# Limit Circle/Limit Point Criteria for Second-Order Superlinear Differential Equations with a Damping Term

Authors: Lihong Xing; Wei Song; Zhengqiang Zhang; Qiyi Xu
Journal: Journal of Applied Mathematics
 (2012)

---

## Abstract

The purpose of the present paper is to establish some new criteria for the classifications of superlinear differential equations as being of the nonlinear limit circle type or of the nonlinear limit point type. The criteria presented here generalize some known results in literature.


---

## Body

## 1. Introduction

In 1910, Weyl [1] published his now classic paper on eigenvalue problems for second-order linear differential equations of the form (1.1)(a(t)y′)′+r(t)y=θy,θ∈C.
He classified this equation to be of the limit circle type if each solution y(t) is square integrable (belongs to L2), that is,(1.2)∫0∞y2(t)dt<∞,
and to be of the limit point type if at least one solution y(t) does not belong to L2, that is,(1.3)∫0∞y2(t)dt=∞.
He showed that the linear equation (1.1) always has at least one square integrable solution if Im θ≠0. Thus, for second-order linear equations with Im θ≠0, the problem reduces to whether (1.1) has one (limit point type) or two (limit circle type) square integrable solutions. This is known as the Weyl Alternative. Weyl also proved that if (1.1) is of the limit circle type for some θ0∈C, then it is of the limit circle type for all θ∈C. In particular, this is true for θ=0; that is, if we can show that(1.4)(a(t)y′)′+r(t)y=0
is of limit circle type, then (1.1) is of the limit circle type for all values of θ. There is considerable interest in this problem over the years. The classification is important in spectral theory for linear equation (1.1) since it characterizes the number boundary conditions for differential operator generated by (1.1) being a self-adjoint operator. The analogous problem for nonlinear equations is relatively new and not as extensively studied as the linear cases. For a survey of known results on 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 readers to [2–9] and the recent monograph [10]. In this paper, we will discuss the equation with a damping term:(1.5)(a(t)y′)′+p(t)y′+r(t)y2k-1=0,
where a,r:R+↦R and p:R+↦R+ are continuous, a′,r′∈ACloc(R+), a′′,r′′∈Lloc2(R+), a(t)>0, r(t)>0, and k>0 is a positive integer. When p(t)≡0, then (1.5) turns into(1.6)(a(t)y′)′+r(t)y2k-1=0,
which is widely researched by many authors (see [10] and references cited therein).Definition 1.1 (see [2]).
A nontrivial solutiony(t) of (1.5) is said to be of the nonlinear limit circle type if
(1.7)∫0∞y2k(t)dt<∞,
and it is of the nonlinear limit point type otherwise, that is,
(1.8)∫0∞y2k(t)dt=∞.
Equation (1.5) is said to be of the nonlinear limit circle type if all its solutions satisfy (1.7), and it is said to be of the nonlinear limit point type if there is at least one nontrivial solution satisfying (1.8).In this paper, we will give necessary and sufficient conditions to guarantee the nonlinear limit circle type or nonlinear point type for (1.5).


## 2. Main Results

To simplify notations, letα=1/2(k+1), andβ=(2k+1)/2(k+1). We make the transformation(2.1)s=∫0t[rα(u)aβ(u)]du,x(s)=y(t).
Then (1.5) becomes(2.2)ẍ+A(t)ẋ+B(t)x2k-1=0,
where A(t)=p(t)/aα(t)rα(t)+α((a(t)r(t))′/aα(t)rα+1(t)), and B(t)=(a(t)r(t))β-α.Moreover, (2.2) can be rewritten as the system(2.3)ẋ=z-A(t)x,ż=Ȧ(t)x-B(t)x2k-1.
We are now ready to prove the first result for system (2.3).Theorem 2.1.
Assume that(2.4)∫0∞A′(u)B1/2(u)du<∞,(2.5)∫0∞A′(u)B1/2(u)du<∞,(2.6)∫0∞1B(u)du<∞.
Then (1.5) is of the nonlinear limit circle type; that is, any solution y(t) of (1.5) satisfies
(2.7)∫0∞y2k(t)dt<∞.Proof.
Define(2.8)V(x,z,s)=z22+B(t)x2k2k+∫0tp(ξ)rβ-α(ξ)a2α(ξ)y2k(ξ)dξ.
Then we have
(2.9)V̇=zż+[B(t)x2k2k]+p(t)[a(t)r(t)]β-2αy2k(t)=z[Ȧ(t)x-B(t)x2k-1]+B(t)x2k-1ẋ+Ḃ(t)x2k2k+p(t)[a(t)r(t)]β-2αy2k(t)=Ȧ(t)xz-B(t)x2k-1z+B(t)x2k-1[z-A(t)x]+Ḃ(t)x2k2k+p(t)[a(t)r(t)]β-2αy2k(t)=Ȧ(t)xz-A(t)B(t)x2k+Ḃ(t)x2k2k+p(t)[a(t)r(t)]β-2αy2k(t)=Ȧ(t)xz+[Ḃ(t)2k-A(t)B(t)]x2k+p(t)[a(t)r(t)]β-2αy2k(t)=Ȧ(t)xz-p(t)[a(t)r(t)]β-2αx2k(s)+p(t)[a(t)r(t)]β-2αy2k(t)=Ȧ(t)xz.
Since |xz|=|B1/2(t)xz/B1/2(t)|≤(B(t)x2/2+z2/2)/B1/2(t)≤(B(t)[x2/2k+K1]+z2/2)/B1/2(t)≤V(s)/B1/2(t)+K1B1/2(t) for some constant K1≥0, we have
(2.10)V̇(s)≤|Ȧ(t)|V(s)B1/2(t)+K1|Ȧ(t)|B1/2(t).
Since
(2.11)Ȧ(t)=A′(t)dtds=A′(t)aβ(t)rα(t),
letting τ(s) denote the inverse function of s(t), we obtain
(2.12)∫0s|Ȧ(τ(v))|B1/2(τ(v))dv=∫0tA′(u)B1/2(u)du.
Moreover,
(2.13)∫0s|Ȧ(τ(v))|B1/2(τ(v))dv=∫0tA′(u)B1/2(u)du.
Integrating (2.10) from 0 to s, we obtain
(2.14)V(s)≤V(0)+∫0s|Ȧ(τ(v))|B1/2(τ(v))V(v)dv+K1∫0s|Ȧ(τ(v))|B1/2(τ(v))dv.
Condition (2.5) implies that the second integral on the right-hand side of (2.10) is convergent. By Gronwall’s inequality, we have
(2.15)V(s)≤M1exp⁡∫0s|Ȧ(τ(v))|B1/2(τ(v))dv,
for some constant M1>0. By condition (2.4), the aforementioned integral is convergent, and we have that V(s) is bounded, say, V(s)≤M2 for some M2>0. Therefore,
(2.16)B(t)y2k(t)=B(t)x2k(s)≤2kM2,
from which it follows that
(2.17)∫0∞y2k(u)du≤2kM2∫0∞1B(u)du<∞
by condition (2.6), so all solutions of (1.5) are of the nonlinear limit circle type, and this completes the proof of Theorem 2.1.Ifa(t)≡1 in (1.5), then it turns into(2.18)y′′+p(t)y′+r(t)y2k-1=0,
and we have the following corollary.Corollary 2.2.
Assume that(2.19)∫0∞|[p(u)/rα(u)+αr′(u)/r1+α(u)]′|r(β-α)/2(u)du<∞,∫0∞|[p(u)rα(u)+αr′(u)r1+α(u)]′|r(β-α)/2(u)du<∞,∫0∞1rβ-α(u)du<∞.
Then (2.18) is of the nonlinear limit circle type.The aforementioned theorem and corollary offer sufficient conditions to guarantee (1.5) and (2.18) to be of the nonlinear limit circle type, respectively. The next theorem gives necessary conditions to guarantee that (1.5) is of the nonlinear limit circle type.Lemma 2.3 (see [10]).
Assume that there existsN1>0 such that
(2.20)|[a(t)r(t)]′a1/2(t)r3/2(t)|≤N1,(2.21)∫0∞[(a(t)r(t))′]2a(t)r3(t)dt<∞.
If y(t) is a nonlinear limit circle type solution of (1.5), then
(2.22)∫0∞[(a(t)r(t))′]2y2(t)a(t)r3(t)dt<∞.Proof.
We have(2.23)∫0∞[(a(t)r(t))′]2y2(t)a(t)r3(t)dt≤N12∫0∞y2k(t)dt+∫0∞[(a(t)r(t))′]2a(t)r3(t)dt<∞
by (1.7) and (2.21).Theorem 2.4.
Suppose that there exist constantsN2>0 and  M3>0 such that
(2.24)|a1/2(t)r′(t)r3/2(t)|≤N2,(2.25)∫0∞a(u)[r′(u)]2r3(u)du<∞,(2.26)p2(t)a(t)r(t)≤M3,(2.27)∫0∞p2(t)a(t)r(t)dt<∞.
If y(t) is a nonlinear limit circle type solution of (1.5), then
(2.28)∫0∞a(u)[y′(u)]2r(u)<∞.Proof.
If we multiply (1.5) by y(t)/r(t), use the identity
(2.29)(a(t)y′)′=(a(t)y′y)′-a(t)[y′]2,
and to integrate by parts, we obtain
(2.30)a(t)y′(t)y(t)r(t)-a(t1)y′(t1)y(t1)r(t1)+∫t1ta(u)y′(u)y(u)r′(u)r2(u)du+∫t1ty2k(u)du+∫t1tp(u)y′(u)y(u)r(u)du-∫t1ta(u)[y′(u)]2r(u)du=0
for any t1≥0. Now conditions (2.26), (2.27), and (1.7) imply
(2.31)∫t1∞p2(u)[y2k(u)+1]a(u)r(u)du≤M3∫t1∞y2k(u)du+∫t1∞p2(u)a(u)r(u)du<∞.
Therefore, there exists a constant k0>0,  subject to
(2.32)∫t1tp2(u)[y2k(u)+1]a(u)r(u)du≤M3∫t1ty2k(u)du+∫t1tp2(u)a(u)r(u)du<k02.
Applying Schwartz inequality, we obtain
(2.33)∫t1tp(u)y′(u)y(u)r(u)du≤(∫t1ta(u)[y′(u)]2r(u)du)1/2(∫t1tp2(u)[y2k(u)+1]a(u)r(u)du)1/2≤k0(∫t1ta(u)[y′(u)]2r(u)du)1/2.
By the Schwartz inequality,
(2.34)∫t1ta(u)y′(u)y(u)r′(u)r2(u)du≤[∫t1ta(u)[y′(u)]2r(u)du]1/2[∫t1ta(u)y2(u)[r′(u)]2r3(u)du]1/2.
Now (2.24) implies
(2.35)a(t)y2(t)[r′(t)]2r3(t)≤a(t)[r′(t)]2r3(t)[y2k(t)+1]≤N22y2k(t)+a(t)[r′(t)]2r3(t).
So integrating the previous inequality and applying (2.25) and (1.7), we obtain
(2.36)∫t1∞a(u)y2(u)[r′(u)]2r3(u)du≤K22<∞
for some constant K2>0.
Ify(t) is not eventually monotonic, let {tj}→∞ be an increasing sequence of zeros of y′(t). Then from (2.30), we have(2.37)(k0+K2)H1/2(tj)+K3≥H(tj),
where
(2.38)H(t)=∫t1ta(u)[y′(u)]2r(u)du,
and K3>0 is a constant. It follows that H(tj)≤K4<∞ for all j and for some constant K4>0, so (2.28) holds.
Ify(t) is eventually monotonic, then y(t)y′(t)≤0 for all t≥t1 for sufficiently large t1≥0 since otherwise (1.7) would be violated. Using this fact in (2.30), we can repeat the type of argument used previously to obtain that (2.28) holds.The following theorem gives sufficient conditions to ensure that (1.5) is of the nonlinear limit point type.Theorem 2.5.
Suppose that conditions (2.4), (2.5), and (2.20)–(2.27) hold. If(2.39)∫0∞1B(u)du=∞,
then (1.5) is of the nonlinear limit point type.Proof.
As in the proof of Theorem2.1, define
(2.40)V(x,z,s)=z22+B(t)x2k2k
and differentiate it to obtain
(2.41)V̇≥-|Ȧ(t)|V(s)B1/2(t)-K1|Ȧ(t)|B1/2(t),
and so we then have
(2.42)V̇+|Ȧ(t)|V(s)B1/2(t)≥-K1|Ȧ(t)|B1/2(t).
If we define functions H and h:R+↦R by H(t)=|Ȧ(t)|/B1/2(t), and h(t)=K1|Ȧ(t)|B1/2(t), then we have
(2.43)dds(exp⁡∫0sH(τ(ξ))dξ)≥-h(t)exp⁡∫0sH(τ(ξ))dξ.
Now condition (2.4) guarantees that
(2.44)exp⁡∫0∞H(τ(ξ))dξ≤K5<∞,
for some constant K5>0, while condition (2.5) implies that
(2.45)K5∫0∞h(τ(ξ))dξ≤K6<∞,
for some K6>0. Let y(t) be any solution of (1.5) such that V(x(0),z(0),0)>K6+1. Integrating (2.43), we get
(2.46)V(s)exp⁡∫0sH(τ(ξ))dξ≥V(0)-K6>1,
and so
(2.47)V(s)≥1K5
for s≥0. Dividing both sides of this last inequality by B(t) and rewriting the left-hand side in terms of t, we have
(2.48)a(t)[y′(t)]22r(t)+y2k(t)2k+[p(t)r(t)+α(a(t)r(t))′r2(t)]y(t)y′(t)+[p2(t)2a(t)r(t)+αp(t)(a(t)r(t))′a(t)r2(t)+α2[(a(t)r(t))′]22a(t)r3(t)]y2(t)≥1K5[a(t)r(t)]β-α.
If y(t) is a limit circle type solution of (1.5), then Theorem 2.4 implies
(2.49)∫0∞a(t)[y′(t)]2r(t)dt<∞,
and Lemma 2.3 implies
(2.50)∫0∞[(a(t)r(t))′]2y2(t)a(t)r3(t)dt<∞.
By the Schwartz inequality,
(2.51)|∫0∞(a(t)r(t))′r2(t)y(t)y′(t)dt|≤[∫0∞[(a(t)r(t))′]2y2(t)a(t)r3(t)dt]1/2×[∫0∞a(t)[y′(t)]2r(t)dt]1/2<∞.
From condition (2.27), using Schwartz inequality, we get
(2.52)∫0∞p(t)(a(t)r(t))′a(t)r2(t)dt≤[∫0∞[(a(t)r(t))′]2y2(t)a(t)r3(t)dt]1/2×[∫0∞p2(t)a(t)r(t)dt]1/2<∞.
So that if y(t) is a limit circle solution of (1.5), from conditions (2.26) and (2.27), we obtain
(2.53)∫0∞p2(t)y2(t)2a(t)r(t)dt<∫0∞p2(t)[y2k(t)+1]2a(t)r(t)dt<∞,∫0∞αp(t)(a(t)r(t))′y2(t)a(t)r2(t)dt≤α(∫0∞p2(t)y2(t)a(t)r(t)dt)1/2(∫0∞[(a(t)r(t))′]2y2(t)a(t)r3(t)dt)1/2<∞.
Furthermore,
(2.54)∫0∞p(t)y′(t)y(t)r(t)dt≤(∫0∞a(t)[y′(t)]2r(t)dt)1/2(∫0∞p2(t)[y2(t)]a(t)r(t)dt)1/2<∞.
Consequently, integrating both sides of (2.48), we see that the integrand of the left side of (2.48) is bounded, but the integrand of the right side of (2.48) tends to infinity according to condition (2.39). This leads to a contradiction, so y(t) is a limit point type solution of (1.5), and (1.5) is of the nonlinear limit point type.The last theorem and corollary give the sufficient and necessary conditions to guarantee (1.5) and (2.18) to be of the nonlinear limit circle type, respectively.Theorem 2.6.
Assume that conditions (2.4), (2.5), (2.20), (2.21), (2.24), (2.25), (2.26), and (2.27) hold. Then (1.5) is of the nonlinear limit circle type if and only if
(2.55)∫0∞1(a(t)r(t))k/(k+1)dt<∞.When one specializes this theorem to (2.18), that is, a(t)≡1 in (1.5), one obtains the following corollary.Corollary 2.7.
Assume that(2.56)∫0∞[p(t)/rα(t)+αr′(t)/rα+1(t)]′r(β-α)/2(t)dt<∞,∫0∞[p(t)rα(t)+αr′(t)rα+1(t)]′r(β-α)/2(t)dt<∞,|r′(t)|r3/2(t)≤N1,∫0∞[r′(t)]2r3(t)dt<∞,p2(t)r(t)≤M3,∫0∞p2(t)r(t)dt<∞.
Then (2.18) is of the nonlinear limit circle type if and only if
(2.57)∫0∞1r(t)k/(k+1)dt<∞.



---

*Source: 361961-2012-04-17.xml*