Summation Characterization of the Recessive Solution for Half-Linear Difference Equations
© O. Došlý and S. Fišnarová. 2009
Received: 24 June 2009
Accepted: 24 August 2009
Published: 11 October 2009
We consider the second-order half-linear difference equation
Qualitative theory of (1.1) was established in the series of the papers of ehák [1–5] and it is summarized in [6, Chapter 3]. It was shown there that the oscillation theory of (1.1) is very similar to that of the linear equation
The concept of the recessive solution of (1.1) has been introduced in . There are several attempts in literature to find a summation characterization of this solution, see  and also related references [9, 10], which are based on the asymptotic analysis of solutions of (1.1). However, this approach requires the sign restriction of the sequence and additional assumptions on the convergence (divergence) of certain infinite series involving sequences and see Proposition 2.1 in the following section. Here we use a different approach which is based on estimates for a certain nonlinear function which appears in the Picone-type identity for (1.1).
The recessive solution of (1.1) is a discrete counterpart of the concept of the principal solution of the half-linear differential equation
Let us recall the main result of  whose discrete version we are going to prove in this paper.
The paper is organized as follows. In Section 2 we recall elements of the oscillation theory of (1.1). Section 3 is devoted to technical statements which we use in the proofs of our main results which are presented in Section 4. Section 5 contains formulation of open problems in our research.
Oscillatory properties of (1.1) are defined using the concept of the generalized zero which is defined in the same way as for (1.2), see, for example, [6, Chapter 3],or [16, Chapter 7]. A solution of (1.1) has a generalized zero in an interval if and . Since we suppose that (oscillation theory of (1.1) generally requires only ), a generalized zero of in is either a "real" zero at or the sign change between and . However, (1.1) is said to be disconjugate in a discrete interval if the solution of (1.1) given by the initial condition , has no generalized zero in . However, (1.1) is said to be nonoscillatory if there exists such that it is disconjugate on for every and is said to be oscillatory in the opposite case.
where is the inverse function of and is the conjugate number to . Moreover, if has no generalized zero in then , . If we suppose that (1.1) is nonoscillatory, among all solutions of (2.1) there exists the so-calleddistinguished solution which has the property that there exists an interval such that any other solution of (2.1) for which , , satisfies , . Therefore, the distinguished solution of (2.1) is, in a certain sense, minimal solution of this equation near and sometimes it is called theminimal solution of (2.1). If is the distinguished solution of (2.1), then the associated solution of (1.1) given by the formula
is said to be the recessive solution of (1.1), see . Note that in the linear case a solution of (1.2) is recessive if and only if
At the end of this section, for the sake of comparison, we recall the main results of [8, 17], where summation characterizations of recessive solutions of (1.1) are investigated using the asymptotic analysis of the solution space of (1.1).
In cases (i) and (iii), the previous proposition givesnecessary and sufficient condition for a solution to be recessive. The reason why under assumptions in (i) or (iii) it is possible to formulate such a condition is that there is a substantial difference in asymptotic behavior of recessive and dominant solutions (i.e., solutions which are linearly independent of the recessive solution). This difference enables to "separate" the recessive solution from dominant ones and to formulate for it a necessary and sufficient condition (2.4). We refer to [8, 17] and also to [9, 10] for more details.
3. Technical Results
and define the function
The statements (i), (ii) are consequences of [18, Lemma 2.5].
that is, the statement holds according to the statement (iii) of this lemma.
4. Main Results
Condition (4.1) implies that there exists such that and either or is not defined. This means that (from (4.4)). On the other hand, (4.3) together with (4.4) and the fact that is increasing with respect to on imply that for Since for we have for a contradiction.
5. Applications and Open Problems
By a similar computation we find that
(ii)A typical example of (1.1) to which Theorems 4.1 and 4.2 apply is (1.1) with
since under these assumption all positive solutions of (1.1) are decreasing, see . However, one can applyindirectly Theorems 4.1 and 4.2 also to (1.1) with
(and , otherwise (1.1) would be oscillatory, see [16, Theorem 8.2.14] ), even if all positive solutions of (1.1) areincreasing in this case. The method which enables to overcome this difficulty is the so-calledreciprocity principle, which can be explained as follows.
Moreover, if does not change its sign for large , (1.1) is nonoscillatory if and only if (5.14) is nonoscillatory, see . The following statement relates recessive solutions of (1.1) and (5.14). A similar statement can be found in , but our proof differs from that given in .
Now suppose that (5.12) holds. Then all solutions of (2.1) satisfying for large are negative (see ), that is, . Then using the same argument as in the first part of the proof we have for large for any solution of (5.15), that is, is the recessive solution of (5.14).
(iii)In , we posed the question whether the sequence is the recessive solution of the difference equation
Now we can give the affirmative answer to this question for . It is shown in  that
assumption (5.12) is satisfied (with , , and instead of , , and , resp.), hence positive solutions of (5.21) are decreasing, that is, Theorems 4.1 and 4.2 apply to this case. By a direct computation, we have
This research is supported by the Grant 201/07/0145 of the Czech Grant Agency of the Czech Republic, and the Research Project MSM0022162409 of the Czech Ministry of Education.
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