This paper is concerned with second-order nonlinear damped dynamic equations on time
scales of the following more general form (p(t)k1(x(t),xΔ(t)))Δ+r(t)k2(x(t),xΔ(t))xΔ(t)+f(t,x(σ(t)))=0. New oscillation results are given to handle some cases not covered by known criteria. An illustrative
example is also presented.
1. Introduction
Let ℝ denote the set of real numbers and 𝕋 a time scale, that is, a nonempty closed subset of ℝ with the topology and ordering inherited from ℝ. The theory of time scales was introduced by Hilger in his Ph.D. thesis [1] in 1988, and for a comprehensive treatment of the subject, see [2]. Much recent attention has been concerned with the oscillation of dynamic equations on time scales; see, for example, [1–15]. In [9], Došlý and Hilger studied the second-order dynamic equation
(1)(p(t)xΔ(t))Δ+q(t)x(σ(t))=0.
The authors gave a necessary and sufficient condition for the oscillation of all solutions of (1) on time scales. In [7, 8], Del Medico and Kong used the Riccati transformation as
(2)u(t)=p(t)xΔ(t)x(t)
and obtained some sufficient conditions for oscillation of (1). In [14], Wang considered the nonlinear second-order damped differential equation
(3)(a(t)ψ(x(t))k(x′(t)))′+p(t)k(x′(t))+q(t)f(x(t))=0,t≥t0,
and established new oscillation criteria. In [13], Tiryaki and Zafer considered the second-order nonlinear differential equation with nonlinear damping
(4)(r(t)k1(x,x′))′+p(t)k2(x,x′)x′+q(t)f(x)=0
and gave interval oscillation criteria of (4). In [10], Huang and Wang considered the second-order nonlinear dynamic equation
(5)(p(t)xΔ(t))Δ+f(t,x(σ(t)))=0.
The authors gave some new oscillation criteria of (5) and extended the results in [7, 8]. In [11], Qiu and Wang studied the second-order nonlinear dynamic equation
(6)(p(t)ψ(x(t))xΔ(t))Δ+f(t,x(σ(t)))=0.
By employing the Riccati transformation as
(7)u(t)=A(t)p(t)ψ(x(t))xΔ(t)x(t)+B(t),
where A∈Crd1(𝕋,(0,∞)), B∈Crd1(𝕋,ℝ), the authors established interval oscillation criteria for (6). And in [12], Qiu and Wang obtained some new Kamenev-type oscillation criteria for dynamic equations of the following more general form:
(8)(p(t)ψ(x(t))k∘xΔ(t))Δ+f(t,x(σ(t)))=0,
by using the transformation
(9)u(t)=A(t)p(t)ψ(x(t))k∘xΔ(t)x(t)+B(t).
In this paper, we consider second-order nonlinear damped dynamic equations of the form
(10)(p(t)k1(x(t),xΔ(t)))Δ+r(t)k2(x(t),xΔ(t))xΔ(t)+f(t,x(σ(t)))=0
on a time scale 𝕋. We will employ functions of the form H(t,s) and a generalized Riccati transformation as (7) and (9) which was used in [14, 15] and derive oscillation criteria for (10) in Section 2. An example is presented to demonstrate the obtained results in the final section.
Definition 1.
A solution x of (10) 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 (10) is said to be oscillatory if all solutions of (10) are oscillatory.
2. Main Results
In this section, we establish some oscillation criteria for (10). Our work is based on the application of the Riccati transformation. Throughout this paper we will assume that sup𝕋=∞ and
p∈Crd(𝕋,(0,∞));
r∈Crd(𝕋,ℝ);
k1,k2∈C(ℝ2,ℝ), and there exist α1≥α2>0 and α3>0 such that 0<α2vk1(u,v)≤k12(u,v)≤α1vk(u,v) and α3k12(u,v)≤uvk2(u,v) for all (u,v)∈(ℝ∖{0})2;
for p, r, α1, α3 above, we always have α1α3r(t)+p(t)>0;
f∈C(𝕋×ℝ,ℝ).
Preliminaries about time scale calculus can be found in [3–6] and are omitted here. For simplicity, we denote (a,b)∩𝕋=(a,b)𝕋 throughout this paper, where a,b∈ℝ and [a,b]𝕋, [a,b)𝕋, (a,b]𝕋 are denoted similarly.
Now, we give the first theorem.
Theorem 2.
Assume that (C1)–(C5) hold and that there exists a function q∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Also, suppose that x(t) is a solution of (10) satisfying x(t)>0 for t∈[t0,∞)𝕋 with t0∈𝕋. For t∈[t0,∞)𝕋, define
(11)u(t)=A(t)p(t)k1(x(t),xΔ(t))x(t)+B(t),
where A∈Crd1(𝕋,(0,∞)), B∈Crd1(𝕋,ℝ), and (α1A-(α1-α2)Aσ)p+α1α2α3rAσ>0 for t∈[t0,∞)𝕋. Then, u(t) satisfies
(12)μ(t)u(t)-μ(t)B(t)+α1A(t)p(t)>0,(13)uΔ(t)+Φ0(t)+Φ1(t)u2(t)-Φ2(t)u(t)+Φ3(t)α1A(t)p(t)(μ(t)u(t)-μ(t)B(t)+α1A(t)p(t))≤0,
where
(14)Φ0(t)=Aσ(t)(q(t)-(B(t)A(t))Δ),Φ1(t)=(α1A(t)-(α1-α2)Aσ(t))p(t)+α1α2α3r(t)Aσ(t),Φ2(t)=((2α2-α1)Aσ(t)+α1A(t))p(t)B(t)+α12p2(t)AΔ(t)A(t)+2α1α2α3r(t)Aσ(t)B(t),Φ3(t)=α2(α1α3r(t)+p(t))Aσ(t)B2(t),Aσ(t)=A(σ(t)).
Proof.
By (C3) we see that xΔ and k1(x,xΔ) are both positive or both negative or both zero. When xΔ>0, which implies that k1(x,xΔ)>0, it follows that
(15)μu-μB+α1Ap≥μApk12(x,xΔ)xk1(x,xΔ)+α2Ap≥α2μApxΔk1(x,xΔ)xk1(x,xΔ)+α2Ap=α2Apxσx>0.
When xΔ<0, which implies that k1(x,xΔ)<0, it follows that
(16)μu-μB+α1Ap=μApk12(x,xΔ)xk1(x,xΔ)+α1Ap≥μApα1xΔk1(x,xΔ)xk1(x,xΔ)+α1Ap=α1Apxσx≥α2Apxσx>0.
When xΔ=0, which implies that k1(x,xΔ)=0 and x=xσ, it follows that
(17)μu-μB+α1Ap=α1Ap≥α2Apxσx>0.
Hence, we always have
(18)μu-μB+α1Ap>0,xxσ≥α2Apμu-μB+α1Ap,
so (12) holds. Then differentiating (11) and using (10), it follows that
(19)uΔ=AΔ(pk1(x,xΔ)x)+Aσ(pk1(x,xΔ)x)Δ+BΔ=AΔA(u-B)+Aσ(pk1(x,xΔ))Δx-pk1(x,xΔ)xΔxxσ+BΔ=AΔAu+BΔ-AΔAB-Aσf(t,xσ)xσ-Aσrk2(x,xΔ)xΔxσ-Aσpk1(x,xΔ)xΔxxσ≤AΔAu+Aσ(BA)Δ-Aσq-Aσrk2(x,xΔ)xxΔxxσ-Aσpk12(x,xΔ)α1xxσ≤AΔAu-Φ0-α3Aσrk12(x,xΔ)x2xxσ-1α1Aσpk12(x,xΔ)x2xxσ≤AΔAu-Φ0-(α3r+pα1)Aσ(u-BAp)2×α2Apμu-μB+α1Ap=AΔAu-Φ0-α2(α1α3r+p)α1pAσA(u-B)2μu-μB+α1Ap=-Φ1u2+Φ2u-Φ3α1Ap(μu-μB+α1Ap)-Φ0,
so (13) holds. Theorem 2 is proved.
Remark 3.
In Theorem 2, the condition (α1A-(α1-α2)Aσ)p+α1α2α3rAσ>0 ensures that the coefficient of u2 in (13) is always negative. The condition is obvious and easy to be fulfilled. For example, when AΔ(t)≤0 for all t∈[t0,∞)𝕋, we have Aσ=A+μAΔ≤A; by (C4) we see that
(20)(α1A-(α1-α2)Aσ)p+α1α2α3rAσ≥(α1Aσ-(α1-α2)Aσ)p+α1α2α3rAσ=α2Aσ(α1α3r+p)>0.
Let D0={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 derivatives of f with respect to s. For E⊂ℝ, denote by L(E) the space of functions which are integrable on any compact subset of E. Define
(21)(𝒜,ℬ)={(A,B):A(s)∈Crd1(D0,(0,∞)),B(s)∈Crd1(D0,ℝ),(α1A(s)-(α1-α2)Aσ(s))p(s)+α1α2α3r(s)Aσ(s)>0,α1A(s)p(s)±μ(s)B(s)>0,s∈D0Crd1(D0,(0,∞))};ℋ={H(t,s)∈C1(D,[0,∞)):H(t,t)=0,H(t,s)>0,H2Δ(t,s)≤0,t>s≥0C1(D,[0,∞))}.
These function classes will be used throughout this paper. Now, we are in a position to give the second theorem.
Theorem 4.
Assume that (C1)–(C5) hold and that there exists a function q∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Also, suppose that there exist (A,B)∈(𝒜,ℬ) and H∈ℋ such that M(t,·)∈L([0,ρ(t)]𝕋) and for any t0∈𝕋,
(22)limsupt→∞1H(t,t0)[∫t0tH(t,σ(s))Φ0(s)Δs-∫t0ρ(t)M(t,s)Δs+H2Δ(t,ρ(t))×(-μ(ρ(t))B(ρ(t)))α1A(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t)))∫t0tH(t,σ(s))Φ0(s)Δs]=∞,
where Φ0 is defined as before, and
(23)M(t,s)=Φ42(t,s)4α1A(s)p(s)min{Φ5(t,s),Φ6(t,s)},Φ4(t,s)=α1p(s)H(t,s)A(s)B(s)+((2α2-α1)p(s)+2α1α2α3r(s))×H(t,σ(s))Aσ(s)B(s)+α12p2(s)A(s)(H(t,s)A(s))Δs,Φ5(t,s)=α2H(t,σ(s))Aσ(s)(α1α3r(s)+p(s))×(α1A(s)p(s)+μ(s)B(s)),Φ6(t,s)=(+α1α2α3r(s)H(t,σ(s))Aσ(s))α1p(s)H(t,s)A(s)-(α1-α2)p(s)H(t,σ(s))Aσ(s)+α1α2α3r(s)H(t,σ(s))Aσ(s))×(α1A(s)p(s)-μ(s)B(s)).
Then, (10) is oscillatory.
Proof.
Assume that (10) is not oscillatory. Without loss of generality we may assume that there exists t0∈[0,∞)𝕋 such that x(t)>0 for t∈[t0,∞)𝕋. Let u(t) be defined by (11). Then by Theorem 2, (12) and (13) hold.
For simplicity in the following, we let Hσ=H(t,σ(s)), H=H(t,s), and H2Δ=H2Δ(t,s) and omit the arguments in the integrals. For s∈𝕋, Hσ-H=μH2Δ.
Multiplying (13), where t is replaced by s, by Hσ and integrating it with respect to s from t0 to t with t∈𝕋 and t≥σ(t0), we obtain
(24)∫t0tHσΦ0Δs≤-∫t0t(HσuΔ+HσΦ1u2-Φ2u+Φ3α1Ap(μu-μB+α1Ap))Δs,
where Φ1, Φ2, Φ3 are defined as before.
Noting that H(t,t)=0, by the integration by parts formula we have
(25)∫t0tHσΦ0Δs≤H(t,t0)u(t0)+∫t0t(H2Δu-HσΦ1u2-Φ2u+Φ3α1Ap(μu-μB+α1Ap))Δs≤H(t,t0)u(t0)+∫t0t(H2Δu-HσΦ1u2-Φ2uα1Ap(μu-μB+α1Ap))Δs=H(t,t0)u(t0)+∫ρ(t)tH2ΔuΔs+∫t0ρ(t)(H2Δu-HσΦ1u2-Φ2uα1Ap(μu-μB+α1Ap))Δs.
Since H2Δ≤0 on D, from (12) we see that, for t≥σ(t0),
(26)∫ρ(t)tH2ΔuΔs=H2Δ(t,ρ(t))u(ρ(t))μ(ρ(t))≤-H2Δ(t,ρ(t))(-μ(ρ(t))B(ρ(t)))α1A(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t))).
Since H2Δ≤0 on D, we see that Hσ≤H. For t≥σ(t0), s∈[t0,ρ(t))𝕋, from (α1A-(α1-α2)Aσ)p+α1α2α3rAσ>0 and (C4), we have
(27)α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσ≥α1pHσA-(α1-α2)pHσAσ+α1α2α3rHσAσ=((α1A-(α1-α2)Aσ)p+α1α2α3rAσ)Hσ>0.
For t≥σ(t0), s∈[t0,ρ(t))𝕋, and u(s)≤0, from (27) we have
(28)H2Δu-HσΦ1u2-Φ2uα1Ap(μu-μB+α1Ap)=-[α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσ]u2+Φ4uα1Ap(μu-μB+α1Ap)=-α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσα1Ap(μu-μB+α1Ap)u2+Φ4α1Ap(α1Ap-μB)u-Φ4α1Ap(α1Ap-μB)μu2μu-μB+α1Ap=-Φ5α1Ap(α1Ap-μB)(μu-μB+α1Ap)u2+Φ4α1Ap(α1Ap-μB)u≤-Φ5α1Ap(α1Ap-μB)2u2+Φ4α1Ap(α1Ap-μB)u=-Φ5α1Ap(α1Ap-μB)2(u-(α1Ap-μB)Φ42Φ5)2+Φ424α1ApΦ5≤Φ424α1Apmin{Φ5,Φ6}=M.
For t≥σ(t0), s∈[t0,ρ(t))𝕋, and u(s)>0, from (27) we have
(29)H2Δu-HσΦ1u2-Φ2uα1Ap(μu-μB+α1Ap)=-[α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσ]u2+Φ4uα1Ap(μu-μB+α1Ap)=-α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσα1Ap(μu-μB+α1Ap)×(u-Φ42(α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσ))2+Φ424α1ApΦ6≤Φ424α1Apmin{Φ5,Φ6}=M.
Therefore, for all t≥σ(t0), s∈[t0,ρ(t))𝕋, we have
(30)H2Δu-HσΦ1u2-Φ2uα1Ap(μu-μB+α1Ap)≤M.
Then, from (25), (26), and (30) we obtain that, for t∈𝕋 and t>σ(t0),
(31)∫t0tHσΦ0Δs≤H(t,t0)u(t0)+∫t0ρ(t)MΔs-H2Δ(t,ρ(t))(-μ(ρ(t))B(ρ(t)))α1A(ρ(t))p(ρ(t))hh-μ(ρ(t))B(ρ(t))).
Hence,
(32)1H(t,t0)[∫t0tH(t,σ(s))Φ0(s)Δs-∫t0ρ(t)M(t,s)Δshhhh+H2Δ(t,ρ(t))hhhh×(α1A(ρ(t))p(ρ(t))-μ(ρ(t))B(ρ(t)))∫t0tH(t,σ(s))Φ0(s)Δs]≤u(t0)<∞,
which contradicts (22) and completes the proof.
Remark 5.
If we change the condition (α1A-(α1-α2)Aσ)p+α1α2α3rAσ>0 in the definition of (𝒜,ℬ) to a stronger one AΔ(t)≤0, (27) in the proof of Theorem 4 will be changed to
(33)α1pHA-(α1-α2)pHσAσ+α1α2α3rHσAσ≥α1pHσAσ-(α1-α2)pHσAσ+α1α2α3rHσAσ=α2HσAσ(α1α3r+p)>0.
Then the definition of M can be simplified as
(34)M(t,s)=(Φ42(t,s))×(4α1α2p(s)H(t,σ(s))Aσ(s)A(s)×(α1α3r(s)+p(s))min{Φ7(s),Φ8(s)})-1,
where
(35)Φ7(s)=α1A(s)p(s)-μ(s)B(s),Φ8(s)=α1A(s)p(s)+μ(s)B(s).
When (A,B)=(1,0), Theorem 4 can be simplified as Corollary 6.
Corollary 6.
Assume that (C1)–(C5) hold and that there exists a function q∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Also, suppose that there exists H∈ℋ such that, for any t0∈𝕋,
(36)limsupt→∞1H(t,t0)×[∫t0tH(t,σ(s))q(s)Δs-α124α2∫t0ρ(t)(p(s)H2Δ(t,s))2(α1α3r(s)+p(s))H(t,σ(s))Δs+α1H2Δ(t,ρ(t))p(ρ(t))∫t0tH(t,σ(s))q(s)Δs]=∞.
Then, (10) is oscillatory.
When r(t)≡0, (10) will be simplified as
(37)(p(t)k1(x(t),xΔ(t)))Δ+f(t,x(σ(t)))=0.
Then Theorem 4 can be simplified as Corollary 7.
Corollary 7.
Assume that (C1)–(C5) hold and that there exists a function q∈Crd(𝕋,ℝ) such that uf(t,u)≥q(t)u2. Also, suppose that there exist (A,B)∈(𝒜,ℬ) and H∈ℋ such that, for any t0∈𝕋,
(38)limsupt→∞1H(t,t0)[∫t0tH(t,σ(s))Φ0(s)Δs-∫t0ρ(t)M1(t,s)Δshh+H2Δ(t,ρ(t))hh×(α1A(ρ(t))p(ρ(t))hh-μ(ρ(t))B(ρ(t)))∫t0tH(t,σ(s))Φ0(s)Δs]=∞,
where
(39)M1(t,s)=Φ92(t,s)4α1A(s)min{Φ10(t,s),Φ11(t,s)},Φ9(t,s)=α1H(t,s)A(s)B(s)+(2α2-α1)H(t,σ(s))Aσ(s)B(s)+α12p(s)A(s)(H(t,s)A(s))Δs,Φ10(t,s)=α2H(t,σ(s))Aσ(s)(α1A(s)p(s)+μ(s)B(s)),Φ11(t,s)=(α1H(t,s)A(s)-(α1-α2)H(t,σ(s))Aσ(s))×(α1A(s)p(s)-μ(s)B(s)).
Then, (37) is oscillatory.
Remark 8.
When r(t)≡0, k1(u,v)=Ψ(u)k(v), and (C3) is replaced by
ψ∈C(ℝ,(0,η]), where η is a fixed positive constant;
k∈C(ℝ,ℝ), and there exists γ1≥γ2>0 such that 0<γ2yk(y)≤k2(y)≤γ1yk(y) for all y≠0.
Theorem 4 is reduced to [12, Theorem 4].
3. Example
In this section, we will give an example to demonstrate Corollary 7.
Example 1.
Consider the equations
(40)[1t22+x2(t)1+x2(t)xΔ(t)]Δ+t2(2+sint)x(σ(t))=0,(41)[1t21+2x2(t)(xΔ(t))21+x2(t)(xΔ(t))2xΔ(t)]Δ+t2(2+sint)x(σ(t))=0,
where p(t)=1/t2, r(t)≡0, q(t)=t2, k1(u,v)=((2+u2)/(1+u2))v in (40), and k1(u,v)=((1+2u2v2)/(1+u2v2))v in (41), so we have both α1=2, α2=1. Letting H(t,s)=(t-s)2, we have
That is, (38) holds. By Corollary 7 we see that (40) and (41) are oscillatory.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This project was supported by the NNSF of China (no. 11271379).
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