# Limit-Point/Limit-Circle Results for Superlinear Damped Equations

Authors: M. Bartušek; John R. Graef
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

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


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## Body

## 1. Introduction

In this paper, we continue the study of the nonlinear limit-point and limit-circle properties for the second-order damped equation(1)(a(t)|y′|p-1y′)′+b(t)|y′|q-1y′+r(t)|y|λ-1y=0,
where ℝ+=[0,∞), ℝ=(-∞,∞), 0<q≤p≤λ, a∈C1(ℝ+), b∈C0(ℝ+), r∈C1(ℝ+), a(t)>0, and r(t)>0. We also consider the special case of (1) with p=q, namely,
(2)(a(t)|y′|p-1y′)′+b(t)|y′|p-1y′+r(t)|y|λ-1y=0.
Previous results of this type for damped equations can be found in the papers of Shao and Song [1], Xing et al. [2], and the present authors [3]. Later in this paper we will compare the results here with those previously known.The limit-point/limit-circle problem has its origins in the work of Weyl [4] over 100 years ago. Weyl considered the second-order linear eigenvalue problem
(C)y′′+r(t)y=θy,θ∈ℂ
and classified this equation to be of the limit-circle type if every solution belongs to L2, that is,
(3)∫0∞‍y2(σ)dσ<∞,
and to be of the limit-point type if at least one solution y(t) does not belong to L2, that is,
(4)∫0∞‍y2(σ)dσ=∞.
The limit-point/limit-circle problem then becomes that of determining conditions on the coefficient function r that allows us to distinguish between these two cases. Weyl also proved that the linear equation (C) always has at least one square integrable solution provided Imθ≠0. The problem then reduces to whether (C) has one (limit-point case) or two (limit-circle case) square integrable solutions; this has come to be known as the Weyl Alternative. Weyl also showed that if (C) is limit-circle for some θ0∈ℂ, then it is limit-circle for all θ∈ℂ. In particular, this is true for θ=0, so that if we can show that the equation
(L)y′′+r(t)y=0
is limit-circle, then (C) is limit-circle for all values of θ, and if (L) is not limit-circle, then (C) is not limit-circle for any value of θ. However, for (L) we are not guaranteed that there is at least one square integrable solution.In the years since Weyl’s original work there has been a great deal of interest in this problem due to its relationship with the solution of certain boundary value problems. By comparison, the analogous problem for nonlinear equations is relatively new and has not been as extensively studied as the linear case.In what follows, we will only consider solutions defined on their maximal interval of existence to the right. We next define what we mean by a proper solution.Definition 1.
A solution of (1) is said to be proper if it is defined on ℝ+ and is nontrivial in any neighborhood of ∞.Remark 2.
Under the covering assumptions here, the functionsa, b, and r are smooth enough so that all solutions of (1) are defined for large t (see [5, Theorem 2(i)]). Moreover, all nontrivial solutions of (1) are proper if either b≤0 on ℝ+ or q=p (see [5, Theorem 4]).The nonlinear limit-point/limit-circle problem originated in the work of Graef [6, 7] and Graef and Spikes [8]. The history and a survey of what is known about the linear and nonlinear problems as well as their relationships with other properties of solutions such as boundedness, oscillation, and convergence to zero, can be found in the monograph by Bartušek et al. [9] as well as the recent papers of Bartušek and Graef [10–14]. The nonlinear limit-point and limit-circle properties of solutions are defined as follows (see [9] and the papers [3, 6, 7, 10–18]).Definition 3.
A solutiony of (1) 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) will be said to be of the nonlinear limit-circle type if every solution y of (1) satisfies (NLC) and to be of the nonlinear limit-point type if there is at least one solution y for which (NLP) holds.
We can write (1) as the equivalent system
(5)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) and a solution (y1,y2) of the system (5) is given by
(6)y1(t)=y(t),y2(t)=a(t)|y′(t)|p-1y′(t).
Also of interest here is what we call the strong nonlinear limit-point and strong nonlinear limit-circle properties of solutions of (1) as can be found in the following definitions. These notions were first introduced in [17, 18], respectively, and further studied, for example, in [10, 11]. We define the function R:ℝ+→ℝ by
(7)R(t)=a1/p(t)r(t)
and the constant δ by
(8)δ=p+1p.Definition 4.
A solutiony of (1) is said to be of the strong nonlinear limit-point type if
(9)∫0∞‍|y(t)|λ+1dt=∞,∫0∞‍|y2(t)|δR(t)dt=∞.
Equation (1) is said to be of the strong nonlinear limit-point type if every proper solution is of the strong nonlinear limit-point type and there is at least one proper solution.Definition 5.
A solutiony of (1) is said to be of the strong nonlinear limit-circle type if
(10)∫0∞‍|y(t)|λ+1dt<∞,∫0∞‍|y2(t)|δR(t)dt<∞.
Equation (1) is said to be of the strong nonlinear limit-circle type if every solution is of the strong nonlinear limit-circle type.Notice that ifb(t)≡0, (1) reduces to
(11)(a(t)|y′|p-1y′)′+r(t)|y|λ-1y=0,
and moreover, if λ=p, then (11) takes the form
(12)(a(t)|y′|p-1y′)′+r(t)|y|p-1y=0.
This is the well-known half-linear equation, a general discussion of which can be found in the monograph by Došlý and Rehák [19]. Using the terminology introduced by the authors in [3, 11, 16], if λ>p, we say that (1) or (11) 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 super-half-linear cases. In [3], we considered the sub-half-linear case of (1) and (11).The limit-point/limit-circle problem for the damped equation(13)(a(t)y′)′+b(t)y′+r(t)yλ=0
with b(t)≥0 was studied in [1] with λ≤1 being the ratio of odd positive integers and in [2] with λ≥1 being an odd integer. The results in both of these papers tend to be modifications of results in [6–8] to accommodate the damping term. We will say more about the relationship between the results in [1, 2] and the present paper in Section 5.It will be convenient to define the following constants:(14)α=p+1(λ+2)p+1,β=(λ+1)p(λ+2)p+1,γ=p+1p(λ+1),α1=αγ-1/(λ+1),β1=p(λ+2)p+1,aaaaβ2=(λ+1)(p+1)λ(p-q)+(λ-q)foreitherp>qorλ>q,ω=1λ+1+pp+1,ω1=q+1p+1,ω2=1λ+1+qp+1,v=qp≤1,v1=q-p(λ+2)p+1.
Notice that α=1-β, ω2≤ω1, and ω≤1. We define the function g:ℝ+→ℝ by
(15)g(t)=-a1/p(t)R′(t)Rα+1(t),
and sometimes we will make use of the assumption that
(16)limt→∞g(t)=0,∫0∞‍|g′(s)|ds<∞.
If (16) holds, we define the constants
(17)γ1=αγ-1/(λ+1)sups∈ℝ+|g(s)|,γ2=δ+γ1.
For any solution y:ℝ+→ℝ of (1), we let
(18)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).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).In Section2, we give some preliminary lemmas. Section 3 contains our main results on (1), and in Section 4 we study (2). Examples to illustrate our results and to compare our results to previously known ones are given in Section 5.


## 2. Lemmas

In this section we establish some lemmas that will be needed to prove the main results in this paper.Lemma 1.
Let eitherp=q or b≤0 on ℝ+. Then for every nontrivial solution y of (1) one has F(t)>0 for large t.Proof.
Lety be a nontrivial solution of (1). Then (18) implies the existence of T≥0 such that F(T)>0. Suppose, to the contrary, that F(t0)=0 for some t0>T. Then (18) implies y(t0)=y′(t0)=0 and so (1) has the solution y- defined by
(19)y-(t)=y(t)fort∈[0,t0],y-(t)=0fort≥t0.
But this contradicts Remark 2 and proves the lemma.Lemma 2.
Lety be a solution of (1). Then(i)
fort∈ℝ+, one has
(20)|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, one has
(21)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)-αg(s)y(s)|y2(s)|1/p]b(s)av(s)|y2(s)|v-1y2(s)ds,(22)|∫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). Then it is a solution of the equation
(23)(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). The expressions (20) and (21) follow from Lemma  1.2 in [16] applied to (23). Inequality (22) follows from (20).The next two lemmas give us sufficient conditions for the boundedness ofF from above and from below by positive constants.Lemma 3.
In addition to (16), assume that
(24)∫0∞‍|b(t)|av(t)Rv1(t)dt<∞
and one of the following conditions holds:(i)
λ=p=q;(ii)
b≤0 for large t and
(25)liminft→∞R-β(t){∫t∞‍[|g′(s)|+|b(s)|av(s)Rv1(s)]ds}β2×exp{∫0t‍R+′(s)R(s)ds}=0;or(iii)
p=q<λ and (25) holds.
Then for any nontrivial solutiony of (1) defined on ℝ+, the function F is bounded from below for large t by a positive constant depending on y.Proof.
Suppose, to the contrary, that there is a nontrivial solution of (1) such that
(26)liminft→∞F(t)=0.
By Lemma 1, F(t)>0 for large t. Let t-∈ℝ+ be such that F(t)>0 for t≥t- and
(27)2α1sups∈[t-,∞)|g(s)|+α1∫t-∞‍|g′(s)|ds+γ2∫t-∞‍|b(s)|av(s)Rv1(s)ds≤14;
the existence of such a t- follows from (16) and (24). Then, for any t0≥t- such that F(t0)≤1, there exist τ and σ such that t0≤σ<τ and
(28)2F(τ)=F(σ)=F(t0)>0,F(τ)≤F(t)≤F(σ)
for σ≤t≤τ. Then (20) implies
(29)|y(t)y2(t)|≤γ-1/(λ+1)Fω(t)
on ℝ+. From (21) (with τ=σ and t=τ), inequalities (22) and (29), and the fact that F(σ)≤1, we have
(30)F(σ)2=F(σ)-F(τ)≤[α1|g(τ)|+α1|g(σ)|+α1∫σ∞‍|g′(s)|ds]Fω(σ)+∫σ∞‍|b(s)|av(s)Rv1(s)ds(γ1Fω2(σ)+δFω1(σ)).
Since min(ω,ω1,ω2)=ω2≤1, (27) and (30) imply
(31)F(σ)≤12Fω2(σ).
It is easy to see that ω2=1 if and only if p=λ=q.
In this case (31) and F>0 give us a contradiction and the statement of the lemma holds in case (i).
Suppose thatp=λ=q does not hold; this implies ω2≠1. Since g is of bounded variation and limt→∞g(t)=0, we see that
(32)|g(σ)|=|g(σ)-g(∞)|≤∫σ∞‍|g′(s)|ds.
From this, (28), and (30), we obtain
(33)F(σ)≤2{∫σ∞‍|b(s)|av(s)3α1∫σ∞‍|g′(s)|ds+γ2∫σ∞‍|b(s)|av(s)Rv1(s)ds}Fω2(σ),
or
(34)F(t0)=F(σ)≤K{∫σ∞‍[|g′(s)|+|b(s)|av(s)Rv1(s)]ds}β2
with K=(6α1+2γ2)β2. Hence,
(35)F(t)≤K{∫t∞‍[|g′(s)|+|b(s)|av(s)Rv1(s)]ds}β2
for all t≥t- such that F(t)≤1. At the same time, (31) implies F(t)≤2-β2<1 for these values of t. Thus, (35) holds for all t≥t- and so limt→∞F(t)=0.
Now in cases (ii) and (iii) we can actually estimate a bound from below onF. Let
(36)Z(t)=F(t)R-β(t).
Then (18) and (36) imply
(37)Z′(t)=(R-1(t))′|y2(t)|δ-δr-1(t)b(t)|y′(t)|q+1.
Define
(38)S(t)={-(R-1(t))-′,incase(ii),-(R-1(t))-′-δb+(t)r(t)a-δ(t),incase(iii).
Suppose case (ii) holds. Then (36) and (37) imply
(39)Z′(t)≥S(t)R(t)Z(t).
Suppose case (iii) holds. Then (36) and (37) imply
(40)Z′(t)≥-(R-1(t))-′|y2(t)|δ-δb+(t)r(t)a-δ(t)|y2(t)|δ≥S(t)R(t)Z(t).
Hence, (39) holds in both cases (ii) and (iii) for t≥t-. So
(41)Z(t)≥Z(t-)exp{∫t-t‍S(σ)R(σ)dσ},
and in view of (35) and the fact that S(t)≤0,
(42)Z(t-)Rβ(t)exp{∫t-t‍S(σ)R(σ)dσ}≤F(t)≤K{∫t∞‍[|g′(s)|+|b(s)|av(s)Rv1(s)]ds}β2,
which contradicts (25). Notice that
(43)exp{-∫t-t‍S(σ)R(σ)dσ}≤Cexp∫0t‍R+′(s)R(s)ds
with C=1 in case (ii) and C=exp{δ∫0∞‍(|b(s)|/a(s))ds} in case (iii).Lemma 4.
Let (16) and (24) hold. Then for every solution of (1), the function F is bounded.Proof.
Lety be a solution of (1) and let t-≥0 be such that (27) holds. Suppose that F is not bounded, that is,
(44)limsupt→∞F(t)=∞.
Then for any t0≥t- with F(t0)≥1, there exist σ and τ such that t0≤σ<τ, F(τ)/2=F(σ)=F(t0), and
(45)1≤F(σ)≤F(t)≤F(τ)forσ≤t≤τ.
Settingτ=σ and t=τ in (20)–(22), we have (29) and
(46)F(τ)2=F(τ)-F(σ)≤[α1|g(σ)|+α1|g(τ)|+α1∫στ‍|g′(s)|ds]Fω(τ)+∫σ∞‍|b(s)|av(s)Rv1(s)ds(γ1Fω2(τ)+δFω1(τ)).
Since max(ω1,ω2,ω3)≤1, (27) and (45) imply
(47)F(τ)≤2[∫t-∞‍|b(s)|av(s)2α1maxs∈[t-,∞)|g(s)|+α1∫t-∞‍|g′(s)|ds+γ2∫t-∞‍|b(s)|av(s)Rv1(s)dsα1maxs∈[t-,∞)|g(s)|+α1∫t-∞‍|g′(s)|d]F(τ)≤F(τ)2.
This contradiction proves that F is bounded.Lemma 5.
Let (16) and (24) hold. Then there exists a solution y of (1) and positive constants c1 and c2 and t0≥0 such that
(48)0<c1≤F(t)≤c2fort∈[t0,∞).Proof.
Assumption (16) implies that g is bounded, so we can choose t0∈ℝ+ such that
(49)M=14[3α1+δ+γ1]-1,|g(t)|≤Mfort≥t0,∫t0∞|g′(s)|ds≤M∫t0∞|b(s)|av(s)Rv1(s)ds≤M.
Consider a solutiony of (1) such that F(t0)=1. First, we will show that
(50)F(t)≤32fort≥t0.
Suppose (50) does not hold. Then there exist t2>t1≥t0 such that
(51)F(t2)=32,F(t1)=1,1<F(t)<32
for t∈(t1,t2). Lemma 2 (with τ=t1 and t=t2), together with the facts that ω≤1 and ω2≤ω1≤1, implies
(52)12=F(t1)2=F(t2)-F(t1)≤3α1MFω(t2)+δMFω1(t2)+γ1MFω2(t2)≤32M[3α1+δ+γ1]=38.
This contradiction shows that (50) holds.
Now, Lemma2 (with t=t, τ=t0) similarly implies
(53)|F(t)-1|≤3α1M(32)ω+δM(32)ω1+γ1M(32)ω2≤32M[3α1+δ+γ1]<12
for t≥t0. From this and from (50) we have
(54)12≤F(t)≤32fort≥t0.
Next, we prove thaty is defined on [0,t0]. Suppose to the contrary that y is defined on (a,t0) with 0≤a<t0 and y cannot be extended to t=a. Then limsupt→a+|y2(t)|=∞. The change of variables x=t0-t and y(t)=Y(x) transforms (1) into
(55)(a(t0-x)|Y˙|p-1Y˙)·-b(t0-x)|Y˙|q-1Y˙+r(t0-x)|Y|λ-1Y=0,“˙”=dds,
with the noncontinuable solution Y(x) defined on [0,t0-a) and limsupx→(t0-a)-|Y˙(x)|=∞. This contradicts Remark 2 applied to (55) and proves that y is defined on ℝ+. The conclusion of the lemma then follows from (54).Lemma 6.
Suppose that (16) and (24) hold and
(56)∫0∞‍R-β(t)dt=∞.
In addition, assume that either
(57)limt→∞Rβ(t)[|(1r(t))′|+|b(t)|av(t)r(t)Rv1(t)]=0
or
(58)limt→∞Rβ(t)[|a′(t)|a(t)r(t)+|b(t)|av(t)r(t)Rv1(t)]=0
holds. If y is a solution of (1) with
(59)c1≤F(t)≤c2
for large t and some positive constants c1 and c2, then
(60)∫0∞‍|y(t)|λ+1dt=∞.
Moreover, if
(61)limsupt→∞r(t)∫0t‍R-β(s)ds=∞,
then
(62)∫0∞‍|y2(t)|δR(t)dt=∞.
That is, y is of the strong nonlinear limit-point type.Proof.
Lety be a nontrivial solution of (1) satisfying (59) on [T,∞)⊂ℝ+. Then, (20) and (59) imply the existence of M1 such that
(63)|y(t)y2(t)|≤γ-1/(λ+1)c2ω≤M1,|y(t)||y2(t)|v≤γ-1/(λ+1)c2ω2Rv1(t)≤M1Rv1(t)
for t≥T.
Lett0≥T be such that
(64)|g(t)|≤c1min(1,γ)(4M1)-1
holds for t≥t0. It follows from (5) that
(65)∫t0t‍|y(s)|λ+1ds=-∫t0t‍y(s)y2′(s)r(s)ds-∫t0t‍b(s)av(s)r(s)y(s)|y2(s)|vsgny2(s)ds=-y(t)y2(t)r(t)+D+∫t0t‍|y2(s)|δR(s)ds+J(t),
where D=y(t0)y2(t0)r-1(t0) and
(66)J(t)=∫t0t‍[(1r(s))′y(s)y2(s)-b(s)av(s)r(s)y(s)|y2(s)|vsgny2(s)(1r(s))′]ds.
From (63) and (66)
(67)|J(t)|≤M1∫t0t‍[|(1r(s))′|+|b(s)|av(s)r(s)Rv1(s)]ds.
Moreover, (18) and (59) imply
(68)c3∫t0t‍R-β(s)ds≤∫t0t‍|y(s)|λ+1ds+∫t0t‍|y2(s)|δR(s)ds
for t≥t0 with c3=c1min(1,γ).
Suppose (57) holds. By l’Hôspital’s Rule, there exists t1>t0 such that
(69)∫t0t‍[|(1r(s))′|+|b(s)|av(s)r(s)Rv1(s)]ds≤c32M1∫t0t‍R-β(s)ds
for t≥t1. From this, (65) and (67), we have
(70)|∫t0t‍|y(s)|λ+1ds-∫t0t‍|y2(s)|δR(s)ds+y(t)y2(t)r(t)|≤|D|+c32∫t0t‍R-β(s)ds
for t≥t1.
Now let (58) hold. Applying l’Hôspital’s Rule, there exists t1>t0 such that
(71)∫t0t‍(|a′(s)|pa(s)r(s)+|b(s)|av(s)r(s)Rv1(s))ds≤c34M1∫t0t‍R-β(s)ds
for t≥t1. Then using (64), (71), and the fact that 1/r=a1/p/R, we have
(72)∫t0t‍[|(1r(s))′|+|b(s)|av(s)r(s)Rv1(s)]ds≤∫t0t‍(|a′(s)|pa(s)r(s)+|b(s)|av(s)r(s)Rv1(s))ds+∫t0t‍|g(s)|Rβ(s)ds≤c32M1∫t0t‍R-β(s)ds
for t≥t1. This together with (65) and (67) implies that inequality (70) again holds. Thus, (70) holds if either (57) or (58) does. Moreover,
(73)-|D|-c32∫t0t‍R-β(s)ds≤∫t0t‍|y(s)|λ+1ds-∫t0t‍|y2(s)|δR(s)ds+y1(t)y2(t)r(t)≤|D|+c32∫t0t‍R-β(s)ds
for t≥t1. Adding (68) and the left-hand inequality in (73) gives
(74)2∫t0t‍|y(s)|λ+1ds+y(t)y2(t)r(t)≥c32∫t0t‍R-β(s)ds-|D|⟶∞
as t→∞, while adding (68) and the right-hand inequality in (73) gives
(75)2∫t0t‍|y2(s)|δR(s)ds-y(t)y2(t)r(t)≥c32∫t0t‍R-β(s)ds-|D|⟶∞
as t→∞.
Ify′ is oscillatory, let {tk}k=1∞→∞ be a sequence of zeros of y′. Then letting t=tk in (74) and (75), we see that y is a strong nonlinear limit-point type solution of (1).
Ify′ is nonoscillatory, then either
(76)y(t)y2(t)>0
or
(77)y(t)y2(t)<0
for large t. First we show (60) holds. Clearly (60) holds if (76) does, so suppose (77) holds. Then y(t)y2(t)r-1(t)<0 for large t and (60) follows from (74).
Finally, assume (61) holds. From (61) and (63), it follows that there is a t2≥t1 such that
(78)|y(t)y2(t)r(t)|≤M1r(t)≤c34∫0t‍R-β(s)ds
for t≥t2; hence, (62) follows from this and from (75).Remark 6.
Lemma6 actually holds for all positive p, λ, and q regardless of their relative size.


## 3. LP/LC Problem for (1)

In this section we present our main results for (1).Theorem 7.
Assume that (16) and (24) hold. Then (1) is of the strong nonlinear limit-circle type if and only if
(79)∫0∞‍R-β(t)dt<∞.Proof.
Suppose (79) holds and let y be any nontrivial solution of (1) defined on ℝ+. Then, by Lemma 4, there is a positive constant c such that 0≤F(t)≤c. Hence, from this and (18),
(80)0≤γ∫0∞‍|y(t)|λ+1dt+∫0∞‍|y2(t)|δR(t)dt=∫0∞‍F(t)R-β(t)dt≤c∫0∞‍R-β(t)dt<∞,
so y is of the strong nonlinear limit-circle type. Thus, (1) is of the strong nonlinear limit-circle type.
Now suppose that (79) does not hold, that is,
(81)∫0∞‍R-β(t)dt=∞.
Let y be a solution of (1) given by Lemma 5. Then there is c1>0 such that c1≤F(t) on [t0,∞)⊂ℝ+. Hence, from this and (18),
(82)γ∫t0∞‍|y(t)|λ+1dt+∫t0∞‍|y2(t)|δR(t)dt=∫t0∞‍F(t)R-β(t)dt≥c1∫t0∞‍R-β(t)dt=∞.
Thus, either
(83)∫0∞‍|y(t)|λ+1dt=∞or∫0∞‍|y2(t)|δR(t)dt=∞,
and so y and (1) are not of the strong nonlinear limit-circle type.Theorem 8.
Let (16), (24), and either (57) or (58) hold. If
(84)∫0∞‍R-β(t)dt=∞,
then (1) is of the nonlinear limit-point type.Proof.
The hypotheses of Lemmas5 and 6 are satisfied; so if y is a solution given by Lemma 5, then (60) holds and the conclusion follows.Theorem 9.
Let conditions (16), (24), and either (57) or (58) hold. Assume, in addition, that either (i) λ=p=q; (ii) b≤0 for large t and (25) holds; or (iii) p=q<λ and (25) holds. If (84) holds, then every nontrivial solution of (1) is of the nonlinear limit-point type. If, moreover, (61) holds, then (1) is of the strong nonlinear limit-point type.Proof.
Note that the hypotheses of Lemmas3–6 are satisfied. Let y be a nontrivial solution of (1) defined on ℝ+. Then by Lemmas 3 and 4, there are positive constants c1 and c2 such that
(85)0<c1≤F(t)≤c2forlarget.
Thus, by Lemma 6, (60) holds; moreover, if (61) holds, then so does (62). The existence of a nontrivial solution of (1) is given by Lemma 5. This proves the theorem.Remark 10.
IfR is nondecreasing for large t, then condition (25) becomes
(86)liminft→∞Rα(t){∫t∞‍[|g′(s)|+|b(s)|av(s)Rv1(s)]ds}β2=0.
If R is nonincreasing for large t, then (25) becomes
(87)liminft→∞R-β{∫t∞‍[|g′(s)|+|b(s)|av(s)Rv1(s)]ds}β2=0.


## 4. LP/LC Problem for (2)

One of the main assumptions is Section3 is condition (24), which takes the form
(88)∫0∞‍|b(t)|a(t)dt<∞
for (2). It is possible to remove this condition when studying (2). The technique to accomplish this is contained in the following lemma; a direct computation proves it (or see [3, Lemma 7]).Lemma 11.
Equation (2) and the equation
(89)(a-(t)|y′|p-1y′)′+r-(t)|y|λ-1y=0
are equivalent where
(90)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 (2) is a solution of (89) and vice versa.Based on this lemma, we can obtain results for (2) by combining Lemma 11 with known results for (89), such as those in [10–14, 16, 18]. Here we only present a sample of the many possibilities.Set(91)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}.Theorem 12.
(i) Letλ>p and
(92)∫0∞‍R--β(t)(∫0t‍|g-′(s)|ds+1)(λ+1)(p+1)/(λ-p)dt<∞.
Then (2) is of the nonlinear limit-circle type.
(ii) Assume that(93)limt→∞g-(t)=0,∫0∞‍|g-′(t)|dt<∞.
Then (2) is of the strong nonlinear limit-circle type if and only if
(94)∫0∞‍R--β(t)dt<∞.Proof.
(
i
) If g- is not identically a constant on ℝ+ the result follows from Lemma 11 and from [13, Theorem 1] applied to (89). If g-≡const.on ℝ+, then the statement follows from Theorem 7 applied to (89) and Lemma 11. Part (ii) follows from Lemma 11 and Theorem 7 applied to (89).The next result follows from Lemma11 and Theorem 8 applied to (89).Theorem 13.
Let (93) hold and assume that either
(95)limt→∞Rβ(t)(|r′(t)|r2(t)+|b(t)|a(t)r(t))×exp{λ-p(λ+2)p+1∫0t‍b(s)a(s)ds}=0
or
(96)limt→∞Rβ(t)|a′(t)|+|b(t)|a(t)r(t)×exp{λ-p(λ+2)p+1∫0t‍b(s)a(s)ds}=0.
If
(97)∫0∞‍(R-(t))-βdt=∞,
then (2) is of the nonlinear limit-point type.Our final theorem is a strong nonlinear limit-point result for (2).Theorem 14.
Assume that (93), (97), and either (95) or (96) hold. In addition, assume that either (i)   λ=p; (ii)   b≤0 on ℝ+ and
(98)liminft→∞(R-(t))-β{∫t∞‍|g-′(s)|ds}(λ+1)(p+1)/(λ-p)×exp{R-+′(s)R-(s)ds}=0;
or (iii)   λ>p and (98) holds. Then every nontrivial solution of (2) is of the nonlinear limit-point type. If, moreover,
(99)limsupt→∞r-(t)∫0t‍(R-(s))-βds=∞,
then (2) is of the strong nonlinear limit-point type.Proof.
This follows from Lemma11 and Theorem 9 applied to (89).


## 5. Examples

In this section we present some examples to illustrate our results.Example 1.
Consider the equation(100)y′′+tsy′+tσ|y|λ-1y=0,t≥1,
with s∈ℝ and λ≥1. We have the following results.(i)
Ifs<-1 and σ>(λ+3)/(λ+1), then (100) is of the strong nonlinear limit-circle type by Theorem 7.(ii)
Ifs<-1 and -(λ+3)/2<σ≤(λ+3)/(λ+1), then (100) is of the strong nonlinear limit-point type by Theorem 8.(iii)
Ifλ=1, s<-1, and -2<σ≤2, then (100) is of the strong nonlinear limit-point type by Theorem 9(i).(iv)
Ifλ>1, s<-1-σ(λ-1)/(λ+3)(λ+1), and 0<σ≤(λ+3)/(λ+1), then (100) is of the strong nonlinear limit-point type by Theorem 9(iii).The following example will allow us to compare our results to those in [2].Example 2.
Again consider (100) with s∈ℝ and λ≥1. The following results hold.(i)
Equation (100) is of the nonlinear limit-circle type if
(a)
s<-1 and σ>(λ+3)/(λ+1) (by Theorem 12(ii));(b)
s=-1 and σ>-(λ-1)/(λ+1) (by Theorem 12(i) if λ>1 and by Theorem 12(ii) if λ=1);(c)
s>-1, λ>1, and σ>(λ+3)/(λ+1) (by Theorem 12(i));(d)
s>-1, λ=1 and s<σ/2 (by Theorem 12(ii)).(ii)
Equation (100) is of the nonlinear limit-point type if
(e)
s<-1 and -(λ+3)/(λ+1)<σ≤(λ+3)/(λ+1) (by Theorem 13);(f)
s=-1 and -2<σ≤-(λ-1)/(λ+1) (by Theorem 13).By [2, Corollary 2.1], (100) is of the nonlinear limit-circle type if either (i) λ=1, σ>2, and s<0, or (ii) λ>1, (λ+3)/(λ+1)<σ<2(λ+3)/(λ-1), and s<-σ((λ-1)/2(λ+3)). This shows that, in the case of nonlinear limit-circle type results, the results in [2] are a special case of those in this paper.Remark 15.
Theorem  2.3 in [2] appears to show that (100) is of the nonlinear limit-point type if -1<σ≤(λ+3)/(λ+1), s≤-σ((λ-1)/2(λ+3)), and s<(σ-1)/2. We have a contradiction to our case (b) in Example 2 above if s=-1, λ=1, and σ=1. The proof of Theorem  2.3 in [2] is incorrect; in their expression for V˙ the term “-p(t)[a(t)r(t)]β-2αx2k(s)” is missing.In our next example we have thatb(t) in (1) (or (2)) is negative.Example 3.
Consider the equation(101)y′′-tsy′+tσ|y|λ-1y=0,t≥1,
with s∈ℝ, λ≥1. Calculations show the following.(i)
Equation (101) is of the nonlinear limit-circle type if
(a)
s<-1 and σ>(λ+3)/(λ+1) (by Theorem 7);(b)
s=-1 and σ>-(λ-1)/(λ+1) (by Theorem 12(ii)).(ii)
Equation (101) is of the nonlinear limit-point type if
(c)
s<-1 and -(λ+3)/2<σ≤(λ+3)/(λ+1) (by Theorem 8);(d)
s=-1 and -2(λ+1)/(λ+3)<σ≤-(λ-1)/(λ+1) (by Theorem 13);(e)
s>-1 and λ>1 (by Theorem 13);(f)
s>-1, λ=1, and σ>max{2s,s} (by Theorem 13).



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