤p0‖x-x0‖+M+sup⁡t≥T|L-∑i=1mpi(t)x0(τi(t))+(-1)n(n-1)!∫t+∞(s-t)n-1g(s)ds|≤p0(M+N)+M+L+p0N+∫T+∞sn-1|g(s)|ds≤(2+p0)M+2p0N+L,∀x∈U(M)¯,
that is, DL(U(M)¯) is uniformly bounded in [γ,+∞).
Put(2.25)S1={x∈CB([ν,+∞),R):N≤x(t)≤M,∀t≥ν,‖x‖=M},S2={x∈CB([ν,+∞),R):N≤x(t)≤M,∀t≥νandthereexistst*≥νsatisfyingx(t*)=N}.
It is easy to verify that ∂U(M)=S1∪S2.
Next, we show that (A2) in Lemma 1.1 does not hold. Otherwise, there exist x∈∂U(M) and λ∈(0,1) satisfying x=(1-λ)p⋆+λDLx. We have to discuss the following possible cases. Case 1.
Letx∈S1. By means of (2.1), (2.2), and (2.4)–(2.8), we get that, for t≥T,
(2.26)x(t)=(1-λ)p⋆+λ[(ALx)(t)+(BLx)(t)]≤(1-λ)(M-ϵ0)+λ[L+∑i=1m|pi(t)|x(τi(t))+1(n-1)!∫t+∞sn-1|g(s)|ds+∫t+∞|f(s,x(α1(s)),…,x(αk(s)))|ds+1(n-1)!∫t+∞sn-1|h(s,x(β1(s)),…,x(βk(s)))|ds]≤(1-λ)(M-ϵ0)+λ[L+p0M+∫t+∞F(s)ds+1(n-1)!∫t+∞sn-1[|g(s)|+H(s)]ds]<(1-λ)(M-ϵ0)+λ[L+p0M+min⁡⁡{L-p0M-N,(1-p0)M-L,M-N2}-ϵ0]≤M-ϵ0,
which implies that
(2.27)M=‖x‖=sup⁡t≥ν|x(t)|≤M-ϵ0<M,
which is a contradiction.Case 2.
Letx∈S2. It follows from (2.1), (2.2), and (2.4)–(2.8) that
(2.28)N=x(t*)=(1-λ)p⋆+λ[(ALx)(t*)+(BLx)(t*)]=(1-λ)(M-ϵ0)+λ[(ALx)(max⁡{t*,T})+(BLx)(max⁡{t*,T})]≥(1-λ)(M-ϵ0)+λ[L-∑i=1m|pi(max⁡{t*,T})|x(τi(max⁡{t*,T}))-1(n-1)!∫max⁡{t*,T}+∞sn-1|g(s)|ds-∫max⁡{t*,T}+∞|f(s,x(α1(s)),…,x(αk(s)))|ds-1(n-1)!∫max⁡{t*,T}+∞sn-1|h(s,x(β1(s)),…,x(βk(s)))|ds]≥(1-λ)(M-ϵ0)+λ[L-p0M-∫max⁡{t*,T}+∞F(s)ds-1(n-1)!∫max⁡{t*,T}+∞sn-1[|g(s)|+H(s)]ds]≥(1-λ)(M-ϵ0)+λ[L-p0M-min⁡{L-p0M-N,(1-p0)M-L,M-N2}+ϵ0]≥(1-λ)(M-ϵ0)+λ(N+ϵ0)≥min⁡{M-ϵ0,N+ϵ0}=N+ϵ0,
which is absurd.
Thus, Lemma1.1 ensures that DL has a fixed point x∈U(M)¯; that is,
(2.29)x(t)=L-∑i=1mpi(t)x(τi(t))+(-1)n(n-1)!∫t+∞(s-t)n-1g(s)ds+∫t+∞f(s,x(α1(s)),…,x(αk(s)))ds+(-1)n-1(n-1)!∫t+∞(s-t)n-1h(s,x(β1(s)),…,x(βk(s)))ds,∀t≥T,
which yields that
(2.30)dndtn[x(t)+∑i=1mpi(t)x(τi(t))]+dn-1dtn-1f(t,x(α1(t)),…,x(αk(t)))+h(t,x(β1(t)),…,x(βk(t)))=g(t),∀t≥T,
which means that x∈U(M)¯ is a bounded positive solution of (1.1).
LetL1,L2∈(p0M+N,(1-p0)M) with L1≠L2. Similarly, we can prove that, for each r∈{1,2}, there exist a constant Tr>1+|t0|+|t1|+|ν| and two mappings ALr,BLr:U(M)¯→CB([ν,+∞),ℝ) satisfying (2.6)–(2.8), where T,L,AL, and BL are replaced by Tr,Lr,ALr, and BLr, respectively, and ALr+BLr has a fixed point zr∈U(M)¯, which is a bounded positive solution of (1.1) in U(M)¯. In order to prove that (1.1) possesses uncountably many bounded positive solutions in U(M)¯, we need only to prove that z1≠z2. By means of (2.1)–(2.3), we know that there exists T3>max{T1,T2} satisfying
(2.31)∫T3+∞F(s)ds+∫T3+∞sn-1H(s)ds<|L1-L2|4.
It follows from (2.1), (2.2), (2.4), (2.7), (2.8), and (2.31) that for t≥T3(2.32)|z1(t)-z2(t)|=|L1-L2-∑i=1mpi(t)z1(τi(t))+∑i=1mpi(t)z2(τi(t))+∫t+∞f(s,z1(α1(s)),…,z1(αk(s)))ds-∫t+∞f(s,z2(α1(s)),…,z2(αk(s)))ds+(-1)n-1(n-1)!∫t+∞(s-t)n-1h(s,z1(β1(s)),…,z1(βk(s)))ds-(-1)n-1(n-1)!∫t+∞(s-t)n-1h(s,z2(β1(s)),…,z2(βk(s)))ds|≥|L1-L2|-p0‖z1-z2‖-2∫T3+∞F(s)ds-2∫T3+∞sn-1H(s)ds≥|L1-L2|-p0‖z1-z2‖-|L1-L2|2,which implies that
(2.33)‖z1-z2‖≥|L1-L2|2(1+p0)>0,
that is, z1≠z2. This completes the proof.Theorem 2.2.
Assume that there exist constantsM,N,p0,t1 and functions F,H∈C([t0,+∞),ℝ+) satisfying (2.1)–(2.3) and
(2.34)N<(1+p0)M,max⁡{pi(t):1≤i≤m}≤0,∑i=1mpi(t)≥p0>-1,∀t≥t1≥t0.
Then, (1.1) has uncountably many bounded positive solutions in U(M)¯.Proof.
LetL∈(N,(1+p0)M). It follows from (2.3) and (2.34) that there exists a constant T>1+|t0|+|t1|+|ν| satisfying
(2.35)∫T+∞F(s)ds+∫T+∞sn-1[|g(s)|+H(s)]ds<min⁡⁡{L-N,(1+p0)M-L,M-N2}.
Take ϵ0∈(0,min{L-N,(1+p0)M-L,(M-N/2)}) such that
(2.36)∫T+∞F(s)ds+∫T+∞sn-1[|g(s)|+H(s)]ds≤min⁡⁡{L-N,(1+p0)M-L,M-N2}-ϵ0.
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.Theorem 2.3.
Assume that there exist constantsM,N,p0,t1 and functions F,H∈C([t0,+∞),ℝ+) satisfying (2.1)–(2.3) and
(2.37)N<(1-p0)M,min⁡⁡{pi(t):1≤i≤m}≥0,∑i=1mpi(t)≤p0<1,∀t≥t1≥t0.
Then, (1.1) has uncountably many bounded positive solutions in U(M)¯.Proof.
LetL∈(p0M+N,M). It follows from (2.3) and (2.37) that there exists a constant T>1+|t0|+|t1|+|ν| satisfying
(2.38)∫T+∞F(s)ds+∫T+∞sn-1[|g(s)|+H(s)]ds<min⁡⁡{L-p0M-N,M-L,M-N2}.
Choose ϵ0∈(0,min{L-p0M-N,M-L,(M-N/2)}) such that
(2.39)∫T+∞F(s)ds+∫T+∞sn-1[|g(s)|+H(s)]ds≤min⁡⁡{L-p0M-N,M-L,M-N2}-ϵ0.
The rest of the proof is similar to that of Theorem 2.1 and is omitted. This completes the proof.Remark 2.4.
Theorems2.1–2.3 extend, improve, and unify the theorem in [1], Theorem  2.2 in [2], Theorem  1 in [4], Theorems  2.1 and 2.3 in [6], Theorems  1 and 3 in [7], and Theorems  1 and 3 in [8].


## 3. Examples and Applications

Now, we construct three nontrivial examples to show the superiority and applications of Theorems2.1–2.3, respectively.Example 3.1.
Consider the higher-order nonlinear neutral delay differential equation:(3.1)dndtn[x(t)-t2cos⁡⁡t1+6t2x(t+2)+(-1)ntsin⁡⁡(1-t3)1+4tx(t3-t)]+dn-1dtn-1[1+t2x4(t-1)-tx2(t2+2)1+t5+tx2(t2+2)sin⁡2⁡(t-t3x2(t-2))1+x2(t-2)+t3]+t3+(1/t)1+tn+4cos⁡3⁡(x(t2ln⁡t))+ln⁡(1+x2(2t))+t2sin⁡⁡(x(t2))1+tn+3+t2=1-t3tn+4ln⁡(1+t3),t≥3.
Let t0=t1=3,p0=5/12,m=2,k=3,M=36,N=3,ν=1,
(3.2)p1(t)=-t2cos⁡⁡t1+6t2,p2(t)=(-1)ntsin⁡⁡(1-t3)1+4t,τ1(t)=t+2,τ2(t)=t3-t,α1(t)=t-1,α2(t)=t2+2,α3(t)=t-2,β1(t)=t2ln⁡t,β2(t)=2t,β3(t)=t2,f(t,u,v,w)=1+t2u4-tv21+t5+tv2sin⁡2⁡(t-t3w2)1+w2+t3,h(t,u,v,w)=t3+(1/t)1+tn+4cos⁡3⁡u+ln⁡⁡(1+v2)+t2sin⁡⁡w1+tn+3+t2,F(t)=1+t2M4+tM21+t5+tM21+N2+t3,H(t)=t3+(1/t)1+tn+4+ln⁡⁡(1+M2)+t21+tn+3+t2,g(t)=1-t3tn+4ln⁡⁡(1+t3),∀(t,u,v,w)∈[t0,+∞)×R3.
It is clear that (2.1)–(2.4) hold. Consequently, Theorem 2.1 ensures that (3.1) has uncountably many bounded positive solutions in U(M)¯. But Theorem in [1], Theorems  2.1 and 2.3 in [6], Theorems  1 and 3 in [7], and Theorems  1 and 3 in [8] are null for (3.1).Example 3.2.
Consider the higher-order nonlinear neutral delay differential equation:(3.3)dndtn[x(t)-t21+3t2x(t+t2)-2t21+7t2x(t2-4t)]+dn-1dtn-1[2+t4x2(t-(1/t))1+t6+t2x2(t-(1/t))(1+t4)(1+tx2(2t2-t))]+t3-x2(t2-t)-t2x3(tln⁡⁡(1+t))(1+tn+4)[2+sin⁡2⁡(t3x(t2-t)x4(tln⁡⁡(1+t)))]=1+t1+tn+4(1-t+t2ln⁡⁡(1+2|t|)),t≥1.
Let t0=t1=1,p0=-13/21,m=2,k=2,M=400,N=100,ν=-4,
(3.4)p1(t)=-t21+3t2,p2(t)=-2t21+7t2,τ1(t)=t+t2,τ2(t)=t2-4t,α1(t)=t-1t,α2(t)=2t2-t,β1(t)=t2-t,β2(t)=tln⁡⁡(1+t),f(t,u,v)=2+t4u21+t6+t2u2(1+t4)(1+tv2),h(t,u,v)=t3-u2-t2v3(1+tn+4)(2+sin⁡2⁡(t3uv4))F(t)=2+t4M21+t6+t2M2(1+t4)(1+tN2),H(t)=t3+M2+t2M32+2tn+4,g(t)=1+t1+tn+4(1-t+t2ln⁡⁡(1+2|t|)),∀(t,u,v)∈[t0,+∞)×R2.
It is easy to verify that (2.1)–(2.3) and (2.34) hold. Consequently, Theorem 2.2 guarantees that (3.3) has uncountably many bounded positive solutions in U(M)¯. But Theorem in [1], Theorem  1 in [4], Theorem  2.3 in [6], Theorem  3 in [7], and Theorem  3 in [8] are useless for (3.3).Example 3.3.
Consider the higher-order nonlinear neutral delay differential equation:(3.5)dndtn[x(t)+t41+t2+4t4x(t4+1)+2ln⁡⁡(1+t2)1+3ln⁡⁡(1+t2)x(2+ln⁡2⁡t)]+dn-1dtn-1[t+x3(t3+2t2)-(-1)nt2sin⁡⁡(x(t2))1+t5+t2x4(t2)]+t3-x2(t-2)+t2x5(t-2)(1+tn+5)(1+tx2(t2ln⁡⁡(1+t)))=t-(-1)nln⁡⁡1+1+t2tn+(5/2)+1+cos⁡2⁡t,t≥2.
Let t0=t1=2,p0=11/12,m=2,k=2,M=24,N=1,ν=0,
(3.6)p1(t)=t41+t2+4t4,p2(t)=2ln⁡⁡(1+t2)1+3ln⁡⁡(1+t2),τ1(t)=t4+1,τ2(t)=2+ln⁡2⁡t,α1(t)=t3+2t2,α2(t)=t2,β1(t)=t-2,β2(t)=t2ln⁡⁡(1+t),f(t,u,v)=t+u3-(-1)nt2sin⁡⁡v1+t5+t2v4,h(t,u,v)=t3-u2+t2u5(1+tn+5)(1+tv2),F(t)=t+M3+t21+t5+t2N4,H(t)=t3+M2+t2M5(1+tn+5)(1+tN2),g(t)=t-(-1)nln⁡⁡(1+1+t2)tn+(5/2)+1+cos⁡2⁡t,∀(t,u,v)∈[t0,+∞)×R2.
Obviously, (2.1)–(2.3) and (2.37) hold. It follows from Theorem 2.3 that (3.5) has uncountably many bounded positive solutions in U(M)¯. But Theorem in [1], Theorem  2.2 in [2], Theorem  2.1 in [4], Theorem  2.1 in [6], Theorem  1 in [7], and Theorem  1 in [8] are inapplicable for (3.5).



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*Source: 328956-2011-10-11.xml*