# Oscillations of a Class of Forced Second-Order Differential Equations with Possible Discontinuous Coefficients

Authors: Siniša Miličić; Mervan Pašić; Darko Žubrinić
Journal: International Journal of Differential Equations
 (2013)

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## Abstract

We study the oscillation of all solutions of a general class of forced second-order differential equations, where their second derivative is not necessarily a continuous function and the coefficients of the main equation may be discontinuous. Our main results are not included in the previously published known oscillation criteria of interval type. Many examples and consequences are presented illustrating the main results.


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## Body

## 1. Introduction

Lett0>0 and let ACloc([t0,∞),ℝ) denote the set of all real functions absolutely continuous on every bounded interval [a,b]⊂[t0,∞). We study the oscillatory behaviour of all solutions x=x(t) of the following class of forced second-order differential equations:
(1)(r(t)Φ(x(t),x′(t)))′+q(t)f(x(t))=e(t),a.e.in[t0,∞),x,rΦ(x,x′)∈ACloc([t0,∞),ℝ),
where the functions Φ:ℝ2→ℝ, Φ=Φ(u,v), f:ℝ→ℝ, and f=f(u) satisfy some general conditions given in Section 2. A continuous function x=x(t) is said to be oscillatory if there is a sequence tn∈[t0,∞), such that x(tn)=0 for all n∈ℕ and tn→∞ as n→∞. A differential equation is oscillatory if all its solutions are oscillatory.The forcing terme(t) is a sign-changing function (possibly discontinuous). This can be formulated by the following hypothesis: for every T≥t0 there exist two intervals (a1,b1) and (a2,b2), T≤a1<b1≤a2<b2, such that
(2)e(t)≥0,t∈(a1,b1),e(t)≤0,t∈(a2,b2).
The coefficient q(t) may be a discontinuous function on [t0,∞) and the case x∉C2((t0,∞),ℝ) occurs in our main results and examples too. Two important classes of functions Φ(u,v) are included in the differential operator (r(t)Φ(x,x′))′ as
(3)Φ(u,v)=ϕ(u)v,Φ(u,v)=ϕ(u)v1+v2,(u,v)∈ℝ2.
The first one is the classic second-order differential operator which is linear in x′ and the second one is the so-called one-dimensional mean curvature differential operator; see Examples 1 and 2.Depending onq(t), we propose the following four simple models for (1): (i)
q(t) is strictly positive and continuous on [t0,∞) as
(4)x′′+4m2f(x)=h(sin(mt)),a.e.in[t0,∞),x,x′∈ACloc([t0,∞),ℝ),x∉C2((t0,∞),ℝ);(ii)
q(t) is nonnegative and continuous on [t0,∞) as
(5)x′′+m2π2[cos(mt)]+f(x)=h(sin(mt)),a.e.in[t0,∞),x,x′∈ACloc([t0,∞),ℝ);(iii)
q(t) is nonnegative and discontinuous on [t0,∞) as
(6)x′′+4m2[sign(cos(mt))]+f(x)=h(sin(mt)),a.e.in[t0,∞),x,x′∈ACloc([t0,∞),ℝ);(iv)
q(t) is sign changing and discontinuous on [t0,∞) as
(7)x′′+4m2sign(cos(mt))f(x)=h(sin(mt)),a.e.in[t0,∞),x,x′∈ACloc([t0,∞),ℝ),where m∈ℕ and h(s) is an arbitrary function such that h(s)s>0 for all s≠0, for instance, h(s)=s or h(s)=sign(s). According to Corollaries 7 and 10, we will show that (4)–(7) are oscillatory provided the function f=f(u) satisfies f(u)/u≥K≥1 for all u≠0; see Examples 8–13. It is interesting that in particular for f(u)=u and h(s)=2m2sign(s), (4) allows an explicit oscillatory solution x(t)=|sin(mt)|sin(mt) as shown in Figures 1 and 2.Figure 1
Functionx(t)=|sin(t)|sin(t) is a solution of (4).Figure 2
x
′
(
t
)
=
2
|
sin
(
t
)
|
cos
(
t
) and hence x′′(t) is not a continuous function.Moreover, as a consequence of Corollary7, one can show that all solutions of (4) are oscillatory; for details see Example 8. The main goal of this paper is to give some sufficient conditions on functions Φ(u,v), f(u) and the coefficients r(t), q(t), and e(t) such that (1) is oscillatory; see Theorems 3 and 4. It will also cover the model equations (4)–(7) as well as some other examples presented in Section 2.To the best of our knowledge, it seems that there are only few papers which study the oscillation of the second-order differential equations with nonsmooth (local integrable) coefficients; see [1–3]. More precisely, in [1] the author studied the interval oscillation criteria for the following second-order half-linear differential equation:
(8)(r(t)|x′(t)|σ-1x′(t))′+q(t)|x(t)|σ-1x(t)=0,a.e.in(0,∞),x,r|x′|σ-1x′∈ACloc((0,∞),ℝ),
where σ>1 and 1/r, q∈Lloc((0,∞),ℝ) such that ∫0∞‍r-1/σ(t)dr=∞. See also [2] but with the solution space C1((0,∞),ℝ) instead of ACloc((0,∞),ℝ), that is, x and r|x′|σ-1x′∈C1((0,∞),ℝ).In [3], the authors consider the following second-order differential equation:
(9)(r(t)x′(t))′+Q(t,x(t),x′(t))=0,t≥t0,
where r(t)>0 a.e. in [t0,∞), 1/r∈Lloc([t0,∞),ℝ), and Q(t,y,z) is locally integrable function in t and continuous in (y,z). Equation (9) allows the forcing term e(t) in the next sense as follows:
(10)yQ(t,y,z)≥q(t)yf(y)-e(t)y∀(t,y,z)∈[t0,∞)×ℝ2,
where e=e(t) satisfies (2), but the functions q=q(t) and f=f(y) are smooth enough in their variables, that is, q∈C([t0,∞),ℝ) and f∈C1(ℝ,ℝ).On certain oscillation criteria for various classes of forced second-order differential equations with continuous coefficients, we refer the reader to [4–13]. Our method modifies a recently used one in [14, 15] and it contains the classic Riccati transformation of the main equation, a blow-up argument and pointwise comparison principle. The comparison principle applies to all sub- and supersolutions of a class of the generalized Riccati differential equations with nonlinear terms that are supposed to be locally integrable in the first variable and locally Lipschitz continuous in the second variable.


## 2. Hypotheses, Results, and Consequences

First of all, the functionΦ(u,v) which appears in the second-order differential operator of (1) satisfies
(11)|u|γ-2vΦ(u,v)≥g(|Φ(u,v)|)∀u,v∈ℝ,
where γ≥2 and g:ℝ+→ℝ+ is a locally Lipschitz function g0:ℝ+→ℝ+ satisfying
(12)g(cs)≥cγg0(s)∀c>0,s>0,g0(s)+M0≥s2forsomeM0≥0andalls∈ℝ+.
In most cases, g(s)=g0(s)g1(s), s>0, where g0(s)=sγ, γ≥2, and g1(s) is an arbitrary function satisfying g1(s)≥1. Thus, for such g(s) with g1(s)≡1, condition (11) became:
(13)|u|γ-2vΦ(u,v)≥|Φ(u,v)|γ∀u,v∈ℝ.
It is not difficult to check that if g0∈C1(ℝ+) or g0(s) is a convex function, then it is locally Lipschitz on ℝ+ too; see for instance [16, Theorem 1.3.3].Two essential classes of the second-order differential operators(r(t)Φ(x,x′))′ satisfy condition (13), as is shown in the next examples.Example 1.
We consider the second-order differential operator which is linear inx′ as follows:
(14)(r(t)Φ(x,x′))′=α(ϕ(x)x′)′,
where α>0 and 0≤ϕ(u)≤1 for all u∈ℝ. Obviously, the function Φ(u,v)=ϕ(u)v satisfies condition (13) in particular for γ=2. Two usual choices for ϕ(u) are ϕ(u)=|sinu| and ϕ(u)=|u|/(1+|u|).Example 2.
We consider a quasilinear differential operator (the so-called one-dimensional prescribed mean curvature operator) as follows:(15)(r(t)Φ(x,x′))′=α(ϕ(x)x′1+x′2)′,
where α>0 and 0≤ϕγ-1(u)≤|u|γ-2 for all u∈ℝ. It is not difficult to check that condition (13) is satisfied in particular for Φ(u,v)=ϕ(u)v/(1+v2)1/2 and for any γ>1. For ϕ(u), we can take the same choice as in the previous example.Next, we suppose the existence of a constantK such that
(16)f(u)u≥K>0∀u≠0.
In order to simplify our consideration here, in many examples we often use f(u)=Ku.Condition (2) means that there exists a sequence of pairs of intervals J1j=[a1j,b1j] and J2j=[a2j,b2j], j∈ℕ, contained in (t0,∞), such that the sequences (a1j)j≥1, (b1j)j≥1, (a2j)j≥1, and (b2j)j≥1 are increasing, a1j<b1j≤a2j<b2j for each j, and
(17)e(t)≥0onJ1j,e(t)≤0onJ2jforeachj∈ℕ,limj→∞a1j=∞.On the intervalsJ1j and J2j, the coefficient r(t) satisfies
(18)r(t)>0onJij,r1-γ∈L1(Jij)∀i∈{1,2},j∈ℕ.Let there be a real functionC=C(t), C∈Lloc1((t0,∞),ℝ), and let there exist a sequence of positive real numbers (λj)j∈ℕ such that
(19)C(t)≥0onJij,cij≔∫Jij‍C(τ)dτ>0∀i∈{1,2},j∈ℕ,1cijC(t)≤1πmin{(λjr(t))1-γ,KM0+1λjq(t)}∀t∈Jij,i∈{1,2},j∈ℕ,
where γ,  M0, and K are constants defined in (11), (12), and (16), respectively.The proof of the following main result will be presented in Section4.Theorem 3.
Let the functionsΦ(u,v), f(u), e(t), and r(t) satisfy (11), (12), (16), (17), and (18), respectively. Let q(t)≥0 and q(t)≢0 on each interval Jij, i∈{1,2}, j∈ℕ. If (19) is fulfilled, then (1) is oscillatory.Condition (19) can be replaced by an equivalent one, which has a more practical value and takes a simpler form since we do not need a sequence of auxiliary parameters (λj)j∈ℕ: let there be a real function C=C(t), C∈Lloc1((t0,∞),ℝ) such that
(20)C(t)≥0onJij,cij≔∫Iij‍C(τ)dτ>0∀i∈{1,2},j∈ℕ,supt∈Jij[C(t)q(t)]supt∈Jij[r(t)C(t)1/(γ-1)]≤KM0+1(cijπ)γ/(γ-1)∀i∈{1,2},j∈ℕ,
where γ, M0, and K are constants defined in (11), (12), and (16), respectively. Since we will show that (19) and (20) are equivalent, see page 8, the next oscillation criterion immediately follows from Theorem 3.Theorem 4.
Let the functionsΦ(u,v), f(u), e(t), and r(t) satisfy (11), (12), (16), (17), and (18), respectively. Let q(t)≥0 and q(t)≢0 on each interval Jij, i∈{1,2}, j∈ℕ. If (20) is fulfilled, then (1) is oscillatory.Remark 5.
Assuming thatL:=limj→∞a1j<∞, we can ensure the oscillation in the point L. Note that L=limj→∞a2j since a1j<a2j<a1j+1. Thus, we can generate a one-sided (right) limit.Now, we consider some consequences of Theorem4, which depend on the qualitative properties of the coefficient q(t).SubstitutingC(t)≡1 in (20), Theorem 4 implies the following result involving lower bounds on the lengths of intervals |Jij|=bij-aij.Corollary 6 (q(t) is positive).
Let the functionsΦ(u,v), f(u), e(t), and r(t) satisfy (11), (12), (16), (17), and (18), respectively. Let inft∈Jijq(t)>0 for each i∈{1,2}, j∈ℕ. If
(21)|Jij|≥π((M0+1)supt∈Jijr(t)Kinft∈Jijq(t))(γ-1)/γ∀i∈{1,2},j∈ℕ,
then (1) is oscillatory.Corollary 7 (q(t) is bounded from below by a positive constant).
Let the functionsΦ(u,v), f(u), e(t), and r(t) satisfy (11), (12), (16), (17), and (18), respectively. Let there be two constants r0, q0 satisfying
(22)0<r(t)≤r0,q(t)≥q0>0∀i∈{1,2},j∈ℕ.
If
(23)|Jij|≥π((M0+1)r0Kq0)(γ-1)/γ∀i∈{1,2},j∈ℕ,
then (1) is oscillatory, where γ, M0, and K are constants defined in (11), (12), and (16), respectively.Example 88 (oscillation of (4)).
We know thatx(t)=|sin(mt)|sin(mt) is an oscillatory solution of (4). However, according to Corollary 7, we can show that all solutions of (4) are oscillatory. Indeed, since Φ(u,v)≡v, the conditions (11) and (12) are satisfied especially for γ=2, g(s)=g0(s)=s2 and M0=0. Next, f(u)≡u implies that condition (16) is satisfied especially for K=1. Since r(t)≡1 and q(t)≡4m2, it is clear that conditions (18) and (22) are also satisfied in particular for r0=1 and q0=4m2. Moreover, since e(t)=h(sin(mt)) and h(s)s>0, s≠0, we have that (17) is fulfilled for a1j=2jπ/m, b1j=(2j+1)π/m=a2j and b2j=(2j+2)π/m. Moreover,
(24)|Jij|=bi-ai=πm>0,i∈{1,2}.
Hence, we conclude that the required condition (23) is fulfilled, that is,
(25)|Jij|=πm≥π4m2=π((M0+1)r0q0)(γ-1)/γ≥π((M0+1)r0Kq0)(γ-1)/γ.
Thus, all conditions of Corollary 7 are satisfied and hence (4) is oscillatory.Example 9.
We consider the following class of equations:(26)x′′-2S′′(t)S(t)x=2sign(S(t))S′2(t),a.e.in[t0,∞),x,x′∈ACloc([t0,∞),ℝ),x∉C2((t0,∞),ℝ),
where S=S(t), S∈C2(ℝ) is an oscillatory function such that the zeros tn of the function sign(S(t))S′2(t) satisfy tn→∞, there is a τ0∈ℝ such that tn+1-tn≥τ0>0 for all n∈ℕ, and S(t)≠0 on (tn,tn+1). This equation allows an explicitly given oscillatory solution x(t)=|S(t)|S(t). Moreover, if there is a constant s0>0 such that
(27)-2S′′(t)S(t)≥s0,t∈(tn,tn+1),τ0≥πs0,
then by Corollary 7 we conclude that (26) is oscillatory. Indeed, conditions (11), (12), and (16) are satisfied by the same reasons as in Example 8. Condition (18) is satisfied because of r(t)≡1. Also, from (27) it follows that (22) and (23) are fulfilled in particular for γ=2, M0=0, r0=1, q0=s0, and K≥1, that is,
(28)|Jij|=|tj+1-tj|≥τ0≥πs0=π((M0+1)r0q0)(γ-1)/γ≥π((M0+1)r0Kq0)(γ-1)/γ.
Hence Corollary 7 proves this result.As the second consequence of Theorem3 is unlike the first one, we consider the case when the coefficient q(t) is not a strictly positive function. Here by {q=0} we denote the set of all t∈ℝ such that q(t)=0.Corollary 10 (q(t) is nonnegative, but not ≡0).
Let the functionsΦ(u,v), f(u), e(t), and r(t) satisfy (11), (12), (16), (17), and (18), respectively. Let q(t)≥0 on each interval Jij, i∈{1,2}, j∈ℕ, such that
(29)qij:=∫Jij‍q(τ)dτ>0,i∈{1,2},j∈ℕ.
If
(30)qijγq(t)≥πγ((M0+1)r(t)K)γ-1,t∈Jij∖{q=0},i∈{1,2},j∈ℕ,
then (1) is oscillatory.Proof.
It suffices to show that (30) is equivalent to the existence of a real number λj such that
(31)λj≥π(M0+1)K1qij,1qijq(t)≤1π(λjr(t))1-γ,t∈Jij,i∈{1,2},j∈ℕ.
The claim will then follow from Theorem 3. Inequality (31) is for any t∈Jij∖{q=0} equivalent to
(32)π(M0+1)K1qij≤λj≤(qijπq(t))1/(γ-1)1r(t),
that is, to
(33)π(M0+1)K1qij≤(qijπq(t))1/(γ-1)1r(t).
This inequality is easily seen to be equivalent to (30) for any t∈Jij∖{q=0}. Note that if t∈{q=0}, then the second inequality in (31) is trivially satisfied.The following three examples are simple consequences of Corollary10.Example 1111 (oscillation of (5)).
Equation (5) as well as equation
(34)x′′+m2π2[sin(mt)]+f(x)=h(cos(mt)),a.e.in[t0,∞),x,x′∈ACloc([t0,∞),ℝ),
are oscillatory, where f(u) satisfies (16) with K≥1 and h(s)s>0, s≠0. Indeed, in (5) we have r(t)≡1, q(t)=m2π2[cos(mt)] and e(t)=h(sin(mt)), and so, (17) and (18) are fulfilled, and q(t)≥0, q(t)≢0 on each interval Jij, i∈{1,2}, j∈ℕ, where J1j=[a1j,b1j], J2j=[a2j,b2j], and
(35)a1j=2jπm,b1j=(2j+1/2)πm,a2j=(2j+3/2)πm,b2j=(2j+2)πm.
Moreover,
(36)qij=∫Jij‍q(τ)dτ=m2π2∫aijbij‍[cos(mτ)]+dτ=mπ2,i∈{1,2},j∈ℕ,
and since γ=2, r(t)≡1, K≥1, and M0=0, for λj=1/(mπ) we have
(37)λj=1mπ≥1K1mπ=π(M0+1)K1mπ2=π(M0+1)K1qij,i∈{1,2},j∈ℕ,1qijq(t)=1mπ2m2π2[cos(mt)]+≤m=1π1(1/mπ)=1π1(λjr(t))γ-1,t∈Jij,i∈{1,2},j∈ℕ.
Thus, the required condition (30) is fulfilled in this case and hence we may apply Corollary 10 to (5) to verify that this equation is oscillatory. Analogously, we can show that (34) is oscillatory too.In the next example, the coefficientq(t) is a discontinuous function on [t0,∞).Example 12 (oscillation of (6)).
Iff(u) satisfies (16) with K≥1, then (6) is oscillatory since the required condition (30) is satisfied especially for q(t)=4m2sign(cos(mt)) on Jij, λj=1/(2m), and Jij as in (35), γ=2, r(t)≡1, K≥1, and M0=0. Indeed,
(38)qij=∫Jij‍q(τ)dτ=4m2∫Jij‍[sign(cos(mτ))]+dτ=4m2π2m=2mπ,i∈{1,2},j∈ℕ,λj=12m=π2mπ=π(M0+1)1qij≥π(M0+1)K1qij,i∈{1,2},j∈ℕ,1qiq(t)=12mπ4m2[sign(cos(mt))]+≤2mπ=1π11/(4m2)=1π(λjr(t))γ-1,t∈Jij,i∈{1,2},j∈ℕ,
which shows that (30) is satisfied.At the end of this section, we consider the case whenq(t) changes sign on [t0,∞) but with the help of Corollary 10 since q(t) is strictly positive on all intervals Jij.Example 13 (oscillation of (7)).
Iff(u) satisfies (16) with K≥1, then model equation (7) is oscillatory. In fact, let Jij be as in (35). If h(s)s>0, s≠0, it is clear that e(t)=h(sin(mt)) satisfies (2), that is, e(t)≥0 on J1j and e(t)≤0 on J2j. On the other hand, we have q(t)=4m2sign(cos(mt)) on Jij and so q(t)≥0, q(t)≢0 on Jij for all i∈{1,2}, j∈ℕ. Hence, the proof of the fact that all assumptions of Corollary 7 are fulfilled is the same as in the preceding example.


## 3. Proofs of Theorems3 and 4

LetACloc([a,b),ℝ) denote the set of all real functions which are absolutely continuous on every interval [a,b-ε], where ε∈(0,b-a). For arbitrary numbers T0<T* and functions G=G(t,u) and b=b(t), we consider the ordinary differential equation
(39)w′=G(t,w)+b(t),a.e.in[T0,T*),
which generalizes the classic Riccati equation w′=a(t)w2+b(t), where a(t) is an arbitrary function. We associate to (39) the corresponding sub- and supersolutions w_,w¯∈ACloc([T0,T*),ℝ) which satisfy, respectively,
(40)w_′≤G(t,w_)+b(t),w¯′≥G(t,w¯)+b(t)a.e.in[T0,T*).
We are interested in studying the following property:
(41)w_(T0)≤w¯(T0)impliesw_(t)≤w¯(t)∀t∈[T0,T*).
In this way, we introduce the following definition.Definition 14.
We say thatcomparison principle holds for(39) on an interval[T0,T*), T≤T0<T*, if property (41) is fulfilled for all sub- and supersolutions w_,w¯∈ACloc([T0,T*),ℝ) of (39) on [T0,T*).Now, we are able to state the next result.Lemma 15.
LetT≤T0<T*. Let G=G(t,u) and b=b(t) be two arbitrary function. For any M>0, let there be a function L=L(t)>0 depending on T0,T*,M such that
(42)L(t)>0on[T0,T*),L∈L1(T0,T*),|G(t,u1)-G(t,u2)|≤L(t)|u1-u2|∀t∈[T0,T*),u1,u2∈[-M,M].
Then comparison principle (41) holds for (39) on an interval [T0,T*), T≤T0<T*.Proof.
It is enough to use the idea of the proof of [14, Lemma 4.1]. Hence, this proof is left to the reader.Corollary 16.
Letλ>0, and let g0:ℝ+→ℝ+ be a locally Lipschitz function on ℝ+. Let K be an arbitrary real number and q(t) an arbitrary function. Let T≤T0<T*. If r(t)>0 on [T0,T*) and r-1+γ∈L1(T0,T*), then comparison principle (41) holds for the Riccati differential equation
(43)w′=1(λr(t))γ-1g0(|w|)+Kλq(t),a.e.in[T0,T*).Proof.
It is clear that (43) is a particular case of (39) in particular for
(44)G(t,u)=1(λr(t))γ-1g0(|u|),b(t)=Kλq(t).
Since g0 is a locally Lipschitz function on ℝ+, for every M>0 there is an L0>0 depending on M such that
(45)|g0(s1)-g0(s2)|≤L0|s1-s2|,∀s1,s1∈[-M,M].
Hence, for any M>0 and all u1,u2∈[-M,M], we obtain:
(46)|G(t,u1)-G(t,u2)|=1(λr(t))γ-1|g0(|u1|)-g0(|u2|)|≤L0(λr(t))γ-1||u1|-|u2||≤L0(λr(t))γ-1|u1-u2|.
Thus, according to assumption r-1+γ∈L1(T0,T*), the required condition (42) is fulfilled in particular for L(t)=L0(λr(t))-γ+1 and so Lemma 15 proves this corollary.Before we give the proof of Theorem3, we state and prove the next two propositions. In the first one, by a nonoscillatory solution x(t) of the main equation (1), we get the existence of a supersolution w¯(t) of the Riccati differential equation (43) on the interval (a1,b1) or (a2,b2). In the second one, we construct two subsolutions w_1(t) and w_2(t) of (43) which blow up on intervals (a1,T1*)⊆(a1,b1) and (a2,T2*)⊆(a2,b2), respectively.Proposition 17.
LetΦ(u,v) and f(u) satisfy (11), (12), and (16), respectively, and let e(t) satisfy (2). Let r(t)>0, q(t)≥0 and q(t)≢0 on [a1,b1]∪[a2,b2]. Let x=x(t) be a nonoscillatory solution of (1) and let, for some T≥t0 and λ>0, the function w¯=w¯(t) be defined by
(47)w¯(t)=-λr(t)Φ(x(t),x′(t))x(t),t≥T.
Then w¯∈ACloc([T,∞),ℝ) and w¯(t) satisfies the inequality
(48)w¯′≥(λr(t))1-γg0(|w¯|)+Kλq(t)a.e.inJ,
where J=(a1,b1) if x(t)<0 and J=(a2,b2) if x(t)>0.Proof.
Sincex=x(t) is a nonoscillatory solution of (1), there is a T≥t0 such that x(t)≠0 on [T,∞). Hence, w¯(t) is well defined by (47).
Because of (2), we have e(t)/x(t)≤0 for all t∈J, where J=(a1,b1) if x(t)<0 and J=(a2,b2) if x(t)>0 and since λ>0 we have
(49)-λe(t)x(t)≥0inJ.
Next, sincex and r(t)Φ(x(t),x′(t)) are from ACloc([t0,∞),ℝ), we can take the first derivative of w¯(t) for almost everywhere in J, which together with r(t)>0, q(t)≥0 and q(t)≢0 on J gives
(50)w¯′(t)=-λ(r(t)Φ(x(t),x′(t)))′x(t)+λr(t)Φ(x(t),x′(t))x2(t)x′(t)=λr(t)|x(t)|γΦ(x(t),x′(t))x′(t)|x(t)|γ-2+λq(t)f(x(t))x(t)-λe(t)x(t)≥λr(t)|x(t)|γg(|Φ(x(t),x′(t))|)+Kλq(t)-λe(t)x(t)=λr(t)|x(t)|γg(|x(t)|λr(t)|w¯(t)|)+Kλq(t)-λe(t)x(t)≥λr(t)|x(t)|γ|x(t)|γ(λr(t))γg0(|w¯(t)|)+Kλq(t)-λe(t)x(t)a.e.inJ,
that is,
(51)w¯′≥(λr(t))1-γg0(|w¯|)+Kλq(t)-λe(t)x(t)a.e.inJ.
Hence, (49) and (51) prove the desired inequality (48).Proposition 18.
Let (19) be satisfied. Let R1 and R2 be two arbitrary real numbers. Then there are two points T1*∈(a1,b1) and T2*∈(a2,b2) and two continuous functions w_1(t) and w_2(t), such that
(52)w_i(ai)≤Ri,limt→Ti*w_i(t)=∞,i∈{1,2},w_i′≤(λr(t))1-γg0(|w_i|)+Kλq(t)fora.e.t∈(ai,Ti*),i∈{1,2}.Proof.
Since the functiony(s)=tan(s) is a bijection from the interval (-π/2,π/2) into ℝ, we observe that there are two s1,s2∈(-π/2,π/2) such that
(53)tan(si)≤Ri,i∈{1,2}.
Next, we define two functions V1(t) and V2(t) by
(54)Vi(t)=si+πci∫ait‍C(τ)dτ,t∈[ai,bi],i∈{1,2},
where the real numbers c1, c2 and the function C(t) are defined in (19). From (19) and (54), one can immediately conclude that
(55)Vi(ai)=si<π2,Vi(bi)=si+π>π2,i∈{1,2}.
Since C∈Lloc1((t0,∞),ℝ), it follows that C∈L1((t0,∞),ℝ), which implies that Vi∈AC([ai,bi],ℝ), i∈{1,2}. Therefore, exploiting the fact that in (55) we have Vi∈C([ai,bi],ℝ), we obtain the existence of two points T1*∈(a1,b1) and T2*∈(a2,b2) such that
(56)Vi(Ti*)=π2,-π2<Vi(t)<π2∀t∈[ai,Ti*),i∈{1,2}.
Now, we are able to define the following two functions ω_1(t) and ω_2(t) by
(57)ω_i(t)=tan(Vi(t)),t∈[ai,Ti*),i∈{1,2}
That are well defined because of (56). Moreover, from (53), (54), and (56) we obtain
(58)ω_i(ai)=tan(si)≤Ri,limt→Tm*ω_i(t)=tan(Vi(Ti*))=tan(π2)=∞.
Also, since Vi∈AC([ai,bi],ℝ), we can take the first derivative of ω_i(t) and hence from (19) and (57) we obtain
(59)w_i′(t)=πciC(t)1cos2(Vi(t))=πciC(t)(|ω_i(t)|2+1)≤πciC(t)[g0(|ω_i(t)|)+M0+1]≤πciC(t)g0(|ω_i(t)|)+πciC(t)(M0+1)≤(λr(t))1-γg0(|ω_i(t)|)+Kλq(t),a.e.in(ai,Ti*),
which together with (58) proves this proposition.Now we are able to present the proof of Theorem3 based on Lemma 15 and Propositions 17 and 18.Proof of Theorem3.
Assuming the contrary, then there is a nonoscillatory solutionx(t) and T≥t0 such that x(t)≠0 for all t≥T, x(t) and r(t)Φ(x,x′) are from ACloc([t0,∞),ℝ). In order to simplify notation, let every j∈ℕ be fixed and omitted in the notation. For instance, instead of (a1j,b1j) and (a2j,b2j) we write (a1,b1) and (a2,b2), respectively, and so on.
On one hand, we observe by Proposition17 that the function w¯(t) defined by (47) is a supersolution of the following Riccati differential equation:
(60)w′=(λr(t))1-γg0(|w|)+Kλq(t)a.e.inJ,
where w¯∈ACloc(J,ℝ) and J=(a1,b1) if x(t)<0 and J=(a2,b2) if x(t)>0; see the proof of Proposition 17. Let for instance x(t)<0 and thus J=(a1,b1).
On the other hand, by Proposition18 there are a number T1*∈(a1,b1) and subsolutions w_1(t), w_1∈AC([a1,T1*),ℝ) of the Riccati differential equation (60) such that
(61)w_1(a1)≤w¯(a1),limt→T1*w_1(t)=∞,
(in the case when J=(a2,b2), then we work with T2*∈(a2,b2) and the other subsolution w_2(t), w_2∈AC([a2,T2*),ℝ)). Hence by Corollary 16 and (61) we conclude that w_1(t)≤w¯(t) for all t∈[a1,T1*) and therefore,
(62)∞=limt→T1*w_1(t)≤limt→T1*w¯(t),
which contradicts w¯∈ACloc([T,∞),ℝ). Thus, the assumption that x(t) is nonoscillatory is not possible and hence every solution of (1) is oscillatory.Proof of Theorem4.
There is only one difference between Theorems3 and 4, and it is the difference between conditions (19) and (20). Hence, Theorem 4 immediately follows from Theorem 3 provided that we prove the equivalence between (19) and (20).First of all, we need the following two lemmas.Lemma 19.
Leta, b, and c be positive real numbers, and γ>1. Then the existence of a positive real number λ such that
(63)c≤min{aλγ-1,bλ}
is equivalent to
(64)c≤a1/γb1/γ′.
Here γ′:=γ/(γ-1) is the conjugate exponent of γ.Proof.
Let us defineg(λ)=min{a/λγ-1,bλ}. Since λ↦a/λγ-1 is decreasing and λ↦bλ is increasing, there exists a unique point of maximum of g. It is achieved at λ=λ0 which is a solution of a/λ0γ-1=bλ0, hence λ0=(a/b)1/γ.
If there is a positive real numberλ such that c≤g(λ), then c≤g(λ0)=bλ0=b(a/b)1/γ=a1/γb1/γ′. Conversely, if c≤a1/γb1/γ′, then c≤g(λ0), and the equivalence in the lemma is proved.The preceding lemma is a special case of the following more general statement.Lemma 20.
Assume thata(t), b(t), and c(t) are positive real functions defined on a subset J⊆ℝ. Then the existence of a positive real number λ such that for all t∈J,
(65)c(t)≤min{a(t)λγ-1,b(t)λ},
is equivalent to
(66)supt∈Jc(t)b(t)≤(inft∈Ja(t)c(t))1/(γ-1).Proof.
The condition that there existsλ>0 such that (65) is true for all t∈J is equivalent with the following two inequalities, which have to hold simultaneously:
(67)c(t)≤a(t)λγ-1,c(t)≤b(t)λ,
that is, with
(68)c(t)b(t)≤λ≤(a(t)c(t))1/(γ-1)∀t∈J.
Taking the supremum of the left-hand side and the infimum of the right-hand side, we obtain (66). Conversely, if (66) is satisfied, then, since the left-hand side in (66) cannot be equal to zero, while the right-hand side is less than ∞, then inequality (66) defines a nonempty closed interval in ℝ (possibly reducing to just one point), in which we can choose any positive λ. Hence, (67) is satisfied for all t∈J, and therefore (65) as well.Remark 21.
Ifa=a(t), b=b(t), and c=c(t), appearing in Lemma 20, are constant functions, then condition (66) is equivalent to c/b=(a/c)1/(γ-1), that is, to (64).Proof of Equivalence of (19) and (20).
We will use Lemma20. Let us fix Jij, where i∈{1,2} and j∈ℕ. We see that that the inequality appearing in the second line of condition (19) is of the form (65), where
(69)c(t):=1cijC(t),a(t):=1πr(t)1-γ,b(t):=Kq(t)π(M0+1).
Condition (66) is equivalent to
(70)supt∈JijC(t)q(t)≤Kcijγ′πγ′(M0+1)inft∈Jij1r(t)C(t)γ-1=Kcijγ′(M0+1)supt∈Jijr(t)C(t)γ-1(cijπ)γ′.
This inequality is equivalent to the corresponding one in (19), and the claim follows from Lemma 19.Proof of Corollary6.
Substitutinginft∈Jijq(t) instead of q(t), and supt∈Jijr(t) instead of r(t) in the inequality appearing in the second line of (20), after a short computation we obtain a stronger inequality than (20) as
(71)1cijsupt∈JijC(t)≤1π(Kinft∈Jijq(t)(M0+1)supt∈Jijr(t))(γ-1)/γ∀i∈{1,2},j∈ℕ.
Substituting C(t)≡1 we obtain that the left-hand side of (71) is equal to |Jij|-1, and the resulting inequality is equivalent to (21). Since (21) implies (20), the claim is proved.Remark 22.
Here we show that the choice ofC(t)≡1 in the proof of Corollary 6 is the best possible. To prove this, note that on the left-hand side of (71) we have the expression depending on an auxiliary function C(t), and this function is absent on the right-hand side. Therefore, it has sense to try to find a function C(t), t∈Jij, such that the value of Q(C):=(1/cij)supt∈JijC(t) is minimal (note that cij depends on the function C=C(t) as well). It is easy to see that the minimum is achieved for C(t)≡1 (or any positive constant). Indeed, since Q(C)=Q(μc) for any μ>0, by taking μ=(supt∈JijC(t))-1 it suffices to assume that supt∈JijC(t)=1. The value of Q(C) is minimal if cij=∫Jij‍C(τ)dτ is maximal possible, and since 0≤C(t)≤1, it is clear that the maximum of cij=cij(C) is achieved when C(t)≡1.Proof of Corollary7.
It is clear that the condition (23) implies (21). Furthermore, since the lengths |Jij| are uniformly bounded from below by a positive constant, see (23), then obviously a1j→∞ as j→∞. Thus, the claim follows immediately from Corollary 6.Proof of Corollary10.
LetC(t):=q(t). Then the required condition (19) is clearly satisfied because of assumption (30). Hence, this corollary immediately follows from Theorem 3.


## 4. An Extension of Condition (12) to the Case of γ>1

In this section, we consider the oscillation of (1) in the case when γ∈(1,2) is allowed in the assumption (12). In this way, the assumption (11) is slightly modified by a real number p>1 such that
(72)|u|(p-1)γ-pvΦ(u,v)≥g(|Φ(u,v)|),
where (12) is supposed, that is, g(cs)≥cγg0(s) for every c,s>0 and for a locally Lipschitz function g0:[0,∞)→[0,∞) for which there exists a constant M0 such that g0(s)+M0≥sp for all s∈[0,∞).For the functionf, we assume that there exists a constant K>0 such that
(73)f(u)sign(u)≥K|u|p-1foreveryu∈ℝ.We will need the following two lemmas.Lemma 23.
Leta<b, and let f:[a,b]→[0,∞) be a measurable function. Then,
(74)∫ab‍f(t)dt≥1
if and only if there exists a function g∈L1([a,b],[0,∞)), ∥g∥1=1, such that
(75)f(t)≥g(t),∀t∈[a,b].Proof.
We assume (74). If ∫ab‍f(t)dt is finite, we may choose g(x):=f(x)/∥f∥1.
For∫ab‍f(t)dt=∞, we may define for every natural number n a function
(76)fn(t)={f(t),f(t)≤n,n,f(t)>n.
Since fn(t)≤n, we have fn∈L1([a,b],ℝ). Obviously, limn∫ab‍fn(t)dt is also infinite. This implies that there exists an n0 such that ∫ab‍fn0(t)dt≥1. We define g(t):=fn0(t)/∥fn0∥1.
Now, we assume (75). Integrating over the whole interval, we get
(77)∫ab‍f(t)dt≥∫ab‍g(t)dt=1.Lemma 24.
Letp>1. There exists a unique function y=y(s) that satisfies the following Cauchy problem:
(78)y′(s)=1+|y(s)|p,s∈ℝ,(79)y(0)=0.
Furthermore, there exists a finite real number S* such that
(80)lims→±S*y(s)=±∞,
and y∈C1((-S*,S*),ℝ) is injective and monotonous. We have S*=(π/p)(1/(sin(π/p))) and y is odd.We will denote bytanp the solution of (78)-(79).Proof.
Obviously every solution to (78) is injective and monotonous on a connected set, since y′=1+|y|p≥1>0.
Letf(x)=1/(1+|x|p). Obviously 1≥f(x)>0 for all x∈ℝ and Cp>|f′(x)|>0 for some constants Cp∈ℝ. We may assume Cp>1. Hence, z′=f(t), z(0)=0 has (by [17, Theorem 3.1]) a locally unique solution on (-1/Cp,1/Cp). By the same argument z can be extended to the whole of ℝ, interval by interval of type ((n-1)/Cp,(n+1)/Cp) for all n∈ℤ.
We know from [18] that the for the integral of the left-hand side we have
(81)∫0∞‍f(x)dx=∫0∞‍dx1+xp=πp1sin(π/p)=S*,
so limt→±∞z(t)=±S*.
Sincez′(t)≠0, z is injective, monotonous, and of class C1(ℝ,(-S*,S*)) and so we define y(s)=z-1(s). It immediately follows that y∈C1((-S*,S*),ℝ) and that y is injective and monotonous. Also, by continuity of z, we have lims→±S*y(s)=limw→±∞y(z(w))=limw→±∞w=±∞.
Differentiating, it also follows thaty′=1+|y|p and by definition of y, y(0)=z-1(0)=0. So, y is a solution of (78)-(79). Since any other such solution must have z as its inverse, y is unique on its domain.Proposition 25.
Leta<b and γ>p>1. Assume (72) and (73) and let q(t)≥0 and e(t)≤0 (or  e(t)≥0 ) for all t∈[a,b]. If there exists a constant λ>0 such that
(82)p2πsin(πp)∫ab‍min{p-1rγ-1(t)λγ-1,Kλq(t)1+M0}dt≥1,
then for any solution x(t) of (1) with x(a)≥0  (or x(a)≤0) there exists a number t*∈[a,b] such that x(t*)=0.Proof.
We assume thatx(t)≠0 for all t∈[a,b]. Then we may define the function
(83)ω¯(t):=-λr(t)Φ(x(t),x′(t))sign(x(t))|x(t)|p-1.
This function is well defined on [a,b] and its derivative is
(84)w¯′(t)=-λ(r(t)Φ(x(t),x′(t)))′sign(x(t))|x(t)|p-1+(p-1)λr(t)Φ(x(t),x′(t))|x(t)|px′(t).
Using the fact that x(t) is a solution of (1) and omitting t from x and x′ for clarity, we get
(85)w¯′(t)=λ(q(t)f(x)-e(t))sign(x)|x|p-1+(p-1)λr(t)Φ(x,x′)xpx′=(p-1)λr(t)Φ(x,x′)|x|px′+λq(t)f(x)sign(x)|x|p-1-λe(t)sign(x)|x|p-1=(p-1)λr(t)|x|γ(p-1)Φ(x,x′)x′|x|(p-1)γ-p+λq(t)f(x)sign(x)|x|p-1-λe(t)sign(x)|x|p-1≥(72)(p-1)λr(t)|x|γ(p-1)g(|Φ(x,x′)|)+λq(t)f(x)sign(x)|x|p-1-λe(t)sign(x)|x|p-1=(p-1)λr(t)|x|γ(p-1)g(|x|p-1λr(t)|ω¯(t)|)+λq(t)f(x)sign(x)|x|p-1-λe(t)sign(x)|x|p-1≥(12)(p-1)λr(t)|x|γ(p-1)(|x|p-1λr(t))γg0(|ω¯(t)|)+λq(t)f(x)sign(x)|x|p-1-λe(t)sign(x)|x|p-1=p-1(λr(t))γ-1g0(|ω¯(t)|)+λq(t)f(x)sign(x)|x|p-1-λe(t)sign(x)|x|p-1≥(73)p-1(λr(t))γ-1g0(|ω¯(t)|)+λq(t)K-λe(t)sign(x)|x|p-1.
Since e(t)/(sign(x)|x|p-1)≤0, we have
(86)w¯′(t)≥p-1(λr(t))γ-1g0(|ω¯(t)|)+λKq(t).
We apply Lemma23 to (82) to obtain a function C(t) such that
(87)∫ab‍C(t)dt=1,
and by (75)
(88)0≤C(t)≤p2πsin(πp)min{p-1rγ-1(t)λγ-1,Kλq(t)1+M0}.
Let s0∈(-(π/p)(1/sin(π/p)),(π/p)(1/sin(π/p))) be such that tanp(s0)≤ω¯(a). We define the function
(89)V(t)=s0+2πp1sin(π/p)∫at‍C(τ)dτ.
Note that (-(π/p)(1/sin(π/p)))<s0=V(a)<(π/p)(1/sin(π/p)) and V(b)=s0+(2π/p)(1/sin(π/p))>(π/p)(1/sin(π/p)). Since V is continuous, there exists a T*∈(a,b) such that V(t)<(π/p)(1/sin(π/p)) for t∈[a,T*) and V(T*)=(π/p)(1/sin(π/p)).
We define(90)ω_(t):=tanp(V(t)),t∈[a,T*),
and note that by Lemma 24, limt→T*ω_(t)=+∞, where tanp(s)=y(s) and y(s) is determined by Lemma 24.
The derivative ofω_ then satisfies
(91)ω_′(t)=tanp′(V(t))V′(t)=(1+tanpp(V(t)))V′(t)=2πp1sin(π/p)(1+ω_p(t))C(t)≤(1+M0+g0(|ω_(t)|))min{p-1rγ-1(t)λγ-1,Kλq(t)1+M0}≤p-1rγ-1(t)λγ-1g0(|ω_(t)|)+Kλq(t).
Thus, we may apply the comparison principle of Lemma15 to ω_(a)≤ω¯(a) and their respective differential inequalities to conclude that ω_(t)≤ω¯(t) for all t∈[a,b]. But, since ω_→+∞ as t→T*<b and ω¯(t) is well defined on the whole segment [a,b], we arrive at a contradiction. Hence, x(t)>0 cannot hold for all t∈[a,b], and since x(t) is continuous, there exists a t*∈[a,b] such that x(t*)=0.Theorem 26.
Letγ>p>1. Assume (72) and (73) and let for any T>t0 there exist T≤a1<b1≤a2<b2 such that g(t)≥0 on [a1,b1]∪[a2,b2] and e(t)≤0 on [a1,b1] and e(t)≥0 on [a2,b2]. If for both i∈{1,2} there exist constants λi>0 such that
(92)p2πsin(π/p)∫aibi‍min{p-1rγ-1(t)λiγ-1,Kλiq(t)1+M0}dt≥1,
then (1) is oscillatory.Proof.
For (1) to be oscillatory it is sufficient that for every n∈ℕ there exists a tn>n such that x(tn)=0.
Letn∈ℕ and let n<a1<b1≤a2<b2 as assumed in the theorem. It is enough to show that every solution x(t) of (1) has a zero on [a1,b2]. If x(a1)≥0, we apply Proposition 25 to the segment [a1,b1] to obtain the sought zero. If x(a2)≤0 we again apply Proposition 25 to the segment [a2,b2]. Suppose x(a1)<0 and x(a2)>0. Then, since x is continuous, there exists a number t1∈[a1,a2] such that x(t1)=0.



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*Source: 297085-2013-05-12.xml*