# New Oscillation Results of Second-Order Damped Dynamic Equations withp-Laplacian on Time Scales

Authors: Yang-Cong Qiu; Qi-Ru Wang
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

By employing a generalized Riccati technique and functions in some function classes for integral averaging, we derive new oscillation criteria of second-order damped dynamic equation withp-Laplacian on time scales of the form (rtφγ(xΔ(t)))Δ+ptφγ(xΔ(t))+f(t,x(g(t)))=0, where the coefficient function p(t) may change sign. Two examples are given to demonstrate the obtained results.


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## Body

## 1. Introduction

In 1988, the theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in order to unify continuous and discrete analysis; see also [2]. In recent years, there has been much research activity concerning the oscillation of solutions of dynamic equations on time scales; for example, see [3–19] and the references therein.Došlý and Hilger [10] considered the second-order dynamic equation
(1)ptxΔtΔ+qtxσt=0,
and they gave some necessary and sufficient conditions for the oscillation of all solutions as (1) on unbounded time scales. Erbe et al. [11] studied the second-order nonlinear dynamic equation
(2)ptxΔtΔ+qtf∘xσ=0.
By means of generalized exponential functions and generalized Riccati transformation techniques, some oscillation criteria were given for (2). Del Medico and Kong [8, 9] employed the Riccati transformation and gave sufficient conditions for Kamenev-type oscillation criteria of (1) on a measure chain. Wang [18] considered the second-order nonlinear damped differential equation
(3)atψxtkx′t′+ptkx′t+qtfxt=0,t≥t0,
used the generalized Riccati transformations, and established new oscillation criteria of (3). Saker et al. [16] studied the second-order damped dynamic equation with damping
(4)a(t)xΔ(t)Δ+p(t)xΔσ(t)+qtf∘xσ=0,
and they gave some oscillation criteria. Agarwal et al. [5] studied the second-order half-linear dynamic equation
(5)r(t)xΔtγΔ+p(t)xγ(t)=0,t∈a,b,
and established Philos-type oscillation criteria. Huang and Wang [13] considered the second-order nonlinear dynamic equation
(6)p(t)xΔ(t)Δ+ft,xσt=0.
The authors extended the results in [8, 9], and established some new Kamenev-type oscillation criteria. Qiu and Wang [14] studied the dynamic equations of a more general form
(7)ptψxtk∘xΔ(t)Δ+ft,xσt=0
and established some similar oscillation criteria.Şenel [17] discussed the oscillation of second-order damped dynamic equations of the following form:
(8)r(t)xΔtγΔ+p(t)xΔtγ+ft,xgt=0
on a time scale T satisfying inf⁡T=t0 and sup⁡T=∞, where p∈Crd(T,[0,∞)). However, it seemed that several mistakes had been made and the obtained theorems and corollaries were incorrect. Qiu and Wang [15] corrected some mistakes in [17] and established some Kamenev-type oscillation criteria of (8) by employing functions in some function classes and a similar generalized Riccati transformation as used in [13, 18, 19].Hassan and Kong [12] considered the second-order dynamic equation with p-Laplacian and damping
(9)r(t)φγxΔ(t)Δ+p(t)φγxΔσt+qtfxσt=0,
where φγ(u)∶=uγ-1u, γ>0. The authors used the notations
(10)Rt=rtat,Q(t)=q(t)a(t),
where a(t)∶=ep(t)/rσ(t)(t,t0), by which (9) can be written into the form
(11)R(t)φγxΔ(t)Δ+Qtfxσt=0fort∈t0,∞T,
and established some oscillation criteria of (9).In this paper, we will consider the second-order damped dynamic equation withp-Laplacian
(12)r(t)φγxΔ(t)Δ+p(t)φγxΔt+ft,xgt=0
on a time scale T satisfying inf⁡T=t0 and sup⁡T=∞, where the coefficient function p(t) may change sign. Throughout this paper, we will assume that(C1)
r∈Crd(T,(0,∞)), φγ(u)∶=uγ-1u, γ>0;(C2)
p∈Crd(T,R), -p(t)/r(t) is positively regressive, which means 1-μ(t)p(t)/r(t)>0, and
(13)∫t0∞‍e-p/r(t,t0)r(t)1/γΔt=∞;(C3)
g∈C(T,T), and for any t∈T,
(14)g(t)≥σ(t),0<γ<1,t,γ≥1;(C4)
f∈C(T×R,R), and there exists a function q∈Crd(T,[0,∞)) such that uf(t,u)≥q(t)uγ+1.The rest of this paper is organized as follows. In Section2, we give some basic lemmas. In Section 3, we derive new oscillation criteria for (12). In Section 4, two examples are included to show the significance of the results.


## 2. Basic Lemmas

Preliminaries about time scale calculus can be found in [3, 4, 6, 7] and hence we omit them here. Note that we have the following properties for some typical time scales, respectively:(1)
T=[0,∞),
(15)σ(t)=ρ(t)=t,fΔ(t)=f′(t),∫ab‍f(t)Δt=∫ab‍f(t)dt;(2)
T=N0,
(16)σ(t)=t+1,ρ(t)=t-1,fΔ(t)=f(t+1)-f(t),∫ab‍f(t)Δt=∑k=ab-1‍f(k),a≤b;(3)
T={2n,n∈N0},
(17)σ(t)=2t,ρ(t)=t2,fΔ(t)=f(2t)-f(t)t,∫ab‍f(t)Δt=∑k=log⁡2alog⁡2b-1‍f(2k)2k,a≤b.Definition 1.
A solutionx of (12) is said to have a generalized zero at t*∈T if x(t*)x(σ(t*))≤0, and it is said to be nonoscillatory on T if there exists t0∈T such that x(t)x(σ(t))>0 for all t>t0. Otherwise, it is oscillatory. Equation  (12) is said to be oscillatory if all solutions of (12) are oscillatory.For simplicity, throughout this paper, we denote(a,b)∩T=a,bT, where a,b∈R, and a,bT,a,bT,a,bT are denoted similarly.To establish new oscillation criteria of (12), we give three lemmas in this section.Lemma 2.
Assume that (C1)–(C4) hold, and there exists a sufficiently larget1∈t0,∞T such that x(t) is a solution of (12) satisfying x(t)>0 for t∈t1,∞T. Then for t∈t1,∞T, we have
(18)xΔ(t)>0,r(t)xΔtγe-p/r(t,t1)Δ<0.Proof.
Lett1∈t0,∞T such that x(t) is a solution of (12) satisfying x(t)>0 for t∈t1,∞T; then we also have x(g(t))>0. It is obvious that (12) is equivalent to
(19)r(t)φγxΔ(t)Δe-p/r(t,t1)+p(t)φγxΔ(t)e-p/r(t,t1)e-p/r(t,t1)e-p/r(σ(t),t1)+ft,xgte-p/r(σ(t),t1)=0,
which is simplified as
(20)r(t)φγxΔ(t)e-p/r(t,t1)Δ+ft,xgte-p/r(σ(t),t1)=0.
By (12) and (C4), we have
(21)r(t)φγxΔ(t)e-p/r(t,t1)Δ=-ft,xgte-p/r(σ(t),t1)<0,t∈t1,∞T.
We claim thatxΔ(t)>0 for t∈t1,∞T. Assume the contrary; that is, there exists t2∈t1,∞T such that xΔ(t2)<0. Also, for t∈t2,∞T we have
(22)r(t)φγxΔ(t)e-p/r(t,t2)Δ<0.
Integrating (22) from t2 to t∈σ(t2),∞T, we obtain
(23)r(t)φγxΔ(t)e-p/r(t,t2)-r(t2)φγxΔ(t2)e-p/r(t2,t2)=rtφγxΔte-p/rt,t2-r(t2)φγxΔt2<0.
That is,
(24)r(t)φγxΔ(t)<r(t2)φγxΔ(t2)e-p/r(t,t2),
which implies that
(25)xΔ(t)<r1/γ(t2)xΔ(t2)e-p/r(t,t2)r(t)1/γ.
Integrating (25) from t2 to t∈σ(t2),∞T and letting t→∞, by (C2) we obtain
(26)x(t)<x(t2)+r1/γ(t2)xΔ(t2)∫t2t‍e-p/r(s,t2)r(s)1/γΔs⟶-∞,
which contradicts x(t)>0. Hence xΔ(t)>0 for t∈t1,∞T, and φγ(xΔ(t))=(xΔ(t))γ. By (21), we obtain
(27)r(t)xΔtγe-p/r(t,t1)Δ<0.
Lemma2 is proved.Lemma 3.
Assume that (C1)–(C4) hold andx(t) is a solution of (12) satisfying x(t)>0 for t∈t1,∞T with t1∈t0,∞T. Then for t∈t1,∞T, if 0<γ<1, we have
(28)xΔ(t)xσ(t)1-γ≥α1(t),
and if γ≥1, we have
(29)x(t)xΔ(t)γ-1≥α2(t),
where
(30)α1(t)=e-p/r(t,t1)r(t)∫t∞‍q(s)e-p/r(σ(s),t1)Δs(1-γ)/γ,α2(t)=r(t)e-p/r(t,t1)1/γ∫t1t‍e-p/r(s,t1)r(s)1/γΔsγ-1.Proof.
Sincex(t) is a solution of (12) satisfying x(t)>0 for t∈t1,∞T with t1∈T, by Lemma 2 we have
(31)xΔ(t)>0,r(t)xΔtγe-p/r(t,t1)Δ<0,t∈t1,∞T.
From (21) it follows that
(32)r(t)xΔtγe-p/r(t,t1)Δ=-ft,xgte-p/r(σ(t),t1).
Integrating (32) from ∞ to t with t∈T, we obtain
(33)r(t)xΔtγe-p/r(t,t1)≥∫∞t‍r(s)xΔsγe-p/r(s,t1)ΔΔs=∫t∞‍fs,xgse-p/r(σ(s),t1)Δs≥∫t∞‍qsxγgse-p/r(σ(s),t1)Δs≥xγ(g(t))∫t∞‍q(s)e-p/r(σ(s),t1)Δs≥xσtγ∫t∞‍q(s)e-p/r(σ(s),t1)Δs.
Hence when 0<γ<1, we have
(34)xΔ(t)xσ(t)1-γ≥α1(t),t∈t1,∞T.
Since[r(t)(xΔt)γ/e-p/r(t,t1)]Δ<0, t∈t1,∞T, for γ≥1, we get
(35)xt≥x(t)-x(t1)=∫t1t‍r(s)xΔsγ/e-p/r(s,t1)1/γr(s)/e-p/r(s,t1)1/γΔs≥r(t)xΔtγe-p/r(t,t1)1/γ∫t1t‍e-p/r(s,t1)r(s)1/γΔs,
and we obtain
(36)x(t)xΔ(t)γ-1≥α2(t),t∈t1,∞T.
Lemma3 is proved.Remark 4.
In Lemma3, when 0<γ<1 and for any t∈t1,∞T, the integral ∫t∞‍q(s)Δs in α1(t) must be convergent, which means
(37)∫t∞‍q(s)Δs<∞.Lemma 5.
Assume that (C1)–(C4) hold andx(t) is a solution of (12) satisfying x(t)>0 for t∈t1,∞T with t1∈t0,∞T. For t∈t1,∞T, define
(38)u(t)=A(t)r(t)xΔtγe-p/r(t,t1)xγ(t)+B(t),
where A∈Crd1(T,(0,∞)), B∈Crd1(T,R). Then u(t) satisfies
(39)uΔ(t)+Φ0(t)-AΔ(t)u(t)-B(t)A(t)σ+γA(t)α(t)r(t)u(t)-B(t)A(t)σ2≤0,
where
(40)Φ0(t)=A(t)q(t)e-p/t(σ(t),t1)-BΔ(t),α(t)=α1(t),0<γ<1,α2(t),γ≥1.Proof.
Without loss of generality we may assume there existst1∈t0,∞T such that x(t)>0 for t∈t1,∞T; then Lemmas 2 and 3 hold. Let u(t) be defined by (38). Then, differentiating (38) and using (12), it follows that
(41)uΔ=AxγrxΔγe-p/rΔ+AxγΔrxΔγe-p/rσ+BΔ=-Axγft,xgte-p/rσ+AΔxγ-AxγΔxγxγσrxΔγe-p/rσ+BΔ.
Using the fact that f(t,x(g(t)))≥q(t)xγ(g(t))≥q(t)xγ(t), we obtain
(42)uΔ≤-Aqe-p/rσ+BΔ+AΔrxΔγe-p/rxγσ-AxγΔxγrxΔγe-p/rxγσ=-Aqe-p/rσ+BΔ+AΔu-BAσ-AxγΔxγu-BAσ.
When0<γ<1, using the Pötzsche chain rule, we have
(43)xγΔ=γ∫01‍x+hμxΔγ-1dh·xΔ≥γxσγ-1xΔ,
and it follows that
(44)xγΔxγ≥γxσγ-1xΔxγ=γxΔxσxσxγ.
By Lemmas 2 and 3, for t∈t1,∞T, we obtain
(45)xΔxσ=1rrxΔγxσγxΔxσ1-γ≥α1rrxΔγσxγσ=α1ru-BAσ,xσx≥1.
So (42) becomes
(46)uΔ≤-Aqe-p/rσ+BΔ+AΔu-BAσ-γAα1ru-BAσ2.
Whenγ≥1, we have
(47)xγΔ=γ∫01‍x+hμxΔγ-1dh·xΔ≥γxγ-1xΔ,
and it follows that
(48)xγΔxγ≥γxγ-1xΔxγ=γxΔx.
By Lemmas 2 and 3, for t∈t1,∞T, we obtain
(49)xΔx=1rrxΔγxγxxΔγ-1≥α2rrxΔγσxγσ=α2ru-BAσ.
So (42) becomes
(50)uΔ≤-Aqe-p/rσ+BΔ+AΔu-BAσ-γAα2ru-BAσ2.
By (46) and (50), (39) holds. Lemma 5 is proved.


## 3. Main Results

In this section, we will derive new oscillation criteria of (12). Our approach to oscillation results of (12) is based on the application of the generalized Riccati transformation. Firstly, we give some definitions.LetD0={s∈T:s≥0} and D={(t,s)∈T2:t≥s≥0}. For any function f(t,s):T2→R, denote by f2Δ the partial derivative of f with respect to s. Define
(51)A,B=(A,B):A(s)∈Crd1(D0,(0,∞)),hhhB(s)∈Crd1(D0,R),s∈D0;H=H(t,s)∈C1(D,[0,∞)):H(t,t)=0,hhhH(t,s)>0,H2Δ(t,s)≤0,t>s≥0.
These function classes will be used throughout this paper. Now, we are in a position to give our first theorem.Theorem 6.
Assume that (C1)–(C4) hold and that there exist(A,B)∈(A,B) and H∈H such that, for any t1∈T,
(52)limsup⁡t→∞1H(t,t1)×∫t1t‍-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)H(t,s)Φ0(s)-H2Δ(t,s)Bσ(s)hhhhhh-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)Δs=∞,
where Φ0 and α are defined as before. Then, (12) is oscillatory.Proof.
Assume that (12) is not oscillatory. Without loss of generality we may assume there exists t1∈t0,∞T such that x(t)>0 for t∈t1,∞T. Let u(t) be defined by (38). Then by Lemma 5, (39) holds.
For simplicity in the following, we letHσ=H(t,σ(s)), H=H(t,s), H2Δ=H2Δ(t,s), and omit the arguments in the integrals.
Multiplying (39), where t is replaced by s, by H, and integrating it with respect to s from t1 to t∈[σt1,∞)T, we obtain
(53)∫t1t‍HΦ0Δs≤-∫t1t‍HuΔ-AΔu-BAσ+γAαru-BAσ2Δs.
Noting that H(t,t)=0, by the integration by parts formula we have
(54)∫t1t‍HΦ0Δs≤H(t,t1)u(t1)+∫t1t-γHAαru-BAσ2H2Δuσ+HAΔu-BAσ‍hhhhhhhhhhhhhhhhhhhh-γHAαru-BAσ2Δs=H(t,t1)u(t1)+∫t1t‍H2ΔBσΔs+∫t1t‍u-BAσ2H2ΔAσ+HAΔu-BAσhhhhhhhhhh-γHAαru-BAσ2Δs.
Since(55)(H2ΔAσ+HAΔ)u-BAσ-γHAαru-BAσ2=rH2ΔAσ+HAΔ24γHAα-γHAαru-BAσ-rH2ΔAσ+HAΔ2γHAα2≤rH2ΔAσ+HAΔ24γHAα,
by (54) we have
(56)∫t1t‍HΦ0Δs≤H(t,t1)u(t1)+∫t1t‍H2ΔBσ+rH2ΔAσ+HAΔ24γHAαΔs.
Hence
(57)1H(t,t1)∫t1t‍HΦ0-H2ΔBσ-rH2ΔAσ+HAΔ24γHAαΔs≤u(t1)<∞,
which contradicts (52) and completes the proof.WhenB=0, (38) is simplified as
(58)u(t)=A(t)r(t)xΔtγe-p/r(t,t1)xγ(t),t∈t1,∞T.
Now we have the following theorem.Theorem 7.
Assume that (C1)–(C4) hold and that there existsA∈Crd1(D0,(0,∞)) such that, for any t1∈T,
(59)limsup⁡t→∞∫t1t‍A(s)q(s)e-p/r(σ(s),t1)-r(s)AΔ(s)24γA(s)α(s)Δs=∞,
where α is defined as before. Then, (12) is oscillatory.Proof.
Assume that (12) is not oscillatory. Without loss of generality we may assume there exists t1∈t0,∞T such that x(t)>0 for t∈t1,∞T. Let u(t) be defined by (58). Then by Lemma 5, we have
(60)uΔ(t)+A(t)q(t)e-p/r(σ(t),t1)-AΔ(t)u(t)A(t)σ+γA(t)α(t)r(t)u(t)A(t)σ2≤0,
which implies that
(61)uΔ≤-Aqe-p/rσ+AΔuAσ-γAαruAσ2=-Aqe-p/rσ+rAΔ24γAα-γAαruAσ-AΔr2γAα2≤-Aqe-p/rσ+rAΔ24γAα.
So we obtain
(62)A(t)q(t)e-p/r(σ(t),t1)-r(t)AΔt24γA(t)α(t)≤-uΔ(t).
Letting t be replaced by s, and integrating (62) with respect to s from t1 to t∈[σt1,∞)T, we obtain
(63)∫t1t‍A(s)q(s)e-p/r(σ(s),t1)-r(s)AΔs24γA(s)α(s)Δs≤-∫t1t‍uΔ(s)Δs=u(t1)-u(t)<u(t1)<∞,
which contradicts (59) and completes the proof.Whenp≥0, it is easy to obtain -p/r≤0, so we have
(64)1e-p/r(σ(t),t1)≥1.
Then, Theorems 6 and 7 can be simplified as the following corollaries, respectively.Corollary 8.
Assume that (C1)–(C4) hold and that there exist(A,B)∈(A,B) and H∈H such that, for any t1∈T,
(65)limsup⁡t→∞1H(t,t1)×∫t1t‍-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)H(t,s)A(s)q(s)-BΔ(s)-H2Δ(t,s)Bσ(s)hhhhhhi-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)Δs=∞.
Then, (12) is oscillatory.Corollary 9.
Assume that (C1)–(C4) hold and that there existsA∈Crd1(D0,(0,∞)) such that, for any t1∈T,
(66)limsup⁡t→∞∫t1t‍A(s)q(s)-r(s)AΔ(s)24γA(s)α(s)Δs=∞.
Then, (12) is oscillatory.Remark 10.
Compared to the theorems and corollaries in [15], the conclusions of Theorems 6 and 7 and Corollaries 8 and 9 are much simpler, and the proofs of Theorems 6 and 7 are more convenient. Furthermore, p(t) may change sign in Theorems 6 and 7, for the employment of the function e-p/r(t,t1).Remark 11.
Consider the equations as the following form:(67)r(t)φγxΔ(t)Δ+p(t)φγxΔσt+ft,xgt=0.
It is obvious that (67) is equivalent to
(68)r(t)φγxΔ(t)Δep/rσ(t,t1)+p(t)φγxΔ(t)ep/rσ(t,t1)+f(t,x(g(t)))ep/rσ(t,t1)=0,
which is simplified as
(69)r(t)ep/rσ(t,t1)φγxΔ(t)Δ+f(t,x(g(t)))ep/rσ(t,t1)=0.
We can get the similar oscillation results as before, where
(70)r(t)e-p/r(t,t1),q(t)e-p/r(σ(t),t1)
are replaced by
(71)r(t)ep/rσ(t,t1),q(t)ep/rσ(t,t1),
respectively, in Lemmas 2, 3, and 5, Theorems 6 and 7, and Corollaries 8 and 9.


## 4. Examples

In this section, we will show the application of our oscillation results in two examples. Firstly, we give an example to demonstrate Theorem6.Example 12.
Consider the equation(72)tφγxΔ(t)Δ-φγxΔ(t)+t+1txγ(g(t))=0,
where r(t)=t, p(t)=-1, q(t)=1/t, γ>2, so we have -p(t)/r(t)=1/t. Letting H(t,s)=t-s2,(A,B)=(s2,1), we have the following results.(i)T=N. In this case, there exists t1∈1,∞T such that
(73)e-p/r(t,t0)=e1/t(t,1)=exp⁡∫1t‍ξ11τΔτ=exp⁡∫1t‍Log1+1τΔτ=∏k=1t-1‍1+1k=t,e-p/r(t,t1)=e1/t(t,t1)=exp⁡∫t1t‍ξ11τΔτ=exp⁡∫t1t‍Log1+1τΔτ=∏k=t1t-1‍1+1k=tt1,∫t0∞‍e-p/r(t,t0)r(t)1/γΔt=∫1∞‍Δt=∞,α2t=r(t)e-p/r(t,t1)1/γ∫t1t‍e-p/r(s,t1)r(s)1/γΔsγ-1=t11/γ∫t1t‍1t11/γΔsγ-1=∫t1t‍Δsγ-1=t-t1γ-1.
Hence
(74)limsup⁡t→∞1H(t,t1)×∫t1t‍-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)H(t,s)Φ0(s)-H2Δ(t,s)Bσ(s)hhhhhh-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)Δs=limsup⁡t→∞1t-t12×∫t1t‍-s-(2t-2s-1)s+12+t-s22s+124γt-s2s2s-t1γ-1t-s2s2·s-1(s+1)·t1-1+2t-2s-1hhhhhh-s-2t-2s-1s+12+t-s22s+124γt-s2s2s-t1γ-1Δs=∞.
That is, (52) holds. By Theorem 6 we see that (72) is oscillatory.(ii)T={2n,n∈N0}. In this case, there exists t1∈1,∞T such that
(75)e-p/r(t,t0)=e1/t(t,1)=exp⁡∫1t‍ξτ1τΔτ=exp⁡∫1t‍1τLog1+τ·1τΔτ=exp⁡Log2∫1t‍1τΔτ=2log⁡2t=t,e-p/r(t,t1)=e1/t(t,t1)=exp⁡∫t1t‍ξτ1τΔτ=exp⁡∫t1t‍1τLog1+τ·1τΔτ=exp⁡Log2∫t1t‍1τΔτ=2log⁡2t/t1=tt1,∫t0∞‍e-p/r(t,t0)r(t)1/γΔt=∫1∞‍Δt=∞,α2t=r(t)e-p/r(t,t1)1/γ∫t1t‍e-p/r(s,t1)r(s)1/γΔsγ-1=t11/γ∫t1t‍1t11/γΔsγ-1=∫t1t‍Δsγ-1=t-t1γ-1.
Hence
(76)limsup⁡t→∞1H(t,t1)×∫t1t‍-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)H(t,s)Φ0(s)-H2Δ(t,s)Bσ(s)hhhhhh-r(s)H2Δ(t,s)Aσ(s)+H(t,s)AΔ(s)24γH(t,s)A(s)α(s)Δs=limsup⁡t→∞1t-t12×∫t1t‍-s-(2t-3s)2s2+t-s2·3s24γt-s2s2s-t1γ-1t-s2s2·s-12s·t1-1+(2t-3s)hhhhhhh-s-(2t-3s)2s2+t-s2·3s24γt-s2s2s-t1γ-1Δs=∞.
That is, (52) holds. By Theorem 6 we see that (72) is oscillatory.The second example illustrates Theorem7 (or Corollary 9).Example 13.
Consider the equation(77)1tφγxΔ(t)Δ+λt2φγxΔ(t)+1tλ+2xγgt=0,
where r(t)=1/t, p(t)=λ/t2, q(t)=1/tλ+2=t-λ-2, γ>λ, where λ is a fixed constant and satisfies 0≤λ≤2, so we have -p(t)/r(t)=-λ/t. Let H(t,s)=t-s2, A(s)=sλ+2, when T=[1,∞); then there exists t1∈[1,∞) such that
(78)e-p/r(t,t0)=e-λ/t(t,1)=exp⁡∫1t‍ξ0-λτdτ=exp⁡∫1t‍-λτdτ=1tλ,e-p/r(t,t1)=e-λ/t(t,t1)=exp⁡∫t1t‍ξ0-λτdτ=exp⁡∫t1t‍-λτdτ=t1tλ,∫t0∞‍e-p/r(t,t0)r(t)1/γΔt=∫1∞‍1tλ-11/γdt=∫1∞‍1t(λ-1)/γdt=∞,α1t=e-p/r(t,t1)r(t)∫t∞‍q(s)e-p/r(σ(s),t1)Δs(1-γ)/γ=t1λtλ-1∫t∞‍s-λ-2·sλt1λds(1-γ)/γ=1tλ-1∫t∞‍1s2ds(1-γ)/γ=t-λ(1-γ)/γ,α2t=r(t)e-p/r(t,t1)1/γ∫t1t‍e-p/r(s,t1)r(s)1/γΔsγ-1=tλ-1t1λ1/γ∫t1t‍t1λsλ-11/γdsγ-1=O(tγ-1).
So we obtain
(79)limsup⁡t→∞∫t1t‍A(s)q(s)-r(s)AΔ(s)24γA(s)α(s)Δs=limsup⁡t→∞∫t1t‍sλ+2·s-λ-2-s-1(λ+2)sλ+124γsλ+2α(s)ds=limsup⁡t→∞∫t1t‍1-λ+22sλ-14γα(s)ds.(i)0<γ<1, α(s)=α1(s)=s-λ(1-γ)/γ. Since
(80)λ-1--λ(1-γ)γ=λγ-1<0,
it follows that
(81)limsup⁡t→∞∫t1t‍A(s)q(s)-r(s)AΔ(s)24γA(s)α(s)Δs=∞.(ii)γ≥1, α(s)=α2(s)=O(sγ-1). Since
(82)λ-1-(γ-1)=λ-γ<0,
it follows that
(83)limsup⁡t→∞∫t1t‍A(s)q(s)-r(s)AΔ(s)24γA(s)α(s)Δs=∞.That is, (66) holds. By Corollary 9 we see that (77) is oscillatory.Similarly, we can get the same conclusion whenT={n∣n≥3,n∈N}.



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*Source: 709242-2015-02-22.xml*