- Research Article
- Open Access
Regularly Varying Solutions of Second-Order Difference Equations with Arbitrary Sign Coefficient
© Matucci and P. Řehák. 2010
- Received: 15 June 2010
- Accepted: 25 October 2010
- Published: 27 October 2010
Necessary and sufficient conditions for regular or slow variation of all positive solutions of a second-order linear difference equation with arbitrary sign coefficient are established. Relations with the so-called -classification are also analyzed and a generalization of the results to the half-linear case completes the paper.
- Difference Equation
- Regular Variation
- Vary Solution
- Nonoscillatory Solution
- Linear Difference Equation
on , where is an arbitrary sequence.
The principal aim of this paper is to study asymptotic behavior of positive solutions to (1.1) in the framework of discrete regular variation. Our results extend the existing ones for (1.1), see , where the additional condition was assumed. We point out that the relaxation of this condition requires a different approach. At the same time, our results can be seen as a discrete counterpart to the ones for linear differential equations, see, for example, . As a byproduct, we obtain new nonoscillation criterion of Hille-Nehari type. We also examine relations with the so-called -classification (i.e., the classification of monotone solutions with respect to their limit behavior and the limit behavior of their difference). We point out that such relations could be established also in the continuous case, but, as far as we know, they have not been derived yet. In addition, we discuss relations with the sets of recessive and dominant solutions. A possible extension to the case of half-linear difference equations is also indicated.
The paper is organized as follows. In the next section we recall the concept of regularly varying sequences and mention some useful properties of (1.1) which are needed later. In the main section, that is, Section 3, we establish sufficient and necessary conditions guaranteeing that (1.1) has regularly varying solutions. Relations with the -classification is analyzed in Section 4. The paper is concluded by the section devoted to the generalization to the half-linear case.
In this section we recall basic properties of regularly and slowly varying sequences and present some useful information concerning (1.1).
The theory of regularly varying sequences (sometimes called Karamata sequences), initiated by Karamata  in the thirties, received a fundamental contribution in the seventies with the papers by Seneta et al. (see [4, 5]) starting from which quite many papers about regularly varying sequences have appeared, see  and the references therein. Here we make use of the following definition, which is a modification of the one given in , and is equivalent to the classical one, but it is more suitable for some applications to difference equations, see .
If , then is said to be slowly varying. Let us denote by the totality of regularly varying sequences of index and by the totality of slowly varying sequences. A positive sequence is said to be normalized regularly varying of index if it satisfies . If , then is called a normalized slowly varying sequence. In the sequel, and will denote, respectively, the set of all normalized regularly varying sequences of index , and the set of all normalized slowly varying sequences. For instance, the sequence , and the sequence , for every ; on the other hand, the sequence .
The main properties of regularly varying sequences, useful to the development of the theory in the subsequent sections, are listed in the following proposition. The proofs of the statements can be found in , see also [4, 5].
A sequence if and only if , where tends to a positive constant and tends to 0 as . Moreover, if and only if , where .
A sequence if and only if , where tends to a positive constant and tends to as .
If a sequence , then in the representation formulae given in (i) and (ii), it holds const , and the representation is unique. Moreover, if and only if , where .
Let . If one of the following conditions holds (a) and , or (b) and , or (c) and , then .
Let . Then and for every .
Let and . Then and . The same holds if is replaced by .
If , , is strictly convex, that is, for every , then is decreasing provided , and it is increasing provided . If , , is strictly concave for every , then is increasing and .
If , then .
with for . Note that, dealing with nonoscillatory solutions of (1.1), we may restrict our considerations just to eventually positive solutions without loss of generality.
We end this section recalling the definition of recessive solution of (1.1). Assume that (1.1) is nonoscillatory. A solution of (1.1) is said to be a recessive solution if for any other solution of (1.1), with , , it holds . Recessive solutions are uniquely determined up to a constant factor, and any other linearly independent solution is called a dominant solution. Let be a solution of (1.1), positive for . The following characterization holds: is recessive if and only if ; is dominant if and only if .
In this section we prove conditions guaranteeing that (1.1) has regularly varying solutions. Hereinafter, means , where and are arbitrary positive sequences.
Let and denote by , the (real) roots of the quadratic equation . Note that , , , and .
where with as . Moreover, is a recessive solution, is a dominant solution, and every eventually positive solution of (1.1) is normalized regularly varying, with .
Now we prove the main statement.
as , that is, (3.1) holds. The same approach shows that implies (3.1).
Noting that for defined in (3.27) it holds, , we get for . This implies , where by virtue of (3.27).
Now, thanks to the contraction mapping theorem, there exists a unique element such that . Thus is a solution of (3.16), and hence of (3.11), and is positively decreasing towards zero. Clearly, defined by (3.9) is such that and therefore for large . This implies that defined by (3.8) is a nonoscillatory (positive) solution of (1.1). Since , we get , see Proposition 2.2. By the same proposition, can be written as , where .
Thanks to this identity, since and , we obtain , which implies .
(i) In the above proof, the contraction mapping theorem was used to construct a recessive solution . A dominant solution resulted from by means of the known formula for linearly independent solutions. A suitable modification of the approach used for the recessive solution leads to the direct construction of a dominant solution . This can be useful, for instance, in the half-linear case, where we do not have a formula for linearly independent solutions, see Section 5.
As a direct consequence of Theorem 3.1 we have obtained the following new nonoscillation criterion.
then (1.1) is nonoscillatory.
then (1.1) is nonoscillatory. Corollary 3.3 extends this result in case exists.
Throughout this section we assume that is eventually of one sign. In this case, all nonoscillatory solutions of (1.1) are eventually monotone, together with their first difference, and therefore can be a priori classified according to their monotonicity and to the values of the limits at infinity of themselves and of their first difference. A classification of this kind is sometimes called -classification, see, for example, [9–12] for a complete treatment including more general equations. The aim of this section is to analyze the relations between the classification of the eventually positive solutions according to their regularly varying behavior, and the -classification. The relations with the set of recessive solutions and the set of dominant solutions will be also discussed. We point out that all the results in this section could be established also in the continuous case and, as far as we know, have never been derived both in the discrete and in the continuous case.
if then ;
if then .
By observing that every solution satisfies and that , we get the following result.
If and , then , with .
If and , then .
If then .
If and , then and .
If and , then and .
If and , then and .
Notice that, being negative for large , it results . The following holds.
Theorem 4.2 (see ).
If and , then and .
If and , then and .
If , then and .
If and , then , and .
If and , then , and .
If , then , and .
We end this section by remarking that in this case positive solutions are convex and therefore they can exhibit also a rapidly varying behavior, unlike the previous case in which positive solutions are concave. We address the reader interested in this subject to the paper , in which the properties of rapidly varying sequences are described and the existence of rapidly varying solutions of (1.1) is completely analyzed for the case .
where and , , for every . For basic information on qualitative theory of (5.1) see, for example, .
Let and denote by , the (real) roots of the equation . Note that and .
The main idea of the proof is the analogous of the linear case, apart from some additional technical problems. We omit all the details, pointing out only the main differences.
and . The proof can then proceed analogously to the linear case.
(compare with (3.8)), where and is such that is a solution of (5.3). All the other details are left to the reader.
Similarly as in the linear case, as a direct consequence of Theorem 5.1 we obtain the following new nonoscillation criterion. Recall that a Sturm type separation theorem holds for equation (5.1), see , hence one solution of (5.1) is nonoscillatory if and only if every solution of (5.1) is nonoscillatory.
then (5.1) is nonoscillatory.
This work was supported by the grants 201/10/1032 and 201/08/0469 of the Czech Grant Agency and by the Institutional Research Plan No. AV0Z10190503.
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