# Limit-Point/Limit-Circle Results for Equations with Damping

Authors: M. Bartušek; John R. Graef
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The authors study the nonlinear limit-point and limit-circle properties for the second order nonlinear damped differential equation(a(t)|y'|p-1y')'+b(t)|y'|q-1y'+r(t)|y|λ-1y=0, where 0<λ≤p≤q, a(t)>0, and r(t)>0. Some examples are given to illustrate the main results.


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## Body

## 1. Introduction

In this paper, we study the nonlinear equation(1.1)(a(t)|y′|p-1y′)′+b(t)|y′|q-1y′+r(t)|y|λ-1y=0
and its special case
(1.2)(a(t)|y′|p-1y′)′+b(t)|y′|p-1y′+r(t)|y|λ-1y=0.
We set ℝ+=[0,∞) and ℝ=(-∞,∞) and assume throughout that 0<λ≤p≤q, a∈C1(ℝ+), b∈C0(ℝ+), a1/pr∈C1(ℝ+), a(t)>0, and r(t)>0.We will only consider solutions defined on their maximal interval of existence to the right.Remark 1.1.
The functionsa, b, and r are smooth enough so that all nontrivial solutions of (1.1) defined on ℝ+ are nontrivial in any neighborhood of ∞ (see Theorem 13(i) in [1]). Moreover, if either q=p or b(t)≥0 on ℝ+, then all nontrivial solutions of (1.1) are defined on ℝ+.We can write (1.1) as the equivalent system
(1.3)y1′=a-1/p(t)|y2|1/psgny2,y2′=-b(t)a-q/p(t)|y2|q/psgny2-r(t)|y1|λsgny1,
where the relationship between a solution y of (1.1) and a solution (y1,y2) of the system (1.3) is given by
(1.4)y1(t)=y(t),y2(t)=a(t)|y′(t)|p-1y′(t).We are interested in what is known as the nonlinear limit-point and nonlinear limit-circle properties of solutions as given in the following definition (see the monograph [2] as well as the papers [3–14]).Definition 1.2.
A solutiony of (1.1) defined on ℝ+ is said to be of the nonlinear limit-circle type if
(NLC)∫0∞|y(t)|λ+1dt<∞,
and it is said to be of the nonlinear limit-point type otherwise, that is, if
(NLP)∫0∞|y(t)|λ+1dt=∞.
Equation (1.1) will be said to be of the nonlinear limit-circle type if every solution y of (1.1) defined on ℝ+ satisfies (NLC) and to be of the nonlinear limit-point type if there is at least one solution y for which (NLP) holds.The properties defined above are nonlinear generalizations of the well-known linear limit-point/limit-circle properties introduced by Weyl [15] more than 100 years ago. For the history and a survey of what is known about the linear and nonlinear problems as well as their relationships to other properties of solutions such as boundedness, oscillation, and convergence to zero, we refer the reader to the monograph by Bartušek et al. [2] as well as the recent papers of Bartušek and Graef [4, 6, 9–11].Here, we are also interested in what we call the strong nonlinear limit-point and strong nonlinear limit-circle properties of solutions of (1.1) as given in the following definitions. These notions were first introduced in [7] and [8], respectively, and further studied, for example, in [4, 6]. We define the function R:ℝ+→ℝ by
(1.5)R(t)=a1/p(t)r(t)
and let constant δ be given by
(1.6)δ=p+1p.Definition 1.3.
A solutiony of (1.1) defined on ℝ+ is said to be of the strong nonlinear limit-point type if
(SNLP)∫0∞|y(t)|λ+1dt=∞,∫0∞|y2(t)|δR(t)dt=∞.
Equation (1.1) is said to be of the strong nonlinear limit-point type if every nontrivial solution defined on ℝ+ is of the strong nonlinear limit-point type and there is at least one nontrivial solution defined on ℝ+.Definition 1.4.
A solutiony of (1.1) defined on ℝ+ is said to be of the strong nonlinear limit-circle type if
(SNLC)∫0∞|y(t)|λ+1dt<∞,∫0∞|y2(t)|δR(t)dt<∞.
Equation (1.1) is said to be of the strong nonlinear limit-circle type if every solution defined on ℝ+ is of the strong nonlinear limit-circle type.From the above definitions we see that for an equation to be of the nonlinear limit-circle type, every solution must satisfy(NLC); whereas for an equation to be of the nonlinear limit-point type, there needs to be only one solution satisfying (NLP). For an equation to be of the strong nonlinear limit-point type, every solution defined on ℝ+ must satisfy (SNLP) and there must be at least one such nontrivial solution.Ifb(t)≡0, (1.1) becomes
(1.7)(a(t)|y′|p-1y′)′+r(t)|y|λ-1y=0,
and moreover, if λ=p, then (1.7) reduces to the well-known half-linear equation,
(1.8)(a(t)|y′|p-1y′)′+r(t)|y|p-1y=0,
a general discussion of which can be found in the monograph by Došlý and Řehák [16]. Using terminology introduced by the authors in [5–7], if λ>p, we say that (1.1) is of the super-half-linear type, and if λ<p, we will say that it is of the sub-half-linear type. Since in this paper we are assuming that λ≤p, we are in the half-linear and sub-half-linear cases.The limit-point/limit-circle problem for the damped equation(1.9)(a(t)y′)′+b(t)y′+r(t)yλ=0
with b(t)≥0 was considered in the papers [17, 18], where λ≤1 is the ratio of odd positive integers and λ≥1 is an odd integer, respectively. The results in both of these papers tend to be modifications of results in [12–14] to accommodate the damping term.It will be convenient to define the following constants:(1.10)α=p+1(λ+2)p+1,β=(λ+1)p(λ+2)p+1,γ=p+1p(λ+1),β1=p(λ+2)p+1,ω2=1λ+1+vpp+1,α1=αγ-1/(λ+1),β2=(λ+1)(p+1)p-λ+(v-1)p(λ+1)foreitherp>λorv>1,ω1=vp+1p+1,v1=β1(v-1),ω=1λ+1+pp+1,v=qp≥1.
Notice that α=1-β,  ω2-1=1/β2,  ω2≥ω1, and ω≥1. We define the function g:ℝ+→ℝ by
(1.11)g(t)=-a1/p(t)R′(t)Rα+1(t),
and in the reminder of this paper we will make use of the assumption that
(H)limt→∞g(t)=0,∫0∞|g′(s)|ds<∞.
If (H) holds, we define the constants
(1.12)γ1=αγ-1/(λ+1)sups∈ℝ+|g(s)|,γ2=δ+γ1.
For any solution y:ℝ+→ℝ of (1.1), we let
(1.13)F(t)=Rβ(t)[a(t)r(t)|y′(t)|p+1+γ|y(t)|λ+1]=Rβ(t)(|y2(t)|δR(t)+γ|y(t)|λ+1).
Note that F≥0 on ℝ+ for every solution of (1.1).For any continuous functionh:ℝ+→ℝ, we let h+(t)=max{h(t),0} and h-(t)=max{-h(t),0} so that h(t)=h+(t)-h-(t).


## 2. Lemmas

In this section we present a number of lemmas that will facilitate proving our main results.Lemma 2.1.
For every nontrivial solutiony of (1.1) defined on ℝ+, F(t)>0 for t≥0.Proof.
Suppose, to the contrary, that (1.1) has a nontrivial solution y such that F(t0)=0 for a number t0∈ℝ+. Then (1.13) implies y(t0)=y′(t0)=0 and so (1.1) has the solution y- defined by
(2.1)y-(t)=y(t)fort∈[0,t0],y-(t)=0fort≥t0.
But this contradicts Remark 1.1 and proves the lemma.Lemma 2.2.
Lety be a solution of (1.1). Then: (i)
fort∈ℝ+, we have
(2.2)|y(t)|≤γ-1/(λ+1)R-β1(t)F1/(λ+1)(t),|y2(t)|≤Rβ1(t)Fp/(p+1)(t).(ii)
for0≤τ<t, we have
(2.3)F(t)=F(τ)-αg(τ)y(τ)y2(τ)+αg(t)y(t)y2(t)-α∫τtg′(s)y(s)y2(s)ds-∫τt[δR-α(s)|y2(s)|1/psgny2(s)-∫τt‍-αg(s)y(s)|y2(s)|1/p]b(s)av(s)|y2(s)|v-1y2(s)ds,(2.4)|∫τt[δR-α(s)|y2(s)|1/psgny2(s)-αg(s)y(s)]b(s)av(s)|y2(s)|v-1y2(s)ds|≤δ∫τt|b(s)|av(s)Rv1(s)Fω1(s)ds+γ1∫τt|b(s)|av(s)Rv1(s)Fω2(s)ds.Proof.
Lety be a solution of (1.1). Then it is a solution of the equation
(2.5)(a(t)|z′|p-1z′)′+r(t)|z|λ-1z=e(t)
with e(t)=-b(t)|y'(t)|q-1y′(t)=-(b(t)/av(t))|y2(t)|v-1y2(t). Then (2.2) and (2.3) follow from Lemma  1.2 in [5] applied to (2.5). Relation (2.4) follows from (2.2).The following two lemmas give us sufficient conditions for the boundedness ofF from above and from bellow by positive constants.Lemma 2.3.
Let(H) hold and assume that
(2.6)∫0∞|b(s)|av(s)Rv1(s)ds<∞.
Then for any nontrivial solution y of (1.1) defined on ℝ+, the function F is bounded from bellow on ℝ+ by a positive constant depending on y.Proof.
Suppose, to the contrary, that there is a nontrivial solution of (1.1) such that
(2.7)liminft→∞F(t)=0.
By Lemma 2.1, F(t)>0 on ℝ+. Let t-∈ℝ+ be such that
(2.8)2α1sups∈[t-,∞)|g(s)|+α1∫t-∞|g′(s)|ds+(γ1+δ)∫t-∞|b(s)|av(s)Rv1(s)ds≤12,
the existence of such a t- follows from (H) and (2.6). Then, for any t0≥t- such that F(t0)≤1, there exist τ and σ such that t0≤σ<τ and
(2.9)2F(τ)=F(σ)=F(t0)>0,F(τ)≤F(t)≤F(σ)
for σ≤t≤τ. Then (2.2) implies
(2.10)|y(t)y2(t)|≤γ-1/(λ+1)Fω(t)
on ℝ+. From this, (2.3) (with τ=σ and t=τ), (2.4), (2.9), and the fact that F(σ)≤1, we have
(2.11)F(σ)2=F(σ)-F(τ)≤[α1|g(τ)|+α1|g(σ)|+α1∫t-∞|g′(s)|ds]Fω(σ)+γ1∫t-∞|b(s)|av(s)Rv1(s)dsFω2(σ)+δ∫t-∞|b(s)|av(s)Rv1(s)dsFω1(σ).
Hence, using (2.8) and the facts that ω2≥ω1≥1,  ω≥1, and F(σ)≤1, we obtain
(2.12)F(σ)≤12F(σ).
This contradiction toF(σ)>0 proves the lemma.Lemma 2.4.
Assume thatb≥0 for large t, (H) holds,
(2.13)∫0∞b(t)av(t)Rv1(t)dt<∞,
and either (i) λ=p=q, or (ii) q>λ and
(2.14)liminft→∞Rβ(t)(∫t∞‍[|g′(s)|+b(s)av(s)Rv1(s)]ds)β2exp{∫0tR-′(s)R(s)ds}=0.
Then for every solution of (1.1) the function F is bounded on ℝ+.Proof.
Lety be a nontrivial solution of (1.1). Then according to Remark 1.1 and Lemma 2.1, y is defined on ℝ+ and F(t)>0 on ℝ+. In view of (H) and (2.13), we can choose t-∈ℝ+ such that
(2.15)∫t-∞[|g′(s)|+b(s)av(s)Rv1(s)]ds≤12[3α1+2ω1δ+2ω2γ1]-1.
Suppose that F is not bounded, that is,
(2.16)limsupt→∞F(t)=∞.
Then, for any t0≥t- with F(t0)≥1, there exist σ and τ such that t0≤σ<τ,  (1/2)F(τ)=F(σ)=F(t0), and
(2.17)1≤F(σ)≤F(t)≤F(τ)forσ≤t≤τ.
Since g is of bounded variation and limt→∞g(t)=0, we see that
(2.18)|g(σ)|=|g(σ)-g(∞)|≤∫σ∞|g′(s)|ds.
Setting τ=σ and t=τ in (2.2)–(2.4), we have (2.10) and
(2.19)F(σ)=F(τ)-F(σ)≤[α1|g(σ)|+α1|g(τ)|+α1∫στ|g′(s)|dsF(τ)-F(σ)≤+γ1∫στb(s)av(s)Rv1(s)ds]Fω2(τ)+δ∫σ∞b(s)av(s)Rv1(s)dsFω1(τ).
From this, (2.8), (2.15), and (2.18), we obtain
(2.20)F(σ)≤[3α1∫σ∞|g′(s)|ds+(2ω2γ1+2ω1δ)∫σ∞b(s)av(s)Rv1(s)ds]Fω2(σ)≤K∫σ∞[|g′(s)|+b(s)av(s)Rv1(s)]dsFω2(σ)≤12Fω2(σ),
where K=3α1+2ω1δ+2ω2γ1.
Ifλ=p=q, then ω2=1 and (2.20) gives us a contradiction.
Now letq>λ and (2.14) hold. Then ω2>1 and (2.20) implies
(2.21)F(t0)=F(σ)≥K-β2(∫σ∞[|g′(s)|+b(s)av(s)Rv1(s)]ds)-β2.
Hence,
(2.22)F(t)≥K1(∫t∞[|g′(s)|+b(s)av(s)Rv1(s)]ds)-β2
for all t≥t- such that F(t)≥1, where K1=K-β2. At the same time, (2.20) implies F(t)≥2β2>1 for these values of t. Thus, (2.22) holds for all t≥t-. On the other hand, if z(t)=F(t)R-β(t), then (1.13) implies
(2.23)z′(t)=(R-1(t))′|y2(t)|δ-δr-1(t)y′(t)b(t)|y′(t)|qsgny′(t)≤(R-1(t))′|y2(t)|δ≤R-′(t)R(t)R-β(t)F(t)=R-′(t)R(t)z(t)
for t≥t-. So,
(2.24)z(t)≤z(t-)exp∫0tR-′(s)R(s)ds.
From this and (2.22),
(2.25)K1{∫t∞[|g′(s)|+b(s)av(s)Rv1(s)]ds}-β2≤F(t)=Rβ(t)z(t)≤z(t-)Rβ(t)exp∫0tR-′(s)R(s)ds,
which contradicts (2.14). Hence, F is bounded from above on ℝ+. Since F>0 on ℝ+, the conclusion follows.Lemma 2.5.
Let(H) and (2.6) hold. Then there exists a solution y of (1.1) defined on ℝ+, a constant c0>0, and t0∈ℝ+ such that
(2.26)0<34c0≤F(t)≤32c0fort≥t0.
Moreover, c0 can be chosen arbitrary small.Proof.
Condition(H) implies that g is bounded, so we can choose M>0,  t0∈ℝ+, and c0 such that
(2.27)|g(t)|≤Mfort≥t0,∫t0∞|g′(s)|ds≤M,∫t0∞|b(s)|av(s)Rv1(s)ds≤M,M≤14(32)-ω2[3αγ-1/(λ+1)+δ+γ1]-1,0<c0≤23.
Consider a solutiony of (1.1) such that F(t0)=c0. First, we will show that
(2.28)F(t)≤32c0≤1fort≥t0.
Suppose (2.28) does not hold. Then there exist t2>t1≥t0 such that
(2.29)F(t2)=32c0,F(t1)=c0,c0<F(t)<32c0
for t∈(t1,t2). Lemma 2.2 (with τ=t1 and t=t2), and the facts that ω≤ω2,  ω1≤ω2, and c0<1 imply
(2.30)c02=F(t1)2=F(t2)-F(t1)≤3αγ-1/(λ+1)M(32c0)ω+M(δFω1(t2))+γ1MFω2(t2)≤M(3αγ-1/(λ+1)+δ+γ1)(32)ω2c0ω≤14c0ω.
Hence, c0ω-1≥2 which contradicts the choice of c0, and so (2.28) holds.
Now, Lemma2.2 (with t=t,   τ=t0) similarly implies
(2.31)|F(t)-c0|≤3αγ-1/(λ+1)M(32c0)ω+M[(32)ω1δc0ω1+(32)ω2γ1c0ω2]≤Mc0[3(32)ωαγ-1/(λ+1)+(32)ω1δ+(32)ω2γ1]≤c04,
and the statement of the lemma is proved.Lemma 2.6.
Suppose that(H) and (2.6) hold and
(2.32)∫0∞R-β(t)dt=∞.
In addition, assume that either
(2.33)∫0∞(|(1r(s))′|+|b(s)|av(s)r(s)Rβ1(v-1)(s))ds<∞
or
(2.34)∫0∞(|a′(s)|a(s)r(s)+|b(s)|av(s)r(s)Rv1(s))ds<∞
holds. If y is a solution of (1.1) with
(2.35)c1≤F(t)≤c2
on ℝ+ for some positive constants c1 and c2, then
(2.36)∫0∞|y(t)|λ+1dt=∞.
Moreover, if r does not tend to zero as t→∞, then
(2.37)∫0∞|y2(t)|δR(t)dt=∞.Proof.
Lety be a nontrivial solution of (1.1) satisfying (2.35). Then in view of (1.13), (2.32), and (2.35)
(2.38)γ∫0t|y(s)|λ+1ds+∫0t|y2(s)|δR(s)ds=∫0tF(s)Rβ(s)ds⟶∞
as t→∞. Now, (2.2) and (2.35) imply
(2.39)|y(t)y2(t)|≤γ-1/(λ+1)c2ω=defM1
for t≥0 so there exists t0≥0 such that
(2.40)|g(t)|≤c12M1max(1,γ)
for t≥t0. It follows from (1.1) that
(2.41)∫0t|y(s)|λ+1ds=-∫0ty(s)y2′(s)r(s)ds-∫0tb(s)av(s)r(s)y(s)|y2(s)|v-1y2(s)ds=-y(t)y2(t)r(t)+D+∫0t|y2(s)|δR(s)ds+J(t),
where D=y(0)y2(0)r-1(0) and
(2.42)J(t)=∫0t[(1r(s))′-b(s)av(s)r(s)|y2(s)|v-1]y(s)y2(s)ds.
Hence, from (2.39),
(2.43)|J(t)|≤M1D1∫0t[|(1r(s))′|+|b(s)|av(s)r(s)Rβ1(v-1)(s)]ds
with D1=1+c2(q-p)/(p+1) for t≥t0. Moreover, (1.13), (2.35), and (2.38) imply
(2.44)c3∫t0tR-β(s)ds≤∫t0t|y(s)|λ+1ds+∫t0t|y2(s)|δR(s)ds≤c4∫t0tR-β(s)ds
for t≥t0 with c3=c1/max(1,γ) and c4=c2/min(1,γ).
If (2.33) holds, then (2.43) implies J is bounded on ℝ+, and in view of (2.41), we have
(2.45)|∫t0t|y(s)|λ+1ds-∫t0t|y2(s)|δR(s)ds+y(t)y2(t)r(t)|≤J1(t)+m∫t0tR-β(s)ds
for t≥t0 with J1(t)=|J(t)|+(|y(0)y2(0)|/r(0)) and m=0. Note, that J1 is bounded on ℝ+.
Now let (2.34) hold. Then, using (2.2), (2.35), and (2.40) and setting c5=max(1,c2(q-p)/(p+1)), we have
(2.46)∫t0t[|(1r(s))′|+|b(s)|av(s)r(s)|y2(s)|v-1]ds≤∫t0t(|(a1/pR(s))′|+|b(s)|av(s)r(s)Rβ1(v-1)(s)c2(q-p)/(p+1))ds≤c5∫0t(|a′(s)|pa(s)r(s)+|b(s)|av(s)r(s)Rβ1(v-1)(s))ds+∫t0t|g(s)|Rβ(s)ds≤M2+c32M1∫t0tR-β(s)ds
for t≥t0, where M2=c5∫0∞‍((|a′(s)|/pa(s)r(s))+((|b(s)|/av(s)r(s))Rβ1(v-1)(s)))ds<∞ by (2.34). From (2.46) together with (2.39), (2.41), and (2.42), inequality (2.45) holds with m=c3/2 and J1(t)=M1M2. Thus, (2.45) holds with m=c3/2 if either (2.33) or (2.34) holds, and J1 is bounded for t≥t0. Moreover,
(2.47)-J1(t)-c32∫t0tR-β(s)ds≤∫t0t|y(s)|λ+1ds-∫t0t|y2(s)|δR(s)ds+y(t)y2(t)r(t)≤J1(t)+c32∫t0tR-β(s)ds
for t≥t0. Adding the left hand inequalities in (2.44) and (2.47) gives
(2.48)2∫t0t|y(s)|λ+1ds+y(t)y2(t)r(t)≥c32∫t0tR-β(s)ds-J1(t)→∞
as t→∞, and subtracting gives
(2.49)2∫t0t|y2(s)|δR(s)ds-y(t)y2(t)r(t)≥c32∫t0tR-β(s)ds-J1(t)→∞
as t→∞.
Ify' is oscillatory, let {tk}k=1∞→∞ be a sequence of zeros of y′. Then letting t=tk in (2.48) and (2.49), it is clear that the conclusion of the lemma holds.
Lety' be nonoscillatory. Then either
(2.50)y(t)y2(t)>0forlarget
or
(2.51)y(t)y2(t)<0forlarget.
We first prove (2.36). It clearly holds if (2.50) does. So suppose (2.51) holds. Then y(t)y2(t)r-1(t)<0 for large t and (2.48) gives us the contradiction. Hence, (2.36) holds.
Finally, we prove (2.37). From (2.49), (2.37) holds if (2.50) does. Let (2.51) hold and assume that (2.37) does not. Then (2.49) implies
(2.52)limt→∞y(t)y2(t)r(t)=-∞.
In view of (2.39) and (2.51),  y(t)y2(t)≥-M1, so limt→∞r(t)=0. This contradicts the assumptions of the lemma and completes the proof.


## 3. LP/LC Problem for (1.1)

In this section we present our main results for (1.1) and give some examples to illustrate them.Theorem 3.1.
Letb≥0 for large t and assume that (H) and (2.13) hold. In addition, if q>λ, assume that (2.14) also holds. Then (1.1) is of the strong nonlinear limit-circle type if and only if
(3.1)∫0∞R-β(t)dt<∞.Proof.
Lety be a nontrivial solution of (1.1). By Remark 1.1, y is defined on ℝ+. The hypotheses of Lemmas 2.3 and 2.4 are satisfied, so there are constants c and c1 such that
(3.2)0<c≤F(t)≤c1
on ℝ+. Hence, from this and (1.13),
(3.3)c∫0∞R-β(t)dt≤γ∫0∞|y(t)|λ+1dt+∫0∞|y2(t)δ|R(t)dt=∫0∞F(t)R-β(t)dt≤c1∫0∞R-β(t)dt.
The conclusion of the theorem then follows from (3.1).In caseb(t)≡0, the results in this paper reduce to previously known results by the present authors except that the necessary part of Theorem 3.1 is new.Theorem 3.2.
Let(H), (2.6), and either (2.33) or (2.34) hold. If
(3.4)∫0∞R-β(t)dt=∞,
then (1.1) is of the nonlinear limit-point type.Proof.
The hypotheses of Lemmas2.5 and 2.6 are satisfied, so if y is a solution given by Lemma 2.5, then (2.36) holds, and the conclusion follows.Theorem 3.3.
Letb≥0 for large t and let conditions (H), (2.13), and either (2.33) or (2.34) hold. In addition, if q>λ, assume that (2.14) holds. If
(3.5)∫0∞R-β(t)dt=∞,
then every nontrivial solution of (1.1) is of the nonlinear limit-point type. If, moreover, r does not tend to zero as t→∞, then (1.1) is of the strong nonlinear limit-point type.Proof.
Note that the hypotheses of Lemmas2.3, 2.4, and 2.6 are satisfied. Let y be a nontrivial solution of (1.1). Then Remark 1.1 implies y is defined on ℝ+, and by Lemmas 2.3 and 2.4, there are positive constants C1 and C2 such that
(3.6)0<C1≤F(t)≤C2onℝ+.
Thus, by Lemma 2.6, (2.36) holds, and if r does not tend to zero as t→∞, then (2.37) holds. This proves the theorem.Remark 3.4.
Note that Lemmas2.1, 2.2, and 2.6 are valid without the assumption that λ≤p.


## 4. LP/LC Problem for (1.2)

One of the main assumptions in Section3 is (2.6), which takes the form
(4.1)∫0∞|b(t)|a(t)dt<∞
for (1.2). It is possible to remove this condition when studying (1.2). The technique to accomplish this is contained in the following lemma; a direct computation proves the lemma.Lemma 4.1.
Equation (1.2) and the equation
(4.2)(a-(t)|y′|p-1y′)′+r-(t)|y|λ-1y=0
are equivalent where
(4.3)a-(t)=a(t)exp{∫0tb(s)a(s)ds},r-(t)=r(t)exp{∫0tb(s)a(s)ds}.
That is, every solution of (1.2) is a solution of (4.2) and vice versa.Based on this lemma, results for (1.2) can be obtained by combining Lemma 4.1 and known results for (4.2), such as those that can be found, for example, in [3, 5, 7, 9, 10]. Here we only state a sample of the many such possible results.Define(4.4)R-(t)=a-1/p(t)r-(t)=R(t)exp{δ∫0tb(s)a(s)ds},g-(t)=-a-1/p(t)R-′(t)R-α+1(t)=-a1/p(t)Rα+1(t)[R′(t)+δb(t)a(t)R(t)]×exp{λ-p(λ+2)p+1∫0tb(s)a(s)ds},ω3=(λ+1)(p+1)p-λforp>λ,ω3=∞forp=λ.Theorem 4.2.
Assume that(4.5)limt→∞g-(t)=0,∫0∞|g-′(s)|ds<∞,
and either (i) λ=p or (ii) λ<p and
(4.6)liminft→∞R-β(t)(∫t∞|g-′(s)|ds)ω3exp{∫0t(R--1(σ))+′R-(σ)dσ}=0.
Then (1.2) is of the strong nonlinear limit-circle type if and only if
(4.7)∫0∞R--β(σ)dσ<∞.Proof.
The conclusion follows from Theorem  2.11 in [5] applied to (4.2) and Lemma 4.1.Our next result follows from Theorem3.2 being applied to (4.2) and Lemma 4.1.Theorem 4.3.
Let (4.5) and either
(4.8)∫0∞|(1r-(t))′|dt<∞or∫0∞|a-′(r)|a-(t)r-(t)dt<∞
hold. If
(4.9)∫0∞R--β(σ)dσ=∞,
then (1.2) is of the nonlinear limit-point type.Our final theorem is a strong nonlinear limit-point result for (1.2).Theorem 4.4.
Assume that (4.5) holds and
(4.10)limt→∞a′(t)+b(t)a1-β/p(t)r(t)exp{1ω3∫0tb(s)a(s)ds}=0.
If ∫0∞‍R--β(σ)dσ=∞, then (1.1) is of the nonlinear limit-point type. If, in addition, r does not tend to zero as t→∞, then (1.2) is of the strong nonlinear limit-point type.Proof.
This result follows from Theorem  2.16 in [5] and Lemma 4.1 above. Note that Theorem  2.16 in [5] is proved for r(t)≥r0>0 for t∈ℝ+, but it is easy to see from (2.34) in [5] and the end of its proof that Theorem 2.16 holds as long as r does not tend to zero as t→∞.We conclude the paper with some examples to illustrate our main results.Example 4.5.
Consider the equation(4.11)(|y′|p-1y′)′+b(t)|y′|q-1y′+tσ|y|λ-1y=0,t≥1.
Assume that σα>-1 and βσ≤1. If b satisfies either
(4.12)σ≥0,∫1∞tσv1|b(t)|dt<∞,
or
(4.13)σ<0,∫1∞tσ(v1-1)|b(t)|dt<∞,
then the conditions of Theorem 3.2 are satisfied, so (4.11) is of the nonlinear limit-point type.Example 4.6.
Consider the special case of (4.11) with p=1, namely,
(4.14)y′′+b(t)|y′|q-1y′+tσ|y|λ-1y=0,t≥1.
Then we have 0<λ≤1≤q, so if
(4.15)-λ+32<σ≤λ+3λ+1
and either
(4.16)σ≥0,∫1∞tσ(q-1)/(λ+3)|b(t)|dt<∞
or
(4.17)σ<0,∫1∞t((q-λ-4)/(λ+3))σ|b(t)|dt<∞
holds, (4.14) is of the nonlinear limit-point type.Example 4.7.
Consider the equation(4.18)y′′+b(t)y′+tσy=0,t≥1,
with b(t)≥0. Note that here p=q=λ=1. Assume that
(4.19)∫1∞b(t)dt<∞.
Then by Theorem 3.1, (4.18) is of the strong nonlinear limit-circle type if and only if σ>2. By Theorem 3.3, (4.18) is of the strong nonlinear limit-point type if 0<σ≤2. It is worth noting that this agrees with the well-known limit-circle criteria
(4.20)∫0∞r-1/2(t)dt<∞
of Dunford and Schwartz [19, page 1414] (also see the discussion in [2]).For our next example we consider the case wherep=q=λ. It may be convenient to refer to this case as the fully half-linear equation.Example 4.8.
Consider the equation(4.21)(ta|y′|ϵ-1y′)′+tb|y′|ϵ-1y′+tσ|y|ϵ-1y=0,t≥1,
where ϵ>0. If σ+ϵ+1>a>max{b+1,ϵ(1-σ)+1}, then (4.21) is of the strong nonlinear limit-circle type by Theorem 3.1. On the other hand, if σ>0 and b+1<a<min{σ+ϵ+1,ϵ(1-σ)+1}, then (4.21) is of the strong nonlinear limit-point type by Theorem 3.3.Our final example will illustrate several of our theorems as well allow us to compare our results to those in [17, 18].Example 4.9.
Consider the equation(4.22)y′′+tsy′+tσ|y|λ-1y=0,t≥1,
with s∈ℝ, σ≥0, and 0<λ≤1. Calculations show the following.(i)
Equation (4.22) is of the nonlinear limit-circle type if
(a)
λ=1 and s>-1 (by Theorem 4.2);(b)
λ<1 and -1<s<σ/2 (by Theorem 4.2);(c)
s=-1 and σ>(1-λ)/(λ+1) (by Theorem 4.2);(d)
λ=1, s<-1, and σ>(λ+3)/(λ+1) (by Theorem 3.1);(e)
λ<1, σ>(λ+3)/(λ+1), and s<-1-σ((1-λ)/(λ+3)) (by Theorem 3.1).(ii)
Equation (4.22) is of the nonlinear limit-point type if
(f)
s=-1 and σ≤(1-λ)/(λ+1) (by Theorem 4.3);(g)
s<-1 and σ≤(λ+3)/(λ+1) (by Theorem 3.2).
Now by [17, Corollary 2.3], (4.22) is of the nonlinear limit-circle type if s≤-σ(1-λ)/2(λ+3) and σ>(λ+3)/(λ+1). The nonlinear limit-point result [17, Theorem 2.6] does not apply to (4.22). This shows that our results substantially extend the ones in [17] in the case of nonlinear limit-circle type results and are new in the case of nonlinear limit-point results. The results in [17] follow from ours if s≥-1 and for s<-1 and λ=1. There are errors in the proofs of the results in [18].
More specifically, for (4.22) with λ=1, that is,
(4.23)y′′+tsy′+tσy=0,t≥1,
the results in [17, 18] show that (4.23) is of the nonlinear limit-circle type if s≤0 and σ>2. and by results in the present paper (4.23) is of the nonlinear limit-point type if and only if(h)
s<-1 and σ>2;(i)
s=-1 and σ>0;(j)
s>-1 and σ is arbitrary.
Hence, the results in [17, 18] follow from ours, and our results are substantially better; note that we obtain necessary and sufficient condition for (4.23) to be of the nonlinear limit-circle type.



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