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

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.

1. Introduction

In this paper, we continue the study of the nonlinear limit-point and limit-circle properties for the second-order damped equation 𝑎(𝑡)𝑦^′𝑝−1𝑦^′′+𝑏(𝑡)𝑦^′𝑞−1𝑦^′+𝑟(𝑡)𝑦^𝜆−1𝑦=0, (1) 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, 𝑎(𝑡)𝑦^′𝑝−1𝑦^′′+𝑏(𝑡)𝑦^′𝑝−1𝑦^′+𝑟(𝑡)𝑦^𝜆−1𝑦=0. (2) 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 𝑦^′′+𝑟(𝑡)𝑦=𝜃𝑦,𝜃∈ℂ (C) and classified this equation to be of the limit-circle type if every solution belongs to 𝐿², that is, ^∞₀𝑦²(𝜎)𝑑𝜎<∞, (3) and to be of the limit-point type if at least one solution 𝑦(𝑡) does not belong to 𝐿², that is, ^∞₀𝑦²(𝜎)𝑑𝜎=∞. (4) 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 𝑦^′′+𝑟(𝑡)𝑦=0 (L) 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 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 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 solution 𝑦 of (1) is said to be of the nonlinear limit-circle type if ^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡<∞, (NLC) and it is said to be of the nonlinear limit-point type otherwise, that is, if ^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡=∞. (NLP) 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 𝑦^′₁=𝑎^−1/𝑝(𝑡)𝑦₂^1/𝑝sgn𝑦₂,𝑦^′₂=−𝑏(𝑡)𝑎^−𝑞/𝑝(𝑡)𝑦₂^𝑞/𝑝sgn𝑦₂−𝑟(𝑡)𝑦₁^𝜆sgn𝑦₁, (5) where the relationship between a solution 𝑦 of (1) and a solution (𝑦₁,𝑦₂) of the system (5) is given by 𝑦₁(𝑡)=𝑦(𝑡),𝑦₂(𝑡)=𝑎(𝑡)𝑦^′(𝑡)^𝑝−1𝑦^′(𝑡). (6)
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 𝑅(𝑡)=𝑎^1/𝑝(𝑡)𝑟(𝑡) (7) and the constant 𝛿 by 𝛿=𝑝+1𝑝. (8)

Definition 4. A solution 𝑦 of (1) is said to be of the strong nonlinear limit-point type if ^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡=∞,^∞₀𝑦₂(𝑡)^𝛿𝑅(𝑡)𝑑𝑡=∞. (9) 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 ^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡<∞,^∞₀𝑦₂(𝑡)^𝛿𝑅(𝑡)𝑑𝑡<∞. (10) 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 𝑎(𝑡)𝑦^′𝑝−1𝑦^′′+𝑟(𝑡)𝑦^𝜆−1𝑦=0, (11) and moreover, if 𝜆=𝑝, then (11) takes the form 𝑎(𝑡)𝑦^′𝑝−1𝑦^′′+𝑟(𝑡)𝑦^𝑝−1𝑦=0. (12) 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 𝑎(𝑡)𝑦^′′+𝑏(𝑡)𝑦^′+𝑟(𝑡)𝑦^𝜆=0 (13) 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: 𝛼=𝑝+1()()𝑝𝜆+2𝑝+1,𝛽=𝜆+1(),𝜆+2𝑝+1𝛾=𝑝+1𝑝(𝜆+1),𝛼₁=𝛼𝛾^{−1/(𝜆+1)},𝛽₁=𝑝(),𝛽𝜆+2𝑝+1₂=(𝜆+1)𝑝+1𝜆+1𝑝−𝑞𝜆−𝑞foreither𝑝>𝑞or𝜆>𝑞,𝜔=+𝑝𝜆+1𝑝+1,𝜔₁=𝑞+1,𝜔𝑝+1₂=1+𝑞𝜆+1,𝑞𝑝+1𝑣=𝑝≤1,𝑣₁=𝑞−𝑝().𝜆+2𝑝+1 (14) Notice that 𝛼=1−𝛽, 𝜔₂≤𝜔₁, and 𝜔≤1. We define the function 𝑔∶ℝ₊→ℝ by 𝑔(𝑡)𝑎=−^1/𝑝(𝑡)𝑅^′(𝑡)𝑅^𝛼+1(𝑡), (15) and sometimes we will make use of the assumption that lim_𝑡→∞𝑔(𝑡)=0,^∞₀𝑔^′(𝑠)𝑑𝑠<∞. (16) If (16) holds, we define the constants 𝛾₁=𝛼𝛾^{−1/(𝜆+1)}sup_𝑠∈ℝ+𝑔(𝑠),𝛾₂=𝛿+𝛾₁. (17) For any solution 𝑦∶ℝ₊→ℝ of (1), we let 𝐹(𝑡)=𝑅^𝛽(𝑡𝑎(𝑡))𝑟(𝑡)𝑦^′(𝑡)^𝑝+1+𝛾𝑦(𝑡)^𝜆+1=𝑅^𝛽(𝑡)𝑦₂(𝑡)^𝛿𝑅(𝑡)+𝛾𝑦(𝑡)^𝜆+1. (18) Note that 𝐹≥0 on ℝ₊ for every solution of (1).

For any continuous function ℎ∶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 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 𝑏≤0 on ℝ₊. 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 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 𝑦(𝑡)=𝑦(𝑡)for𝑡∈0,𝑡₀,𝑦(𝑡)=0for𝑡≥𝑡₀. (19) But this contradicts Remark 2 and proves the lemma.

Lemma 2. Let 𝑦 be a solution of (1). Then(i)for 𝑡∈ℝ₊, one has 𝑦(𝑡)≤𝛾^{−1/(𝜆+1)}𝑅^−𝛽1(𝑡)𝐹^1/(𝜆+1)(𝑡),𝑦₂(𝑡)≤𝑅^𝛽1(𝑡)𝐹^{𝑝/(𝑝+1)}(𝑡); (20)(ii)for 0≤𝜏<𝑡, one has 𝐹(𝑡)=𝐹(𝜏)−𝛼𝑔(𝜏)𝑦(𝜏)𝑦₂(𝜏)+𝛼𝑔(𝑡)𝑦(𝑡)𝑦₂(𝑡)−𝛼^𝑡_𝜏𝑔^′(𝑠)𝑦(𝑠)𝑦₂(𝑠)−𝑑𝑠_𝑡𝜏𝛿𝑅^−𝛼(𝑠)𝑦₂(𝑠)^1/𝑝sgn𝑦₂(𝑠)(𝑠)𝑦(𝑠𝑏(𝑠)−𝛼𝑔)𝑎^𝑣(𝑠)𝑦₂(𝑠)^𝑣−1𝑦₂(𝑠)𝑑𝑠,_𝑡𝜏𝛿𝑅^−𝛼(𝑠)𝑦₂(𝑠)^1/𝑝sgn𝑦₂(𝑠)(𝑠)𝑦(𝑠×𝑏(𝑠)−𝛼𝑔)𝑎^𝑣(𝑠)𝑦₂(𝑠)^𝑣−1𝑦₂(𝑠)𝑑𝑠≤𝛿^𝑡_𝜏|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝐹^𝜔1(𝑠)𝑑𝑠+𝛾₁^𝑡_𝜏|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝐹^𝜔2(𝑠)𝑑𝑠. (21)(22)

Proof. Let 𝑦 be a solution of (1). Then it is a solution of the equation 𝑎(𝑡)𝑧^′𝑝−1𝑧^′′+𝑟(𝑡)|𝑧|^𝜆−1𝑧=𝑒(𝑡) (23) with 𝑒(𝑡)=−𝑏(𝑡)|𝑦^′(𝑡)|^𝑞−1𝑦^′(𝑡)=−(𝑏(𝑡)/𝑎^𝑣(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 of 𝐹 from above and from below by positive constants.

Lemma 3. In addition to (16), assume that ^∞₀|𝑏(𝑡)|𝑎^𝑣(𝑡)𝑅^𝑣1(𝑡)𝑑𝑡<∞ (24) and one of the following conditions holds:(i)   𝜆=𝑝=𝑞;(ii)   𝑏≤0 for large 𝑡 and liminf_𝑡→∞𝑅^−𝛽(𝑡)^∞_𝑡𝑔^′(𝑠+|𝑏(𝑠))|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠^𝛽2×exp^𝑡₀𝑅^′₊(𝑠)𝑅(𝑠)𝑑𝑠=0; (25) or(iii)  𝑝=𝑞<𝜆 and (25) holds.
Then for any nontrivial solution 𝑦 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 liminf_𝑡→∞𝐹(𝑡)=0. (26) By Lemma 1, 𝐹(𝑡)>0 for large 𝑡. Let 𝑡∈ℝ₊ be such that 𝐹(𝑡)>0 for 𝑡≥𝑡 and 2𝛼₁sup_{𝑠∈[𝑡,∞)}𝑔(𝑠)+𝛼₁^∞_𝑡𝑔^′(𝑠)𝑑𝑠+𝛾₂^∞_𝑡|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)1𝑑𝑠≤4; (27) 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 2𝐹(𝜏)=𝐹(𝜎)=𝐹𝑡₀>0,𝐹(𝜏)≤𝐹(𝑡)≤𝐹(𝜎) (28) for 𝜎≤𝑡≤𝜏. Then (20) implies 𝑦(𝑡)𝑦₂(𝑡)≤𝛾^{−1/(𝜆+1)}𝐹^𝜔(𝑡) (29) on ℝ₊. From (21) (with 𝜏=𝜎 and 𝑡=𝜏), inequalities (22) and (29), and the fact that 𝐹(𝜎)≤1, we have 𝐹(𝜎)2(𝜎)(𝜏)≤𝛼=𝐹−𝐹₁𝑔(𝜏)+𝛼₁𝑔(𝜎)+𝛼₁^∞_𝜎𝑔^′(𝑠)𝑑𝑠𝐹^𝜔(𝜎)+^∞_𝜎|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠𝛾₁𝐹^𝜔2(𝜎)+𝛿𝐹^𝜔1(𝜎). (30) Since min(𝜔,𝜔₁,𝜔₂)=𝜔₂≤1, (27) and (30) imply 𝐹(𝜎)≤12𝐹^𝜔2(𝜎). (31) 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 𝜔₂≠1. Since 𝑔 is of bounded variation and lim_𝑡→∞𝑔(𝑡)=0, we see that 𝑔(𝜎)=𝑔(𝜎)−𝑔(∞)≤^∞_𝜎𝑔^′(𝑠)𝑑𝑠. (32) From this, (28), and (30), we obtain 𝐹(𝜎)≤23𝛼₁^∞_𝜎𝑔^′(𝑠)𝑑𝑠+𝛾₂^∞_𝜎|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠𝐹^𝜔2(𝜎), (33) or 𝐹𝑡₀=𝐹(𝜎)≤𝐾^∞_𝜎𝑔^′(𝑠+|𝑏(𝑠))|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠^𝛽2 (34) with 𝐾=(6𝛼₁+2𝛾₂)^𝛽2. Hence, 𝐹(𝑡)≤𝐾^∞_𝑡𝑔^′(𝑠+|𝑏(𝑠))|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠^𝛽2 (35) 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 lim_𝑡→∞𝐹(𝑡)=0.
Now in cases (ii) and (iii) we can actually estimate a bound from below on 𝐹. Let 𝑍(𝑡)=𝐹(𝑡)𝑅^−𝛽(𝑡). (36) Then (18) and (36) imply 𝑍^′(𝑡)=𝑅⁻¹(𝑡)^′𝑦₂(𝑡)^𝛿−𝛿𝑟⁻¹(𝑡)𝑏(𝑡)𝑦^′(𝑡)^𝑞+1. (37) Define 𝑆(𝑡)=−(𝑅⁻¹(𝑡))^′₋(),−𝑅,incaseii₋₁(𝑡)^′₋𝑏−𝛿⁺(𝑡)𝑟(𝑡)𝑎^−𝛿(𝑡),incase(iii). (38)
Suppose case (ii) holds. Then (36) and (37) imply 𝑍^′(𝑡)≥𝑆(𝑡)𝑅(𝑡)𝑍(𝑡). (39)
Suppose case (iii) holds. Then (36) and (37) imply 𝑍^′(𝑡)≥−𝑅⁻¹(𝑡)^′₋𝑦₂(𝑡)^𝛿𝑏−𝛿⁺(𝑡)𝑟(𝑡)𝑎^−𝛿(𝑡)𝑦₂(𝑡)^𝛿≥𝑆(𝑡)𝑅(𝑡)𝑍(𝑡). (40)
Hence, (39) holds in both cases (ii) and (iii) for 𝑡≥𝑡. So 𝑍(𝑡)≥𝑍𝑡exp^𝑡_𝑡𝑆(𝜎)𝑅(𝜎)𝑑𝜎, (41) and in view of (35) and the fact that 𝑆(𝑡)≤0, 𝑍𝑡𝑅^𝛽(𝑡)exp^𝑡_𝑡𝑆(𝜎)𝑅(𝜎)(𝑡)𝑑𝜎≤𝐹≤𝐾^∞_𝑡𝑔^′(𝑠+|𝑏(𝑠))|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠^𝛽2, (42) which contradicts (25). Notice that exp−^𝑡_𝑡𝑆(𝜎)𝑅(𝜎)𝑑𝜎≤𝐶exp^𝑡₀𝑅^′₊(𝑠)𝑅(𝑠)𝑑𝑠 (43) 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. Let 𝑦 be a solution of (1) and let 𝑡≥0 be such that (27) holds. Suppose that 𝐹 is not bounded, that is, limsup_𝑡→∞𝐹(𝑡)=∞. (44) 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 1≤𝐹(𝜎)≤𝐹(𝑡)≤𝐹(𝜏)for𝜎≤𝑡≤𝜏. (45)
Setting 𝜏=𝜎 and 𝑡=𝜏 in (20)–(22), we have (29) and 𝐹(𝜏)2(𝜏)(𝜎)≤𝛼=𝐹−𝐹₁𝑔(𝜎)+𝛼₁𝑔(𝜏)+𝛼₁^𝜏_𝜎𝑔^′(𝑠)𝑑𝑠𝐹^𝜔(𝜏)+^∞_𝜎|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠𝛾₁𝐹^𝜔2(𝜏)+𝛿𝐹^𝜔1(𝜏). (46) Since max(𝜔₁,𝜔₂,𝜔₃)≤1, (27) and (45) imply 𝐹(𝜏)≤22𝛼₁max_{𝑠∈[𝑡,∞)}𝑔(𝑠)+𝛼₁^∞_𝑡𝑔^′(𝑠)𝑑𝑠+𝛾₂^∞_𝑡|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝐹(𝜏)≤𝐹(𝜏)𝑑𝑠2. (47) 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 0<𝑐₁≤𝐹(𝑡)≤𝑐₂𝑓𝑜𝑟𝑡∈𝑡₀,∞. (48)

Proof. Assumption (16) implies that 𝑔 is bounded, so we can choose 𝑡₀∈ℝ₊ such that 1𝑀=43𝛼₁+𝛿+𝛾₁⁻¹,𝑔(𝑡)≤𝑀for𝑡≥𝑡₀,^∞_𝑡0𝑔^′(𝑠)𝑑𝑠≤𝑀_∞𝑡0|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠≤𝑀. (49)
Consider a solution 𝑦 of (1) such that 𝐹(𝑡₀)=1. First, we will show that 𝐹(𝑡)≤32for𝑡≥𝑡₀. (50) Suppose (50) does not hold. Then there exist 𝑡₂>𝑡₁≥𝑡₀ such that 𝐹𝑡₂=32,𝐹𝑡₁(𝑡)<3=1,1<𝐹2 (51) for 𝑡∈(𝑡₁,𝑡₂). Lemma 2 (with 𝜏=𝑡₁ and 𝑡=𝑡₂), together with the facts that 𝜔≤1 and 𝜔₂≤𝜔₁≤1, implies 12=𝐹𝑡₁2=𝐹𝑡₂−𝐹𝑡₁≤3𝛼₁𝑀𝐹^𝜔𝑡₂+𝛿𝑀𝐹^𝜔1𝑡₂+𝛾₁𝑀𝐹^𝜔2𝑡₂≤32𝑀3𝛼₁+𝛿+𝛾₁=38. (52) This contradiction shows that (50) holds.
Now, Lemma 2 (with 𝑡=𝑡, 𝜏=𝑡₀) similarly implies |𝐹(𝑡)−1|≤3𝛼₁𝑀32^𝜔3+𝛿𝑀2^𝜔1+𝛾₁𝑀32^𝜔2≤32𝑀3𝛼₁+𝛿+𝛾₁<12 (53) for 𝑡≥𝑡₀. From this and from (50) we have 12(𝑡)≤3≤𝐹2for𝑡≥𝑡₀. (54)
Next, we prove that 𝑦 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 limsup_𝑡→𝑎+|𝑦₂(𝑡)|=∞. The change of variables 𝑥=𝑡₀−𝑡 and 𝑦(𝑡)=𝑌(𝑥) transforms (1) into 𝑎𝑡₀−𝑥𝑌̇^𝑝−1𝑌̇^⋅−𝑏𝑡₀−𝑥𝑌̇^𝑞−1𝑌̇+𝑟𝑡₀−𝑥|𝑌|^𝜆−1𝑑𝑌=0,“”̇=,𝑑𝑠 (55) with the noncontinuable solution 𝑌(𝑥) defined on [0,𝑡₀−𝑎) and limsup_{𝑥→(𝑡0−𝑎)}⁻|𝑌̇(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 ^∞₀𝑅^−𝛽(𝑡)𝑑𝑡=∞. (56) In addition, assume that either lim_𝑡→∞𝑅^𝛽(𝑡1)𝑟(𝑡)^′+|𝑏(𝑡)|𝑎^𝑣(𝑡)𝑟(𝑡)𝑅^𝑣1(𝑡)=0 (57) or lim_𝑡→∞𝑅^𝛽(𝑡)𝑎^′(𝑡)𝑎(𝑡)𝑟(𝑡)+|𝑏(𝑡)|𝑎^𝑣(𝑡)𝑟(𝑡)𝑅^𝑣1(𝑡)=0 (58) holds. If 𝑦 is a solution of (1) with 𝑐₁≤𝐹(𝑡)≤𝑐₂ (59) for large 𝑡 and some positive constants 𝑐₁ and 𝑐₂, then ^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡=∞. (60) Moreover, if limsup_𝑡→∞𝑟(𝑡)^𝑡₀𝑅^−𝛽(𝑠)𝑑𝑠=∞, (61) then ^∞₀𝑦₂(𝑡)^𝛿𝑅(𝑡)𝑑𝑡=∞. (62) That is, 𝑦 is of the strong nonlinear limit-point type.

Proof. Let 𝑦 be a nontrivial solution of (1) satisfying (59) on [𝑇,∞)⊂ℝ₊. Then, (20) and (59) imply the existence of 𝑀₁ such that 𝑦(𝑡)𝑦₂(𝑡)≤𝛾^{−1/(𝜆+1)}𝑐^𝜔₂≤𝑀₁,𝑦(𝑡)𝑦₂(𝑡)^𝑣≤𝛾^{−1/(𝜆+1)}𝑐^𝜔2₂𝑅^𝑣1(𝑡)≤𝑀₁𝑅^𝑣1(𝑡) (63) for 𝑡≥𝑇.
Let 𝑡₀≥𝑇 be such that 𝑔(𝑡)≤𝑐₁min1,𝛾4𝑀₁⁻¹ (64) holds for 𝑡≥𝑡₀. It follows from (5) that ^𝑡_𝑡0𝑦(𝑠)^𝜆+1𝑑𝑠=−^𝑡_𝑡0𝑦(𝑠)𝑦^′₂(𝑠)𝑟(𝑠)−𝑑𝑠_𝑡𝑡0𝑏(𝑠)𝑎^𝑣(𝑠)𝑟(𝑠)𝑦(𝑠)𝑦₂(𝑠)^𝑣sgn𝑦₂(𝑠)𝑦(𝑡)𝑦𝑑𝑠=−²(𝑡)𝑟(𝑡)+𝐷+^𝑡_𝑡0𝑦₂(𝑠)^𝛿𝑅(𝑠)𝑑𝑠+𝐽(𝑡), (65) where 𝐷=𝑦(𝑡₀)𝑦₂(𝑡₀)𝑟⁻¹(𝑡₀) and 𝐽(𝑡)=^𝑡_𝑡01𝑟(𝑠)^′𝑦(𝑠)𝑦₂(𝑠)−𝑏(𝑠)𝑎^𝑣(𝑠)𝑟(𝑠)𝑦(𝑠)𝑦₂(𝑠)^𝑣sgn𝑦₂(𝑠)𝑑𝑠. (66) From (63) and (66) |𝐽(𝑡)|≤𝑀₁^𝑡_𝑡01𝑟(𝑠)^′+|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑟(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠. (67) Moreover, (18) and (59) imply 𝑐₃^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠≤^𝑡_𝑡0𝑦(𝑠)^𝜆+1𝑑𝑠+^𝑡_𝑡0𝑦₂(𝑠)^𝛿𝑅(𝑠)𝑑𝑠 (68) for 𝑡≥𝑡₀ with 𝑐₃=𝑐₁min(1,𝛾).
Suppose (57) holds. By l’Hôspital’s Rule, there exists 𝑡₁>𝑡₀ such that ^𝑡_𝑡01𝑟(𝑠)^′+|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑟(𝑠)𝑅^𝑣1(𝑠≤𝑐)𝑑𝑠₃2𝑀₁^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠 (69) for 𝑡≥𝑡₁. From this, (65) and (67), we have ^𝑡_𝑡0𝑦(𝑠)^𝜆+1𝑑𝑠−^𝑡_𝑡0𝑦₂(𝑠)^𝛿𝑅(𝑠)𝑦(𝑡)𝑦𝑑𝑠+²(𝑡)𝑟(𝑡)≤|𝐷|+𝑐₃2^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠 (70) for 𝑡≥𝑡₁.
Now let (58) hold. Applying l’Hôspital’s Rule, there exists 𝑡₁>𝑡₀ such that ^𝑡_𝑡0𝑎^′(𝑠)(𝑠)𝑟(𝑠)+|𝑏(𝑠𝑝𝑎)|𝑎^𝑣(𝑠)𝑟(𝑠)𝑅^𝑣1(𝑠)≤𝑐𝑑𝑠₃4𝑀₁^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠 (71) for 𝑡≥𝑡₁. Then using (64), (71), and the fact that 1/𝑟=𝑎^1/𝑝/𝑅, we have ^𝑡_𝑡01𝑟(𝑠)^′+|𝑏(𝑠)|𝑎^𝑣(𝑠)𝑟(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠≤^𝑡_𝑡0𝑎^′(𝑠)(𝑠)𝑟(𝑠)+|𝑏(𝑠𝑝𝑎)|𝑎^𝑣(𝑠)𝑟(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠+^𝑡_𝑡0𝑔(𝑠)𝑅^𝛽(𝑠)≤𝑐𝑑𝑠₃2𝑀₁^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠 (72) 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, −|𝐷|−𝑐₃2^𝑡_𝑡0𝑅^−𝛽(𝑠)≤𝑑𝑠_𝑡𝑡0𝑦(𝑠)^𝜆+1𝑑𝑠−^𝑡_𝑡0𝑦₂(𝑠)^𝛿𝑅(𝑠)𝑦𝑑𝑠+¹(𝑡)𝑦₂(𝑡)𝑟(𝑡)≤|𝐷|+𝑐₃2^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠 (73) for 𝑡≥𝑡₁. Adding (68) and the left-hand inequality in (73) gives 2^𝑡_𝑡0𝑦(𝑠)^𝜆+1𝑦(𝑡)𝑦𝑑𝑠+²(𝑡)𝑟(𝑡)≥𝑐₃2^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠−|𝐷|⟶∞ (74) as 𝑡→∞, while adding (68) and the right-hand inequality in (73) gives 2^𝑡_𝑡0𝑦₂(𝑠)^𝛿𝑅(𝑠)𝑦(𝑡)𝑦𝑑𝑠−²(𝑡)𝑟(𝑡)≥𝑐₃2^𝑡_𝑡0𝑅^−𝛽(𝑠)𝑑𝑠−|𝐷|⟶∞ (75) as 𝑡→∞.
If 𝑦^′ is oscillatory, let {𝑡_𝑘}^∞_𝑘=1→∞ be a sequence of zeros of 𝑦^′. Then letting 𝑡=𝑡_𝑘 in (74) and (75), we see that 𝑦 is a strong nonlinear limit-point type solution of (1).
If 𝑦^′ is nonoscillatory, then either 𝑦(𝑡)𝑦₂(𝑡)>0 (76) or 𝑦(𝑡)𝑦₂(𝑡)<0 (77) 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 𝑦(𝑡)𝑦₂(𝑡)𝑟(𝑡)≤𝑀₁𝑟(𝑡)≤𝑐₃4^𝑡₀𝑅^−𝛽(𝑠)𝑑𝑠 (78) for 𝑡≥𝑡₂; 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 (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 ^∞₀𝑅^−𝛽(𝑡)𝑑𝑡<∞. (79)

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), 0≤𝛾^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡+^∞₀𝑦₂(𝑡)^𝛿𝑅(𝑡)=𝑑𝑡_∞0𝐹(𝑡)𝑅^−𝛽(𝑡)𝑑𝑡≤𝑐^∞₀𝑅^−𝛽(𝑡)𝑑𝑡<∞, (80) 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, ^∞₀𝑅^−𝛽(𝑡)𝑑𝑡=∞. (81) 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), 𝛾^∞_𝑡0𝑦(𝑡)^𝜆+1𝑑𝑡+^∞_𝑡0𝑦₂(𝑡)^𝛿𝑅(𝑡)=𝑑𝑡_∞𝑡0𝐹(𝑡)𝑅^−𝛽(𝑡)𝑑𝑡≥𝑐₁^∞_𝑡0𝑅^−𝛽(𝑡)𝑑𝑡=∞. (82) Thus, either ^∞₀𝑦(𝑡)^𝜆+1𝑑𝑡=∞or^∞₀𝑦₂(𝑡)^𝛿𝑅(𝑡)𝑑𝑡=∞, (83) 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 ^∞₀𝑅^−𝛽(𝑡)𝑑𝑡=∞, (84) 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) 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 Lemmas 3–6 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 0<𝑐₁≤𝐹(𝑡)≤𝑐₂forlarge𝑡. (85) 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 liminf_𝑡→∞𝑅^𝛼(𝑡)^∞_𝑡𝑔^′(𝑠+|𝑏(𝑠))|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠^𝛽2=0. (86) If 𝑅 is nonincreasing for large 𝑡, then (25) becomes liminf_𝑡→∞𝑅^−𝛽∞_𝑡𝑔^′(𝑠+|𝑏(𝑠))|𝑎^𝑣(𝑠)𝑅^𝑣1(𝑠)𝑑𝑠^𝛽2=0. (87)

4. LP/LC Problem for (2)

One of the main assumptions is Section 3 is condition (24), which takes the form ^∞₀|𝑏(𝑡)|𝑎(𝑡)𝑑𝑡<∞ (88) 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 𝑎(𝑡)𝑦^′𝑝−1𝑦^′′+𝑟(𝑡)𝑦^𝜆−1𝑦=0 (89) are equivalent where 𝑎(𝑡)=𝑎(𝑡)exp^𝑡₀𝑏(𝑠)𝑎(𝑠)𝑑𝑠,𝑟(𝑡)=𝑟(𝑡)exp^𝑡₀𝑏(𝑠)𝑎(𝑠)𝑑𝑠. (90) 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 𝑅(𝑡)=𝑎^1/𝑝(𝑡)𝑟(𝑡)=𝑅(𝑡)exp𝛿^𝑡₀𝑏(𝑠)𝑎(𝑠)𝑑𝑠,𝑔(𝑡)=−𝑎^1/𝑝(𝑡)𝑅^′(𝑡)𝑅^𝛼+1(𝑡)𝑎=−^1/𝑝(𝑡)𝑅^𝛼+1(𝑡)𝑅^′(𝑡)𝑏(𝑡)+𝛿𝑎(𝑡)𝑅(𝑡)×exp𝜆−𝑝(𝜆+2)𝑝+1^𝑡₀𝑏(𝑠)𝑎(𝑠)𝑑𝑠. (91)

Theorem 12. (i) Let 𝜆>𝑝 and ^∞₀𝑅^−𝛽(𝑡)^𝑡₀𝑔^′(𝑠)𝑑𝑠+1^{(𝜆+1)(𝑝+1)/(𝜆−𝑝)}𝑑𝑡<∞. (92) Then (2) is of the nonlinear limit-circle type.
(ii) Assume that lim_𝑡→∞𝑔(𝑡)=0,^∞₀𝑔^′(𝑡)𝑑𝑡<∞. (93) Then (2) is of the strong nonlinear limit-circle type if and only if ^∞₀𝑅^−𝛽(𝑡)𝑑𝑡<∞. (94)

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 Lemma 11 and Theorem 8 applied to (89).

Theorem 13. Let (93) hold and assume that either lim_𝑡→∞𝑅^𝛽(𝑡)𝑟^′(𝑡)𝑟²(𝑡)+|𝑏(𝑡)|𝑎(𝑡)𝑟(𝑡)×exp𝜆−𝑝(𝜆+2)𝑝+1^𝑡₀𝑏(𝑠)𝑎(𝑠)𝑑𝑠=0 (95) or lim_𝑡→∞𝑅^𝛽(𝑡)𝑎^′(𝑡)+|𝑏(𝑡)|𝑎(𝑡)𝑟(𝑡)×exp𝜆−𝑝(𝜆+2)𝑝+1^𝑡₀𝑏(𝑠)𝑎(𝑠)𝑑𝑠=0. (96) If ^∞₀𝑅(𝑡)^−𝛽𝑑𝑡=∞, (97) 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 liminf_𝑡→∞𝑅(𝑡)^−𝛽∞_𝑡𝑔^′(𝑠)𝑑𝑠^{(𝜆+1)(𝑝+1)/(𝜆−𝑝)}×exp𝑅^′₊(𝑠)𝑅(𝑠)𝑑𝑠=0; (98) or (iii)   𝜆>𝑝 and (98) holds. Then every nontrivial solution of (2) is of the nonlinear limit-point type. If, moreover, limsup_𝑡→∞𝑟(𝑡)^𝑡₀𝑅(𝑠)^−𝛽𝑑𝑠=∞, (99) 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 𝑦^′′+𝑡^𝑠𝑦^′+𝑡^𝜎𝑦^𝜆−1𝑦=0,𝑡≥1, (100) with 𝑠∈ℝ and 𝜆≥1. We have the following results.(i)If 𝑠<−1 and 𝜎>(𝜆+3)/(𝜆+1), then (100) is of the strong nonlinear limit-circle type by Theorem 7.(ii)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.(iii)If 𝜆=1, 𝑠<−1, and −2<𝜎≤2, then (100) is of the strong nonlinear limit-point type by Theorem 9(i).(iv)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 [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 𝑉̇ the term “−𝑝(𝑡)[𝑎(𝑡)𝑟(𝑡)]^𝛽−2𝛼𝑥^2𝑘(𝑠)” is missing.

In our next example we have that 𝑏(𝑡) in (1) (or (2)) is negative.

Example 3. Consider the equation 𝑦^′′−𝑡^𝑠𝑦^′+𝑡^𝜎𝑦^𝜆−1𝑦=0,𝑡≥1, (101) 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).

Acknowledgment

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