AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 784761 10.1155/2013/784761 784761 Research Article Limit-Point/Limit-Circle Results for Superlinear Damped Equations Bartušek M. 1 Graef John R. 2 Peterson Allan 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 2013 7 10 2013 2013 15 04 2013 06 09 2013 2013 Copyright © 2013 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 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<qpλ, a(t)>0, and r(t)>0. Examples to illustrate the main results are included.

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 +=[0,), =(-,), 0<qpλ, aC1(+), bC0(+), rC1(+), 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 , Xing et al. , and the present authors . 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  over 100 years ago. Weyl considered the second-order linear eigenvalue problem (C)y′′+r(t)y=θy,θ and classified this equation to be of the limit-circle type if every solution belongs to L2, that is, (3)0y2(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)0y2(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 Imθ0. 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, then it is limit-circle for all θ. 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 + and is nontrivial in any neighborhood of .

Remark 2.

Under the covering assumptions here, the functions a, 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 b0 on + 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 . 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.  as well as the recent papers of Bartušek and Graef . The nonlinear limit-point and limit-circle properties of solutions are defined as follows (see  and the papers [3, 6, 7, 1018]).

Definition 3.

A solution y 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:+ by (7)R(t)=a1/p(t)r(t) and the constant δ by (8)δ=p+1p.

Definition 4.

A solution y 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 solution y 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 if b(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 . 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 λp, we are in the half-linear and super-half-linear cases. In , 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  with λ1 being the ratio of odd positive integers and in  with λ1 being an odd integer. The results in both of these papers tend to be modifications of results in  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)for  either  p>qorλ>q,ω=1λ+1+pp+1,ω1=q+1p+1,ω2=1λ+1+qp+1,v=qp1,v1=q-p(λ+2)p+1. Notice that α=1-β, ω2ω1, and ω1. We define the function g:+ by (15)g(t)=-a1/p(t)R(t)Rα+1(t), and sometimes we will make use of the assumption that (16)limtg(t)=0,0|g(s)|ds<. If (16) holds, we define the constants (17)γ1=αγ-1/(λ+1)sups+|g(s)|,γ2=δ+γ1. For any solution y:+ 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 F0 on + for every solution of (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).

In Section 2, 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 either p=q or b0 on +. Then for every nontrivial solution y of (1) one has F(t)>0 for large t.

Proof.

Let y be a nontrivial solution of (1). Then (18) implies the existence of T0 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)for  t[0,t0],y-(t)=0for  tt0. But this contradicts Remark 2 and proves the lemma.

Lemma 2.

Let y be a solution of (1). Then

for t+, one has (20)|y(t)|γ-1/(λ+1)R-β1(t)F1/(λ+1)(t),|y2(t)|Rβ1(t)Fp/(p+1)(t);

for 0τ<t, one has (21)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)-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|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). 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  applied to (23). Inequality (22) follows from (20).

The next two lemmas give us sufficient conditions for the boundedness of F 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:

λ=p=q;

b0 for large t and (25)liminftR-β(t){t[|g(s)|+|b(s)|av(s)Rv1(s)]ds}β2×exp{0tR+(s)R(s)ds}=0;

or

p=q<λ and (25) holds.

Then for any nontrivial solution y of (1) defined on +, 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)liminftF(t)=0. By Lemma 1, F(t)>0 for large t. Let t-+ be such that F(t)>0 for tt- and (27)2α1sups[t-,)|g(s)|+α1t-|g(s)|ds+γ2t-|b(s)|av(s)Rv1(s)ds14; the existence of such a t- follows from (16) and (24). Then, for any t0t- such that F(t0)1, there exist τ and σ such that t0σ<τ and (28)2F(τ)=F(σ)=F(t0)>0,F(τ)F(t)F(σ) for σtτ. Then (20) implies (29)|y(t)y2(t)|γ-1/(λ+1)Fω(t) on +. 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)|+α1σ|g(s)|ds]Fω(σ)+σ|b(s)|av(s)Rv1(s)ds(γ1Fω2(σ)+δFω1(σ)). Since min(ω,ω1,ω2)=ω21, (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 that p=λ=q does not hold; this implies ω21. Since g is of bounded variation and limtg(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)3α1σ|g(s)|ds+γ2σ|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 tt- 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 tt- and so limtF(t)=0.

Now in cases (ii) and (iii) we can actually estimate a bound from below on F. 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))-,in  case  (ii),-(R-1(t))--δb+(t)r(t)a-δ(t),in  case  (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)|δS(t)R(t)Z(t).

Hence, (39) holds in both cases (ii) and (iii) for tt-. 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}Cexp0tR+(s)R(s)ds with C=1 in case (ii) and C=exp{δ0(|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.

Let y be a solution of (1) and let t-0 be such that (27) holds. Suppose that F is not bounded, that is, (44)limsuptF(t)=. Then for any t0t- with F(t0)1, there exist σ and τ such that t0σ<τ, F(τ)/2=F(σ)=F(t0), and (45)1F(σ)F(t)F(τ)for  σtτ.

Setting τ=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)|+α1στ|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)|+α1t-|g(s)|ds+γ2t-|b(s)|av(s)Rv1(s)dsα1maxs[t-,)|g(s)|+α1t-|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 t00 such that (48)0<c1F(t)c2for  t[t0,).

Proof.

Assumption (16) implies that g is bounded, so we can choose t0+ such that (49)M=14[3α1+δ+γ1]-1,|g(t)|Mfor  tt0,t0|g(s)|dsMt0|b(s)|av(s)Rv1(s)dsM.

Consider a solution y of (1) such that F(t0)=1. First, we will show that (50)F(t)32for  tt0. Suppose (50) does not hold. Then there exist t2>t1t0 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 ω1 and ω2ω11, 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, Lemma 2 (with t=t, τ=t0) similarly implies (53)|F(t)-1|3α1M(32)ω+δM(32)ω1+γ1M(32)ω232M[3α1+δ+γ1]<12 for tt0. From this and from (50) we have (54)12F(t)32for  tt0.

Next, we prove that y is defined on [0,t0]. Suppose to the contrary that y is defined on (a,t0) with 0a<t0 and y cannot be extended to t=a. Then limsupta+|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 +. The conclusion of the lemma then follows from (54).

Lemma 6.

Suppose that (16) and (24) hold and (56)0R-β(t)dt=. In addition, assume that either (57)limtRβ(t)[|(1r(t))|+|b(t)|av(t)r(t)Rv1(t)]=0 or (58)limtRβ(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)c1F(t)c2 for large t and some positive constants c1 and c2, then (60)0|y(t)|λ+1dt=. Moreover, if (61)limsuptr(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.

Let y be a nontrivial solution of (1) satisfying (59) on [T,)+. Then, (20) and (59) imply the existence of M1 such that (63)|y(t)y2(t)|γ-1/(λ+1)c2ωM1,|y(t)||y2(t)|vγ-1/(λ+1)c2ω2Rv1(t)M1Rv1(t) for tT.

Let t0T be such that (64)|g(t)|c1min(1,γ)(4M1)-1 holds for tt0. 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)|M1t0t[|(1r(s))|+|b(s)|av(s)r(s)Rv1(s)]ds. Moreover, (18) and (59) imply (68)c3t0tR-β(s)dst0t|y(s)|λ+1ds+t0t|y2(s)|δR(s)ds for tt0 with c3=c1min(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)]dsc32M1t0tR-β(s)ds for tt1. 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|+c32t0tR-β(s)ds for tt1.

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))dsc34M1t0tR-β(s)ds for tt1. 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)]dst0t(|a(s)|pa(s)r(s)+|b(s)|av(s)r(s)Rv1(s))ds+t0t|g(s)|Rβ(s)ds  c32M1t0tR-β(s)ds for tt1. This together with (65) and (67) implies that inequality (70) again holds. Thus, (70) holds if either (57) or (58) does. Moreover, (73)-|D|-c32t0tR-β(s)dst0t|y(s)|λ+1ds-t0t|y2(s)|δR(s)ds+y1(t)y2(t)r(t)|D|+c32t0tR-β(s)ds for tt1. Adding (68) and the left-hand inequality in (73) gives (74)2t0t|y(s)|λ+1ds+y(t)y2(t)r(t)c32t0tR-β(s)ds-|D| as t, while adding (68) and the right-hand inequality in (73) gives (75)2t0t|y2(s)|δR(s)ds-y(t)y2(t)r(t)c32t0tR-β(s)ds-|D| as t.

If y 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).

If y 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 t2t1 such that (78)|y(t)y2(t)r(t)|M1r(t)c340tR-β(s)ds for tt2; hence, (62) follows from this and from (75).

Remark 6.

Lemma 6 actually holds for all positive p, λ, and q regardless of their relative size.

3. LP/LC Problem for (<xref ref-type="disp-formula" rid="EEq1">1</xref>)

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)0R-β(t)dt<.

Proof.

Suppose (79) holds and let y be any nontrivial solution of (1) defined on +. Then, by Lemma 4, there is a positive constant c such that 0F(t)c. Hence, from this and (18), (80)0γ0|y(t)|λ+1dt+0|y2(t)|δR(t)dt=0F(t)R-β(t)dtc0R-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)0R-β(t)dt=. Let y be a solution of (1) given by Lemma 5. Then there is c1>0 such that c1F(t) on [t0,)+. Hence, from this and (18), (82)γt0|y(t)|λ+1dt+t0|y2(t)|δR(t)dt=t0F(t)R-β(t)dtc1t0R-β(t)dt=. Thus, either (83)0|y(t)|λ+1dt=or0|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)0R-β(t)dt=, then (1) is of the nonlinear limit-point type.

Proof.

The hypotheses of Lemmas 5 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) b0 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 Lemmas 36 are satisfied. Let y be a nontrivial solution of (1) defined on +. Then by Lemmas 3 and 4, there are positive constants c1 and c2 such that (85)0<c1F(t)c2for  large  t. 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.

If R is nondecreasing for large t, then condition (25) becomes (86)liminftRα(t){t[|g(s)|+|b(s)|av(s)Rv1(s)]ds}β2=0. If R is nonincreasing for large t, then (25) becomes (87)liminftR-β{t[|g(s)|+|b(s)|av(s)Rv1(s)]ds}β2=0.

4. LP/LC Problem for (<xref ref-type="disp-formula" rid="EEq2">2</xref>)

One of the main assumptions is Section 3 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 [1014, 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{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}.

Theorem 12.

(i) Let λ>p and (92)0R--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)limtg-(t)=0,0|g-(t)|dt<. Then (2) is of the strong nonlinear limit-circle type if and only if (94)0R--β(t)dt<.

Proof.

( i ) If g- is not identically a constant on + the result follows from Lemma 11 and from [13, Theorem 1] applied to (89). If g-const.  on +, 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 Lemma 11 and Theorem 8 applied to (89).

Theorem 13.

Let (93) hold and assume that either (95)limtRβ(t)(|r(t)|r2(t)+|b(t)|a(t)r(t))×exp{λ-p(λ+2)p+10tb(s)a(s)ds}=0 or (96)limtRβ(t)|a(t)|+|b(t)|a(t)r(t)×exp{λ-p(λ+2)p+10tb(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)   b0 on + 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)limsuptr-(t)0t(R-(s))-βds=, then (2) is of the strong nonlinear limit-point type.

Proof.

This follows from Lemma 11 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,t1, with s and λ1. We have the following results.

If s<-1 and σ>(λ+3)/(λ+1), then (100) is of the strong nonlinear limit-circle type by Theorem 7.

If s<-1 and -(λ+3)/2<σ(λ+3)/(λ+1), then (100) is of the strong nonlinear limit-point type by Theorem 8.

If λ=1, s<-1, and -2<σ2, then (100) is of the strong nonlinear limit-point type by Theorem 9(i).

If λ>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 .

Example 2.

Again consider (100) with s and λ1. The following results hold.

Equation (100) is of the nonlinear limit-circle type if

s<-1 and σ>(λ+3)/(λ+1) (by Theorem 12(ii));

s=-1 and σ>-(λ-1)/(λ+1) (by Theorem 12(i) if λ>1 and by Theorem 12(ii) if λ=1);

s>-1, λ>1, and σ>(λ+3)/(λ+1) (by Theorem 12(i));

s>-1, λ=1 and s<σ/2 (by Theorem 12(ii)).

Equation (100) is of the nonlinear limit-point type if

s<-1 and -(λ+3)/(λ+1)<σ(λ+3)/(λ+1) (by Theorem 13);

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  are a special case of those in this paper.

Remark 15.

Theorem  2.3 in  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  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 that b(t) in (1) (or (2)) is negative.

Example 3.

Consider the equation (101)y′′-tsy+tσ|y|λ-1y=0,t1, with s, λ1. Calculations show the following.

Equation (101) is of the nonlinear limit-circle type if

s<-1 and σ>(λ+3)/(λ+1) (by Theorem 7);

s=-1 and σ>-(λ-1)/(λ+1) (by Theorem 12(ii)).

Equation (101) is of the nonlinear limit-point type if

s<-1 and -(λ+3)/2<σ(λ+3)/(λ+1) (by Theorem 8);

s=-1 and -2(λ+1)/(λ+3)<σ-(λ-1)/(λ+1) (by Theorem 13);

s>-1 and λ>1 (by Theorem 13);

s>-1, λ=1, and σ>max{2s,s} (by Theorem 13).

Acknowledgment

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

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 ZBL05997825 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 2012 11 361961 10.1155/2012/361961 MR2910921 ZBL1244.34047 Bartušek M. Graef J. R. Limit-point/limit-circle results for equations with damping Abstract and Applied Analysis 2012 2012 19 979138 10.1155/2012/979138 MR3004891 ZBL1260.34053 Weyl H. Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen Mathematische Annalen 1910 68 2 220 269 10.1007/BF01474161 MR1511560 Bartušek M. Pekárková E. On existence of proper solutions of quasilinear second order differential equations Electronic Journal of Qualitative Theory of Differential Equations 2007 5 1 14 MR2295683 ZBL1115.34032 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: Theory, Methods & Applications 1983 7 8 851 871 10.1016/0362-546X(83)90062-7 MR709039 ZBL0535.34023 Bartušek M. Došlá Z. Graef J. R. The Nonlinear Limit-Point/Limit-Circle Problem 2004 Boston, Mass, USA Birkhäuser MR2020682 10.1007/978-0-8176-8218-7 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. 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. 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 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. Asymptotic behaviour of solutions of a differential equation with p-Laplacian and a forcing term Differential Equations and Dynamical Systems 2007 15 1-2 61 87 MR3075347 ZBL1165.34367 Bartušek M. Graef J. R. The strong nonlinear limit-point/limit-circle properties for sub-half-linear equations Dynamic Systems and Applications 2006 15 3-4 585 601 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 Došlý O. Rehák P. Half-Linear Differential Equations 2005 202 Amsterdam, The Netherlands Elsevier North-Holland Mathematics Studies MR2158903