On Conjugacy of Second-order Half-linear Differential Equations on the Real Axis

Some conjugacy criteria are given for the equation |u | α sgn u + p(t)|u| α sgn u = 0, where p : R → R is a locally integrable function and α > 0, which generalise and supplement results known in the existing literature. Illustrative examples justifying applicability of the main results are given, as well. The results obtained are new even for linear differential equations, i.e., if α = 1.


Introduction
On the real axis, we consider the equation |u | α sgn u + p(t)|u| α sgn u = 0, ( where p : R → R is a locally integrable function and α > 0. A function u : I → R is said to be a solution to equation (1.1) on the interval I ⊆ R, if it is continuously differentiable on I, |u | α sgn u is absolutely continuous on every compact subinterval of I, and u satisfies equality (1.1) almost everywhere on I.In [8, Lemma 2.1], Mirzov proved that every solution to equation (1.1) is extendable to the whole real axis.Therefore, speaking about a solution to equation (1.1), we assume that it is defined on R.Moreover, for any a ∈ R, the initial value problem |u | α sgn u + p(t)|u| α sgn u = 0; u(a) = 0, u (a) = 0 has only the solution u ≡ 0 (see [8,Lemma 1.1]).Hence, a solution u to equation (1.1) is said to be non-trivial, if u ≡ 0 on R. B Email: sremr@ipm.czJ. Šremr Definition 1.1.We say that equation (1.1) is conjugate on R if it has a non-trivial solution with at least two zeros, and disconjugate on R otherwise.
It is clear that in the case α = 1, equation (1.1) reduces to the linear equation u + p(t)u = 0. (1.2) As it is mentioned in [2], a history of the problem of conjugacy of (1.2) began in the paper by Hawking and Penrose [6].In [10], Tipler presented an interesting relevance of the study of conjugacy of (1.2) to the general relativity and improved Hawking holds.Later, Peña [9] proved that the same condition is sufficient also for the conjugacy of half-linear equation (1.1).
The study of conjugacy of (1.1) on R is closely related to the question of oscillation of (1.1) on the whole real axis.It is known that Sturm's separation theorem holds for equation (1.1) (see [8,Theorem 1.1]).Therefore, if equation (1.1) possesses a non-trivial solution with a sequence of zeros tending to +∞ (resp.−∞), then any other its non-trivial solution has also a sequence of zeros tending to +∞ (resp.−∞).Definition 1.2.Equation (1.1) is said to be oscillatory in the neighbourhood of +∞ (resp. in the neighbourhood of −∞) if every its non-trivial solution has a sequence of zeros tending to +∞ (resp.to −∞).We say that equation (1.1) is oscillatory on R if it is oscillatory in the neighbourhood of either +∞ or −∞, and non-oscillatory on R otherwise.
Clearly, if equation (1.1) is oscillatory on R, then it is conjugate on R, as well.It is well known that oscillations of (1.1) in the neighbourhood of +∞ (resp.−∞) can be described by means of behaviour of the Hartman-Wintner type expression 1 |t| t 0 s 0 p(ξ)dξ ds (1.4) in the neighbourhood of +∞ (resp.−∞), see [7,Theorem 12.3].However, expression (1.4) is very useful also in the study of conjugacy of (1.1) on R. In particular, efficient conjugacy and disconjugacy criteria for linear equation (1.2) formulated by means of expression (1.4) are given in [2].Abd-Alla and Abu-Risha [1] observed that for the study of conjugacy on whole real axis, it is more convenient to consider a Hartman-Wintner type expression in a certain symmetric form, where all values of the function p are involved simultaneously.They proved in [1], among other things, that equation (1.1) with a continuous p is conjugate on R provided that p ≡ 0 and lim inf which obviously improves Peña's criterion (1.3).In the present paper, we generalise and supplement criterion (1.5) (see Theorems 2.1 and 2.3 below), and we establish further statements, which can be applied in the cases not covered by Theorems 2.1 and 2.3 (see Subsections 2.1 and 2.2).Moreover, we provide illustrative examples justifying the meaningfulness of the results obtained (see Section 3).In Sections 4 and 5, we establish auxiliary statements and prove the main results in detail.

Main results
For any ν < 1, we put We start with a Hartman-Wintner type result, which guarantees that equation (1.1) is oscillatory on R (not only conjugate).
Theorem 2.1.Let ν < 1 be such that either Then equation (1.1) is oscillatory on R and consequently, conjugate on R.
Remark 2.2.Using integration by parts, it is easy to verify that for any ν 1 , ν 2 < 1, we have According to the above said, we conclude that neither of Theorems 2.1 and 2.3 can be applied in the following two cases: (2.9) In Subsections 2.1 and 2.2 below, we provide some conjugacy criteria in both cases (2.8) and (2.9).It is worthwhile mentioning here that the results obtained therein are new even for linear equation (1.2), i.e., if α = 1.

The case (2.8)
In the first statement, we require that the function c(• ; 1 − α) is at some point far enough from its limit c(+∞).
We conclude this example by the following remark.As we have mentioned above, condition (2.11) is not fulfilled.Therefore, we cannot claim in Proposition 3.5 that equation (1.1) with p satisfying (3.2) is oscillatory on R (see Remark 2.7).
Example 3.6.Let p : R → R be a locally integrable function such that e.g., Then it is clear that Therefore, for any ν < 1 we get We first show that lim inf Consequently, condition (3.7) holds and thus, neither of Theorems 2.1 and 2.3 can be applied.For any κ > α we put Observe that Therefore, Proposition 2.10 with κ := 1 yields the following proposition.

Now we show that lim
However, by direct calculation, one can verify that and thus, we have Consequently, lim Observe that Let M > 0 be arbitrary.In view of (3.9), there exists t 0 > π such that . Then, using previous calculations and (3.10), one gets Since M > 0 was arbitrary, from the latter inequality we get lim t→+∞ where n ∈ {4, 5, . . .}.Then, using previous calculations and (3.10), one gets If we integrate by parts the first term on the right-hand side of the latter inequality, for any t ≥ t 0 we obtain Since M > 0 was arbitrary, from the latter inequality we get lim t→+∞ Therefore, it follows from (3.9), (3.11), and (3.12) that condition (3.8) holds and thus, Proposition 2.10 cannot be applied if α ≥ 1.
On the other hand, we have Assuming α ≥ 1, the latter integral can be estimated from below as follows Hence, for any α ≥ 1, we have Therefore, if α ≥ 1 and 16(α + 3) Consequently, Theorem 2.11 with κ := 1 + α yields the following statement.
Proof.Assume on the contrary that inequality (4.3) is violated.Then there exist a 0 ∈ [a, τ[ and Then the functions σ 1 , σ 2 are absolutely continuous on every compact subinterval of [a, τ[ and it follows from (1.1) that  (4.4).
Analysis similar to that in the proof of Lemma 4.3 shows that the following statement holds.
Lemma 4.4.Let a ∈ R, τ < a, and u 1 , u 2 be solutions to equation (1.1) satisfying the inequalities Proof.Let a ∈ R and b > a be arbitrary and w be a solution to equation (1.1) satisfying the initial conditions ) In particular, we have 0 where In view of (   Then it is clear that the function v is a solution to equation (4.32) on R satisfying conditions (4.33).Now we show that v satisfies also inequality (4.34).Indeed, assume on the contrary that (4.34) is violated.Then, by virtue of (4.33) and (4.39), there exists t 0 > 0 such that Passing to the limit t → −∞ in the latter inequality, we obtain 1+α α ds for t ≥ t v which, by virtue of Lemma 4.2, yields that because: Then, by using relations (4.62), (4.63), and (4.64), from inequality (4.61) we get for t ≥ t v , where Assume on the contrary that inequality (4.56) is violated, i.e., Therefore, where Now it follows from equality (4.70) that Observe that for any τ ≥ t v , we have for τ ≥ t v and thus, we have (4.73) Furthermore, by using Hölder's inequality, we get Consequently, by virtue of relations (4.73) and (4.74), from equality (4.72) we obtain Therefore, in view of (2.1) and (4.71), the function c(• ; 1 − α) has a finite limit (4.67) and c(+∞) satisfies (4.69).To finish the proof it is sufficient to mention that desired equality (4.68) now follows from (4.70), (4.71), and the above-proved equality (4.69).Lemma 4.11.Let ν < 1.If inequality (2.5) holds, then there exists κ > α such that Proof.Let n ∈ N be such that n > max{1, α}.Using integration by parts, one gets Assume that inequality (2.5) holds.Then there exist A ∈ R and t 0 ≥ 0 such that c(t; ν) ≥ A for t ≥ t 0 .

Proofs of main results
Proof of Theorem

Concluding remark
All results presented in Section 2 can be easily generalised for the equation However, the case, when condition (6.1) is violated, deserves a further investigation because it is related to the question of conjugacy of equation (1.1) either on a finite interval or on a half-line.

Lemma 4 . 5 .
Let equation (1.1) be disconjugate on R. Then for any a ∈ R and b > a, there exists a solution u to equation (1.1) such that ( r(t)|u | α sgn u + p(t)|u| α sgn u = 0 with a positive function r, continuous on R and such that 0 ds for t ≥ 0, whence we get the following assertions.The following statement provides a conjugacy criterion covering this case.Corollary 2.4 generalises several conjugacy criteria known in the existing literature.In particular, [4, Theorem 2.2] can be derived from Corollary 2.4.Moreover, conjugacy criterion (1.5) given in [1, Theorem 2.2] follows immediately from Corollary 2.4 with ν := 0. Corollary 2.4 also yields the following half-linear extension of [2, Theorem 1].
Assume that equation (1.1) is disconjugate on R. By virtue of Lemma 4.5, for any n ∈ N, there are solutions u n , z n to equation (1.1) such that 1) is disconjugate on R, we havew(t) sgn(t − a) > 0 for t ∈ R \ {a} resp.w(t)sgn(t−b) < 0 for t ∈ R \ {b} .Ris a solution to equation (1.1) satisfying desired conditions (4.11).Proposition 4.6.If equation (1.1) is disconjugate on R, then it has a solution u such that u(t) > 0 for t ∈ R.(4.12)Proof.
.15)We first show that u n (0) > z k (0) for n, k ∈ N.(4.16) R are continuous functions.Now we show that u is a solution to equation (1.1).Indeed, (1.1) yields that h n (t) = h n (0) − (s)| α sgn u n (s)ds for t ∈ R, n ∈ N.