“A Study On Oscillation Criteria for Linear and
Non-Linear Neutral Delay Differential Equations"
CHAPTER - I
INTRODUCTION
This dissertation entitled “A STUDY ON OSCILLATION CRITERIA FOR LINEAR AND
NON-LINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS’’ contains four chapters it is meant to study the basic concept and various interesting result on oscillation of second order Emden-fowler nonlinear neutral delay differential equations.
The chapter-1 is meant to recall the basic definition and various well known results relevant to this dissertation.
In chapter-2, we study about the basic definitions, and the most important results are:
Employing Riccati techniques and the integral averaging method,we establish interval oscillation criteria for the second-order Emden-Fowler neutral delay differential equation
[|𝑥′(𝑡)|𝛾−1𝑥′(𝑡)]′+ 𝑞1(𝑡)|𝑦(𝑡 − 𝜎)|𝛼−1𝑦(𝑡 − 𝜎) + 𝑞
2(𝑡)𝑦(𝑡 − 𝜎)=0
where t ≥ t0 and x(t) = y(t) + p(t)y(t−𝜏).
Consider the second-order Emden-Fowler neutral delay differential equation
[|𝑥′(𝑡)|𝛾−1𝑥′(𝑡)]′+ 𝑞1(𝑡)|𝑦(𝑡 − 𝜎)|𝛼−1𝑦(𝑡 − 𝜎) + 𝑞
2(𝑡)|𝑦(𝑡 − 𝜎)|𝛽−1𝑦(𝑡 − 𝜎) = 0 (1.1)
where t ≥ t0 and x(t) = y(t) + p(t)y(t−𝜏).
We assume that
(A1) 𝜏 and 𝜎 are nonnegative constants, α,β and γ are positive constants with 0 <α<γ<β (A2) q1, q2 ϵC ([t0, ∞), R+), R+ = (0, ∞)
(A3) p ϵ C ([t0, ∞), R), and −1 < p0 ≤ p(t) ≤1, p0 is a constant.
For any 𝜑ϵC ([t0−θ, t0],R), θ = max{τ, σ}, (1.1) has a solution y(t) extendable on [t0,∞) satisfying the initial
condition y(t) ≡𝜑(t) for [t0-θ, t0]
𝑦′′(𝑡) + 𝑞(𝑡)|𝑦(𝑡)|𝛾−1𝑦(𝑡) = 0, 𝑞𝜖∁([𝑡
0,∞), 𝑅)𝑎𝑛𝑑 𝛾 > 0 (1.2)
have been extended to the second order neutral delay differential equation
[𝑦(𝑡) + 𝑝(𝑡)𝑦(𝑡 − 𝜏)]′′+ 𝑞(𝑡)𝑓((𝑦 − 𝜎)) = 0 (1.3)
Under the assumption that the nonlinear function f satisfies The sub linear condition
0 < ∫0∈+𝑓(𝑢)𝑑𝑢 ,∫
𝑑𝑢 𝑓(𝑢)<∞ −∈
0− 𝑓𝑜𝑟𝑎𝑙𝑙 ∈> 0 , as well as the super linear condition
0 < ∫ 𝑑𝑢
𝑓(𝑢) , ∫ 𝑑𝑢
𝑓(𝑢)< ∞, −∞
−∈ 𝑓𝑜𝑟 𝑎𝑙𝑙 ∈> 0
∞
Also it will be of great interest to find some oscillation criteria for special case for (1.3), even for the Emden-Fowler neutral delay differential equation
[y(t) + p(t)y(t −τ)]'' + q(t)|y(t −σ)|γ-1y(t-σ) = 0, γ> 0. (1.4)
The first beautiful interval criteria in this direction for some interval criteria for the oscillation of the second order linear ordinary differentia equation
(r(t)y'(t))'(t) + q(t)y(t) = 0 (1.5) Recently, Kong-type interval criteria to certain neutral differential equations.
In chapter-3, we study about the oscillation criteria for second order nonlinear neutral delay differential equations.
The oscillation of second order neutral differential equation
(x(t) + p(t)x(τ (t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, (1.6)
Where p,q Є C([t0,∞),R), f Є C(R,R):
Throughout this dissertation, we assume that 0 ≤ p(t) ≤ p0<+∞, q(t) ≥ 0, and q(t) is not identically zero on
any ray of the form [t*,∞) for any t* ≥ t0, where p0 is a constant, The oscillation of the second-order linear
ordinary differential equation
x''(t) + p(t)x(t) = 0, (1.7) and used the class of functions as follows: Suppose there exist continuous functions
H, h : D ≡ {(t, s) : t ≥ s ≥ t0 }→R such that H(t, t) = 0, t ≥ t0,
H(t, s) >0, t > s ≥ t0, and H has a continuous and non positive partial derivative on D with respect to
the second variable.
then every solution of Eq. (1.7) oscillates. The oscillation criteria for second order linear equations for nonlinear analysis, we have
(r(t)x'(t))'+ p(t)x(t) = 0, (1.8) and used the generalized Riccati substitution and established some new sufficient conditions for oscillation. If there exists a positive function Є C1([t
0,∞),R+) such that
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡, 𝑡0) ∫[𝑎(𝑠)𝑟(𝑠)ℎ(𝑡, 𝑠)]𝑑𝑠 <∞,
𝑡
𝑡0
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡, 𝑡0) ∫ 𝑎(𝑠)[𝐻(𝑡, 𝑠)𝜓(𝑠) − 1
4𝑟(𝑠)ℎ
2(𝑡, 𝑠)]𝑑𝑠 = ∞, 𝑡
𝑡0
Where a(s) = exp{− ∫ 𝑔(𝑢)𝑑𝑢} 0𝑠 and ψ(s) = {p(s)+r(s)g2(s)-(r(s)g(s))'}, then every solution of (1.8)
(1
𝑡𝑥
′(𝑡))′+ 1
𝑡3𝑥(𝑡)=0 is oscillatory.
An important tool in the study of oscillation is the integral averaging technique. Say a function H = H(t, s) belongs to a function class H, denoted by
H Є H, if H Є C(D,R+∪ {0}),
Where D = {(t, s) :t0 ≤ s ≤ t < ∞ and R+ = (0,∞), which satisfies
H(t, t) = 0, H(t, s) >0, for t > s, and has partial derivatives ∂H/∂t and ∂H/∂s on D such that
𝜕𝐻(𝑡,𝑠)
𝜕𝑡 = ℎ1(𝑡, 𝑠)√𝐻(𝑡, 𝑠)and 𝜕𝐻(𝑡,𝑠)
𝜕𝑠 = −ℎ2(𝑡, 𝑠)√𝐻(𝑡, 𝑠)
Sun defined another type of function class X and considered the oscillation of the second-order nonlinear damped differential equation
(r(t)y'(t))'+ p(t)y'(t) + q(t)f(y(t)) = 0, (1.9) The oscillation of the second-order neutral delay differential equation
[r(t)(y(t) + p(t)y(σ(t)))']'+∑𝑛𝑖=1𝑞𝑖(𝑡)𝑓𝑖(𝑦(𝜏𝑖(𝑡))) = 0 (1.10)
𝜔(𝑡) = 𝑟(𝑡)𝑧
′(𝑡)
𝑧(𝑡), 𝑧(𝑡) = 𝑦(𝑡) + 𝑝(𝑡)𝑦(𝜎(𝑡)),
CHAPTER-II
PRELIMINARIES
Definition: 2.1
An equation involving derivatives or differential of one or more dependent variables with respect to one or more independent variables is called differential equation.
Examples:
1. 𝑑𝑦
𝑑𝑥 = (𝑥 + 𝑠𝑖𝑛𝑥)
2. 𝑑 4𝑥
𝑑𝑡4 +
𝑑2𝑥 𝑑𝑡2 + [
𝑑𝑥 𝑑𝑡]
5= 𝑒𝑡
3. 𝜕 2𝑢
𝜕𝑥2+
𝜕2𝑢
𝜕𝑦2+
𝜕2𝑢
𝜕𝑧2 = 0
Definition:2.2
A differential equation involving derivatives with respect to single independent variable is called an Ordinary Differential Equation.
Equation 1 & 2 are Ordinary Differential Equations.
Definition: 2.3
A partial differential equation is an equation which equation contains one or more partial derivatives. Equation (3) is second order partial differential equation.
Definition: 2.4
A non trivial solution x(t) is said to be oscillatory, if it has arbitrarily large zeros for t ≥ t0,
That is, there exist a sequence of zeros {tn}(x(tn)=0) of x(t) such that lim
𝑛→∞𝑠 𝑡𝑛 = +∞
Definition: 2.5
A non trivial solution x(t) is said to be non oscillatory, if there exist a t1, such that x(t) ≠ 0 for all t ≥
t1
Definition: 2.6
defined by
𝑇𝑛[𝑔, 𝑙, 𝑡] = ∫𝑡Φ𝑛(𝑡, 𝑠, 𝑙)𝑔(𝑠)𝑑𝑠
𝑙 ,
for n ≥ 1, t ≥ s ≥ l ≥ t0and g Є C([t0,∞),R).s
Definition: 2.7
The function φ = φ(t, s, l) is defined by
𝜕Φ(t, s, 𝑙)
∂s = 𝜑(𝑡, 𝑠, 𝑙)Φ(t, s, 𝑙)
It is easy to verify that Tn[∙;l, t] is a linear operator and that it satisfies
CHAPTER-III
OSCILLATION CRITERIA FOR SECOND ORDER NONLINEAR
NEUTRAL DELAY DIFFERENTIAL EQUATIONS
In this chapter, we give some new oscillation results for (1.6). We start with the following oscillation results.
Theorem:3.1
Assume that σ(t) ≤τ(t) for t ≥ t0: Further, suppose that there exists a function g Є C1 ([t0, ∞), R)such that for
some β ≥ 1 and some H Є H, one has
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡,𝑡0)∫ [𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 +
𝑝0
𝜏0)
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ
2(𝑡, 𝑠)] 𝑑𝑠 𝑡
𝑡0 = ∞ (3.1)
where ψ(t) := u(t){k Q(t)+(1 + p0/τ0)[σ'(t)g2(t)-g'(t)]},
Q(t):= min {q(t), q (τ(t))}, u (t) :=exp {2∫ 𝜎𝑡𝑡 ′(𝑠)𝑔(𝑠)𝑑𝑠}
0 .
Then every solution of
(x(t) + p(t)x (τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, is oscillatory.
Proof:
Let x be a non oscillatory solution of
(x(t) + p(t)x(τ (t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0,.
Without loss of generality, we assume that there exists t1≥ t0 such that
x(t) >0, x(τ(t)) >0, x(σ(t)) >0, for all t ≥ t1: Define
z(t) = x(t) + p(t)x(τ(t)) for t ≥ t0, then z(t) >0 for t ≥ t1.
From (1.6), we have
z''(t)+q(t)f(x(σ(t)))=0, t≥t1
then by (b), we have
z''(t)=-q(t)f(x(σ(t)))
z''(t) ≤ -kq(t)x(σ(t)) ≤ 0, t ≥ t1 (3.2)
It is obvious that z''(t) ≤0 and z(t) >0 for t ≥ t1
Implies z'(t) >0 for t ≥ t1.
Using (3.2) and the condition (b), there exists t2≥ t1 such that for t ≥ t2,
We get
= z''(t) + q(t)f(x(σ(t))) + p0[z''(τ (t)) + q(τ (t))f(x(σ(τ (t))))]
=[z(t) + p0z(τ (t))]''+ q(t)f(x(σ(t))) + p0q(τ (t))f(x(σ(τ (t))))
≥[z(t) +𝑝0
𝜏0z(τ (t))]
′′+k[q(t)x(σ(t)) + p
0q(τ (t))x(τ (σ(t)))]
≥ [z(t) +𝑝0
𝜏0z(τ (t))]'' +kQ(t)[x(σ(t)) + p0x(τ(σ(t)))] ≥[z(t) +𝑝0
𝜏0z(τ (t))]'' + kQ(t)z(σ(t)). (3.3) We introduce a generalized Riccati transformation
ω(t) = u(t)[ 𝑧′(𝑡)
𝑧(𝜎(𝑡))+ 𝑔(𝑡)] (3.4)
Note that σ (t) ≤ t. Then we have z''(t) ≤ z'(σ(t)) Differentiating (3.4), Thus, there exists t3≥ t1 such that for
all t ≥ t3,
ω'(t)= u'(t)[ 𝑧′(𝑡)
𝑧(𝜎(𝑡))+g(t)]
+u(t)[𝑧
′′(𝑡)𝑧(𝜎(𝑡))−𝑧′(𝑡)𝑧′(𝜎(𝑡))𝜎′(𝑡)
[𝑧(𝜎(𝑡))]2 + 𝑔
′(𝑡)]
≤𝑢(𝑡) 𝑧′′(𝑡)
𝑧(𝜎(𝑡))− 𝑢(𝑡) [𝜎
′(𝑡)( 𝑧′(𝑡)
𝑧(𝜎(𝑡)))
2+ 𝑔′(𝑡)] + 𝑢′(𝑡)𝑤(𝑡)
𝑢(𝑡)
ω'(t)≤-2σ'(t)g(t)ω(t) + u(t){ 𝑧′′(𝑡)
𝑧(𝜎(𝑡))− 𝜎
′(𝑡) [𝜔(𝑡)
𝑢(𝑡)− 𝑔(𝑡)] 2
+ 𝑔′(𝑡)} = 𝑢(𝑡) 𝑧′′(𝑡)
𝑧(𝜎(𝑡))+ 𝑢(𝑡)[−𝜎
′(𝑡)𝑔2(𝑡) + 𝑔′(𝑡)] − 𝜎′(𝑡)𝜔2(𝑡)
𝑢(𝑡) (3.5)
Similarly, we introduce another generalized Riccati transformation υ(t) = u(t)[𝑧′(𝜏(𝑡))
𝑧(𝜎(𝑡))+ 𝑔(𝑡)] (3.6)
Differentiating (3.6), note that σ(t) ≤ τ (t), by (3.2) we have
z'(σ(t)) ≥ z'(τ (t)), then for all sufficiently large t, one has
v'(t) = u'(t)[𝑧′(τ(𝑡))
𝑧(𝜎(𝑡))+g(t)]+u(t)[
𝑧′′(τ(𝑡))𝜏′(𝑡)𝑧(𝜎(𝑡))−𝑧′(𝜎(𝑡))𝑧′(τ(𝑡))𝜎′(𝑡)
[𝑧(𝜎(𝑡))]2 + 𝑔
′(𝑡)]
υ'(t) ≤ -2σ'(t)g(t)υ(t) + u(t){𝜏0𝑧′′(𝜏(𝑡))
𝑧(𝜎(𝑡)) − 𝜎
′(𝑡) [υ(𝑡)
𝑢(𝑡)− 𝑔(𝑡)] 2
+ 𝑔′(𝑡)}
= τ0𝑢(𝑡)
𝑧′′(𝜏(𝑡))
𝑧(𝜎(𝑡)) + 𝑢(𝑡)[−𝜎
′(𝑡)𝑔2(𝑡) + 𝑔′(𝑡)] − 𝜎′(𝑡)υ2(𝑡)
𝑢(𝑡) (3.7)
From (3.5) and (3.7), we have
[ω(t) +p0
τ0 υ(t)] ′
≤ u(t)
z(σ(t))[z(t) + p0
τ0 z(τ(t))]"
+(1 +p0
τ0)𝑢(𝑡)[−𝜎′(𝑡)𝑔2(𝑡) + 𝑔′(𝑡)] −
𝜎′(𝑡)ω2(𝑡)
𝑢(𝑡) −
p0
τ0
σ′(t)υ2(t)
u(t)
[ω(t) +p0 τ0υ(t)]
′
≤ −ψ(t) −𝜎′(𝑡)ω2(𝑡)
𝑢(𝑡) −
p0 τ0
σ′(t)υ2(t)
u(t) (3.8)
Multiplying (3.8) by H(t, s) and integrating from T to t, we have, for any β ≥ 1 and for all t ≥ T ≥ t3,
∫ 𝐻(𝑡, 𝑠)𝜓(𝑠)𝑑𝑠 ≤ − ∫ H(t, s)ω′(s)ds − ∫ H(t, s)𝜎′(𝑠)ω2(𝑠)
𝑢(𝑠) 𝑑𝑠 t T t T 𝑡 𝑇
−p0
τ0 ∫ H(t, s)υ′(s)ds −
p0
τ0 ∫ H(t, s)
𝜎′(𝑠)υ2(𝑠)
𝑢(𝑠) 𝑑𝑠
t
T t
T
Using integration by parts we get,
∫ 𝐻(𝑡, 𝑠)ψ(s)ds = −H(t, s)ω(s)|Tt 𝑡
𝑇
− ∫ [−∂H(t, s)
∂(s) ω(s) + H(t, s)
𝜎′(𝑠)ω2(𝑠)
𝑢(𝑠) ] 𝑑𝑠
t
T
−p0
τ0H(t, s)υ(s)|Tt −
p0
τ0∫[−
∂H(t, s)
∂(s) υ(s) + H(t, s)
𝜎′(𝑠)υ2(𝑠)
𝑢(𝑠) ]𝑑𝑠
t
T
Since H(t,t)=0
= 𝐻(𝑡, 𝑇)𝜔(𝑇) − ∫[ℎ(𝑡, 𝑠)√𝐻(𝑡, 𝑠)𝜔(𝑠) +
𝑡
𝑇
H(t, s)𝜎
′(𝑠)ω2(𝑠)
𝑢(𝑠) ]𝑑𝑠
+ p0
τ0
H(t, T)υ(T)p0
τ0
∫[h(t, s)√H(t, s)υ(s) + H(t, s)𝜎
′(𝑠)υ2(𝑠)
𝑢(𝑠) ]𝑑𝑠
t
T
= 𝐻(𝑡, 𝑇)𝜔(𝑇) − ∫[√𝐻(𝑡, 𝑠)𝜎
′(𝑠)
𝛽𝑢(𝑠)
𝑡
𝑇
𝜔(𝑠) + √𝛽𝑢(𝑠)
4𝜎′(𝑠)ℎ(𝑡, 𝑠)]2𝑑𝑠
+ ∫𝛽𝑢(𝑠)
4𝜎′(𝑠)ℎ
2(𝑡, 𝑠)𝑑𝑠 − ∫(𝛽 − 1)𝜎
′(𝑠)𝐻(𝑡, 𝑠)
𝛽𝑢(𝑠) 𝜔 2(𝑠)𝑑𝑠 𝑡 𝑇 𝑡 𝑇
+p0
τ0𝐻(𝑡, 𝑇)υ(𝑇) −
p0
τ0 ∫[√
𝐻(𝑡, 𝑠)𝜎′(𝑠)
𝛽𝑢(𝑠)
𝑡
𝑇
υ(𝑠) + √𝛽𝑢(𝑠)
4𝜎′(𝑠)ℎ(𝑡, 𝑠)]2𝑑𝑠 +p0
τ0∫
𝛽𝑢(𝑠) 4𝜎′(𝑠)ℎ
2(𝑡, 𝑠)𝑑𝑠 −p0 τ0 ∫
(𝛽−1)𝜎′(𝑠)𝐻(𝑡,𝑠)
𝛽𝑢(𝑠) υ
2(𝑠)𝑑𝑠 𝑡
𝑇 𝑡
𝑇 (3.9)
From the above inequality and using monotonicity of H, for all t ≥ t3, we obtain
∫[𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 +p0
τ0)
𝑡
𝑡3
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ2(𝑡, 𝑠)]𝑑𝑠 ≤ 𝐻(𝑡, 𝑡0)|𝜔(𝑡3)| +
p0
τ0 H(t, t0)
∫[𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 +p0
τ0)
𝑡
𝑡0
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ2(𝑡, 𝑠)]𝑑𝑠 ≤ 𝐻(𝑡, 𝑡0)[∫ |𝜓(𝑠)|𝑑𝑠 +𝑡𝑡3
0 |𝜔(𝑡3)| +
p0
τ0|υ(t3)|] (3.10) By (3.10),
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡, 𝑡0) ∫[𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 + p0
τ0)
𝑡
𝑡0
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ2(𝑡, 𝑠)]𝑑𝑠
≤ ∫ |𝜓(𝑠)|𝑑𝑠 +𝑡𝑡3
0 |𝜔(𝑡3)| +
p0
τ0|υ(t3)|< ∞. This contradicts to
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡,𝑡0)∫ [𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 +
𝑝0
𝜏0)
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ
2(𝑡, 𝑠)] 𝑑𝑠 =∞ 𝑡
𝑡0 ,
This completes the proof.
Corollary: 3.1
Suppose that σ(t) ≤τ(t) for t ≥ t0.Furthermore, assume that there exists a function g Є C1([t0,∞),R)
such that for some integer n >2 and some β ≥ 1,
lim
𝑡→∞𝑠𝑢𝑝 𝑡
1−𝑛 ∫(𝑡 − 𝑠)𝑛−3[(𝑡 − 𝑠)2𝜓(𝑠) 𝑡
𝑡0
−(1 +𝑝0 𝜏0)
𝛽(𝑛 − 1)2
4𝜎′(𝑠) 𝑢(𝑠)]𝑑𝑠 =∞,
where, ψ(t) := u(t){kQ(t)+(1 + p0/τ0)[σ'(t)g2(t)-g'(t)]},
Q(t):= min{q(t), q(τ(t))}, u(t) :=exp{2∫ 𝜎𝑡𝑡 ′(𝑠)𝑔(𝑠)𝑑𝑠}
0 .
Then every solution of
(x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0,
Isoscillatory.
Proof:
By theorem (3.1), we have
lim
𝑡→∞𝑠𝑢𝑝
1
(𝑡 − 𝑡0)𝑛−1 ∫[(𝑡 − 𝑠)𝑛−1𝜓(𝑠) 𝑡
𝑡0
− (1 +p0 τ0)
𝛽
4𝜎′(𝑠)(𝑛 − 1)
= lim
𝑡→∞𝑠𝑢𝑝 𝑡
1−𝑛∫ (𝑡 − 𝑠)𝑡 𝑛−3[(𝑡 − 𝑠)2𝜓(𝑠) −
𝑡0 (1 +
𝑝0
𝜏0)
𝛽(𝑛−1)2
4𝜎′(𝑠) 𝑢(𝑠)]𝑑 = ∞
Example: 3.2
Consider the second-order neutral differential equation [x(t) + (3 + sin t)x(t - τ )]''+ γ
𝑡2𝑥(𝑡 − 𝜎) = 0, t ≥ 1, (3.11)
Where σ ≥ τ, γ>0
Let p(t) = 3 + sin t, q(t) =γ
𝑡2, f(x) = x, g(t) = −1 2𝑡
Thenu(t) = exp {−2 ∫ 𝑔(𝑡)𝑑𝑡}𝑡𝑡
0
= exp {−2 ∫ −1
2𝑡 𝑑𝑡}
𝑡
𝑡0
= exp(log t) =t, ψ(t) = 𝑢(𝑡) {𝑘𝑄(𝑡) + (1 +p0
τ0) [𝜎′(𝑡)𝑔
2(𝑡) − 𝑔′(𝑡)]}
= 𝑡 {𝑘 γ
𝑡2+ (1 +
p0
τ0) [(1)
1 4𝑡2−
1 2𝑡2]}
Taking k=1, p=4 and τ0=τ'(t)=1 we have,
Ψ(t) = (γ – 5/4)/t.
Applying Corollary 3.1 with n = 3, for any β ≥ 1,
lim
𝑡→∞𝑠𝑢𝑝 𝑡
1−𝑛 ∫(𝑡 − 𝑠)𝑛−3[(𝑡 − 𝑠)2𝜓(𝑠) 𝑡
𝑡0
−(1 +𝑝0 𝜏0)
𝛽(𝑛 − 1)2
4𝜎′(𝑠) 𝑢(𝑠)]𝑑𝑠 = lim
𝑡→∞𝑠𝑢𝑝 𝑡
−2∫(𝑡 − 𝑠)0[(𝑡 − 𝑠)2(𝛾 − 5 4) 𝑠 − 𝑡 1 𝛽𝑠]𝑑𝑠 = lim
𝑡→∞𝑠𝑢𝑝 𝑡
−2∫[(𝑡2− 2𝑡𝑠 + 𝑠2)(𝛾 − 5 4) 𝑠 − 𝑡 1 𝛽𝑠]𝑑𝑠 = lim
𝑡→∞𝑠𝑢𝑝 𝑡
−2[(𝑡2𝑙𝑜𝑔𝑠 − 2𝑡𝑠 +𝑠2
2) (𝛾 − 5 4) − 𝛽
𝑠2
2]1
𝑡
= lim
𝑡→∞𝑠𝑢𝑝 𝑡
−2[(𝑡2𝑙𝑜𝑔𝑡 − 2𝑡2+𝑡2 2) (𝛾 −
5 4) − 𝛽
𝑡2
2] − [(𝑡
2𝑙𝑜𝑔1 − 2𝑡 +1 2) (𝛾 −
5 4) − 𝛽
= ∞, for γ>5/4. Hence,
[x(t) + (3 + sin t)x(t - τ)]''+ γ
𝑡2𝑥(𝑡 − 𝜎) = 0, t ≥ 1, is oscillatory for γ>5/4. Corollary 3.1 can be applied to the second-order
Euler Differential Equation
x''(t) +γ
𝑡2x(t) = 0, t ≥ 1, (3.12)
where γ>0. Let p(t) = 0, q(t) = γ/t2, f(x) = x, g(t) = -1/(2t)
Then u(t) = t, Ψ(t) =(γ -1/4)/t. Take k =1, p0 = 0.
Applying Corollary 4.1 with n = 3, for any β ≥ 1,
lim sup
𝑡→∞ 𝑡
1−𝑛 ∫(𝑡 − 𝑠)𝑛−3[(𝑡 − 𝑠)2𝜓(𝑠) − 𝑡
𝑡0
𝛽(𝑛 − 1)2(1 + 𝑝 0)
4 𝑢(𝑠)]𝑑𝑠
= lim
𝑡→∞𝑠𝑢𝑝
1
𝑡2∫ [(γ−
1 4)
(𝑡−𝑠)2
𝑠 −βs] ds =∞
𝑡
1 For γ > ¼.
Hence, x’’ (t) +γ
𝑡2x (t) = 0, t ≥ 1is oscillatory for γ > ¼.
It may happen that assumption (3.1) is not satisfied, or it is not easy to verify, consequently, Theorem 3.1 does not apply or is difficult to apply. The following results provide some essentially new oscillation criteria for equation
(x(t) + p(t)x(τ (t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0.
Theorem 3.2
Assume that σ(t) ≤ τ (t) for t ≥ t0, and for some H Є H,
0 < inf
s≥t0
[ lim inf
t→∞
H(t,s)
H(t,t0)] ≤∞. (3.13)
Further, suppose that there exist functions g Є C1([t
0,∞),R) and m Є C([t0,∞),R) such that for all T ≥ t0and
for some β >1,
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡,𝑇)∫ [𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 +
p0 τ0)
𝑡 𝑇
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ
2(𝑡, 𝑠)]𝑑𝑠 ≥ 𝑚(𝑇), (3.14)
Where ψ(t) = u(t){kQ(t)+(1 + p0/τ0)[σ'(t)g2(t)-g'(t)]},
u(t) =exp{2∫ 𝜎𝑡𝑡 ′(𝑠)𝑔(𝑠)𝑑𝑠}
0 .
Suppose further that
lim
𝑡→∞𝑠𝑢𝑝 ∫
𝜎′(𝑠)𝑚+2(𝑠)
𝑢(𝑠) 𝑡
𝑡0 𝑑𝑠 =∞, (3.15) where m+(t) := max{m(t), 0}.Then every solution of
(x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, Isoscillatory.
Proof:
Assuming, without loss of generality, that there exists a solution x of
(x(t) + p(t)x(τ (t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, such that x(t) >0,
x(τ (t)) >0, x(σ(t)) >0, for all t ≥ t1 .
We define the functions
ω(t) = u(t)[ 𝑧′(𝑡)
𝑧(𝜎(𝑡))+ 𝑔(𝑡)]and
υ(t) = u(t)[𝑧′(𝜏(𝑡))
𝑧(𝜎(𝑡))+ 𝑔(𝑡)]·
we arrive at the inequality (3.9), which yields for t > T ≥ t1, sufficiently large 1
𝐻(𝑡, 𝑇)∫[𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 + p0
τ0)
𝑡
𝑇
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ2(𝑡, 𝑠)]𝑑𝑠
≤ 𝜔(𝑡) − 1
𝐻(𝑡, 𝑇)∫
(𝛽 − 1)𝜎′(𝑠)𝐻(𝑡, 𝑠)
𝛽𝑢(𝑠) 𝜔
2(𝑠)𝑑𝑠 𝑡
𝑇
+𝑝0
𝜏0𝜐(𝑡) − 𝑝0 𝜏0
1
𝐻(𝑡, 𝑇)∫
(𝛽 − 1)𝜎′(𝑠)𝐻(𝑡, 𝑠)
𝛽𝑢(𝑠) 𝜐
2(𝑠)𝑑𝑠 𝑡
𝑇
Therefore, for t > T ≥ t1, sufficiently large
lim
𝑡→∞𝑠𝑢𝑝
1
𝐻(𝑡, 𝑇)∫[𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 + p0
τ0
)
𝑡
𝑇
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ
2(𝑡, 𝑠)]𝑑𝑠
≤ 𝜔(𝑡) +𝑝0
𝜏0𝜐(𝑡) − lim inf𝑡→∞
1
𝐻(𝑡, 𝑇)∫
(𝛽 − 1)𝜎′(𝑠)𝐻(𝑡, 𝑠)
𝛽𝑢(𝑠) (𝜔
2(𝑠)
𝑡
𝑇
+ 𝑝0 𝜏0
𝜐2(𝑠))𝑑𝑠
𝜔(𝑡) +𝑝0
𝜏0𝜐(𝑡) ≥ 𝑚(𝑇) + lim inf𝑡→∞
1
𝐻(𝑡, 𝑇)∫
(𝛽 − 1)𝜎′(𝑠)𝐻(𝑡, 𝑠)
𝛽𝑢(𝑠) (𝜔
2(𝑠)
𝑡
𝑇
+𝑝0 𝜏0𝜐
2(𝑠))𝑑𝑠
for all T ≥ t1 and for any β >1.Consequently, for all T ≥ t1, we obtain
ω(T) + 𝑝0
𝜏0υ(T) ≥ m(T), and (3.16)
lim inf
𝑡→∞
1 𝐻(𝑡, 𝑡1) ∫
𝜎′(𝑠)𝐻(𝑡, 𝑠)
𝑢(𝑠) (𝜔
2(𝑠) +𝑝0
𝜏0𝜐
2(𝑠))𝑑𝑠 𝑡
𝑡1
≤ 𝛽
𝛽−1(𝜔(𝑡1) +
𝑝0
𝜏0𝜐(𝑡1) − 𝑚(𝑡1)) < ∞ (3.17) In order to prove that
∫ 𝜎′(𝑠)
𝑢(𝑠)(𝜔
2(𝑠) +𝑝0
𝜏0𝜐
2(𝑠))𝑑𝑠
∞
𝑡1 < ∞ (3.18) suppose the contrary, that is,
∫ 𝜎𝑢(𝑠)′(𝑠)(𝜔2(𝑠) +𝑝0
𝜏0𝜐
2(𝑠))𝑑𝑠
∞
𝑡1 =∞ (3.19)
Assumption
0 < inf
s≥t0
[ lim inf
t→∞
H(t, s)
H(t, t0)] ≤∞
Existence of a ρ>0 such that
inf
s≥t0
[ lim inf
t→∞
H(t,s)
H(t,t0)] > 𝜌 (3.20) By (3.20), we have
lim inf
t→∞
H(t, s)
H(t, t0)> 𝜌 > 0,
and there exists a T2≥ T1 such that H(t, T1)/H(t, t0) ≥ ρ, for all t ≥ T2 .
On the other hand, by virtue of (3.19), for any positive number κ, there exists a T1≥ t1 such that, for all t ≥
T1,
∫ 𝜎𝑢(𝑠)′(𝑠)(𝜔2(𝑠) +𝑝0
𝜏0𝜐
2(𝑠))𝑑𝑠 𝑡
𝑡1 ≥
𝜅 𝜌.
Using integration by parts, we conclude that, for all t ≥ T1, 1
𝐻(𝑡,𝑡1)∫
𝐻(𝑡,𝑠)𝜎′(𝑠)
𝑢(𝑠) (𝜔
2(𝑠) +𝑝0
𝜏0𝜐
2(𝑠))𝑑𝑠 𝑡
𝑡1
= 1
𝐻(𝑡, 𝑡1) ∫ [−
𝜕𝐻(𝑡, 𝑠)
𝜕𝑠 ] [ ∫
𝜎′(𝜐) 𝑢(𝜐)
𝑠
𝑡1
(𝜔2(𝜐) +𝑝0 𝜏0𝜐
2(𝜐))𝑑𝑣]𝑑𝑠
𝑡
≥ 𝜅
𝜌 1
𝐻(𝑡,𝑡1)∫ [−
𝜕𝐻(𝑡,𝑠)
𝜕𝑠 ] 𝑑𝑠 =
𝑡 𝑇1
𝜅 𝜌
𝐻(𝑡,𝑇1)
𝐻(𝑡,𝑡1) (3.21) It follows from (3.21) that, for all t ≥ T2,
1 𝐻(𝑡, 𝑡1)∫
𝐻(𝑡, 𝑠)𝜎′(𝑠)
𝑢(𝑠) (𝜔
2(𝑠) +𝑝0
𝜏0𝜐
2(𝑠))𝑑𝑠 𝑡
𝑡1
≥ 𝜅,
Since κis an arbitrary positive constant, we get
lim inf
𝑡→∞
1 𝐻(𝑡, 𝑡1)
∫𝜎
′(𝑠)𝐻(𝑡, 𝑠)
𝑢(𝑠) (𝜔
2(𝑠) +𝑝0
𝜏0
𝜐2(𝑠))𝑑𝑠
𝑡
𝑡1
=∞,
which contradicts (3.17). Consequently, (3.18) holds, so
∫𝜎 ′(𝑠) 𝑢(𝑠) 𝜔 2(𝑠)𝑑𝑠 ∞ 𝑡1
< ∞, ∫𝜎
′(𝑠) 𝑢(𝑠)𝜐 2(𝑠)𝑑𝑠 ∞ 𝑡1 < ∞
and, by virtue of (3.16)
∫𝜎 ′(𝑠)𝑚 + 2(𝑠) 𝑢(𝑠) ∞ 𝑡1
𝑑𝑠 ≤ ∫
𝜎′(𝑠)𝜔2(𝑠) + (𝑝0
𝜏0)
2
𝜎′(𝑠)𝜐2(𝑠) +2𝑝0
𝜏0 𝜎
′(𝑠)𝜔(𝑠)𝜐(𝑠)
𝑢(𝑠)
∞
𝑡1
𝑑𝑠
≤ ∫ 𝜎
′(𝑠)𝜔2(𝑠)+(𝑝0 𝜏0)
2
𝜎′(𝑠)𝜐2(𝑠)+𝑝0
𝜏0𝜎′(𝑠)[𝜔
2(𝑠)+𝜐2(𝑠)]
𝑢(𝑠)
∞
𝑡1 𝑑𝑠< ∞,
which contradicts (3.15). This completes the proof.
Choosing H as in Corollary 3.1, it is easy to verify that condition (3.13) is satisfied because, for any s ≥ t0, lim
t→∞
H(t, s)
H(t, t0)= lim t→∞
(t − s)n−1
(t − t0)n−1 = 1.
Consequently, we have the following result.
Corollary:3.2
Suppose that σ(t) ≤ τ(t) for t ≥ t0.Furthermore, assume that there exist functions g Є C1([t0,∞),R) and
m Є C([t0,∞),R) such that for all T ≥ t0, for some integer n >2
and some β ≥ 1,lim
𝑡→∞𝑠𝑢𝑝 𝑡
−(1 +𝑝0 𝜏0)
𝛽(𝑛 − 1)2
4𝜎′(𝑠) 𝑢(𝑠)]𝑑𝑠 ≥ 𝑚(𝑇)
where u and ψ are as in Theorem 3.1. Suppose further that (3.15) holds, where m+is as in Theorem 3.2. Then
every solution of (x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, isoscillatory.
From Theorem 3.2, we have the following result.
Theorem: 3.3
Assume that σ(t) ≤τ(t) for t ≥ t0.Further, suppose that H Є H, there exist functions
g Є C1([t
0,∞),R) and m Є C([t0,∞),R) such that for all T ≥ 0 and for some β >1, lim
𝑡→∞𝑖𝑛𝑓
1
𝐻(𝑡,𝑇)∫ [𝐻(𝑡, 𝑠)𝜓(𝑠) − (1 +
p0 τ0)
𝑡 𝑇
𝛽
4𝜎′(𝑠)𝑢(𝑠)ℎ
2(𝑡, 𝑠)]𝑑𝑠 ≥ 𝑚(𝑇), (3.22)
Where
ψ(t) = u(t){kQ(t)+(1 + p0/τ0)[σ'(t)g2(t)-g'(t)]},
Q(t)= min{q(t), q(τ(t))}, u(t) =exp{2∫ 𝜎𝑡𝑡 ′(𝑠)𝑔(𝑠)𝑑𝑠}
0 .
Suppose further that (3.15) holds, where m+ is as in Theorem 3.2.
Then every solution of (x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0 ,isoscillatory.
Theorem:3.4
Assume that σ(t) ≤ τ(t) for t ≥ t0.Further, assume that
there exists a function Φ Є X, such that for each l ≥ t0, for some n ≥ 1,
lim
𝑡→∞𝑠𝑢𝑝 𝑇𝑛[𝜓(𝑠) −
𝑛2
4 (1 +
𝑝0
𝜏0)
𝑢(𝑠)𝜑2(𝑠)
𝜎′(𝑠) ; 𝑙, 𝑡] > 0, (3.23)
where ψ, u are defined as in Theorem 3.1, the operator Tn is defined by 𝑇𝑛[𝑔, 𝑙, 𝑡] = ∫ 𝛷𝑡 𝑛(𝑡, 𝑠, 𝑙)𝑔(𝑠)𝑑𝑠
𝑙 , and
φ = φ(t, s, l) is defined by 𝜕𝛷(𝑡,𝑠,𝑙)
𝜕𝑠 = 𝜑(𝑡, 𝑠, 𝑙)𝛷(𝑡, 𝑠, 𝑙)
Then every solution of
(x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0,isoscillatory.
Assuming, without loss of generality, that there exists a solution x of
(x(t) + p(t)x(τ (t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, such that x(t) >0, x(τ (t)) >0, x(σ(t)) >0, for all t ≥ t1.
We define the functions
ω(t) = u(t)[ 𝑧′(𝑡)
𝑧(𝜎(𝑡))+ 𝑔(𝑡)] and
υ(t) = u(t)[𝑧′(𝜏(𝑡))
𝑧(𝜎(𝑡))+ 𝑔(𝑡)]·
we arrive at the inequality (3.8). Applying Tn [∙; l, t] to (3.8), we get
𝑇𝑛[[𝜔(𝑠) +𝑝0 𝜏0𝜐(𝑠)]
′
; 𝑙, 𝑡] ≤ 𝑇𝑛[−𝜓(𝑠) −𝜎
′(𝑠)𝜔2(𝑠)
𝑢(𝑠) −
𝑝0 𝜏0
𝜎′(𝑠)𝜐2(𝑠)
𝑢(𝑠) ; 𝑙, 𝑡].
𝑇𝑛[𝜓(𝑠); 𝑙, 𝑡] ≤ 𝑇𝑛[𝑛𝜑𝜔(𝑠) −𝜎′(𝑠)𝜔2(𝑠)
𝑢(𝑠) + 𝑛
𝑝0
𝜏0𝜑𝜐(𝑠) −
𝑝0
𝜏0
𝜎′(𝑠)𝜐2(𝑠)
𝑢(𝑠) ; 𝑙, 𝑡] (3.24)
Hence, from (3.24) we have
𝑇𝑛[[𝜔(𝑠) +𝑝0 𝜏0𝜐(𝑠)]
′
; 𝑙, 𝑡] ≤ 𝑇𝑛[𝑛
2
4 (1 +
𝑝0 𝜏0)
𝑢(𝑠)𝜑2(𝑠)
𝜎′(𝑠) ; 𝑙, 𝑡]
that is,
𝑇𝑛[𝜓(𝑠) −𝑛2
4 (1 +
𝑝0
𝜏0)
𝑢(𝑠)𝜑2(𝑠)
𝜎′(𝑠) ; 𝑙, 𝑡] ≤ 0,
Taking the super limit in the above inequality, we get
lim
𝑡→∞𝑠𝑢𝑝 𝑇𝑛[𝜓(𝑠) −
𝑛2
4 (1 +
𝑝0
𝜏0)
𝑢(𝑠)𝜑2(𝑠)
𝜎′(𝑠) ; 𝑙, 𝑡] ≤ 0,
which contradicts (3.23). This completes the proof. If we choose
Φ(t, s, l) = ρ(s)(t - s)α(s - l)β (3.25)
for α, β >½, and ρ Є C1([t
0,∞), (0,∞)), then we have 𝜑(𝑡, 𝑠, 𝑙) =𝜌′(𝑠)
𝜌(𝑠) +
𝛽𝑡−(𝛼+𝛽)𝑠+𝛼𝑙
(𝑡−𝑠)(𝑠−𝑙) . (3.26)
Thus by Theorem 3.4, we have the following oscillation result.
Corollary:3.3
Suppose that σ(t) ≤τ(t) for t ≥ t0 .Further, assume that for each l ≥ t0, there exists a function ρ Є
C1([t
0,∞), (0,∞)) and two constants α, β >½,such that for some
lim
𝑡→∞𝑠𝑢𝑝 ∫ 𝜌
𝑛(𝑠)
𝑡
𝑙
(𝑡 − 𝑠)𝑛𝛼(𝑠 − 𝑙)𝑛𝛽[𝜓(𝑠) −𝑛 2
4
(1 +𝑝0
𝜏0)
𝑢(𝑠) 𝜎′(𝑠)(
𝜌′(𝑠) 𝜌(𝑠) +
𝛽𝑡−(𝛼+𝛽)𝑠+𝛼𝑙
(𝑡−𝑠)(𝑠−𝑙) )
2]𝑑𝑠 > 0.
Where ψ(t) = u(t){kQ(t)+(1 + p0/τ0)[σ'(t)g2(t)-g'(t)]},
Q(t)= min{q(t), q(τ(t))},
u(t) =exp{2∫ 𝜎𝑡𝑡 ′(𝑠)𝑔(𝑠)𝑑𝑠}
0 .
Then every solution of
(x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0, isoscillatory.
If we choose
𝛷(𝑡, 𝑠, 𝑙) = √𝐻1(𝑠, 𝑙)𝐻2(𝑡, 𝑠) (3.27)
whereH1, H2 Є H, then we have
𝜑(𝑡, 𝑠, 𝑙) =1
2( ℎ1(1)(𝑠,𝑙) √𝐻1(𝑠,𝑙)−
ℎ2(2)(𝑡,𝑠)
√𝐻2(𝑡,𝑠)), (3.28)
whereℎ1(1)(𝑠, 𝑙), ℎ2(2)(𝑡, 𝑠) are defined as the following:
𝜕𝐻1(𝑠,𝑙)
𝜕𝑠 = ℎ1
(1)
(𝑠, 𝑙)√𝐻1(𝑠, 𝑙)and 𝜕𝐻2(𝑡,𝑠)
𝜕𝑠 = −ℎ2
(2)
(𝑡, 𝑠)√𝐻2(𝑡, 𝑠), (3.29)
According to Theorem 3.4, we have the following oscillation result.
Corollary: 3.4
Suppose thatσ(t) ≤τ(t) for t ≥ t0.Further, assume that for each l ≥ t0, there exist two functions H1,H2ЄH such
that for some n ≥1
lim
𝑡→∞𝑠𝑢𝑝 ∫(√𝐻1(𝑠, 𝑙)𝐻2(𝑡, 𝑠))
𝑛 𝑡
𝑙
[𝜓(𝑠)
𝑛2
16 (1 + 𝑝0
𝜏0)
𝑢(𝑠) 𝜎′(𝑠)(
ℎ1(1)(𝑠,𝑙) √𝐻1(𝑠,𝑙)−
ℎ2(2)(𝑡,𝑠) √𝐻2(𝑡,𝑠))
2
]𝑑𝑠 > 0,
Where ψ(t) = u(t){kQ(t)+(1 + p0/τ0)[σ'(t)g2(t)-g'(t)]},
Q(t)= min{q(t), q(τ(t))}, u(t) =exp{2∫ 𝜎𝑡𝑡 ′(𝑠)𝑔(𝑠)𝑑𝑠}
Then every solution of
(x(t) + p(t)x(τ(t)))''+ q(t)f(x(σ(t))) = 0, t ≥ t0,isoscillatory.
In the following, we give some new oscillation results for equation
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