# Oscillatory Criteria for Higher Order Functional Differential Equations with Damping

Authors: Peiguang Wang; Hai Cai
Journal: Journal of Function Spaces and Applications
 (2013)

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## Abstract

We investigate a class of higher order functional differential equations with damping. By using a generalized Riccati transformation and integral averaging technique, some oscillation criteria for the differential equations are established.


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## Body

## 1. Introduction

In this paper, we consider the following higher order functional differential equations with distributed deviating arguments of the form as follows:(1)x(n)(t)+p(t)x(n-1)(t)+∫ab‍q(t,ξ)×f(x[g1(t,ξ)],…,x[gm(t,ξ)])dμ(ξ)=0,t≥t0,
where n≥2 is an even number, p(t)∈C1([t0,∞),R+), q(t,ξ)∈C([t0,∞)×[a,b],R+), gi(t,ξ)∈C([t0,∞)×[a,b],R), liminft→∞,ξ∈[a,b]gi(t,ξ)=∞ for i∈Im={1,2,…,m}, and f(u1,…,um)∈C(Rm,R) has the same sign as u1,…,um; when they have the same sign, μ(ξ)∈C([a,b],R) is nondecreasing, and the integral of (1) is a Stieltjes one.We restrict our attention to those solutionsx(t) of (1) which exist on same half liner [tμ,∞) with sup{x(t):t≥T}≠0 for any T≥tμ and satisfy (1). As usual, a solution x(t) of (1) is called oscillatory if the set of its zeros is unbounded from above, otherwise, it is called nonoscillatory. Equation (1) is called oscillatory if all solutions are oscillatory.In recent years, there has been an increasing interest in studying the oscillation behavior of solutions for the differential equations with distributed deviating arguments, and a number of results have been obtained (refer to [1–3] and their references). However, to the best of our knowledge, very little is known for the case of higher order differential equations with damping. The purpose of this paper is to establish some new oscillation criteria for (1) by introducing a class of functions Φ(t,s,r) defined in [2] and a generalized Riccati technique.Firstly, we define the following two class functions.We say that a functionΦ=Φ(t,u,v) belongs to the function class X, denoted by Φ∈X, if Φ∈C(E,R), where E={(t,u,v):t0≤v≤u≤t<∞}, which satisfies Φ(t,t,v)=Φ(t,v,v)=0, Φ(t,u,v)≠0, v<u<t and has the partial derivative ∂Φ/∂u on E that is locally integrable with respect to u in E.LetD0={(t,u):t0≤u<t}, D={(t,u):t0≤u≤t}. We say that a function H=H(t,u) belongs to the function class Y, denoted by H∈Y, if H(t,t)=0 for t≥t0, H(t,u)≠0 in D0, H has continuous partial derivative in D0 with respect to t and s.In order to prove the main theorems, we need the following lemmas.Lemma 1 (see [4]).
Letx(t)∈Cn([t0,∞),R+), if x(n)(t) is of constant sign and not identically zero on any ray [tμ,+∞) for tμ≥t0, then there exists a tσ≥tμ, an integer l(0≤l≤n), with n+l even for x(t)x(n)(t)≥0 or n+l odd for x(t)x(n)(t)≤0; and for t≥tσ≥tμ, x(t)x(k)(t)>0, 0≤k≤l, and (-1)k+lx(t)x(k)(t)>0, l≤k≤n.Lemma 2 (see [5]).
If the functionx(t) is as in Lemma 1 and x(n-1)(t)x(n)(t)≤0 for t≥tσ≥tμ, then there exists a constant θ∈(0,1) such that for sufficiently large t, there exists a constant Mθ>0, satisfying
(2)x′(θt)≥Mθtn-2x(n-1)(t).Lemma 3 (see [3]).
Suppose thatx(t) is a nonoscillatory solution of (1). If
(3)limt→∞∫T0t‍exp{-∫T0s‍p(τ)dτ}ds=∞,T0≥t0,
then x(t)x(n-1)(t)>0 for any large t.


## 2. Main Results

Theorem 4.
Assume that (3) holds, and(A1)
there exists a functionσ(t)∈Cn([t0,∞),R+) such that σ(t)=min1≤i≤m{t,infξ∈[a,b]gi(t,ξ)}, limt→∞σ(t)=∞. σ′(t)≥σ>0, σn-2(t)≥ρn-2>0, (Mθ/2)σρn-2=:k>0, where σ, ρ, and k are constants.(A2)
f(u1,…,um) is nondecreasing with ui, i∈Im, and there exist constants N>0 and λ>0 such that
(4)liminfui0→∞f(u1,…,um)ui0≥λ,ui≥N,i≠i0.
If there exists a functionΦ(t,u,v)∈X, such that for any l(t)∈C1([t0,∞),R+), r(t)∈C1([t0,∞),R) and T0≥t0,
(5)limsupt→∞∫T0t‍{×[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]2Φ2(s,u,v)ψ(s)-ϕ(s)k×[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]2}ds>0,
where
(6)ϕ(t)=l(t)exp{-2k∫T0t‍r(s)ds},ψ(t)=ϕ(t){λ∫ab‍q(t,ξ)dμ(ξ)+kr2(t)-r′(t)-p′(t)2k-p2(t)4k}.
Then (1) is oscillatory.Proof.
Suppose to the contrary that (1) has a nonoscillatory solution x(t). Without loss of generality, we may suppose that x(t) is an eventually positive solution. From the conditions of gi(t,ξ) and f(u1,…,um), there exists a T0≥t0, such that
(7)x(t)>0,x[gi(t,ξ)]>0,f(x[g1(t,ξ)],…,x[gm(t,ξ)])>0,t≥T0,i∈Im.
By Lemma3, there exists a t1≥T0 such that x(n-1)(t)>0, t≥t1. Thus, we have
(8)x(n)(t)=-p(t)x(n-1)(t)-∫ab‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])×dμ(ξ)≤0,t≥t1.
By Lemma1, there exists a t2>t1 such that x′(t)>0, t≥t2. Further, by Lemma 2, there exist constant Mθ>0 and a t3≥t2, such that
(9)x′[σ(t)2]≥Mθσn-2(t)x(n-1)[σ(t)]≥Mθσn-2(t)x(n-1)(t),t≥t3.
Set(10)y(t)=ϕ(t){x(n-1)(t)x[σ(t)/2]+r(t)+p(t)2k},
then
(11)y′(t)=ϕ′(t)ϕ(t)y(t)+ϕ(t)×{-∫ab‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])x[σ(t)/2]×dμ(ξ)-p(t)x(n-1)(t)x[σ(t)/2]-σ′(t)x(n-1)(t)x′[σ(t)/2]2x2[σ(t)/2]+r′(t)+p′(t)2k}.
In view of(A1), (A2) and the definition of y(t), ϕ(t), we have
(12)y′(t)≤ϕ′(t)ϕ(t)y(t)+ϕ(t)×{-p(t)x(n-1)(t)x[σ(t)/2]-λ∫ab‍q(t,ξ)dμ(ξ)-k[x(n-1)(t)x[σ(t)/2]]2+r′(t)+p′(t)2k}=-ky2(t)ϕ(t)+l′(t)y(t)l(t)-ψ(t),
where ψ(t)=ϕ(t){λ∫ab‍q(t,ξ)dμ(ξ)+kr2(t)-r′(t)-p′(t)/2k-p2(t)/4k}.
Multiplying (12) by Φ2(t,u,v) and integrating from t3 to t, we have
(13)∫t3t‍Φ2(s,u,v)ψ(s)ds≤∫t3t‍Φ2(s,u,v)[-y′(s)+l′(s)l(s)y(s)]ds-∫t3t‍Φ2(s,u,v)ky2(s)ϕ(s)ds.
Integrating by parts and using integral averaging technique, we have(14)∫t3t‍Φ2(s,u,v)ψ(s)ds≤∫t3t‍ϕ(s)k[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]2ds-∫t3t{Φ(s,u,v)y(s)kϕ(s)-ϕ(s)k×[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]ϕ(s)k}2ds≤∫t3t‍ϕ(s)k[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]2ds,
thus
(15)limsupt→∞∫t3t{[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]Φ2(s,u,v)ψ(s)-ϕ(s)k×[Φs′(s,u,v)+l′(s)Φ(s,u,v)2l(s)]2}ds≤0,
which contradicts (5). This completes the proof of Theorem 4.If we chooseΦ(t,u,v)=H1(t,u)H2(u,v), where H1,H2∈Y. By Theorem 4, we have the following results.Corollary 5.
Assume that (3), (A1), and (A2) hold. If there exist H1,H2∈Y such that for each T0≥t0,
(16)limsupt→∞∫T0t‍H1(s,u)H2(u,v)×{ψ(s)-[h1(s,u)+h2(u,v)]24k}ds>0,
where h1 and h2 are defined by ∂H1(t,u)/∂u=h1(t,u)H1(t,u), ∂H2(u,v)/∂u=h2(u,v)H2(u,v), and
(17)ψ(t)=λ∫ab‍q(t,ξ)dμ(ξ)-p′(t)2k-p2(t)4k.
Then (1) is oscillatory.If we choosel(t)=1, r(t)=0, and let Φ(t,u,v)=(t-u)(u-v)α,α>1/2, by Theorem 4, we have the following corollary.Corollary 6.
Assume that (3), (A1), and (A2) hold. If there exists a constant α>1/2 such that for each T0≥t0,
(18)limsupt→∞1t2α+1∫T0t‍(s-u)2(u-v)2α×kψ(s)ds>α(2α-1)(2α+1),
where ψ(t) is defined as in Corollary 5. Then (1) is oscillatory.Theorem 7.
Assume that (3) holds, and(A3)
there exist functionsγi(t)∈Cn([t0,∞),R+), such that γi(t)≤min1≤i≤m{t,infξ∈[a,b]gi(t,ξ)}, limt→∞γi(t)=∞, γi′(t)≥γi>0, where γi are constants, i∈Im;(A4)
there exist constantsλi∈[0,1] and λ¯>0, such that
(19)∑i=1m‍λiγi(t)>0,f(u1,…,um)λ1u1+⋯+λmum≥λ¯.γin-2(t)≥ρin-2≥0, (Mθ/2)∑i=1m‍λiγiρin-2(t)=:k>0, where ρi and k are constants, i∈Im.If there exists a functionΦ∈X, such that for any l(t)∈C1([t0,∞),R+), r(t)∈C1([t0,∞),R), and T0≥t0, and (5) holds, where ϕ(t) is defined as in Theorem 4:
(20)ψ(t)=ϕ(t){λ¯∫ab‍q(t,ξ)dμ(ξ)+kr2(t)-r′(t)-p′(t)2k-p2(t)4k}.Then (1) is oscillatory.Proof.
Suppose to the contrary that (1) has a nonoscillatory solution x(t). Without loss of generality, we may suppose that x(t) is an eventually positive solution. Similar to the proof of Theorem 4, there exists a T0≥t0, such that x[gi(t,ξ)]>0, x[γi(t)]>0, x′(t)>0, f(x[g1(t,ξ)],…,x[gm(t,ξ)])>0, x(n-1)(t)>0, and x(n)(t)≤0, for t≥T0, i∈Im. Set
(21)y(t)=ϕ(t){x(n-1)(t)∑i=1mλix[γi(t)/2]+r(t)+p(t)2k},
then
(22)y′(t)=ϕ′(t)ϕ(t)y(t)+ϕ(t)×{p′(t)2k-p(t)x(n-1)(t)∑i=1mλix[γi(t)/2]-∫ab‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])∑i=1mλix[γi(t)/2]×dμ(ξ)-x(n-1)(t)2{∑i=1mλix[γi(t)/2]}2×∑i=1m‍λix′[γi(t)]γi′(t)+r′(t)+p′(t)2k}.
In view of(A3), (A4) and the definition of y(t), ϕ(t), we have
(23)y′(t)≤-ky2(t)ϕ(t)+l′(t)y(t)l(t)-ψ(t).
The following proof is similar to Theorem4, and we omit the details. This completes the proof of Theorem 7.Similar to Corollaries5 and 6, we have the following corollaries.Corollary 8.
Assume that (3), (A3), and (A4) hold. If there exist H1,H2∈Y such that for each T0≥t0, and (16) holds, where h1,h2 are defined as in Corollary 5:
(24)ψ(t)=λ¯∫ab‍q(t,ξ)dμ(ξ)-p′(t)2k-p2(t)4k.
Then (1) is oscillatory.Corollary 9.
Assume that (3), (A3), and (A4) hold. If there exists a constant α>1/2 such that for each T0≥t0, and (18) holds. where ψ(t) is defined as in Corollary 8, then (1) is oscillatory.For the case of the functionf(u1,…,um) with monotonicity, we have the following theorem.Theorem 10.
Assume that (3), (A3) hold, and(A5)
there exist∂f/∂ui and ∂f/∂ui≥λi≥0, such that ∑i=1m‍λiγi′(t)>0, where λi is constants. γin-2≥ρin-2≥0, (Mθ/2)∑i=1m‍λiγiρin-2(t)=:k>0, in which ρi and k are constants, i∈Im.
If there exists a functionΦ∈X, such that for any l(t)∈C1([t0,∞),R+), r(t)∈C1([t0,∞),R), and (5) holds, where ϕ(t) is defined as in Theorem 4:
(25)ψ(t)=ϕ(t){∫ab‍q(t,ξ)dμ(ξ)+kr2(t)-r′(t)-p′(t)2k-p2(t)4k}.
Then (1) is oscillatory.Proof.
Suppose to the contrary that (1) has a nonoscillatory solution x(t). Without loss of generality, we may suppose that x(t) is an eventually positive solution. Similar to the proof of Theorem 4, there exists a T0≥t0, when t≥T0, and we have x[gi(t,ξ)]>0, x[γi(t)]>0, x′(t)>0, f(x[g1(t,ξ)],…,x[gm(t,ξ)])>0, x(n-1)(t)>0, and x(n)(t)≤0, i∈Im. Set
(26)y(t)=ϕ(t){x(n-1)(t)f(x[γ1(t)/2]+⋯+x[γm(t)/2])+r(t)+p(t)2kx(n-1)(t)f(x[γ1(t)/2]+⋯+x[γm(t)/2])},
then
(27)y′(t)=ϕ′(t)ϕ(t)y(t)+ϕ(t)×{p′(t)2k-p(t)x(n-1)(t)f(x[γ1(t)/2]+⋯+x[γm(t)/2])-∫ab‍q(t,ξ)f(x[g1(t,ξ)],…,x[gm(t,ξ)])f(x[γ1(t)/2]+⋯+x[γm(t)/2])×dμ(ξ)-x(n-1)(t)2{f(x[γ1(t)/2]+⋯+x[γm(t)/2])}2×∑i=1m‍λix′[γi(t)]γi′(t)+r′(t)+p′(t)2kx(n-1)(t)f(x[γ1(t)/2]+⋯+x[γm(t)/2])}.
In view of(A3), (A5) and the definition of y(t), ϕ(t), we have
(28)y′(t)≤-ky2(t)ϕ(t)+l′(t)y(t)l(t)-ψ(t).
The following proof is similar to Theorem4, we omit the details. This completes the proof of Theorem 10.Similar to Corollaries5 and 6, we have the following corollaries.Corollary 11.
Assume that (3), (A3), and (A5) hold. If there exist H1,H2∈Y such that for each T0≥t0, and (16) holds, where h1,h2 are defined as in Corollary 5:
(29)ψ(t)=∫ab‍q(t,ξ)dμ(ξ)-p′(t)2k-p2(t)4k.
Then (1) is oscillatory.Corollary 12.
Assume that (3), (A3), and (A5) hold. If there exists a constant α>1/2 such that for each T0≥t0, and (18) holds. where ψ(t) is defined as in Corollary 11. Then (1) is oscillatory.


## 3. Examples

Example 13.
Consider the following equation(30)x(4)(t)+1tx(3)(t)∫0π/2‍ξt22x(t+sinξ)2-exp(-x2(t+cosξ))dξ=0,t≥1,
where u1=x(t+cosξ), u2=x(t+sinξ), obviously f(u1,u2)/u2≥1=λ. Choosing σ=1, ρ=1, then k=Mθ/2, and
(31)limsupt→∞1t2α+1∫0t‍(s-u)2(u-v)2α(Mθ2s2+14s2)ds=(Mθ2+14)limsupt→∞1t2α+1∫0t‍(s-u)2(u-v)2α1s2ds=(Mθ2+14)1α(2α+1)(2α-1).
Thus, there exists a constantα>1/2, such that (Mθ/2+1/4)>α2, that is,
(32)(Mθ2+14)1α(2α+1)(2α-1)>α(2α+1)(2α-1).
By Corollary6, then (30) is oscillatory.Example 14.
Consider the following equation(33)x(4)(t)+1tx(3)(t)+∫01‍ξt2×[x(t-ξ)+x(t+ξ)+x3(t-ξ)+x5(t+ξ)]dξ=0,
where f(u1,u2)=u1+u2+u13+u25, obviously ∂f/∂u1=1+3u12≥1, ∂f/∂u2=1+5u24≥1. Choosing λi=1, γi=1, ρi=1, and i=1,2, then k=Mθ. By Corollary 12, then (33) is oscillatory.



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