ON THE INTEGRAL CHARACTERIZATION OF PRINCIPAL SOLUTIONS FOR HALF-LINEAR ODE

We discuss a new integral characterization of principal
solutions for half-linear differential equations, introduced in
the recent paper of S. Fisnarova and R. Marik, Nonlinear Anal.
74 (2011), 6427-6433. We study this characterization in the
framework of the existing results and we show when this new
integral characterization with a parameter $\alpha$ is
equivalent with two extremal cases of the integral
characterization used in the literature. We illustrate our
results on the Euler and Riemann-Weber differential equations.


Introduction
Consider the half-linear differential equation (1) (r(t)Φ(x ′ )) ′ + c(t)Φ(x) = 0, Φ(x) := |x| p−1 sgn x, where p > 1, r, c are continuous functions on [t 0 , ∞) and r(t) > 0. Denote by q the conjugate number to p, i.e. q = p/(p − 1) and set It is well-known, see, e.g., [9,Theorem 1.2.3], that if c is positive and both integrals J r and J c are divergent, then (1) is oscillatory.Throughout the paper we suppose that (1) is nonoscillatory, that is, all its solutions are either positive or negative for large t.Since the solution space of (1) is homogeneous, we consider its positive solutions only.Some recent trends in the qualitative theory of ODE's consist in the extension of properties of linear second order Sturm-Liouville equations, see, e.g., [9].One of them is related to the notion of principal solution of (1).More precisely, when (1) is nonoscillatory, following [13,16], a nontrivial solution h of (1) is called a principal solution if for every nontrivial solution x of (1) such that x = λh, λ ∈ R, we have (2) h ′ (t) h(t) < x ′ (t) x(t) for large t.
As in the linear case, a principal solution h exists and it is unique up to a nonzero constant multiplicative factor.For this reason, in the following we will denote it by the principal solution.Any nontrivial solution x = λh is called a nonprincipal solution.The problem of an integral characterization of principal solutions of (1), similar to that in the linear case, was initiated in 1988 when Mirzov's paper [16] was published, see also [7,9] for more details.For instance, in [8], see also [7,Proposition 2], the following integral characterization of the principal solution has been suggested as an extension of that given in the linear case, see [14, Chap.XI].
Theorem A. Let (1) be nonoscillatory and h be its positive solution satisfying h ′ (t) = 0 for large t.Then: holds, then h is the principal solution of (1).
Note that Theorem A-(iii) was stated in [8] without the assumption p ≥ 2. When c(t) > 0, the implication (4) h is the principal solution =⇒ Q = ∞ may fail to hold for p ∈ (1, 2) as Example 2 below shows, see also an example in [7].When c is negative for large t, in [1, Theorem 3.2] it is shown that a solution h of (1) is the principal solution if and only if (5) Later on, in [3,Theorem 7], it is proved that (5) is necessary and sufficient for h to be the principal solution of (1) when J r < ∞ and J c < ∞, independently of the sign of c.As it follows from the proof of [3, Theorem 7], when J r < ∞ and J c < ∞, condition (5) is equivalent with Finally, other contributions to the problem of integral characterizations of principal solutions can be found in [2,4,7,16].Observe that in these papers, under some additional conditions, either sufficient conditions or necessary conditions are presented.As a reaction on the fact that the implication (4) does not generally hold when c is eventually positive and p ∈ (1, 2), the following alternative integral characterization has been proposed recently in [11,Theorem 4.1].
Theorem B. Let (1) be nonoscillatory and h be its positive solution satisfying h ′ (t) = 0 for large t.
EJQTDE, 2013 No. 12, p. 2 For the extremal cases of the parameter α, namely for α = 2 and α = q, we have where Q is given by (3) and N by (6).Hence, the integral characterization Q [α] creates, roughly speaking, a bridge between the characterizations Q and N.Moreover, the role of the parameter α in Theorem B suggests that the situation is different for p ≤ 2 (in which Q [α] may diverge for some α) and p ≥ 2 (in which Q [α] may diverge for every α).
Nevertheless, when the function c changes its sign, (1) can have positive solutions with changing-sign derivatives, as the following example illustrates, and this fact makes Theorem B inapplicable.
Since x(t) = 2 − cos t is a solution of (8) and x ′ does not have a fixed sign, Theorem B cannot be used.
The main aim here is to show that, when c(t) > 0 for large t, the new characterization Q [α] introduced in [11] is equivalent with two extremal cases, that is the integrals Q or N.As a consequence, we will obtain that the opposite implication in the claim (ii) of Theorem B is valid under an additional condition.The obtained results are illustrated by two critical cases, namely the half-linear Euler and Riemann-Weber differential equations.Finally, an application of the obtained results to the so-called reciprocal equation completes the paper.

Main results
Here we study the integral characterizations Q, N and Q [α] of principal solutions defined by ( 3), ( 6) and (7), respectively.
When p > 2, J r = ∞, J c < ∞ and c(t) > 0 for large t, in view of Theorem A-(iii), the integral Q gives a necessary and sufficient condition for h to be a principal solution of (1).For this reason, throughout the paper we assume In view of (H1) and Theorem A, the problem of integral characterization of principal solutions of (1) reduces to the problem whether at least one of the integrals Q, N and Q [α]  diverges when h is the principal solution of (1).EJQTDE, 2013 No. 12, p. 3 Let h be a solution of (1) and denote by h [1] its quasi-derivative, i.e. h [1] (t) = r(t)Φ(h ′ (t)).Consider the function G h (t) = h(t)h [1] (t).
Since J r = ∞, any eventually positive solution h of (1) satisfies h ′ (t) > 0 for large t.Indeed h [1] is decreasing for large t, say t ≥ T ; if there exists t 1 > T such that h [1] (t 1 ) < 0, we obtain h [1] (t) < h [1] (t 1 ) < 0 for t ≥ t 1 , or h(t) < h [1] (t 1 ) which contradicts the positiveness of h.Hence, also the function G h is positive for large t.
The role of the function G h is given by the following result.
Our next lemma shows how the integral characterizations Q, N and Q [α] can be formulated in terms of the function G. Lemma 2. Let (1) be nonoscillatory and h be its positive solution satisfying h ′ (t) = 0 for large t.Then the following identities hold: Proof.The assertion follows by a direct calculation.
Hence, if h is a nonoscillatory solution of (1) such that lim t→∞ G h (t) = c, where c > 0, then integrals Q, N and Q [α] have the same behavior, i.e. either all are divergent or all are convergent.In the remaining cases when lim t→∞ G h (t) is zero or infinity, the following inequalities hold.Theorem 1. Assume (H1).Let (1) be nonoscillatory and h be its solution.
Claim (ii) can be proved using a similar argument.
A partial answer to the question when the implication (4) holds is given by the following theorem.
(i) If h is unbounded and then h is the principal solution and To prove this theorem, the following lemma will be needed.
Proof of Theorem 2. Claim (i).By [4, Lemma 1], h is the principal solution.The second conclusion follows immediately from Lemma 2.
Since h is eventually increasing, there exists k 1 > 0 such that we have for large t .
When (H1) holds, we get the following improvement of Theorem B.
Corollary 1. Assume (H1) and (11).Then a solution h of (1) is the principal solution if and only if The following example is taken from [2,Corrigendum] and shows that the implication (4) may fail to hold not only for Q but also for Q [α] with α ∈ [2, q).Example 2. Consider the equation (14) (Φ(x ′ )) ′ + e −t Φ(x) = 0 with 1 < p < 2. This equation is nonoscillatory and has both bounded and unbounded solutions, as it follows, for instance, from [15,Theorems 4.1,4.2].If h is the principal solution of ( 14), in view of [6, Theorem 2-(i 1 )], h is bounded and h ′ is eventually positive and satisfies lim t→∞ h ′ (t) = 0. Integrating (14) for large t we have Thus, in virtue of the boundedness of h, there exist two positive constants k 1 < k 2 such that for large t k 1 e −t/(p−1) < h ′ (t) < k 2 e −t/(p−1) .Hence, we have as Notice that the same happens for Q [α] with α ∈ [2, q), since a standard calculation gives as t → ∞ h ′ (t) Thus, from Lemma 2 we get On the other hand, N = Q [q] = ∞.This fact also illustrates how, in general, in Corollary 1 the stronger statement "Q [α] = ∞ for every α ∈ [2, q]" can fail.
Remark 2. Let h be a solution of (1).When lim t→∞ G h (t) = 0, in view of Theorem 1 we have Observe that lim t→∞ G h (t) = 0 may occur not only when the principal solution h is bounded, but also when every solution of (1) is unbounded and the Euler half-linear equation discussed in the next section is a typical example.We will show also that the principal solution can satisfy lim t→∞ G h (t) = ∞ and the implication (15) can fail.A typical example of this fact is the Riemann-Weber equation which will also be studied in the following section.EJQTDE, 2013 No. 12, p. 7

Examples
The following example illustrates Theorem 2.
The following example presents a typical equation for which lim t→∞ G h (t) = ∞ and N < ∞ for every solution.