# Fite-Wintner-Leighton-Type Oscillation Criteria for Second-Order Differential Equations with Nonlinear Damping

Authors: Mervan Pašić
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

Some new oscillation criteria for a general class of second-order differential equations with nonlinear damping are shown. Except some general structural assumptions on the coefficients and nonlinear terms, we additionally assume only one sufficient condition (of Fite-Wintner-Leighton type). It is different compared to many early published papers which use rather complex sufficient conditions. Our method contains three items: classic Riccati transformations, a pointwise comparison principle, and a blow-up principle for sub- and supersolutions of a class of the generalized Riccati differential equations associated to any nonoscillatory solution of the main equation.


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## Body

## 1. Introduction

In the paper, we develop some new oscillation criteria for the following class of second-order differential equations with nonlinear damping:(1)(r(t)k1(x,x′))′+p(t)k2(x,x′)x′+q(t)f(x)=0,t≥t0>0,
where the coefficients r∈C1([t0,∞),(0,∞)),p,q∈C([t0,∞),ℝ), and the functions k1(u,v), k2(u,v) are continuous in all their variables, k1∈C1(ℝ2,ℝ) and solution x=x(t), x∈C2((t0,∞),ℝ)∩C([t0,∞),ℝ). A function x(t) is said to be oscillatory if there is a sequence tn≥t0 such that x(tn)=0 and tn→∞ as n→∞.In Section2, we present some basic structural assumptions on the coefficients: r(t), p(t), and q(t) and on the nonlinear functions: k1(u,v) and k2(u,v), which are slightly more general than those of the previously published results such as in Zhao et al. [1, Theorem 2.1] (see also Theorem A, Section 2), [1, Theorems 2.2–2.8], [2, Theorem 2], [3, Theorem 2.1]. In Section 3, we study some new oscillation criteria for (1) based on an additional sufficient condition of Fite-Wintner-Leighton type, which is rather simpler than Kamenev-type conditions or related complex ones. Equation (1) in various different forms has been considered in many several published papers, see, for instance, [4–12] and references therein. In Section 4, we state and prove a pointwise comparison principle between all sub- and supersolutions of the corresponding generalized Riccati differential equation associated with every nonoscillatory solution x(t) of (1). Furthermore, under the main assumption of Fite-Wintner-Leighton type, we construct a subsolution of the Riccati differential equation which blows up in time. It together with classic Riccati transformation gives the proof of the main result.


## 2. Main Assumptions and Remarks

In particular, form=n=1, in [1] authors firstly supposed the next five basic conditions on the coefficients p(t), q(t) and the functions f(u), k1(u,v), and k2(u,v):(2)p(t)≥0∀t≥t0,(3)f(u)u≥Kfor  someK>0and  allu∈ℝ,u≠0,(4)q(t)≥0∀t≥t0,q(t)≢0on[t*,∞)for  anyt*≥t0,(5)k12m(u,v)≤α1u2m-2vk1(u,v)forsomem∈ℕ,α1>0k12m(u,v)and  all(u,v)∈ℝ2,u≠0,(6)u2n-1vk2(u,v)≥α2k12n(u,v)for  somen∈ℕ,α2>0u2n(u,v)and  all(u,v)∈ℝ2,u≠0.
Such a set of assumptions, with slightly different (6) and m=n=1, was introduced for the first time in [2], see also [3]. Just the same as in [1], besides (6) we also consider a similar assumption:
(6)1uvk2(u,v)≥α2uk1(u,v)for  someα2>0and  all(u,v)∈ℝ2.
And, in the case when p(t) and q(t) may change the sign, instead of (2)–(4) and (6), one considers also:
(3)1f∈C1(ℝ,ℝ),uf(u)≠0,f′(u)≥K>0∀u∈ℝ,u≠0,(6)2vk2(u,v)=α2k1(u,v)for  someα2>0and  all(u,v)∈ℝ2.Here, assumptions (5) and (6) are generalized in the following sense, see Theorem 5—(ii) and (iii), respectively,
(5)wvk1(u,v)≥0∀(u,v)∈ℝ2,(6)wuvk2(u,v)≥0∀(u,v)∈ℝ2,
which are weaker than (5) and (6), respectively. One of the reasons for that is presented in the next remarks.Remark 1.
(1)
The most simple second-order differential operator which satisfies assumption (5) for m=1 is linear in variable v; that is,
(7)(r(t)k1(x,x′))′=(A(x)x′)′,
where r(t)≡1, k1(u,v)=A(u)v, and A(u) is an arbitrary function satisfying 0≤A(u)≤α1. It is because k12(u,v)=A2(u)v2≤α1A(u)v2=α1vk1(u,v) for all (u,v)∈ℝ2 and α1≥0, see also Corollary 6. However, it is easy to check that the differential operator from (7) does not satisfy assumption (5) for every m>1.
(2)
Next, we consider the corresponding second-order quasilinear differential operator:(8)(r(t)k1(x,x′))′=(A(x)|x′|β-1x′)′,
where r(t)≡1, k1(u,v):=A(u)|v|β-1v, and A(u) is an arbitrary function satisfying 0≤A(u)≤α1 and in order to ensure that k1∈C1(ℝ2,ℝ), we take β≥1 since ∂k1/∂v=β|v|β-1. Unfortunately, the differential operator from (8) does not satisfy assumption (5) for every m∈ℕ, β>1. It is because k12(u,v)=A2(u)v2β-2v2=A2(u)v2β≤α1A(u)v2β=α1vk1(u,v)|v|β-1, which is different from (5).
(3)
Unlike (5), the differential operator from (8) satisfies assumption (5)w, and hence, (8) is also included in our study of the oscillation of (1), see Corollary 11.
(4)
Although both differential operators from (7) and (8) do not satisfy assumption (5) for every m>1, the so-called generalized prescribed mean curvature-like differential operator:
(9)(r(t)k1(x,x′))′=(A(x)x′(1+x′2)α/2)′
satisfies assumption (5) for every m≥1, where r(t)≡1, k1(u,v):=A(u)v/(1+v2)α/2, α≥1, and A(u) is an arbitrary function satisfying 0≤A2m-1(u)≤α1u2m-2, see Corollary 9.
(5)
The simple casek2(u,v)≡0 is involved in (6)w unlike (6), and hence, the nonlinear equation x′′+q(t)f(x)=0 can be considered as a special case of (1).We pay attention to the recently published paper [13] in which authors show that any generalization of the assumptions (2)–(6) should be done very carefully.Now, we can recall [1, Theorems 2.5].Theorem A.
Let (2)–(6) hold. Assume that there exist ρ∈C1([t0,∞),(0,∞)), H∈ℍ, g∈C1([t0,∞),ℝ), and some t1≥t0 such that for all T≥t1:(10)limsupt→∞1H(t,T)∫Tt‍[{α1ρ(s)v(s)r2(s)4(α2p(s)+r(s))}H(t,s)γ1(s)limsupt→∞1H(t,T)-α1ρ(s)v(s)r2(s)4(α2p(s)+r(s))Q12(t,s)]ds=∞,
where v(t) and γ1(t) are defined, respectively, by
(11)v(t)=exp(-2α1∫t1tg(s)ds),γ1(t)=ρ(t)v(t)[1α1r(t)g2(t)γ1(t)=ρ(t)v(t)+Kq(t)+α2g2(t)p(t)-(r(t)g(t))′{1α1}{1α1}],
and Q1∈C(D,ℝ) satisfies
(12)-∂H(t,s)∂s=(ρ′(s)ρ(s)+2α2p(s)g(s)r(s))H(t,s)+Q1(t,s)H(t,s).
Then, (1) is oscillatory.In Theorem A, the setD={(t,s):t≥s≥t0}. And the assumption H∈ℍ means that H∈C(D,ℝ+), ∂H(t,s)/∂s is continuous on D0={(t,s):t>s≥t0}, H(t,t)=0 for all t≥t0 and H(t,s)>0 for all (t,s)∈D0. It is easy to see that the coefficients: r(t), p(t), and q(t) are involved in the assumptions (10)–(12), often called the general Kamenev-type conditions, about the Kamenev-type conditions and their several generalization we refer the reader, for instance, to [14–18]. The main purpose of supposing the existence of the functions: ρ(t), H(t,s), and g(t) satisfying the corresponding assumptions (10)–(12) is to ensure the nonexistence of continuous function w(t) which satisfies the corresponding Riccati differential inequality:
(13)w′≥α(t)ω2+β(t)ω+γ(t),t≥T,
where α(t), β(t), and γ(t) depend on r(t), p(t), α1, and α2, and T≥t0.Instead of Kamenev-type conditions (10)–(12), we consider the next one (which can be called the Fite-Wintner-Leighton-type condition by a reason given in Remark 2): for the explicitly given two functions a(t) and b(t) which depend on the data r(t), p(t), m, n, α1, α2, and q(t), (see Theorems 5 and 15), let there be a function E(t) and a point T1≥t0 such that
(14)E∈C([T1,∞)),limsupt→∞∫T1t‍E(τ)dτ=∞,E∈C([T1,∞)),lim∀m,n∈ℕ,we  haveE(t)≤{min{a(t),b(t)},t≥T1,ifmin{m,n}=1,min{a(t),b(t)-a(t)},t≥T1,ifmin{m,n}>1.
Combining a pointwise comparison principle and a blow-up argument, which is a different method than that in the case of Kamenev-type conditions, we are able to prove the nonexistence of any continuous function ψ(t) which satisfies the corresponding Riccati differential inequality:
(15)ψ′≥a1(t)ψ2m+a2(t)ψ2n+b(t),t≥T,
where a1(t), a2(t), and b(t) are arbitrary functions. On the various aspects of the comparison principles, we refer the reader, for instance, to [19, 20]—the comparison principles for Volterra integral operators, [21, 22]—the pointwise comparison principle for ODEs and [23]—the abstract form of comparison principles.Remark 2.
It is simple to check that in particular fork1(u,v)≡v, k2(u,v)≡0, and f(u)≡u, the conditions (3), (5) with m=1 and (6)w still hold where the inequality “≥” is replaced by “=.” Then (1) becomes the linear second-order differential equation (LEq): (r(t)x′)′+q(t)x=0. Hence, the inequality in (14) for m=n=1 can be replaced by the corresponding equality, where a(t)=1/r(t) and b(t)=q(t) (see the case (iii) of Theorem 5), and so, we conclude that in this case, (14) is equivalent to:
(16)limsupt→∞∫T1t‍1r(τ)dτ=limsupt→∞∫T1t‍q(τ)dτ=∞,
which presents the classic Fite-Wintner-Leighton oscillation criterion for linear second-order differential equation (LEq), where “lim” appears instead of “limsup.” In Fite [24], Wintner [25], and Leighton [26] equation (LEq) was considered, respectively, with r(t)≡1 and q(t)>0, r(t)≡1 and q(t) may change sign, and arbitrary r(t)>0 and q(t) may change sign. Nonlinear version of such a class of oscillation criteria was due to Wong [27], and Nth-order extension for linear equations can be found in Travis [28].In order to simplify notation, we firstly introduce the following definition for the pointwise comparison principle of the corresponding Riccati differential equation:(17)w′=a1(t)w2m+a2(t)w2n+b(t),t≥T,
where a1(t), a2(t), and b(t) are three arbitrary functions, and T≥t0.Definition 3.
LetT0 and T* be two arbitrary real numbers, T≤T0<T*. Two functions, φ(t) and ψ(t),φ,ψ∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ), are said to be, respectively, subsolution and supersolution of the Riccati differential equation (17) provided that
(18)φ′≤a1(t)φ2m+a2(t)φ2n+b(t),ψ′≥a1(t)ψ2m+a2(t)ψ2n+b(t),t∈(T0,T*).
Moreover, if the statement:
(19)φ(T0)≤ψ(T0)implyingφ(t)≤ψ(t)∀t∈[T0,T*)
is fulfilled for all sub- and supersolutions φ,ψ∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ) of (17), then we say that comparison principle (19) holds for (17) with arbitrary T0 and T*, T≤T0<T*.Remark 4.
The possibilty that (19) holds for all sub- and supersolutions and with arbitrary T0 and T*, T≤T0<T* plays an essential role in some concrete situations. According to it, when the comparison principle (19) holds for the Riccati differential equation (17) with arbitrary T0 and T*, T≤T0<T*, then we can choose some concrete sub- and supersolutions as well as T0 and T* with some suitable properties.Our method contains the next three steps:(i)
at the first step, we give a sufficient condition ona1(t),a2(t) such that comparison principle (19) holds for the Riccati differential equation (17) with arbitrary b(t), T0 and T*, T≤T0<T*;(ii)
at the second step, for a supersolutionψ∈C1((T,∞),ℝ)∩C([T,∞),ℝ) of (17), where a1(t), a2(t), and b(t) are three arbitrary functions, and under assumption (14), we find two real numbers T0 and T*, T≤T0<T*, and construct a subsolution φ(t), φ∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ) of (17) such that the following initial and blow-up arguments are satisfied:
(20)φ(T0)≤ψ(T0),limt→T*φ(t)=∞;(iii)
at the third step, under conditions (2)–(6) or related ones such as (5)w and (6)w, we show that if the main equation (1) allows a nonoscillatory solution x(t), then the function:
(21)ψ(t)=-r(t)k1(x(t),x′(t))x(t),t≥T,
is well defined for some T≥t0, ψ∈C1((T,∞),ℝ)∩C([T,∞),ℝ), and ψ(t) is a supersolution of (17) with some concrete a1(t), a2(t), and b(t); in the case when p(t) and q(t) change the sign, instead of (21), we consider the function:
(22)ψ(t)=-r(t)k1(x(t),x′(t))f(x(t)),t≥T.In conclusion, combining (19) and (20), we obtain the nonexistence of any continuous supersolution of the Riccati differential equation (17), and hence, the function ψ(t) given by (21) or (22) is not possible. Therefore, (1) does not allow any nonoscillatory solution.


## 3. Main Results and Examples

As usual, we recognize two main different cases: the first one is whenp(t) and q(t) are positive and the second one is when they may change the sign. Moreover, in the first case, depending on the combination of assumptions (5), (6), (5)w, (6)w, and (6)1, we consider five subcases such as is done in our first oscillation criterion for (1).Theorem 5 (positive coefficients).
Let assumptions (2)–(4) be fulfilled. Then, (1) is oscillatory if one of the next five cases is met.(i)
Letm,n∈ℕ and (5), (6) hold. One supposes (14) with respect to a(t):=a1(t)+a2(t) provided that m=n=1 or a(t):=min{a1(t),a2(t)}, otherwise,
(23)a1(t)=1α1r2m-1(t),a2(t)=p(t)α2r2n(t),b(t)=Kq(t).(ii)
Letn∈ℕ and (5)w, (6) hold. One supposes (14) with respect to a(t) and b(t) given by
(24)a(t)=p(t)α2r2n(t),b(t)=Kq(t).(iii)
Letm∈ℕ and (5), (6)w hold. One supposes (14) with respect to a(t) and b(t) given by
(25)a(t)=1α1r2m-1(t),b(t)=Kq(t).(iv)
Letm=1 and (5), (6)1 hold. One supposes (14) with respect to a(t) and b(t) given by
(26)a(t)=1α1r(t)e-α2∫(p(τ)/r(τ))dτ,b(t)=Kq(t)eα2∫(p(τ)/r(τ))dτ.(v)
Letp∈C1((t0,∞),ℝ), m=1, and (5), (6)1 hold. One supposes (14) with respect to a(t) and b(t) given by
(27)a(t)=1α1r(t),b(t)=Kq(t)-α1α2p′(t)2-α1α22p2(t)4r(t).For each of the cases (i)–(v) of Theorem5, we derive some consequences and examples, which show the importance of our oscillation criterion.The case (i) of Theorem5 for m=n=1 allows us to consider the following class of equations:
(28)(r(t)A(x)x′)′+p(t)B(x)x′2+q(t)f(x)=0,t≥t0>0,
where the functions A=A(u), A∈C1(ℝ), and B(u) satisfy
(29)0≤A(u)≤α1,uB(u)≥α2A2(u)∀(u,v)∈ℝ2,
for some α1>0 and α2>0. Under assumption (29), it is easy to see that the functions k1(u,v):=A(u)v and k2(u,v):=B(u)v satisfy both required assumptions (5) and (6) with m=n=1. Hence, as an easy consequence of Theorem 5, we obtain the following result.Corollary 6.
Let (2)–(4) and (14) hold with respect to a(t) and b(t) given in case (i) of Theorem 5 with m=n=1. If A(u) and B(u) satisfy (29), then (28) is oscillatory.Example 7.
LetK>0, μ≤1 or ν≥2μ-1, and σ≤1. Then, the equation:
(30)(tμx21+x2x′)′+tνx3x′2+Kt-σx=0,t≥t0>0
is oscillatory. Indeed, it is enough to check that the coefficients r(t)=tμ, p(t)=tν, and q(t)=t-σ and the functions f(u)=Ku, A(u)=u2/(1+u2), and B(u)=u3 satisfy all the assumptions of Corollary 6 with respect to α1=α2=1 and E(t)=c/t for some c>0.Example 8.
LetK>0, μ≤1 or ν≥2μ-1, and σ≤1. Then, the equation:
(31)(tμ(sinx)2x′)′+tνx3x′2+Kt-σx=0,t≥t0>0
is oscillatory. In fact, it is easy to check that the coefficients r(t)=tμ, p(t)=tν, and q(t)=t-σ and the functions f(u)=Ku, A(u)=sin2u, and B(u)=u3 satisfy all the assumptions of Corollary 6 with respect to α1=α2=1 and E(t)=c/t for some c>0.The case (i) of Theorem5 for m,n∈ℕ proposes the following class of differential equations:
(32)(r(t)A(x)x′(1+x′2)α/2)′+p(t)B(x)(x′(1+x′2)α/2)2n+q(t)f(x)=0,t≥t0>0,
where α≥1, n∈ℕ, and the functions A=A(u), A∈C1(ℝ), and B(u) satisfy
(33)0≤A2m-1(u)≤α1u2m-2,u2n-1B(u)≥α2A2n(u),u∈ℝ.As a consequence of Theorem5, we derive the next interesting corollary.Corollary 9.
Let (2)–(4) and (14) hold with respect to a(t) and b(t) given in case (i) of Theorem 5. If A(u) and B(u) satisfy (33), then (32) is oscillatory.Example 10.
Letα≥1, n∈ℕ, K>0, μ≤1 or ν≥2μ-1, and σ≤1. Then, according to Corollary 9, we conclude that the equation:
(34)(tμx21+x2x′(1+x′2)α/2)′+tνx(xx′(1+x2)(1+x′2)α/2)2n+Kt-σx=0,t≥t0>0
is oscillatory.We have pointed out in Remark1 that assumption (5)w unlike (5) allows to consider the oscillation of the following quasilinear differential equation:
(35)(r(t)A(x)|x′|β-1x′)′+p(t)B(x)x′2β+q(t)f(x)=0,t≥t0>0,
where β≥1 and the functions A=A(u), A∈C1(ℝ), and B(u) satisfy
(36)0≤A(u),uB(u)≥α2A2(u)∀(u,v)∈ℝ2,
for some α2>0. It is clear that (28) is a particular case of (35) for β=1. Under assumption (36), the functions
(37)k1(u,v):=A(u)|v|β-1v,k2(u,v):=B(u)v2β-1
satisfy both required assumptions (5)w and (6) with n=1. Therefore, we can derive the following easy consequence of the case (ii) of Theorem 5.Corollary 11.
Let (2)–(4) and (14) hold with respect to a(t) and b(t) given in case (ii) of Theorem 5 with n=1. If A(u) and B(u) satisfy (36), then (35) is oscillatory.Example 12.
Letβ≥1, K>0, ν≥2μ-1 and σ≤1. Then the equation:
(38)(tμ(sinx)2x′β)′+tνx3x′2β+Kt-σx=0,t≥t0>0,
is oscillatory. In fact, it is enough to check that the coefficients r(t)=tμ, p(t)=tν, q(t)=t-σ and the functions f(u)=Ku, A(u)=sin2u and B(u)=u3 satisfy all assumptions of Corollary 11 with respect to E(t)=c/t for some c>0.The case (iii) of Theorem5 allows us to consider the following class of equations:
(39)(r(t)A(x)x′)′+p(t)B(x)C(x′)x′+q(t)f(x)=0,t≥t0>0,
where the functions A=A(u), A∈C1(ℝ), B(u) and C(v) satisfy:
(40)0≤A(u)≤α1,uB(u)≥0,vC(v)≥0∀(u,v)∈ℝ2,
for some α1>0. Under this assumption, it is easy to see that the functions k1(u,v):=A(u)v and k2(u,v):=B(u)C(v) satisfy both required assumptions (5) and (6)w. Hence, as an easy consequence of case (iii) of Theorem 5, we obtain the following result.Corollary 13.
Let (2), (3), and (14) hold with respect to a(t) given in case (iii) of Theorem 5. If A(u), B(u), and C(v) satisfy (40), then (39) is oscillatory.Example 14.
LetK>0, μ≤1, ν≥0, λ≥0, and σ≤1. Then, the equation:
(41)(tμx21+x2x′)′+tν|x|λxsh(x′)x′+Kt-σx=0,t≥t0>0
is oscillatory. In order to show that, it is enough to check that the coefficients: r(t)=tμ, q(t)=t-σ and the functions: f(u)=Ku, A(u)=u2/(1+u2), B(u)=|u|λu, and C(v)=sh(v) satisfy all the assumptions of Corollary 13 with respect to α1=α2=1 and E(t)=c/t for some c>0.Next, we consider the oscillation of (1) in the case when the coefficients p(t) and q(t) may change the sign.Theorem 15 (coefficients may change the sign).
Letm=1 and assumptions (3)1, (5), and (6)2 hold. Then, (1) is oscillatory provided that one of the following two cases is met. (vi) One assumes (14) with respect to a(t) and b(t) given by
(42)a(t)=Kα1r(t)e-α2∫Tt(p(τ)/r(τ))dτ,b(t)=q(t)eα2∫Tt(p(τ)/r(τ))dτ,t≥T.
(vii) Let p∈C1((t0,∞),ℝ). One assumes (14) with respect to a(t) and b(t) given by
(43)a(t)=Kα1r(t),b(t)=q(t)-α1α2p′(t)2K-α1α22p2(t)4Kr(t),t≥T.The case (vi) of Theorem15 allows us to consider the following class of equations:
(44)(r(t)A(x)x′)′+α2p(t)A(x)x′+q(t)f(x)=0,t≥t0>0,
where α2∈ℝ, and the function A(u) satisfies
(45)0≤A(u)≤α1∀(u,v)∈ℝ2and someα1>0.
Under (45), one can easily check that the functions k1(u,v):=A(u)v and k2(u,v):=α2A(u) satisfy both required assumptions (5) and (6)2. Hence, as an easy consequence of case (vi) of Theorem 5, we conclude the next consequence.Corollary 16.
Let(3)1 and (14) hold with respect to a(t) and b(t) given in case (vi) of Theorem 5. If A(u) satisfies (45), then (44) is oscillatory.Example 17.
Letμ≥2 and q0∈ℝ. Then, the equations:
(46)(t-μx21+x2x′)′+t-μ-1x21+x2x′+q0(sint)x=0,t≥t0>0,(t-μ(sinx)2x′)′+t-μ-1(sinx)2x′+q0(sint)x=0,t≥t0>0
are oscillatory. In order to show that, it is enough to check that the coefficients: r(t)=t-μ, p(t)=t-μ-1, and q(t)=q0sint and the functions: f(u)=u, A(u)=u2/(1+u2), and A(u)=sin2u satisfy all the assumptions of Corollary 16 with respect to α1=α2=1 and E(t)=ctsint for some c∈ℝ, c≠0.


## 4. Proofs of the Main Results

In this section, we study the oscillation of (1) in the view of a pointwise comparison principle presented below, which will be shown for the corresponding Riccati differential equation.Definition 18.
A functionh(t,u) is said to be locally Lipschitz in the second variable if for any bounded interval I0⊆[T,∞) and M>0 there is a constant L>0 depending on I0,M,h such that
(47)|h(t,u1)-h(t,u2)|≤L|u1-u2|∀t∈I0,|h(t,u1)-h|u1,u2∈[-M,M].Now, we state and use the following general comparison principle, which will be proved at the end of this section.Lemma 19.
LetT0 and T* be two arbitrary real numbers such that T0<T*. Let φ~(t) and ψ~(t), φ~,ψ~∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ), be two functions satisfying:
(48)φ~′≤h(t,φ~),ψ~′≥h(t,ψ~),t∈(T0,T*),
where h(t,u) is a locally Lipschitz function in the second variable. Then, we have
(49)φ~(T0)≤ψ~(T0)impliesφ~(t)≤ψ~(t)∀t∈[T0,T*).Definition 20.
A functiona(t) is said to be locally bounded on [T,∞), if for any bounded interval I0⊆[T,∞) there is a constant C>0 depending on I0 such that |a(t)|≤C for all t∈I0.According to Lemma19, we are able to give a sufficient condition on the functions: a1(t),a2(t) such that the Riccati differential equation (17) satisfies the comparison principle (19).Lemma 21.
Ifa1(t) and a2(t) are two locally bounded functions on [T,∞), then comparison principle (19) holds for the Riccati differential equation (17) with arbitrary b(t), T0, and T*, where T≤T0<T*.Proof.
Letφ(t) and ψ(t),φ,ψ∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ), be, respectively, sub- and supersolution of (17); that is, they satisfy (18). It is not difficult to check that h(t,u):=a1(t)u2m+a2(t)u2n+b(t) is a locally Lipschitz function in the second variable. Indeed, for any bounded interval I0⊆[t0,∞), M>0, for all t∈I0 and u1,u2∈[-M,M], we have
(50)|h(t,u1)-h(t,u2)|≤|a1(t)||u12m-u22m|+|a2(t)||u12n-u22n|=|a1(t)||u1-u2||∑j=12m‍u12m-ju2j-1|+|a2(t)||u1-u2||∑j=12n‍u12n-ju2j-1|≤2mM2m-1|a1(t)||u1-u2|+2nM2n-1|a2(t)||u1-u2|≤2C(mM2m-1+nM2n-1)|u1-u2|,
where C=max{supI0|a1(t)|,supI0|a2(t)|}. Hence, Lemma 19 can be applied to φ(t) and ψ(t). If we set φ~(t):=φ(t), and ψ~(t):=ψ(t), then statement (48) is fulfilled because of assumption (18), and therefore, the desired conclusion (19) immediately follows from (49).Corollary 22.
Ifa1(t) and a2(t) are two continuous functions on [T,∞), then comparison principle (19) holds for the Riccati differential equation (17) with arbitrary b(t), T0, and T*, where T≤T0<T*.Proof.
Sincea1(t) and a2(t) are two continuous functions on [T,∞), they are also locally bounded functions on [t0,∞), and hence, this corollary immediately follows from Lemma 21.Next, we present an essential lemma in which we construct a subsolutionφ(t) of (17) which has a blow-up desired property.Lemma 23.
Leta1(t)≥0, a2(t)≥0, and b(t) be three arbitrary functions, and let assumption (14) hold, where a(t)=a1(t)+a2(t) if m=n=1 and a(t)=min{a1(t),a2(t)} otherwise. Let ψ∈C1((T,∞),ℝ)∩C([T,∞),ℝ) be a supersolution of the Riccati differential equation (17). Then, there are two real numbers T0 and T*, T≤T0<T*, and a subsolution φ∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ) of (17) satisfying
(51)φ(T0)≤ψ(T0),limt→T*φ(t)=∞.Proof.
In particular from (14), we obtain a sequence tn→∞ as n→∞ such that
(52)∫T1tn‍E(τ)dτ→∞asn→∞,
where T1 (determined in (14)) can be chosen so that T1≥T. From the previous statement, we conclude that there is a T2>T1 such that
(53)∫T1T2‍E(τ)dτ=π.
Since ∫T1t‍E(τ)dτ is a continuous function in the variable t, there is a T0∈[T1,T2) such that
(54)∫T1T0‍E(τ)dτ=0,∫T1t‍E(τ)dτ≥0∀t∈[T0,T2).
Consequently, we derive that
(55)∫T1t‍E(τ)dτ=∫T1T0‍E(τ)dτ+∫T0t‍E(τ)dτ=∫T0t‍E(τ)dτ,t∈[T0,T2),
which together with (53) and (54) shows
(56)∫T0t‍E(τ)dτ≥0∀t∈[T0,T2),∫T0T2‍E(τ)dτ=π.
Next, lets0∈(-π/2,π/2) be such that tan(s0)=ψ(T0), where T0 is from (54)-(56). Such s0 exists since the tangent function is a bijection from (-π/2,π/2) to ℝ. Let
(57)V(t):=∫T0t‍E(τ)dτ,t∈[T0,T2].
Because of (56), we have s0+V(t)>-π/2, t∈[T0,T2), s0+V(T0)<π/2 and s0+V(T2)>π/2. Since V(t) is a continuous function, it implies the existence of a T*∈(T0,T2) such that
(58)s0+V(T*)=π2,-π2<s0+V(t)<π2∀t∈[T0,T*).
As a consequence, the function
(59)φ(t)=tan(s0+V(t)),t∈[T0,T*)
is well defined and obviously satisfies
(60)φ(T0)=tan(s0+V(T0))=tan(s0)=ψ(T0),limt→T*φ(t)=tan(s0+V(T*))=tan(π2)=∞.
Also, since E(t) is continuous on [t0,∞), we have φ∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ). Now, by taking the derivative of φ(t) for every t∈[T0,T*), we obtain
(61)φ′=1cos2(s0+∫T0t‍E(τ)dτ)·E(t)=E(t)(1+tan2(s0+∫T0t‍E(τ)dτ))=E(t)φ2+E(t).
According to (14), we observe that(1)
ifm=n=1, then
(62)E(t)φ2+E(t)≤(a1(t)+a2(t))φ2+b(t)=a1(t)φ2m+a2(t)φ2n+b(t);(2)
if1=min{m,n}<max{m,n}, then
(63)E(t)φ2+E(t)≤a(t)φ2+b(t)≤a(t)(φ2m+φ2n)+b(t)≤a1(t)φ2m+a2(t)φ2n+b(t);(3)
ifmin{m,n}>1, then
(64)E(t)φ2+E(t)≤a(t)φ2+b(t)-a(t)≤a(t)(φ2m+φ2n+1)+b(t)-a(t)≤a1(t)φ2m+a2(t)φ2n+b(t).Thus, in all three cases of m,n∈ℕ, we have
(65)E(t)φ2+E(t)≤a1(t)φ2m+a2(t)φ2n+b(t),t∈[T0,T*).
Putting the previous inequality into (61) and taking into account of (60), we conclude that
(66)φ(T0)≤ψ(T0),φ′≤a1(t)φ2m+a2(t)φ2n+b(t),φ(T0)≤t∈[T0,T*),limt→T*φ(t)=∞.
It proves that φ(t) is a subsolution of the Riccati differential equation (17) which satisfies the statement (51).Next, we are concerned with the following technical but crucial lemma.Lemma 24.
Let the assumptions of Theorem5 in the cases (i)–(iii) hold. If the main equation (1) allows a nonoscillatory solution x(t), then the function ψ(t) given by (21) is well-defined with respect to such an x(t) and some T≥t0, ψ∈C1((T,∞),ℝ)∩C([T,∞),ℝ), and ψ(t) is a supersolution of the Riccati differential equation (17).Proof.
If the main equation (1) allows a nonoscillatory solution x(t), then there is a T≥t0 such that x(t)≠0 for all t≥T. Hence, the function ψ(t) given by (21) is well defined for such an x(t). Next, making the derivative of ψ(t), using that x(t) satisfies (1) and taking common assumptions of Theorem 5 for the functions p(t), r(t), q(t), k1(u,v), and k2(u,v), we obtain
(67)ψ′(t)=-1x(t)(r(t)k1(x(t),x′(t)))′+1x2(t)r(t)k1(x(t),x′(t))x′(t)=p(t)x(t)k2(x(t),x′(t))x′(t)+r(t)x2(t)k1(x(t),x′(t))x′(t)+q(t)f(x(t))x(t)=p(t)x2n(t)[k2(x(t),x′(t))x2n-1(t)x′(t)]+r(t)x2m(t)[k1(x(t),x′(t))x2m-2(t)x′(t)]+q(t)f(x(t))x(t).
Depending on each of the three cases (i)–(iii) of Theorem 5, from the previous equality, we obtain(68)ψ′(t)≥{α2p(t)x2n(t)k12n(x(t),x′(t))+r(t)α1x2m(t)k12m(x(t),x′(t))+Kq(t),in(i)—Theorem5,α2p(t)x2n(t)k12n(x(t),x′(t))+Kq(t),in(ii)—Theorem5,r(t)α1x2m(t)k12m(x(t),x′(t))+Kq(t),in(ii)—Theorem5,Next from (21), we also have
(69)k1(x(t),x′(t))=-x(t)r(t)ψ(t).
Now, from (68) and (69), we immediately obtain: ψ′≥a1(t)ψ2m+a2(t)ψ2n+b(t), t≥T. According to the definition of a supersolution, the previous inequality shows this lemma.Lemma 25.
Let the assumptions of Theorem5 in the cases (iv)-(v) hold. If the main equation (1) allows a nonoscillatory solution x(t), then the function ψ(t) given by
(70)ψ(t)={-r(t)k1(x(t),x′(t))x(t)×eα2∫‍(p(τ)/r(τ))dτ,t≥T,inthecase(iv),-r(t)k1(x(t),x′(t))x(t)-α1α2p(t)2,t≥T,inthecase(v)
is well defined with respect to such an x(t) and some T≥t0 such that ψ∈C1((T,∞),ℝ)∩C([T,∞),ℝ), and ψ(t) is a supersolution of the Riccati differential equation (17).The proof of Lemma25 is omitted because it is very similar to the proof of the following lemma.Lemma 26.
Let assumptions of Theorem15 hold. If the main equation (1) allows a nonoscillatory solution x(t), then the function ψ(t) given by
(71)ψ(t)={-r(t)k1(x(t),x′(t))f(x(t))×eα2∫(p(τ)/r(τ))dτ,t≥T,inthecase(vi),-r(t)k1(x(t),x′(t))f(x(t))-α1α2p(t)2K,t≥T,inthecase(vii)
is well defined with respect to such an x(t) and some T≥t0, ψ∈C1((T,∞),ℝ)∩C([T,∞),ℝ), and ψ(t) is a supersolution of the Riccati differential equation (17), where a(t)=a1(t)+a2(t) and a(t), b(t) are given in the case (vi) of Theorem 5.Proof.
Letx(t) be a nonoscillatory solution of (1), and thus, we can take a T≥t0 such that x(t)≠0 on [T,∞). Let ψ0(t) be a function defined by
(72)ψ0(t)=-r(t)k1(x(t),x′(t))f(x(t)),t≥T.
From the assumptions of Theorem 15 and from equalities (1) and (72), we can easily make the following computation:
(73)ψ0′(t)=-1f(x(t))(r(t)k1(x(t),x′(t)))′+1f2(x(t))r(t)k1(x(t),x′(t))x′(t)f′(x(t))=p(t)f(x(t))[k2(x(t),x′(t))x′(t)]+r(t)f2(x(t))[k1(x(t),x′(t))x′(t)]f′(x(t))+q(t)≥α2p(t)[k1(x(t),x′(t))f(x(t))]+Kr(t)α1[k1(x(t),x′(t))f(x(t))]2+q(t)=-α2p(t)r(t)ψ0(t)+Kα1r(t)ψ02(t)+q(t);
that is,
(74)ψ0′(t)≥Kα1r(t)ψ02(t)-α2p(t)r(t)ψ0(t)+q(t),t>T.
Now, if the middle term on the right-hand side of (74) is moved into the left-hand side, and multiplying such equality by eα2∫‍(p(τ)/r(τ))dτ, we conclude that the function
(75)ψ(t)=ψ0(t)eα2∫p(τ)/r(τ)dτ,t≥T
satisfies the Riccati differential equation (17) with respect to a(t) and b(t) given in the case (vi) of Theorem 15, which proves the first statement of this lemma.
However, if we group the first two terms on the right-hand side of (74) by the purpose of getting the complete square, then from (74) we easily conclude that the function:
(76)ψ(t)=ψ0(t)-α1α2p(t)2K,t≥T
satisfies the Riccati differential equation (17) with respect to a(t) and b(t) given in the case (vii) of Theorem 15, which proves the second statement of this lemma.Now, we are able to present a common proof of the main results of the paper.Proof of Theorems5 and 15.
At first, it is worth pointing out that the functions:a(t), a1(t), a2(t), and b(t), which are appearing at the same time in the main assumption (14) and the Riccati differential equation (17), only depend on the appropriate combination of basic assumptions on the coefficients: r(t)p(t), and q(t) and the functions: k1(u,v) and k2(u,v), which are formulated in one of the five cases of Theorem 5 and one of the two cases of Theorem 15.
Now, if we assume the contrary to the main assertion of the theorem; that is, if (1) is not oscillatory, then there is a nonoscillatory solution x(t) of (1) and a point T≥t0 and T≥T1, where T1 is appearing in (14), such that x(t)≠0 for all t∈[T,∞). Then by Lemmas 24, 25 and 26, the function ψ(t) given by (21) or (70), and (71) is well defined with respect to such an x(t), smooth enough on (T,∞), and it is a supersolution of the Riccati differential equation (17). Taking into account the main results of Lemma 23, we obtain the two numbers T0 and T*, T≤T0<T*, and a subsolution φ(t) of (17) such that the blow-up argument (51) is satisfied. By Corollary 22, we can apply the comparison principle (19) to (17) with arbitrary T0 and T*, where T≤T0<T*. Hence, combining (19) and (51), we get ψ(t)→∞ as t→T*, which contradicts the fact that ψ∈C1((T0,∞),ℝ)∩C([T0,∞),ℝ). Thus, ψ(t) is not possible, and therefore, (1) does not allow any nonoscillatory solution.Proof of Lemma19.
Letd~(t)=ψ~(t)-φ~(t) and φ~(T0)≤ψ~(T0); that is,
(77)d~(T0)≥0.
If statement (49) does not hold, then there is a point T*∈(T0,T*) such that φ~(T*)>ψ~(T*); that is,
(78)d~(T*)<0.
Moreover, since d~∈C1((T0,T*),ℝ)∩C([T0,T*),ℝ) from (77) and (78), we obtain a T1∈[T0,T*) such that
(79)d~(T1)=0,d~(t)<0∀t∈(T1,T*].
Since φ~,ψ~∈C([T1,T*]), we may use (47) in particular for
(80)I0=[T1,T*],M=max{maxt∈I0|φ~(t)|,maxt∈I0|ψ~(t)|}.
Hence, from (47), (48), and (79), we get
(81)d~′(t)=ψ~′(t)-φ~′(t)≥h(t,ψ~(t))-h(t,φ~(t))≥-L|d~(t)|=Ld~(t),t∈(T1,T*).
Multiplying this inequality by e-Lt and denoting by θ(t):=d~(t)e-Lt, we get
(82)θ′(t)=e-Lt(d~′(t)-Ld~(t))≥0,t∈(T1,T*).
Thus, according to (79) and (82), we have that θ(T1)=0, θ(t)<0 and θ′(t)≥0 on (T1,T*), which is not possible. Hence, the hypothesis (78) yields to a contradiction and, thus, φ~(t)≤ψ~(t) for all t∈[T0,T*).



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*Source: 852180-2013-02-21.xml*