# New Oscillation Criteria for Second-Order Forced Quasilinear Functional Differential Equations

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

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## Abstract

We establish some new interval oscillation criteria for a general class of second-order forced
quasilinear functional differential equations withϕ-Laplacian operator and mixed nonlinearities. It
especially includes the linear, the one-dimensional p-Laplacian, and the prescribed mean curvature
quasilinear differential operators. It continues some recently published results on the oscillations of
the second-order functional differential equations including functional arguments of delay, advanced,
or delay-advanced types. The nonlinear terms are of superlinear or supersublinear (mixed) types.
Consequences and examples are shown to illustrate the novelty and simplicity of our oscillation criteria.


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## Body

## 1. Introduction

We study the oscillation of the following three kinds of second-order forced quasilinear functional differential equations of delay, advanced, and delay-advanced types:(1)(r(t)ϕ(x′(t)))′+∑i=1n‍ri(t)f(x(hi(t)))+∑i=1n‍qi(t)|x(hi(t))|αisgnx(hi(t))=e(t),
where t≥t0>0 and r(t), ri(t), qi(t), and e(t) are continuous functions on [t0,∞), and x=x(t), x∈C2((t0,∞),ℝ), is a classic solution of (1). A continuous function x(t)is said to be nonoscillatory if there is a T>t0such that x(t)≠0 on [T,∞). Otherwise, x(t) is said to be oscillatory. Equation (1) is called oscillatory if all of its classic solutions are oscillatory.In the delay casehi(t):=τi(t)≤t,i∈{1,…,n}, and r(t) is nondecreasing; in the advanced case hi(t):=σi(t)≥t, i∈{1,…,n}, and r(t) is nonincreasing; in the delay-advanced case hi(t):=τi(t)≤t, i∈{1,…,m}, hi(t):=σi(t)≥t,i∈{m+1,…,n}, and r(t)≡const.>0. The exponents {αi}satisfy a superlinear or a supersublinear (mixed) condition. On the function ϕ:ℝ→ℝ which appears in the first term of (1), we impose such conditions that the following three main classes of second-order differential operators are especially included: the linear (r(t)x′)′, the one-dimensionalp-Laplacian (r(t)|x′|p-1x′)′, and the prescribed mean curvature operator (r(t)x′(1+x′2)-1/2)′. The function f(u) satisfies a usual growth condition and the coefficients ri(t), and qi(t) are positive only on some intervals where e(t) changes the sign.Recently, Bai and Liu [1] have studied the oscillation of second-order delay differential equation:
(2)(r(t)x′(t))′+∑i=1n‍ri(t)x(t-τi)+∑i=1n‍qi(t)|x(t-τi)|αisgnx(t-τi)=e(t),t≥t0,
where τi≥0. In Murugadass et al. [2], authors have studied the oscillation of the second-order quasilinear delay differential equation:
(3)(r(t)(x′(t))p)′+q(t)xp(t-τ)+∑i=1n‍qi(t)|x(t-τ)|αisgnx(t-τ)=e(t),t≥t0,
where p and {αi} are ratio of odd positive integers. Later, in Hassan et al. [3], authors consider the oscillation of the half-linear functional differential equation:
(4)(r(t)(x′(t))p)′+r0(t)xp(h0(t))+∑i=1n‍qi(t)|x(hi(t))|αisgnx(hi(t))=e(t),t≥t0,
where the function hi=hi(t) is positive continuous functions with limt→∞hi(t)=∞. Moreover, in [1–3] some doubts concerning the proof of the main result of [4] are resolved. The well-known variational technique that uses the generalized Philos’ results based on the so-called H-function has been used in [1, Theorem 2.2], [2, Theorems 2.3, and 2.6], and [3, Theorems 2.5, 2.6, and 2.7] (see also [5–11] and references therein). Since the second-order quasilinear differential operator (r(t)ϕ(x′(t)))′ usually causes some difficulties in many problems, the application of previous method on (1) is stated here as an open problem. In contrast to the preceding, we use a combination of the Riccati classic transformation, a blow-up argument, and a comparison pointwise principle recently established in [12, 13] but for differential equations without functional arguments. It seems that our criteria are slightly simpler to be verified, which is discussed on some examples given in the next section.Among some recently published papers on the oscillation of quasilinear functional differential equations with both delay and advanced terms, we point out an oscillation criterion by Zafer [5] obtained for equation
(5)(|x′(t)|p-1x′(t))′+q1(t)|x(τ(t))|β-1x(τ(t))+q2(t)|x(σ(t))|γ-1x(σ(t))=e(t),t≥0,
where p>0,β≥p, and γ≥p. On an interesting example in [5], the author established the oscillations provided at least one of constants appearing in the coefficients is sufficiently large (see also [6] for p=1), which is presented here in the next section as a particular case of our main results.On the properties of some classes of the one-dimensional quasilinear differential equations withϕ-Laplacian operators we refer reader to [14–20] and the references therein. About the applications of second-order functional differential equations in the mathematical description of certain phenomena in physics, technics, and biology (oscillation in a vacuum tube; interaction of an oscillator with an energy source; coupled oscillators in electronics, chemistry, and ecology; relativistic motion of a mass in a central field; ship course stabilization; moving of the tip of a growing plant; etc.), we suggest reading Kolmanovskii and Myshkis book [21].


## 2. Main Results and Examples

Letϕ=ϕ(v) be a function which appears in the first term of (1) such that
(6)ϕ∈C1(ℝ,ℝ),ϕisoddandincreasingfunctiononℝ,(7)ϕ(v)v≥|ϕ(v)|(p+1)/p∀v∈ℝandsomep>0.These two assumptions are fulfilled, respectively, in the linear case(r(t)x′(t))′, whereϕ(v)≡v and p=1; in the one-dimensionalp-Laplacian quasilinear case (r(t)|x′(t)|p-1x′(t))′, where ϕ(v)≡|v|p-1v and p>0; and in the mean prescribed curvature quasilinear case
(8)[r(t)x′(t)(1+x′2(t))-1/2]′,
where ϕ(v)=v(1+v2)-1/2 and p=1. The importance of the prescribed mean curvature quasilinear differential operators lies in the capillarity type problems in fluid mechanics, flux-limited diffusion phenomena, and prescribed mean curvature problems; see for instance [14, 22, 23].The functionf=f(u) which appears in the second term of (1) satisfies
(9)f(u)isoddfunctiononℝ,f(u)up≥K>0∀u>0,andsomeK∈ℝ,
where p>0is from assumption (7).In the first two theorems below, the exponents{αi} satisfy
(10)α1≥⋯≥αm>p>αm+1≥⋯≥αn>0,m∈ℕ,αi>αi+1,i∈{1,…,n-1};thereexists(n+1)-tuple(η0,η1,…,ηn)suchthat0<ηi<1,∑i=1n‍ηi<1,η0=1-∑i=1n‍ηi,∑i=1n‍αiηi=p,
where p is from assumption (7). For instance if n=2,α1=5/2, p=1, and α2=1/2, then η0=η1=η2=1/3. On assumption (10) see for instance [9, Lemma 1].Unlike recently published oscillation criteria for the linear and half-linear second-order forced functional differential equations, our oscillation criterion is only based on the following elementary integral inequality:(11)12πp∫ajbj‍min{p(λjr(t))1/p,λj(Rj(t)+Qj(t))}dt≥1,mmmmmmmmmmmmmmmmmmmmmmnj∈{1,2},
wherea1<b1≤a2<b2,pis from (7),πp=(p/(p+1))π/sin(pπ/(p+1)),λj>0, and functionsQj(t), and Rj(t)are explicitly expressed by the coefficients of (1) just like it is done in the next three main results.Theorem 1 (delay equation).
One assumes (6), (7), (9), and (10). Let r(t) be a nondecreasing positive function on [t0,∞). Let hi(t)=τi(t)≤t on [t0,∞) and limt→∞τi(t)=∞,i∈{1,2,…,n}. Let for every T≥t0there exist a1,b1,a2,b2, T≤a1<b1≤τmin(a2)≤a2<b2 such that
(12)mmnri(t)≥0,qi(t)≥0on[τmin(a1),b1]∪[τmin(a2),b2],(13)e(t)≤0on[τmin(a1),b1],e(t)≥0on[τmin(a2),b2],
where τmin(t)=min{τ1(t),τ2(t),…,τn(t)}. Equation (1) is oscillatory provided there are two real parameters λ1,λ2>0 such that (11) is fulfilled, where
(14)Rj(t)=K∑i=1n‍ri(t)(τi(t)-τi(aj)t-τi(aj))p,(15)Qj(t)=(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n‍(τi(t)-τi(aj)t-τi(aj))αiηi,
for t∈[aj,bj], j∈{1,2}, and positive constantsp,K,ηiappearing, respectively, in (7), (9), and (10).As we can see in (13), the forcing term e(t)can be an oscillatory function. The main property of any interval oscillation criterion (see [8]) is that the coefficients of the considered equation do not satisfy some conditions on the whole [t0,∞)than only on intervals [a1,b1]∪[a2,b2]. The main consequence of Theorem 1 is the following oscillation criterion for (1) with r(t)≡1 and ri(t)≡0, i∈{1,2,…,n}.Corollary 2.
One assumes (6), (7), (10), (12), and (13). Then equation
(16)(ϕ(x′(t)))′+∑i=1n‍qi(t)|x(τi(t))|αisgnx(τi(t))=e(t)
is oscillatory provided
(17)pp/(p+1)2πp∫ajbj‍Qj(t)dt≥(maxt∈[aj,bj]Qj(t))p/(p+1)>0,mmmmmmmmmmmmmmmmmmmmmj∈{1,2},
where Qj(t)is defined in (15).In particular forp=1, previous corollary takes the following form.Corollary 3.
Let (10) hold with p=1, τi(t)≤ton [t0,∞), and limt→∞τi(t)=∞, i∈{1,2,…,n}. Let for every T≥t0there exist a1,b1,a2,b2, T≤a1<b1≤τmin(a2)≤a2<b2 such that assumptions (12) and (13) hold with ri(t)≡0. Equation
(18)x′′(t)+∑i=1n‍qi(t)|x(τi(t))|αisgnx(τi(t))=e(t)
is oscillatory provided
(19)1π∫ajbj‍Qj(t)dt≥maxt∈[aj,bj]Qj(t)>0,j∈{1,2},
where Qj(t) is defined in (15).In the next examples and remarks, we discuss the application of oscillation criteria from Corollary3 and [1, Theorem 2.2] on the following equation:
(20)x′′+m1sin(t)|x(τ1(t))|α1sgnx(τ1(t))+m2cos(t)|x(τ2(t))|α2sgnx(τ2(t))=-e0cos(2t),
where α1>1>α2>0, τ1(t)=τ2(t)=t-π/8, and m1,  m2, and e0 are positive constants.Example 4.
As a consequence of Corollary3, we show in this example that (20) is oscillatory provided the constants m1,  m2, and e0 satisfy the following simple inequality:
(21)Ijπηe0η0m1η1m2η2≥1,j∈{1,2},
whereηisatisfy (10), η=∏i=02‍ηi-ηi, and I1,I2>0 are two real numbers defined by
(22)I1=∫π/8π/4‍W(t)t-π/8tdt,I2=∫3π/8π/2‍W(t)t-3π/8t-π/4dt,W(t)=|cos2t|η0|sint|η1|cost|η2.
Indeed, letπp=π/2, τmin(t)=τ1(t)=τ2(t)=t-π/8, [a1,b1]=[π/8+2kπ,π/4+2kπ],[a2,b2]=[3π/8+2kπ,π/2+2kπ], k∈ℕ, q1(t)=sin(t), q2(t)=cos(t), and e(t)=-e0cos(2t). Firstly, it is elementary to check that the required inequalities (12) and (13) are satisfied. Next, we prove that required inequality (19) immediately follows from assumption (21). On the first hand, we estimate from above the term on the right hand side in (19) using that α1η1+α2η2=1:
(23)Qj(t)=ηe0η0m1η1m2η2W(t)t-ajt-aj+π/8≤ηe0η0m1η1m2η2,mmmmmmmmmmmmmmmmmmmnnmt∈[aj,bj],
that is
(24)0<maxt∈[aj,bj]Qj(t)≤ηe0η0m1η1m2η2,j∈{1,2}.
On the other hand, we calculate the integral on the left hand side in (19):
(25)∫a1b1‍Q1(t)dt=ηe0η0m1η1m2η2∫π/8+2kππ/4+2kπ‍W(t)t-(π/8+2kπ)t-(π/8+2kπ)+π/8dt=ηe0η0m1η1m2η2I1,∫a2b2‍Q2(t)dt=ηe0η0m1η1m2η2∫3π/8+2kππ/2+2kπ‍W(t)t-(3π/8+2kπ)t-(3π/8+2kπ)+π/8dt=ηe0η0m1η1m2η2I2.
Now, from previous two equalities, and inequalities (21) and (24), we conclude
(26)1π∫ajbj‍Qj(t)dt=Ijπηe0η0m1η1m2η2≥ηe0η0m1η1m2η2≥maxt∈[aj,bj]Qj(t)>0.
Thus, assumption (21) proves desired inequality (19), and thus, Corollary 2 verifies that (20) is oscillatory provided (21) holds.Remark 5.
One way to obtain the inequality (21) is to suppose, for instance, that at least one of the constants m1,  m2, ande0is large enough.Remark 6.
We can consider the slightly more general equation than (20):
(27)(ϕ(x′(t)))′+m1sin(t)|x(t-τ1)|α1sgnx(t-τ1)+m2cos(t)|x(t-τ2)|α2sgnx(t-τ2)=-e0cos(2t),
where τ1=τ2=π/8 and ϕ=ϕ(v)satisfy (6) and (7), andα1>p>α2>0. In a similar way as in Example 4, one can show that (27) is oscillatory provided at least one of the constantsm1,  m2, ande0is large enough.Example 7.
In this example we present an application of [1, Theorem 2.2] for getting another condition on the constantsm1,  m2, ande0for oscillation of (20). Letτ1=τ2=π/8,
(28)[a1,c1]=[π8+2kπ,3π16+2kπ],[c1,b1]=[3π16+2kπ,π4+2kπ],[a2,c2]=[3π8+2kπ,7π16+2kπ],[c2,b2]=[7π16+2kπ,π2+2kπ],q1(t)=m1sin(t), q2(t)=m2cos(t), and e(t)=-e0cos(2t). It is clear thatqi(t)≥0on[a1-τi,b1]∪[a2-τi,b2],i=1,2,e(t)≤0on[a1-τi,b1], ande(t)≥0on[a2-τi,b2],i=1,2. Therefore we may apply [1, Theorem 2.2] on (20) provided the following inequality holds:
(29)1Hj(cj,aj)∫ajcj‍(Qj(t)Hj(t,aj)-1)dt+1Hj(bj,cj)∫cjbj‍(Qj(t)Hj(bj,t)-1)dt>0,mmmmmmmmmmmmmmmmmmmmnnlj∈{1,2}.
Now, if we putH1(t,s)=H2(t,s)=(t-s)2andhj,1(t,s)=hj,2(t,s)=1fort>s, then previous inequality takes the following concrete form:
(30)∫π/8+2kπ3π/16+2kπ‍Q1(t)(t-π8-2kπ)2dt+∫3π/16+2kππ/4+2kπ‍Q1(t)(t-π4-2kπ)2dt>π8,∫3π/8+2kπ7π/16+2kπ‍Q2(t)(t-3π8-2kπ)2dt+∫7π/16+2kππ/2+2kπ‍Q2(t)(t-π2-2kπ)2dt>π8.
Similarly as in Example4, we can calculate previous integrals and conclude by [1, Theorem 2.2] that (20) is oscillatory provided the following inequalities hold:
(31)ηe0η0m1η1m2η2(I11+I12)>π8,ηe0η0m1η1m2η2(I21+I22)>π8,
where
(32)I11=∫π/83π/16‍W(t)(t-π/8)3tdt,I12=∫3π/16π/4‍W(t)(t-π/8)(t-π/4)2tdt,I21=∫3π/87π/16‍W(t)(t-3π/8)3t-π/4dt,I22=∫7π/16π/2‍W(t)(t-3π/8)(t-π/2)2t-π/4dt,
and W(t)=|cos2t|η0|sint|η1|cost|η2.We leave to the reader to compare and verify which one of the two inequalities (21) and (31) is simpler to be verified. Also, similar to the previous example, it is possible to get corresponding inequalities analogously to (31) for other criteria published in the papers cited in the references.Remark 8.
In [1, Section 3] the authors give an application of [1, Theorem 2.2] to (20) whereτ1(t)=t-τ1,τ2(t)=t-τ2,τ1=π/8, andτ2=π/4. However, the following choice (see [1, Section 3])
(33)[a1,c1]=[2kπ,π8+2kπ],[c1,b1]=[π8+2kπ,π4+2kπ],[a2,c2]=[π4+2kπ,3π8+2kπ],[c2,b2]=[3π8+2kπ,π2+2kπ],
is not correct since the desired conditionsqi(t)≥0on[a1-τi,b1]∪[a2-τi,b2],i=1,2are not fulfilled. Hence, we suggest reader to use the intervals proposed in the previous example.Open Question 9.
The well-known variational technique based on the generalized Philos’H-function has been used in [1–11] to obtain some oscillation criteria for (1), where the second-order quasilinear differential operator (r(t)ϕ(x′(t)))′ is linear or half-linear. Is it possible to use this technique in the case of any function ϕ(v) satisfying general conditions (6) and (7)?Theorem 10 (advanced equation).
Under assumptions (6), (7), (9), and (10), letr(t)be a nonincreasing positive function on [t0,∞) and hi(t)=σi(t)≥t on [t0,∞),i∈{1,2,…,n}. Let for everyT≥t0there exist numbers a1,b1,a2,b2, such that T≤a1<b1≤σmax(b1)≤a2<b2 and(34)ri(t)≥0,qi(t)≥0on[a1,σmax(b1)]∪[a2,σmax(b2)],e(t)≤0on[a1,σmax(b1)],e(t)≥0on[a2,σmax(b2)],
whereσmax(t)=max{σ1(t),…,σn(t)}. Equation (1) is oscillatory provided there are two real parametersλ1,λ2>0such that (11) is fulfilled, where
(35)Rj(t)=K∑i=1n‍ri(t)(σi(bj)-σi(t)σi(bj)-t)p,(36)Qj(t)=(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n‍(σi(bj)-σi(t)σi(bj)-t)αiηi,
fort∈[aj,bj],j∈{1,2}, and positive constantsp,K, andηiappearing, respectively, in (7), (9), and (10).Now, analogously with (16), we consider the oscillation of the advanced equation:
(37)(ϕ(x′(t)))′+∑i=1n‍qi(t)|x(σi(t))|αisgnx(σi(t))=e(t).As a consequence of Theorem10, we have the following criterion.Corollary 11.
One assumes (6), (7), (10), and (34). Then (37) is oscillatory provided condition (17) is satisfied, whereQj(t)is defined in (36).As the third case, we consider second-order functional differential equations with both delay and advanced arguments.Theorem 12 (delay-advanced equation).
One assumes (6), (7), and (9),αi>pfori∈{1,2,…,n}, andm∈ℕ,1<m<n. Letr(t)≡const>0on[t0,∞),hi(t)=τi(t)≤tfori∈{1,…,m},  andhi(t)=σi(t)≥tfori∈{m+1,…,n}on[t0,∞). Letri(t),qi(t), ande(t)satisfy (12)-(13) fori∈{1,…,m}and (34) fori∈{m+1,…,n}. Equation (1) is oscillatory provided there are two real parametersλ1,λ2>0such that (11) is fulfilled, where
(38)Rj(t)=K∑i=1m‍ri(t)(τi(t)-τi(aj)t-τi(aj))pRj(t)=+K∑i=m+1n‍ri(t)(σi(bj)-σi(t)σi(bj)-t)p,(39)Qj(t)=∑i=1m‍αipqip/αi(t)(p|e(t)|2(αi-p)m)(αi-p)/αiRj(t)=m×(τi(t)-τi(aj)t-τi(aj))pRj(t)=+∑i=m+1n‍αipqip/αi(t)(p|e(t)|2(αi-p)(n-m-1))(αi-p)/αiRj(t)=mmm×(σi(bj)-σi(t)σi(bj)-t)p,
for t∈[aj,bj], j∈{1,2}, and positive constants p,K appearing, respectively, in (7) and (9).Analogously with (16) and (37), we consider the oscillation of delay-advanced equation:
(40)(ϕ(x′(t)))′+∑i=1m‍qi(t)|x(τi(t))|αisgnx(τi(t))+∑i=m+1nqi(t)|x(σi(t))|αisgnx(σi(t))=e(t).As a consequence of Theorem12 we derive the following oscillation criterion for (40).Corollary 13.
Let all assumptions of Theorem12 hold with respect to ri(t)≡0and f(u)≡0. Equation (40) is oscillatory provided condition (17) is satisfied, where Qj(t)is defined in (39).In Zafer [5] (see also [6]) the author has illustrated its main oscillation result on the following example of the second-order functional differential equations with delay and advanced arguments. Since |x|α-1x=|x|αsgnx, it can be rewritten in the form
(41)(|x′(t)|p-1x′(t))′+m1sin(t)|x(τ(t))|α1sgnx(τ(t))+m2cos(t)|x(σ(t))|α2sgnx(σ(t))=-e0cos(2t),
wherem1,  m2, and e0 are positive constants; τ(t)=t-π/5; σ(t)=t+π/20; and α1,α2>0. It has been proved that previous equation is oscillatory provided at least one of the constants m1,m2, and e0 is large enough. Here according to Corollary 13 we can repeat this interesting conclusion for slightly complicated equation but with α1,α2>1.Example 14.
We consider the followingϕ-Laplacian functional differential equation with delay and advanced arguments:
(42)(ϕ(x′(t)))′+m1sin(t)|x(τ(t))|α1sgnx(τ(t))+m2cos(t)|x(σ(t))|α2sgnx(σ(t))=-e0cos(2t),
where ϕ=ϕ(v) satisfies (6), (7), τ(t)=t-τ0,σ(t)=t+σ0, and τ0,σ0∈[0,π/4). We claim that the previous equation is oscillatory provided at least one of the constants m1,  m2, and e0 is large enough. Indeed, it is enough to show that condition (17) is fulfilled by using Corollary 13. It can be done in a similar way as in Example 4 with (20) and Corollary 3. We leave it to the reader.


## 3. Auxiliary Results: Qualitative Properties of Concave-Like Functions

Leta<b be two arbitrary real numbers. It is known that if x(t) is a concave and smooth enough function on (a,b), that is, if x′′(t)≤0 for all t∈(a,b), then x(t)-x(s)≥x′(t)(t-s) for all s,t∈(a,b), s<t. Moreover, if x(t)>0 for all t∈(a,b), then from previous inequality we obtain
(43)x′(t)x(t)≤1t-s∀s,t∈(a,b),s<t.However, often we do not have any information about the sign of the second-order linear differential operator in (1) except only in a particular case when r(t)≡1 and ϕ(v)≡v. Hence the main goal of this section is to find some sufficient conditions on r(t) and ϕ(v) such that the assumption “(r(t)ϕ(x′(t)))′≤0 for all t∈(a,b)” implies the desired inequality (43). It is done in the following results.Lemma 15.
Letϕ=ϕ(v) satisfy (6). Let 0<r(s)≤r(t) for all s,t∈(a,b), s≤t. For any function x∈C2((a,b),ℝ)∩C([a,b),ℝ) such that x(t)>0 for all t∈(a,b), the following statement holds:
(44)if(r(t)ϕ(x′(t)))′≤0∀t∈(a,b),mmthenx′(t)x(t)≤1t-a∀t∈(a,b).Proof.
From assumption(r(t)ϕ(x′(t)))′≤0 for all t∈(a,b), it follows that
(45)r(t)ϕ(x′(t))≤r(ξ)ϕ(x′(ξ))∀ξ,t∈(a,b),ξ≤t.
To the end of this proof, lett∈(a,b) be fixed. Thanks to (6), if ϕ(x′(t))≤0, then x′(t)≤0 too, becauseϕis odd and increasing. In this case, since r(s)>0 and x(s)>0 on(a,b), it implies that
(46)x′(t)x(t)≤0<1t-a∀t∈(a,b).
Thus, it remains to show (44) for the case of ϕ(x′(t))≥0. From 0<r(ξ)≤r(t) for all ξ∈(a,b),ξ≤t, we obtain r(t)/r(ξ)≥1which together with ϕ(x′(t))≥0 and (45) gives
(47)ϕ(x′(t))≤r(t)r(ξ)ϕ(x′(t))≤ϕ(x′(ξ))∀ξ∈(a,b),ξ≤t.
Acting on (47) with ϕ-1 and using that ϕ-1is an increasing function because of (6), we obtain
(48)x′(ξ)≥x′(t)∀ξ∈(a,b),ξ≤t.
Next, sincex(t)>0 on (a,b)and x∈C([a,b),ℝ), we have x(a)≥0. From the mean-value theorem for x(t)on (a,t), we get a ξ∈(a,t)depending on aand tsuch that x(t)-x(a)=x′(ξ)(t-a), and since x(a)≥0, from (48) we obtain
(49)x(t)≥x′(ξ)(t-a)≥x′(t)(t-a),
which proves the desired inequality in (44).In the advanced case of (1), that is, when hi(t)=σi(t)≥t, we have the analogous result to Lemma 15.Lemma 16.
Letϕ=ϕ(v)satisfy (6). Let r(s)≥r(t)>0 for all s,t∈(a,b), s≤t. For any function x∈C2((a,b),ℝ)∩C((a,b],ℝ) such that x(s)>0 for all s∈(a,b), the following statement holds:
(50)if(r(s)ϕ(x′(s)))′≤0∀s∈(a,b),mthenx′(s)x(s)≥-1b-s∀s∈(a,b).Proof.
From assumption(r(s)ϕ(x′(s)))′≤0 for all s∈(a,b), we have
(51)r(s)ϕ(x′(s))≥r(ξ)ϕ(x′(ξ))∀s,ξ∈(a,b),s≤ξ.
To the end of this proof, lets∈(a,b) be fixed. If ϕ(x′(s))≥0, then x′(s)≥0 because ϕ is odd and increasing, and since by assumption x(t)>0 for all t∈(a,b), we have that
(52)x′(s)x(s)≥0≥-1b-s∀s∈(a,b),
which proves (50) in this case. Let now ϕ(x′(s))≤0. It implies that x′(s)≤0 and ϕ(x′(ξ))≤0 because of (51) and r(s)>0for all s∈(a,b). Hence, from r(ξ)≤r(s) for all ξ∈(a,b),ξ≥s, from (51), and ϕ(x′(ξ))≤0, we especially conclude that
(53)ϕ(x′(s))≥r(ξ)r(s)ϕ(x′(ξ))≥ϕ(x′(ξ))∀ξ∈(a,b),ξ≥s.
Acting on (53) with ϕ-1we obtain
(54)x′(s)=ϕ-1(ϕ(x′(s)))≥ϕ-1(ϕ(x′(ξ)))=x′(ξ)mmmmmmmmmmmmmmmml∀ξ∈(a,b),ξ≥s.
By the Lagrange’s mean-value theorem, there exists aξ∈(s,b) depending on s,bsuch that x(b)-x(s)=x′(ξ)(b-s). Since x(s)>0for all s∈(a,b)and x∈C((a,b],ℝ), from (54) we obtain
(55)-x(s)≤x(b)-x(s)=x′(ξ)(b-s)≤x′(s)(b-s),
which proves the desired inequality.Statement (44) will be frequently used in the following form.Corollary 17.
Letϕ=ϕ(v) satisfy (6), τ(t)≤t, and 0<r(s)≤r(t) for all s,t, t0≤s≤t. Let a1,b1be two arbitrary real numbers such that t0≤τ(a1)≤a1<b1. Then for any function x∈C2((τ(a1),b1),ℝ)∩C([τ(a1),b1),ℝ) such that x(t)>0 for all t∈(τ(a1),b1), the following statement holds:
(56)miif(r(t)ϕ(x′(t)))′≤0,∀t∈(τ(a1),b1),thenx(τ(t))x(t)≥τ(t)-τ(a1)t-τ(a1),∀t∈(a1,b1).Proof.
By assumptions of this corollary, we haveτ(t)≤t, x(t)>0, and (r(t)ϕ(x′(t)))′≤0 for all t∈(τ(a1),b1). In particular from statement (44) applied on (a,b)=(τ(a1),b1), we get
(57)x′(t)x(t)≤1t-τ(a1)∀t∈(τ(a1),b1).
Integrating this inequality over the interval[τ(t),t] for all t∈(a1,b1), we obtain
(58)x(t)x(τ(t))≤t-τ(a1)τ(t)-τ(a1)∀t∈(a1,b1),
which proves the desired inequality in (56).Statement (50) will appear in the following form.Corollary 18.
Leta1<b1be two arbitrary real numbers. Let ϕ=ϕ(v)satisfy (6), σ(t)≥t, r(t)>0, and r(s)≥r(t)for all s,t, s≤t. Then for any function x∈C2((a1,σ(b1)),ℝ)∩C((a1,σ(b1)],ℝ)such that x(t)>0for all t∈(a1,σ(b1)), the following statement holds:
(59)mnif(r(s)ϕ(x′(s)))′≤0∀s∈(a1,σ(b1))thenx(σ(s))x(s)≥σ(b1)-σ(s)σ(b1)-s∀s∈(a1,b1).Proof.
Analogously with the proof of Corollary17, we just need to use the second inequality in (50) on (a,b)=(a1,σ(b1))and integrate it over the interval [s,σ(s)] for all s∈(a1,b1).Next, by Corollary17 and the arithmetic-geometric mean inequality
(60)∑i=0n‍ηiui≥∏i=0n‍uiηi,ui≥0,ηi>0,
we can prove the following proposition.Proposition 19 (with delay arguments).
Let0<r(s)≤r(t) for all s,tsuch that t0≤s<t and let ϕ=ϕ(v) satisfy (6). Let exponents {αi} and the (n+1)-tuple (η0,η1,…,ηn) satisfy (10). Let τi(t)≤t on [t0,∞), i∈{1,…,n}, and τmin(t)=min{τ1(t),…,τn(t)}. Let e(t)≤0and qi(t)≥0on[a1,b1], where t0≤a1<b1. Then for any function x∈C2((τmin(a1),b1),ℝ)∩C([τmin(a1),b1),ℝ) such that x(t)>0 and (r(t)ϕ(x′(t)))′≤0 for all t∈(τmin(a1),b1), we have
(61)1xp(t)[∑i=1n‍qi(t)(x(τi(t)))αi-e(t)]≥(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n‍(τi(t)-τi(a1)t-τi(a1))αiηi,mmmmmmmmmmmmmmmmmmmmmmnnmt∈(a1,b1).Proof.
By inequality (60) in particular for ui=ηi-1qi(t)(x(τi(t)))αi and u0=η0-1e(t), together with (56), we obtain
(62)1xp(t)[∑i=1n‍qi(t)(x(τi(t)))αi-e(t)]=1xp(t)[∑i=1n‍ηi(ηi-1qi(t)(x(τi(t)))αi)+η0(η0-1|e(t)|)]≥1xp(t)∏i=1n‍(ηi-1qi(t)(x(τi(t)))αi)ηi(η0-1|e(t)|)η0=(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n(x(τi(t)))αiηi(x(t))∑i=1nαiηi=(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n(x(τi(t)))αiηi∏i=1n(x(t))αiηi=(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n‍(x(τi(t))x(t))αiηi≥(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n‍(τi(t)-τi(a1)t-τi(a1))αiηi,
which proves this proposition.Proposition 20 (with advanced arguments).
Letr(s)≥r(t)>0 for all s,t such that t0≤s<t and let ϕ=ϕ(v) satisfy (6). Let exponents {αi} and the (n+1)-tuple (η0,η1,…,ηn) satisfy (10). Let σi(t)≥t on [t0,∞), i∈{1,…,n}, and σmax(t)=max{σ1(t),…,σn(t)}. Let e(t)≤0 and qi(t)≥0 on [a1,b1], where t0≤a1<b1. Then for any function x∈C2((a1,σmax(b1)),ℝ)∩C((a1,σmax(b1)],ℝ) such that x(t)>0 and (r(t)ϕ(x′(t)))′≤0 for all t∈(a1,σmax(b1)), we have
(63)1xp(t)[∑i=1n‍qi(t)(x(σi(t)))αi-e(t)]≥(η0-1|e(t)|)η0∏i=1n‍(ηi-1qi(t))ηi∏i=1n‍(σi(b1)-σi(t)σi(b1)-t)αiηi,mmmmmmmmmmmmmmmmmmmmnnmmt∈(a1,b1).Proof.
It is very similar to the proof of Proposition19 but instead of Corollary 17 we need to use Corollary 18.According to Corollary17, we are able to prove the next useful proposition in which we also use the well-known Young inequality X,Y≥0:
(64)Xγ+(γ-1)Yγ≥γXYγ-1,γ>1.Proposition 21 (with delay-advanced arguments).
Letr(t)≡const>0 on [t0,∞), real numbers αi>p>0for all i=1,2,…,n and m∈ℕ,1<m<n. Let ϕ=ϕ(v)satisfy (6), τi(t)≤t,i∈{1,…,m}and σi(t)≥t, i∈{m+1,…,n} on [t0,∞), and τmin(t)=min{τ1(t),…,τm(t)}, σmax(t)=max{σm+1(t),…,σn(t)}. Let e(t)≤0 and qi(t)≥0on [a1,b1], where t0≤a1<b1. Then for any function x∈C2((τmin(a1),σmax(b1)),ℝ)∩C([τmin(a1),σmax(b1)],ℝ) such that x(t)>0 and (r(t)ϕ(x′(t)))′≤0 on (τmin(a1),σmax(b1)), we have
(65)1xp(t)[∑i=1m‍qi(t)(x(τi(t)))αi-12e(t)]+1xp(t)[∑i=m+1n‍qi(t)(x(σi(t)))αi-12e(t)]≥∑i=1m‍αipqip/αi(t)(p|e(t)|2(αi-p)m)(αi-p)/αimmmm×(τi(t)-τi(a1)t-τi(a1))p+∑i=m+1n‍αipqip/αi(t)(p|e(t)|2(αi-p)(n-m-1))(αi-p)/αimmmmm×(σi(b1)-σi(t)σi(b1)-t)p,
for all t∈(a1,b1).Proof.
First of all, by inequality (64) in particular for γ=αi/pand
(66)X=(qi(t))p/αixp(τi(t)),Y=(p|e(t)|2(αi-p)m)p/αi,
together with (56), we obtain
(67)1xp(t)[∑i=1m‍qi(t)(x(τi(t)))αi-12e(t)]=1xp(t)∑i=1m‍{(p|e(t)|2(αi-p)m)p/αi[(qi(t))p/αixp(τi(t))]αi/pmmmmmmmm+(αip-1)[(p|e(t)|2(αi-p)m)p/αi]αi/p}≥1xp(t)∑i=1m‍αip(qi(t))p/αi(p|e(t)|2(αi-p)m)(αi-p)/αixp(τi(t))=∑i=1m‍αip(qi(t))p/αi(p|e(t)|2(αi-p)m)(αi-p)/αi[x(τi(t))x(t)]p≥∑i=1m‍αip(qi(t))p/αi(p|e(t)|2(αi-p)m)(αi-p)/αi×(τi(t)-τi(a1)t-τi(a1))p.
In the same way, one can show that(68)1xp(t)[∑i=m+1n‍qi(t)(x(σi(t)))αi-12e(t)]≥∑i=m+1n‍αipqip/αi(t)(p|e(t)|2(αi-p)(n-m-1))(αi-p)/αi×(σi(b1)-σi(t)σi(b1)-t)p.
The previous two inequalities prove this proposition.


## 4. Proof of Main Results

Proof of Theorem1. If the assertion of this theorem does not hold, then there is a nonoscillatory solution x(t) of (1) such that x(t)≠0 for all t≥T and some T≥t0. Moreover, it is enough to work only with the case x(t)>0and e(t)≤0on [τmin(a1),b1], since the second case x(t)<0 and -e(t)≤0 on [τmin(a2),b2]can be transformed into the first one. Indeed, multiplying equation (1) by -1and using assumptions that ϕ(v) and f(u) are odd functions, we have that (1) is equivalent to
(69)(r(t)ϕ(-x′(t)))′+∑i=1n‍ri(t)f(-x(τi(t)))+∑i=1n‍qi(t)|(-x(τi(t)))|αisgn(-x(τi(t)))=-e(t),mmmmmmmmmmmmmmmmmmmmmlmmmnt≥t0.The proof is outlined in the following three steps.Step  1. Next, for any λ1>0the following function is well defined:
(70)ω¯1(t)=-λ1r(t)ϕ(x′(t))xp(t),t∈[a1,b1],
and ω¯1∈C1([a1,b1),ℝ). Since λ1>0and r(t)>0on [a1,b1], from the previous equality we get
(71)|ϕ(x′(t))|=|ω¯1(t)|λ1r(t)xp(t),ω¯1′(t)=-λ1(r(t)ϕ(x′(t)))′xp(t)+pλ1r(t)ϕ(x′(t))xp+1(t)x′(t).Using (1) and assumptions (7) and (9), from the second equality of (71), we get
(72)ddtω¯1(t)≥p(λ1r(t))1/p|ω¯1(t)|(p+1)/p+λ1K∑i=1n‍ri(t)(x(τi(t))x(t))p+λ1xp(t)[∑i=1n‍qi(t)(x(τi(t)))αi-e(t)],
where t∈(a1,b1). Next, using assumptions (12) and (13) in (1) for all t∈(τmin(a1),b1) we easily conclude that x(t)satisfies all assumptions of Corollary 17 and Proposition 19. Hence from (56), (61), and (72) we observe
(73)ddtω¯1(t)≥p(λ1r(t))1/p|ω¯1(t)|(p+1)/p+λ1(R1(t)+Q1(t)),mmmmmmmmmmmmmmmmmmmmmmmt∈(a1,b1).Step  2. In this step, we need the next elementary proposition.Proposition 22.
Letp>0 and let πp be a positive number defined by
(74)πp=π((p+1)/p)sin(pπ/(p+1)).
Then there is an increasing odd functiony=y(s), y∈C1((-πp,πp),ℝ), such that
(75)y′(s)=1+|y(s)|(p+1)/p,s∈(-πp,πp),y(0)=0,y(πp)=∞.
Moreover, forp=1we have πp=π/2 and we can take y(s)=tan(s), s∈(-π/2,π/2).Proof.
Letz=z(t)be a function defined by
(76)z(t)=∫0t‍11+|τ|(p+1)/pdτ,t∈ℝ.
It is not difficult to see thatz(∞)=πp (see [24]) and z is a bijection from ℝ on interval (-z(∞),z(∞)) and that the function y(s)determined by the formula y(s)=z-1(s), where z-1(s)is the inverse function of z(t), satisfies all properties given in (75).Let nowsj∈(-π/2,π/2)be such that y(sj)=ω¯1(aj)(such an sjexists since y(s)is a bijection from (-π/2,π/2)to (-∞,∞)),J=[a1,b1]∪[a2,b2], and let C0(t)be a function defined by
(77)C0(t)=12πpmin{p(λ1r(t))1/p,λ1(R1(t)+Q1(t))},mmmmmmmmmmmmmmmmmmmmmmmmnt∈J.Because of (11), we have c0j:=∫ajbj‍C0(τ)dτ≥1. Hence,
(78)2πpc0jC0(t)≤min{p(λ1r(t))1/p,λ1(R1(t)+Q1(t))},mmmmmmmmmmlmmmmmmmmmmmmmnt∈J.Next, letV1(t) and V2(t) be two functions defined by
(79)Vj(t)=sj+2πpc0j∫ajt‍C0(τ)dτ,t∈[aj,bj],j∈{1,2}.SinceC0(t)≥0 for all t∈J, we have Vj(aj)=sj<πp and Vj(bj)>2πp+sj. Since Vj(t)is continuous, it gives the existence of Tj*∈(aj,bj) such that Vj(Tj*)=πp. Hence, the function ω_j(t)=tan(Vj(t)), t∈(aj,Tj*), satisfies
(80)ω_j(aj)=y(Vj(aj))=y(sj)=ω¯1(aj),ω_j(Tj*)=y(Vj(Tj*))=y(πp)=∞,
and because of (78), we obtain
(81)ω_j′(t)=y′(Vj(t))Vj′(t)=(1+|y(Vj(t))|(p+1)/p)Vj′(t)=(1+|ω_j(t)|(p+1)/p)Vj′(t)≤(1+|ω_j(t)|(p+1)/p)2πpc0jC0(t)≤(1+|ω_j(t)|(p+1)/p)×min{p(λ1r(t))1/p,λ1(R1(t)+Q1(t))},
that is
(82)ω_j′(t)≤p(λ1r(t))1/p|ω_j(t)|(p+1)/p+λ1(R1(t)+Q1(t)),mmmmmmmmmmmmmmmmmmmmmnt∈[aj,Tj*).Step  3. We claim that
(83)ω_1(t)≤ω¯1(t)on[a1,T1*).In order to prove inequality (83), we need the following proposition.Proposition 23.
Leta<T*and φ,ψ∈C1([a,T*),ℝ) be two functions satisfying on [a,T*), respectively,(84)φ′≤p(λ1r(t))1/p|φ|(p+1)/p+λ1(R1(t)+Q1(t)),ψ′≥p(λ1r(t))1/p|ψ|(p+1)/p+λ1(R1(t)+Q1(t)).
Then the following statement holds:(85)φ(a)≤ψ(a)impliesφ(t)≤ψ(t)∀t∈[a,T*).Proof.
If we denote by(86)F(t,u)=p(λ1r(t))1/pu(p+1)/p+λ1(R1(t)+Q1(t)),
then (84) can be rewritten in the form φ′≤F(t,φ)andψ′≥F(t,ψ)on[a,T*). Also, it is not difficult to check thatF(t,u)is locally Lipschitz function in the second variable and bounded in the first variable on bounded interval[a,T*). Hence, we may apply the pointwise comparison principle from [12, Lemma 19] and, consequently, we conclude that statement (85) holds, which proves this proposition.Next, we proceed with the proof of inequality (83). By (80) we know thatω_j(aj)=ω¯1(aj)which, together with (73) and (82), ensures that we may apply Proposition 23. Now, the desired inequality (83) immediately follows from statement (85).Next, from the second equality of (80) and inequality (83) we conclude that ω¯1(T1*)=∞. It contradicts the fact that ω¯1∈C1([a1,b1)). Hence, x(t)can not be oscillatory as it is supposed at the beginning of this proof and we conclude that (1) is oscillatory.Proof of Corollary2. From (16), we see that r(t)≡1andri(t)≡0. Next, to the end of this proof, let λjbe defined by
(87)λj=(pmax[aj,bj]Qj(t))p/(p+1),j∈{1,2}.Now, we can rewrite (17) in the form
(88)12πp∫ajbj‍λjQj(t)dt≥1.Sincemax[aj,bj]Qj(t)≤Qj(t), it is clear that
(89)pλj1/p≥λjQj(t),t∈[aj,bj],
and so
(90)min{pλj1/p,λjQj(t)}=λjQj(t).Together with assumption (88) we show that
(91)12πp∫ajbj‍min{p(λjr(t))1/p,λj(Rj(t)+Qj(t))}dt=12πp∫ajbj‍min{pλj1/p,λjQj(t)}dt=12πp∫ajbj‍λjQj(t)dt≥1.It proves that desired condition (11) is fulfilled. Thus, this corollary follows from Theorem 1.Proofs of Theorems10 and 12. It follows by the same line of arguments as in the proof of Theorem 1, but instead of Proposition 19 we use Propositions 20 and 21, respectively.



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