Open Access

Summation Characterization of the Recessive Solution for Half-Linear Difference Equations

Advances in Difference Equations20092009:521058

DOI: 10.1155/2009/521058

Received: 24 June 2009

Accepted: 24 August 2009

Published: 11 October 2009

Abstract

We show that the recessive solution of the second-order half-linear difference equation ,   ,  , where are real-valued sequences, is closely related to the divergence of the infinite series .

1. Introduction

We consider the second-order half-linear difference equation

(1.1)

where are real-valued sequences and and we investigate properties of itsrecessive solution.

Qualitative theory of (1.1) was established in the series of the papers of ehák [15] 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

(1.2)

which is the special case in (1.1). We will recall basic facts of the oscillation theory of (1.1) in the following section.

The concept of the recessive solution of (1.1) has been introduced in [7]. There are several attempts in literature to find a summation characterization of this solution, see [8] 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

(1.3)

which attracted considerable attention in recent years, we refer to the work in [1115] and the references given therein.

Let us recall the main result of [11] whose discrete version we are going to prove in this paper.

Proposition 1.1.

Let be a solution of (1.3) such that for large .
  1. (i)

    Let . If

    (1.4)

    then is the principal solution of (1.3).

     
  2. (ii)

    If and , then is not the principal solution of (1.3).

     

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.

2. Preliminaries

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.

If is a solution of (1.1) such that in some discrete interval then is a solution of the associated Riccati type equation

(2.1)

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

(2.2)

is said to be the recessive solution of (1.1), see [7]. Note that in the linear case a solution of (1.2) is recessive if and only if

(2.3)

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).

Proposition 2.1.

Let be a solution of (1.1).
  1. (i)
    Suppose that , then is the recessive solution of (1.1) if and only if
    (2.4)
     
  2. (ii)

    Suppose that , , and

    (2.5)
    If is the recessive solution of (1.1), then
    (2.6)
     
  3. (iii)

    Suppose that , , and . Then is the recessive solution if and only if (2.4) holds.

     

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

Throughout the rest of the paper we suppose that (1.1) is nonoscillatory and is its solution. Denote

(3.1)

and define the function

(3.2)

Lemma 3.1.

Put
(3.3)
where is a solution of (2.1) and is any sequence satisfying . Then the following statements hold:
  1. (i)
    is a solution of (2.1) if and only if is a solution of
    (3.4)
     
  2. (ii)

    for with the equality if and only if

     
  3. (iii)

    if and only if

     
  4. (iv)

    let be a solution of (3.4) and suppose that for some , that is, , then if and only if

     

Proof.

The statements (i), (ii) are consequences of [18, Lemma  2.5].

We have
(3.5)
We have
(3.6)
Denote by the expression in brackets, then
(3.7)
Consequently,
(3.8)

that is, the statement holds according to the statement (iii) of this lemma.

Lemma 3.2.

Let be defined by (3.1), (3.2) and suppose that for large . Then one has the following inequalities for large .

If , then and
(3.9)
If , then and
(3.10)

Proof.

We have (with using the Lagrange mean value theorem)
(3.11)

where and hence .

Thus, if ,
(3.12)
and in the case we obtain
(3.13)
Next we proceed similarly as in [18, Lemma  2.6]. Inequalities (3.9), (3.10) can be written in the equivalent forms:
(3.14)
(3.15)
Denote and let . Then
(3.16)
Consequently, and
(3.17)
Hence, in view of the assumption , . It follows that
(3.18)
in some left neighborhood of and the function is positive, decreasing, and convex for and is negative, increasing, and concave for (with respect to ). Hence, both the inequalities (3.14) and (3.15) are satisfied in some left neighborhood of The proof will be completed by showing that has constant sign on the given intervals. By a direct computation,
(3.19)
where
(3.20)
Hence
(3.21)
and from (3.18)
(3.22)

in some left neighborhood of .

Moreover, for
(3.23)
and (for ) if and only if
(3.24)

Next we distinguish between the cases and .

If , then using (3.12),
(3.25)

hence is decreasing on and in view of (3.22) it means that and consequently from (3.21) also is positive for . Hence, (3.14) holds.

Similarly, if , then
(3.26)

hence is increasing for and from (3.22) we have that and hence also is negative for . This means that (3.15) is satisfied.

4. Main Results

Theorem 4.1.

Suppose and let be a solution of (1.1) such that for large . If
(4.1)

then is the recessive solution.

Proof.

Denote by the associated solution of (2.1) and let be a solution of (2.1) generated by another solution (linearly independent of ) of (1.1). Then, it follows from Lemma 3.1 that is a solution of (3.4), that is,
(4.2)
and suppose that this solution satisfies the condition This means that and to prove that is the recessive solution of (1.1), we need to show that there exists such that , that is, according to Lemma 3.1, . Suppose by contradiction that for . According to Lemma 3.1 (iv), it means that for , that is, . Then we have from Lemma 3.2 that and
(4.3)
Next, consider the equation
(4.4)
and let be its solution satisfying . However, (4.4) is equivalent to
(4.5)
that is,
(4.6)
where we have substituted for from (4.4) in the denominator. Hence
(4.7)
and we obtain
(4.8)

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.

Theorem 4.2.

Suppose and let be a solution of (1.1) such that for large . If
(4.9)

then is not the recessive solution.

Proof.

Similarly, as in the proof of Theorem 4.1, denote and let be a solution of (2.1) generated by another solution (linearly independent of ) of (1.1). Then is a solution of (3.4), that is,
(4.10)

and suppose that this solution satisfies the condition , being sufficiently small (will be specified later). Hence and we have to show that for , that is, for

Let be a solution of (4.4) and suppose that . Hence, similarly as in the proof of Theorem 4.1, we obtain
(4.11)
If is sufficiently small, then condition (4.9) implies that for and from (4.4), we have for . Consequently, from Lemma 3.2 we obtain that and
(4.12)

Moreover, since is increasing with respect to on we obtain from (4.12) that for Hence for and hence also for

5. Applications and Open Problems

(i)Theorems 4.1 and 4.2, as formulated in the previous section, apply only to positive decreasing (or negative increasing) solutions of (1.1). The reason is that we have been able to prove inequalities (3.9), (3.10) only when . We conjecture that Theorems 4.1 and 4.2 remain to hold forevery solution of (1.1) for which for large . To justify this conjecture, consider the function
(5.1)
By an easy computation one can find that inequalities (3.9), (3.10) are equivalent to the inequalities
(5.2)
However, if , that is, , we have
(5.3)

so inequalities (3.9), (3.10) are no longer valid in this case. Numerical computations together with a closer examination of the graph of the function lead to the following conjecture.

Conjecture 5.1.

Let , , and Then for one has
(5.4)

To explain this conjecture in more details, consider the case , the case can be treated analogically. We have (we skip the index , only indices different from are written explicitly)

(5.5)

where If , the direct substitution yields

(5.6)
If , then and we proceed as follows. For , the function is concave for nonnegative arguments, so for we have the inequality
(5.7)

We substitute , , then , that is, . Hence we have

(5.8)

Hence

(5.9)

Next we prove that for . Denote , then we need to prove the inequality for . A standard investigation of the graph of the function shows that the required inequality really holds, so we have

(5.10)

By a similar computation we find that

(5.11)

These computations lead to the conjecture that attains its global minimum at a point in if and at a point in if . Numerical computations suggest that this minimum is , where .

Having proved inequalities (5.4), Theorems 4.1 and 4.2 could be proved for any positive with in the same way as in the previous section, it is only sufficient to replace by .

(ii)A typical example of (1.1) to which Theorems 4.1 and 4.2 apply is (1.1) with

(5.12)

since under these assumption all positive solutions of (1.1) are decreasing, see [19]. However, one can applyindirectly Theorems 4.1 and 4.2 also to (1.1) with

(5.13)

(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.

Suppose that in (1.1) and let . Then by a direct computation one can verify that solves the so-calledreciprocal equation:

(5.14)

Moreover, if does not change its sign for large , (1.1) is nonoscillatory if and only if (5.14) is nonoscillatory, see [9]. The following statement relates recessive solutions of (1.1) and (5.14). A similar statement can be found in [9], but our proof differs from that given in [9].

Theorem 5.2.

Suppose that (1.1) is nonoscillatory and (5.12) or (5.13) holds. If a solution of (1.1) is recessive, then is the recessive solution of (5.14).

Proof.

First suppose that (5.13) holds and let be the distinguished solution of (2.1). Assumption (5.13) implies that for large , see [7]. The solution of the Riccati equation
(5.15)
associated with (5.14) is given by and we have the following relationship between solutions of (5.15) and (2.1) (no index means again the index ):
(5.16)
Since the function
(5.17)
is increasing for the inequality for large and for any solution of (2.1) implies the inequality where
(5.18)

and is any other solution of (5.15). Consequently, is the distinguished solution of (5.15) and hence is the recessive solution of (5.14).

Now suppose that (5.12) holds. Then all solutions of (2.1) satisfying for large are negative (see [19]), 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 [18], we posed the question whether the sequence is the recessive solution of the difference equation

(5.19)

Now we can give the affirmative answer to this question for . It is shown in [18] that

(5.20)

both as . The sequence is a solution of the equation

(5.21)

which is reciprocal to (5.19) and is a solution of the equation

(5.22)

which is reciprocal to (5.21) and differs from (5.19) only by the shift in the sequence . Since

(5.23)

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

(5.24)

This means, by Theorem 4.1, that if , then is the recessive solution of (5.21) and hence is the recessive solution of (5.22). Consequently, is the recessive solution of (5.19) if .

Declarations

Acknowledgments

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.

Authors’ Affiliations

(1)
Department of Mathematics and Statistics, Masaryk University
(2)
Department of Mathematics, Mendel University of Agriculture and Forestry in Brno

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Copyright

© O. Došlý and S. Fišnarová. 2009

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.