We study oscillation properties of -order Sturm-Liouville difference equations. For these equations, we show a conjugacy criterion using the -criticality (the existence of linear dependent recessive solutions at and ). We also show the equivalent condition of -criticality for one term -order equations.

1. Introduction

In this paper, we deal with -order Sturm-Liouville difference equations and operators where is the forward difference operator, that is, , and , are real-valued sequences. The main result is the conjugacy criterion which we formulate for the equation , that is viewed as a perturbation of (1.1), and we suppose that (1.1) is at least -critical for some . The concept of -criticality (a disconjugate equation is said to be -critical if and only if it possesses solutions that are recessive both at and , see Section 3) was introduced for second-order difference equations in [1], and later in [2] for (1.1). For the continuous counterpart of the used techniques, see [35] from where we get an inspiration for our research.

The paper is organized as follows. In Section 2, we recall necessary preliminaries. In Section 3, we recall the concept of -criticality, as introduced in [2], and show the first result—the equivalent condition of -criticality for the one term difference equation (Theorem 3.4). In Section 4 we show the conjugacy criterion for equation and Section 5 is devoted to the generalization of this criterion to the equation with the middle terms

2. Preliminaries

The proof of our main result is based on equivalency of (1.1) and the linear Hamiltonian difference systems where , and are matrices of which and are symmetric. Therefore, we start this section recalling the properties of (2.1), which we will need later. For more details, see the papers [611] and the books [12, 13].

The substitution transforms (1.1) to linear Hamiltonian system (2.1) with the matrices , and given by Then, we say that the solution of (2.1) is generated by the solution of (1.1).

Let us consider, together with system (2.1), the matrix linear Hamiltonian system where the matrices , and are also given by (2.3). We say that the matrix solution of (2.4) is generated by the solutions of (1.1) if and only if its columns are generated by , respectively, that is, . Reversely, if we have the solution of (2.4), the elements from the first line of the matrix are exactly the solutions of (1.1). Now, we can define the oscillatory properties of (1.1) via the corresponding properties of the associated Hamiltonian system (2.1) with matrices , and given by (2.3), for example, (1.1) is disconjugate if and only if the associated system (2.1) is disconjugate, the system of solutions is said to be recessive if and only if it generates the recessive solution of (2.4), and so forth. Therefore, we define the following properties just for linear Hamiltonian systems.

For system (2.4), we have an analog of the continuous Wronskian identity. Let and be two solutions of (2.4). Then, holds with a constant matrix . We say that the solution of (2.4) is a conjoined basis, if Two conjoined bases of (2.4) are called normalized conjoined bases of (2.4) if in (2.5) (where denotes the identity operator).

System (2.1) is said to be right disconjugate in a discrete interval , , if the solution of (2.4) given by the initial condition , satisfies for , see [6]. Here ker, , and ≥ stand for kernel, Moore-Penrose generalized inverse, and nonnegative definiteness of the matrix indicated, respectively. Similarly, (2.1) is said to be left disconjugate on , if the solution given by the initial condition , satisfies System (2.1) is disconjugate on , if it is right disconjugate, which is the same as left disconjugate, see [14, Theorem 1], on for every , . System (2.1) is said to be nonoscillatory at (nonoscillatory at ), if there exists such that it is right disconjugate on for every (there exists such that (2.1) is left disconjugate on for every ).

We call a conjoined basis of (2.4) the recessive solution at , if the matrices are nonsingular, (both for large ), and for any other conjoined basis , for which the (constant) matrix is nonsingular, we have The solution is called the dominant solution at . The recessive solution at is determined uniquely up to a right multiple by a nonsingular constant matrix and exists whenever (2.4) is nonoscillatory and eventually controllable. (System is said to be eventually controllable if there exist such that for any the trivial solution of (2.1) is the only solution for which .) The equivalent characterization of the recessive solution of eventually controllable Hamiltonian difference systems (2.1) is see [12]. Similarly, we can introduce the recessive and the dominant solutions at . For related notions and results for second-order dynamic equations, see, for example, [15, 16].

We say that a pair is admissible for system (2.1) if and only if the first equation in (2.1) holds.

The energy functional of (1.1) is given by Then, for admissible , we have To prove our main result, we use a variational approach, that is, the equivalency of disconjugacy of (1.1) and positivity of ; see [6].

Now, we formulate some auxiliary results, which are used in the proofs of Theorems 3.4 and 4.1. The following Lemma describes the structure of the solution space of

Lemma 2.1 (see [17, Section 2]). Equation (2.13) is disconjugate on and possesses a system of solutions , , such that as , where as for a pair of sequences means that . If (2.14) holds, the solutions form the recessive system of solutions at , while form the dominant system, . The analogous statement holds for the ordered system of solutions as .

Now, we recall the transformation lemma.

Lemma 2.2 (see [14, Theorem 4]). Let and consider the transformation . Then, one has where that is, solves if and only if solves the equation

The next lemma is usually called the second mean value theorem of summation calculus.

Lemma 2.3 (see [17, Lemma 3.2]). Let and the sequence be monotonic for (i.e., does not change its sign for ). Then, for any sequence there exist such that

Now, let us consider the linear difference equation where for some and , and let us recall the main ideas of [18] and [19, Chapter IX].

An integer is said to be a generalized zero of multiplicity of a nontrivial solution of (2.19) if , and . Equation (2.19) is said to be eventually disconjugate if there exists such that no non-trivial solution of this equation has or more generalized zeros (counting multiplicity) on .

A system of sequences is said to form the D-Markov system of sequences for if Casoratians are positive on .

Lemma 2.4 (see [19, Theorem 9.4.1]). Equation (2.19) is eventually disconjugate if and only if there exist and solutions of (2.19) which form a D-Markov system of solutions on . Moreover, this system can be chosen in such a way that it satisfies the additional condition

3. Criticality of One-Term Equation

Suppose that (1.1) is disconjugate on and let and , , be the recessive systems of solutions of at and , respectively. We introduce the linear space

Definition 3.1 (see [2]). Let (1.1) be disconjugate on and let . Then, we say that the operator (or (1.1)) is -critical on . If , we say that is subcritical on . If (1.1) is not disconjugate on , that is, , we say that is supercritical on .

To prove the result in this section, we need the following statements, where we use the generalized power function For reader's convenience, the first statement in the following lemma is slightly more general than the corresponding one used in [2] (it can be verified directly or by induction).

Lemma 3.2 (see [2]). The following statements hold.
(i) Let be any sequence, , and
(ii) The generalized power function has the binomial expansion

We distinguish two types of solutions of (2.13). The polynomial solutions , for which , and nonpolynomial solutions for which . (Using Lemma 3.2 we obtain .)

Now, we formulate one of the results of [20].

Proposition 3.3 (see [20, Theorem 4]). If for some then that is, (2.13) is at least -critical on .

Now, we show that (3.7) is also sufficient for (2.13) to be at least -critical.

Theorem 3.4. Let . Equation (2.13) is at least -critical if and only if (3.7) holds.

Proof. Let and denote the subspaces of the solution space of (2.13) generated by the recessive system of solutions at and , respectively. Necessity of (3.7) follows directly from Proposition 3.3. To prove sufficiency, it suffices to show that if one of the sums in (3.7) is convergent, then . We show this statement for the sum . The other case is proved similarly, so it will be omitted. Particularly, we show Let us denote , and let us consider the following nonpolynomial solutions of (2.13): where . By Stolz-Cesàro theorem, since (using Lemma 3.2 ) , these solutions are ordered, that is, , as well as the polynomial solutions, that is, .
By some simple calculation and by Lemma 3.2 (at first, we use (i), and at the end, we use (ii)), we have Hence, from this and by Stolz-Cesàro theorem, we get thus . We obtained that , which means that we have solutions less than , therefore and (2.13) is at most -critical.

4. Conjugacy of Two-Term Equation

In this section, we show the conjugacy criterion for two-term equation.

Theorem 4.1. Let , be a real-valued sequence, and let there exist an integer and real constants such that (2.13) is at least -critical and the sequence satisfies If , then is conjugate on .

Proof. We prove this theorem using the variational principle; that is, we find a sequence such that the energy functional .
At first, we estimate the first term of . To do this, we use the fact that this term is an energy functional of (2.13). Let us denote it by that is, Using the substitution (2.2), we find out that (2.13) is equivalent to the linear Hamiltonian system (2.1) with the matrix ; that is, and to the matrix system Now, let us denote the recessive solutions of (4.5) at and by and , respectively, such that the first columns of and are generated by the sequences . Let , and be arbitrary integers such that , and (some additional assumptions on the choice of will be specified later), and let and be the solutions of (4.4) given by the formulas where and is the solution of (4.4) generated by . By a direct substitution, and using the convention that , we obtain Now, from (4.1), together with the assumption , we have that there exist and such that . Because the numbers , and have been “almost free” so far, we may choose them such that .
Let us introduce the test sequence where To finish the first part of the proof, we use (4.4) to estimate the contribution of the term Using the definition of the test sequence , we can split into three terms. Now, we estimate two of them as follows. Using (4.4), we obtain where we used the fact that (recall that the last entries of are zeros and that the first columns of and are generated by the solutions ). Similarly, Using property (2.10) of recessive solutions of the linear Hamiltonian difference systems, we can see that as and as . We postpone the estimation of the middle term of to the end of the proof.
To estimate the second term of , we estimate at first its terms For this estimation, we use Lemma 2.3. To do this, we have to show the monotonicity of the sequences Let be the ordered system of solutions of (2.13) in the sense of Lemma 2.1. Then, again by Lemma 2.1, there exist real numbers such that . Because , at least one coefficient is nonzero. Therefore, we can denote , and we replace the solution by . Let us denote this new system again and note that this new system has the same properties as the original one.
Following Lemma 2.2, we transform (2.13) via the transformation , into that is, possesses the fundamental system of solutions Now, let us compute the Casoratians Hence, form the D-Markov system of sequences on , for sufficiently large. Therefore, by Lemma 2.4, (4.17) is eventually disconjugate; that is, it has at most generalized zeros (counting multiplicity) on . The sequence is a solution of (4.17), and we have that this sequence has generalized zeros of multiplicity both at and at ; that is, Moreover, and . Hence, . We can proceed similarly for the sequence .
Using Lemma 2.3, we have that there exist integers such that Finally, we estimate the remaining term of . By (4.9), we have Altogether, we have where for sufficiently small is , for sufficiently large is , and, from (4.1), for and . Therefore, which means that for sufficiently small, and (4.2) is conjugate on .

5. Equation with the Middle Terms

Under the additional condition for large , and by combining of the proof of Theorem 4.1 with the proof of [2, Lemma 1], we can establish the following criterion for the full -order equation.

Theorem 5.1. Let , be a real-valued sequence, and let there exist an integer and real constants such that (1.1) is at least -critical and the sequence satisfies If for large and , then is conjugate on .

Remark 5.2. Using Theorem 3.4, we can see that the statement of Theorem 4.1 holds if and only if (3.7) holds. Finding a criterion similar to Theorem 3.4 for (1.1) is still an open question.

Remark 5.3. In the view of the matrix operator associated to (1.1) in the sense of [21], we can see that the perturbations in Theorem 4.1 affect the diagonal elements of the associated matrix operator. A description of behavior of (1.1), with regard to perturbations of limited part of the associated matrix operator (but not only of the diagonal elements), is given in [2].


The research was supported by the Czech Science Foundation under Grant no. P201/10/1032.