AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 979138 10.1155/2012/979138 979138 Research Article Limit-Point/Limit-Circle Results for Equations with Damping Bartušek M. 1 Graef John R. 2 Braverman Elena 1 Faculty of Science Masaryk University Brno Kotlářská 2 611 37 Brno Czech Republic muni.cz 2 Department of Mathematics University of Tennessee at Chattanooga Chattanooga TN 37403 USA tennessee.edu 2012 23 12 2012 2012 27 09 2012 12 12 2012 2012 Copyright © 2012 M. Bartušek and John R. Graef. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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<λpq, a(t)>0, and r(t)>0. Some examples are given to illustrate the main results.

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 +=[0,) and =(-,) and assume throughout that 0<λpq, aC1(+), bC0(+), a1/prC1(+), 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 functions a, b, and r are smooth enough so that all nontrivial solutions of (1.1) defined on + are nontrivial in any neighborhood of (see Theorem 13(i) in ). Moreover, if either q=p or b(t)0 on +, then all nontrivial solutions of (1.1) are defined on +.

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  as well as the papers ).

Definition 1.2.

A solution y of (1.1) defined on + 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 + 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  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.  as well as the recent papers of Bartušek and Graef [4, 6, 911].

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  and , respectively, and further studied, for example, in [4, 6]. We define the function 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 solution y of (1.1) defined on + 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 + is of the strong nonlinear limit-point type and there is at least one nontrivial solution defined on +.

Definition 1.4.

A solution y of (1.1) defined on + 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 + 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 + must satisfy (SNLP) and there must be at least one such nontrivial solution.

If b(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 . Using terminology introduced by the authors in , 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 λp, 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 λ1 is the ratio of odd positive integers and λ1 is an odd integer, respectively. The results in both of these papers tend to be modifications of results in  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=α  γ-1/(λ+1),β2=(λ+1)(p+1)p-λ+(v-1)p(λ+1)for  either  p>λ  or  v>1,ω1=vp+1p+1,v1=β1(v-1),ω=1λ+1+pp+1,v=qp1. Notice that α=1-β,  ω2-1=1/β2,  ω2ω1, and ω1. We define the function g:+ 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)limtg(t)=0,0|g(s)|ds<. If (H) holds, we define the constants (1.12)γ1=α  γ-1/(λ+1)sups+|g(s)|,γ2=δ+γ1. For any solution y:+ 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 F0 on + for every solution of (1.1).

For any continuous function h:+, 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 solution y of (1.1) defined on +, F(t)>0 for t0.

Proof.

Suppose, to the contrary, that (1.1) has a nontrivial solution y such that F(t0)=0 for a number t0+. 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)for  t[0,t0],y-(t)=0for  tt0. But this contradicts Remark 1.1 and proves the lemma.

Lemma 2.2.

Let y be a solution of (1.1). Then:

for t+, 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).

for 0τ<t, we have (2.3)F(t)=F(τ)-αg(τ)y(τ)y2(τ)+αg(t)y(t)y2(t)-ατtg(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|δτt|b(s)|av(s)Rv1(s)Fω1(s)ds+γ1τt|b(s)|av(s)Rv1(s)Fω2(s)ds.

Proof.

Let y 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  applied to (2.5). Relation (2.4) follows from (2.2).

The following two lemmas give us sufficient conditions for the boundedness of F 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 +, the function F is bounded from bellow on + 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)liminftF(t)=0. By Lemma 2.1, F(t)>0 on +. Let t-+ be such that (2.8)2α1sups[t-,)|g(s)|+α1t-|g(s)|ds+(γ1+δ)t-|b(s)|av(s)Rv1(s)ds12, the existence of such a t- follows from (H) and (2.6). Then, for any t0t- such that F(t0)1, there exist τ and σ such that t0σ<τ and (2.9)2F(τ)=F(σ)=F(t0)>0,F(τ)F(t)F(σ) for σtτ. Then (2.2) implies (2.10)|y(t)y2(t)|γ-1/(λ+1)Fω(t) on +. 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)|+α1t-|g(s)|ds]Fω(σ)+γ1t-|b(s)|av(s)Rv1(s)dsFω2(σ)+δt-|b(s)|av(s)Rv1(s)dsFω1(σ). Hence, using (2.8) and the facts that ω2ω11,  ω1, and F(σ)1, we obtain (2.12)F(σ)12F(σ).

This contradiction to F(σ)>0 proves the lemma.

Lemma 2.4.

Assume that b0 for large t, (H) holds, (2.13)0b(t)av(t)Rv1(t)dt  <, and either (i) λ=p=q, or (ii) q>λ and (2.14)liminftRβ(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 +.

Proof.

Let y be a nontrivial solution of (1.1). Then according to Remark 1.1 and Lemma 2.1, y is defined on + and F(t)>0 on +. In view of (H) and (2.13), we can choose t-+ such that (2.15)t-[|g(s)|+b(s)av(s)Rv1(s)]ds12[3α1+2ω1δ+2ω2γ1]-1. Suppose that F is not bounded, that is, (2.16)limsuptF(t)=. Then, for any t0t- with F(t0)1, there exist σ and τ such that t0σ<τ,  (1/2)F(τ)=F(σ)=F(t0), and (2.17)1F(σ)F(t)F(τ)for  σtτ. Since g is of bounded variation and limtg(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)|+α1στ|g(s)|dsF(τ)-F(σ)+γ1στb(s)av(s)Rv1(s)ds]Fω2(τ)+δσb(s)av(s)Rv1(s)dsFω1(τ). From this, (2.8), (2.15), and (2.18), we obtain (2.20)F(σ)[3α1σ|g(s)|ds+(2ω2γ1+2ω1δ)σb(s)av(s)Rv1(s)ds]Fω2(σ)Kσ[|g(s)|+b(s)av(s)Rv1(s)]ds  Fω2(σ)12Fω2(σ), where K=3α1+2ω1δ+2ω2γ1.

If λ=p=q, then ω2=1 and (2.20) gives us a contradiction.

Now let q>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 tt- 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 tt-. 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)|δR-(t)R(t)R-β(t)F(t)=R-(t)R(t)  z(t) for tt-. So, (2.24)z(t)z(t-)exp0tR-(s)R(s)ds. From this and (2.22), (2.25)K1{t[|g(s)|+b(s)av(s)Rv1(s)]ds}-β2F(t)=Rβ(t)z(t)z(t-)Rβ(t)exp0tR-(s)R(s)ds, which contradicts (2.14). Hence, F is bounded from above on +. Since F>0 on +, the conclusion follows.

Lemma 2.5.

Let (H) and (2.6) hold. Then there exists a solution y of (1.1) defined on +, a constant c0>0, and t0+ such that (2.26)0<34c0F(t)32c0for  tt0. Moreover, c0 can be chosen arbitrary small.

Proof.

Condition (H) implies that g is bounded, so we can choose M>0,  t0+, and c0 such that (2.27)|g(t)|Mfor  tt0,t0|g(s)|dsM,t0|b(s)|av(s)Rv1(s)dsM,M14(32)-ω2[3αγ-1/(λ+1)+δ+γ1]-1,0<c023.

Consider a solution y of (1.1) such that F(t0)=c0. First, we will show that (2.28)F(t)32c01for  tt0. Suppose (2.28) does not hold. Then there exist t2>t1t0 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 ωω2,  ω1ω2, 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)ω2c0ω14c0ω. Hence, c0ω-12 which contradicts the choice of c0, and so (2.28) holds.

Now, Lemma 2.2 (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)0R-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)c1F(t)c2 on + 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.

Let y be a nontrivial solution of (1.1) satisfying (2.35). Then in view of (1.13), (2.32), and (2.35) (2.38)γ0t|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 t0 so there exists t00 such that (2.40)|g(t)|c12M1max(1,γ) for tt0. 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)|M1D10t[|(1r(s))|+|b(s)|av(s)r(s)Rβ1(v-1)(s)]ds with D1=1+c2(q-p)/(p+1) for tt0. Moreover, (1.13), (2.35), and (2.38) imply (2.44)c3t0tR-β(s)dst0t|y(s)|λ+1ds+t0t|y2(s)|δR(s)dsc4t0tR-β(s)ds for tt0 with c3=c1/max(1,γ) and c4=c2/min(1,γ).

If (2.33) holds, then (2.43) implies J is bounded on +, 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)+mt0tR-β(s)ds for tt0 with J1(t)=|J(t)|+(|y(0)y2(0)|/r(0)) and m=0. Note, that J1 is bounded on +.

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]dst0t(|(a1/pR(s))|+|b(s)|av(s)r(s)Rβ1(v-1)(s)c2(q-p)/(p+1))ds  c50t(|a(s)|pa(s)r(s)+|b(s)|av(s)r(s)Rβ1(v-1)(s))ds+t0t|g(s)|Rβ(s)ds  M2+c32M1t0tR-β(s)ds for tt0, where M2=c50((|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 tt0. Moreover, (2.47)-J1(t)-c32t0tR-β(s)dst0t|y(s)|λ+1ds-t0t|y2(s)|δR(s)ds+y(t)  y2(t)r(t)J1(t)+c32t0tR-β(s)ds for tt0. Adding the left hand inequalities in (2.44) and (2.47) gives (2.48)2t0t|y(s)|λ+1ds+y(t)y2(t)r(t)c32t0tR-β(s)ds-J1(t) as t, and subtracting gives (2.49)2t0t|y2(s)|δR(s)ds-y(t)y2(t)r(t)c32t0tR-β(s)ds-J1(t) as t.

If y' 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.

Let y' be nonoscillatory. Then either (2.50)y(t)y2(t)>0for  large  t     or (2.51)y(t)y2(t)<0for  large  t.     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)limty(t)  y2(t)r(t)=-. In view of (2.39) and (2.51),  y(t)  y2(t)-M1, so limtr(t)=0. This contradicts the assumptions of the lemma and completes the proof.

3. LP/LC Problem for (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>)

In this section we present our main results for (1.1) and give some examples to illustrate them.

Theorem 3.1.

Let b0 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)0R-β(t)dt<.

Proof.

Let y be a nontrivial solution of (1.1). By Remark 1.1, y is defined on +. The hypotheses of Lemmas 2.3 and 2.4 are satisfied, so there are constants c and c1 such that (3.2)0<cF(t)c1 on +. Hence, from this and (1.13), (3.3)c0R-β(t)dtγ0|y(t)|λ+1dt+0|y2(t)δ|R(t)dt=0F(t)R-β(t)dtc10R-β(t)dt. The conclusion of the theorem then follows from (3.1).

In case b(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)0R-β(t)dt=, then (1.1) is of the nonlinear limit-point type.

Proof.

The hypotheses of Lemmas 2.5 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.

Let b0 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)0R-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 Lemmas 2.3, 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 +, and by Lemmas 2.3 and 2.4, there are positive constants C1 and C2 such that (3.6)0<C1F(t)C2on  +. 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 Lemmas 2.1, 2.2, and 2.6 are valid without the assumption that λp.

4. LP/LC Problem for (<xref ref-type="disp-formula" rid="EEq2">1.2</xref>)

One of the main assumptions in Section 3 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{δ0tb(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+10tb(s)a(s)ds},ω3=(λ+1)(p+1)p-λfor  p>λ,ω3=for  p=λ.

Theorem 4.2.

Assume that (4.5)limtg-(t)  =0,0|g-(s)|ds<, and either (i) λ=p or (ii) λ<p and (4.6)liminftR-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)0R--β(σ)dσ<.

Proof.

The conclusion follows from Theorem  2.11 in  applied to (4.2) and Lemma 4.1.

Our next result follows from Theorem 3.2 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)0R--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)limta(t)+b(t)a1-β/p(t)r(t)exp{1ω30tb(s)a(s)ds}=0. If 0R--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  and Lemma 4.1 above. Note that Theorem  2.16 in  is proved for r(t)r0>0 for t+, but it is easy to see from (2.34) in  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,t1. Assume that σα>-1 and βσ1. If b satisfies either (4.12)σ0,1tσv1|b(t)|dt<, or (4.13)σ<0,1tσ(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,t1. Then we have 0<λ1q, so if (4.15)-λ+32<σλ+3λ+1 and either (4.16)σ0,1tσ(q-1)/(λ+3)|b(t)|dt< or (4.17)σ<0,1t((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,t1, with b(t)0. Note that here p=q=λ=1. Assume that (4.19)1b(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<σ2. It is worth noting that this agrees with the well-known limit-circle criteria (4.20)0r-1/2(t)dt< of Dunford and Schwartz [19, page 1414] (also see the discussion in ).

For our next example we consider the case where p=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,t1, 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,t1, with s, σ0, and 0<λ1. Calculations show the following.

Equation (4.22) is of the nonlinear limit-circle type if

λ=1 and s>-1 (by Theorem 4.2);

λ<1 and -1<s<σ/2 (by Theorem 4.2);

s=-1 and σ>(1-λ)/(λ+1) (by Theorem 4.2);

λ=1, s<-1, and σ>(λ+3)/(λ+1) (by Theorem 3.1);

λ<1, σ>(λ+3)/(λ+1), and s<-1-σ((1-λ)/(λ+3)) (by Theorem 3.1).

Equation (4.22) is of the nonlinear limit-point type if

s=-1 and σ(1-λ)/(λ+1) (by Theorem 4.3);

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  in the case of nonlinear limit-circle type results and are new in the case of nonlinear limit-point results. The results in  follow from ours if s-1 and for s<-1 and λ=1. There are errors in the proofs of the results in .

More specifically, for (4.22) with λ=1, that is, (4.23)y′′+tsy+tσy=0,t1, the results in [17, 18] show that (4.23) is of the nonlinear limit-circle type if s0 and σ>2. and by results in the present paper (4.23) is of the nonlinear limit-point type if and only if

s<-1 and σ>2;

s=-1 and σ>0;

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.

Acknowledgment

This research is supported by Grant no. 201/11/0768 of the Grant Agency of the Czech Republic.

Bartušek M. Pekárková E. On existence of proper solutions of quasilinear second order differential equations Electronic Journal of Qualitative Theory of Differential Equations 2007 2007 1 1 14 MR2295683 ZBL1115.34032 Bartušek M. Došlá Z. Graef J. R. The Nonlinear Limit–Point/Limit–Circle Problem 2004 Boston, Mass, USA Birkhäuser 10.1007/978-0-8176-8218-7 MR2020682 ZBL1053.34024 Bartušek M. Graef J. R. Nonlinear limit-point/limit-circle properties of solutions of second order differential equations with p-Laplacian International Journal of Pure and Applied Mathematics 2008 45 4 501 518 MR2426228 Bartušek M. Graef J. R. Strong nonlinear limit-point/limit-circle properties for forced Thomas-Fermi equations with p-Laplacian Panamerican Mathematical Journal 2008 18 1 73 88 MR2388598 Bartušek M. Graef J. R. Asymptotic behavior of solutions of a differential equation with p-Laplacian and forcing term Differential Equations and Dynamical Systems 2007 15 61 87 Bartušek M. Graef J. R. The strong nonlinear limit-point/limit-circle properties for super-half-linear equations Panamerican Mathematical Journal 2007 17 1 25 38 MR2292448 ZBL1148.34023 Bartušek M. Graef J. R. The strong nonlinear limit-point/limit-circle properties for sub-half-linear equations Dynamic Systems and Applications 2006 15 3-4 585 602 MR2367665 Bartušek M. Graef J. R. The strong limit-point property for Emden-Fowler equations Differential Equations and Dynamical Systems 2006 14 3-4 383 405 MR2311816 Bartušek M. Graef J. R. Asymptotic properties of solutions of a forced second order differential equation with p-Laplacian Panamerican Mathematical Journal 2006 16 2 41 59 MR2213166 Bartušek M. Graef J. R. The nonlinear limit-point/limit-circle problem for second order equations with p-Laplacian Dynamic Systems and Applications 2005 14 3-4 431 446 MR2179157 Bartušek M. Graef J. R. Some limit-point and limit-circle results for second order Emden-Fowler equations Applicable Analysis 2004 83 5 461 476 10.1080/00036810310001632835 MR2054640 ZBL1053.34024 Graef J. R. Limit circle criteria and related properties for nonlinear equations Journal of Differential Equations 1980 35 3 319 338 10.1016/0022-0396(80)90032-7 MR563385 ZBL0441.34024 Graef J. R. Limit circle type results for sublinear equations Pacific Journal of Mathematics 1983 104 1 85 94 MR683730 10.2140/pjm.1983.104.85 ZBL0535.34024 Graef J. R. Spikes P. W. On the nonlinear limit-point/limit-circle problem Nonlinear Analysis 1983 7 8 851 871 10.1016/0362-546X(83)90062-7 MR709039 ZBL0535.34023 Weyl H. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen Mathematische Annalen 1910 68 2 220 269 10.1007/BF01474161 MR1511560 Došlý O. Řehák P. Half-Linear Differential Equations 2005 202 Amsterdam, The Netherlands Elsevier North-Holland Mathematics Studies MR2158903 Shao J. Song W. Limit circle/limit point criteria for second-order sublinear differential equations with damping term Abstract and Applied Analysis 2011 2011 12 803137 10.1155/2011/803137 MR2854934 Xing L. Song W. Zhang Z. Xu Q. Limit circle/limit point criteria for second-order superlinear differential equations with a damping term Journal of Applied Mathematics 2012 2011 11 361961 10.1155/2012/361961 MR2910921 ZBL1244.34047 Dunford N. Schwartz J. T. Linear Operators; Part II: Spectral Theory 1963 New York, NY, USA John Wiley & Sons MR1009163