Oscillation behavior of second-order self-adjoint $q$-difference equations

1.   Introduction

The theory of $q$-difference equations and related $q$-calculus has a long history, dating back to the work of Jackson in 1910 ^[1]. Recently, studies have gained renewed interest due to their connections with applications in several research areas such as quantum mechanics ^[2], computational biology ^[3], and financial mathematics ^[4]. A comprehensive exploration of fundamental aspects of $q$-calculus can be found in ^[5,6]. Concerning the asymptotic properties and the oscillation of solutions of $q$-difference equations, we may refer to ^{[7,8,9,10,11]} and the references cited therein. On the other hand, $q$-difference equations may be viewed as a special case of dynamic equations on time scales. Recent studies on the oscillation problem of such equations and their connections to the $q$-difference equations can be found in ^[12,13]. To this end, we should note that the strength of the present work lies in the absence of any sign condition on the function $p(t)$ in Eq (1.1), and its applicability to the canonical and noncanonical cases.

We study the oscillation of a second-order linear $q$-difference equation

by exploring a connection with the pioneering work of Leighton ^[14]. Here, $q > 1$ is a real number, $t$ and $t_0$ belong to the set $q^{\mathbb N} = \{q^k: \; k\in \mathbb N\}$ of discrete points, and the $q$-derivative, also known as the Jackson derivative ^[5], of a function $x$ at a point $t\in q^{\mathbb N}$ is defined by

Obviously, if $x$ is differentiable at $t$, then $D_q x(t) = x'(t)$ as $q\to 1^+$. Therefore, Eq (1.1) can be considered the $q$ analog of the second-order ordinary differential equation

As usual, a nontrivial solution of (1.1) or (1.3) is called oscillatory if it is neither eventually positive nor eventually negative in the neighborhood of infinity, and the equation is said to be oscillatory if all its solutions are oscillatory.

Leighton's famous oscillation theorem is the following:

Theorem 1.1 (^[14]). Let $a$ and $p$ be continuous functions on $[t_0, \infty)$ with $a(t) > 0$. If

then Eq (1.3) is oscillatory.

Before we state the $q$-version of Theorem 1.1, we need to recall some background material from $q$-calculus. The $q$ analog of a number $\alpha$, denoted by $[\alpha]_q$, is defined as

Using this and the definition of the $q$-derivative formula (1.1), the so-called power rule becomes:

On the other hand, the $q$-definite integral of a function $f: q^{\mathbb N} \to \mathbb R$ is defined by

Improper integrals are defined as usual by the limits of corresponding proper integrals.

It follows that

and that the improper integral

is convergent if and only if $\alpha > 1$.

The quotient and product rules of $q$-derivatives read as

and

where $f^\sigma = f\circ \sigma$, $\sigma = q\, t$.

In view of the above notations, we now state the $q$ version of the famous Leighton oscillation theorem.

Theorem 1.2 (^[15]). Let $a, p:\, q^{\mathbb N}\to \mathbb R$ with $a(t) > 0$. If

then the $q$-difference equation (1.1) is oscillatory.

Our focus in this work will be on condition (1.10), which cannot be satisfied for most equations. As an example, let us consider the simple second-order $q$-Euler equation

Using (1.6), one can easily see that

which means Theorem 1.2 fails to apply. On the other hand, it is known (see ^[15]) that the $q$-difference equation (1.11) is oscillatory if $c > 1$. Motivated by the above example, our aim in this work is to find an alternative theorem or an improvement that can be applied when condition (1.10) fails. Indeed, we will allow

and

We should note that there is already a large body of research papers in the literature in which the authors extend, generalize, and improve Theorem 1.1, as well as provide alternative theorems when condition (1.4) does not hold; see ^{[16,17,18,19,20,21,22]} and the references therein. Our methodology in this study is fundamentally inspired by the approach utilized in ^[16] to explore the oscillation and boundedness of solutions of (1.3), where the author used an equivalence transformation $x = uz$. Note that if the arbitrary function $u$ is positive within a designated interval of interest, then the zeros of the functions $x$ and $z$ coincide under this transformation.

2.   Main results

Our main theorem is as follows:

Theorem 2.1. Let $a, p:\, q^{\mathbb N}\to \mathbb R$ with $a(t) > 0$. If there exists a function $u:\, q^{\mathbb N}\to (0, \infty)$ such that

and

then the $q$-difference equation (1.1) is oscillatory.

Proof. Let $x$ be a solution of (1.1). Define $z = x/u$. Clearly, $x$ is oscillatory if and only if $z$ is oscillatory. Applying the quotient rule (1.8), we write

and so,

Using the product rule (1.9), we see that

This leads to the second-order $q$-difference equation

where

Using the conditions (2.1) and (2.2), it follows from Theorem 1.2 that Eq (2.4) is oscillatory. Thus the $q$-difference equation (1.1) is also oscillatory. □

Remark 2.1. If $u\equiv 1$, then Theorem 2.1 reduces to Theorem 1.2.

Let us re-consider (1.11). Put $u(t) = t^{1/2}$. Then, from (2.5) and (2.6),

Moreover, if $c > 1$, then $p_1(t)$ becomes positive for all $q > 1$. In view of (1.7), we compute

and

Thus, the conditions (2.1) and (2.2) of Theorem 2.1 are satisfied. Therefore, we may also conclude that (1.11) is oscillatory if $c > 1$. Indeed, we only need in (2.7) that

for each fixed $q > 1$. This is an improvement over condition $c > 1$. For example, if $q = 4$, then $c > 1/3$ is sufficient for the oscillation. In Figure 1, we display the graph of an exemplary solution $x(t)$ of (1.11) with initial conditions $x(0) = 1$ and $x(1) = 2$ to illustrate its oscillation behavior on different intervals, where $q = 4$ and $c = 2/3 < 1$.

The next result is a Leighton-type theorem under condition (1.13). In this case, we may define

Theorem 2.2. Let (1.13) hold. If

then the $q$-difference equation (1.1) is oscillatory.

Proof. It suffices to show that the conditions of Theorem 2.1 are satisfied for $u = \phi$. First, we calculate that

and so, using the definition (1.5), we have $D_q \phi(t) = -1/a(t)$. Then, $D_q\big[a(t) D_q \phi(t)\big] = 0,$ and hence,

Using (2.9), we see that (2.2) is valid. We only need to show that (2.1) is satisfied as well. Indeed,

□

Example 2.1. Consider the equation

where $r$ and $k$ are real numbers such that $r > 1$ and $2r+k\leq 3$.

Note that $a(t) = t^r$ and $p(t) = 1/t^{k}$. Thus, by (1.7), we see that

Also, we calculate from (2.8) that $\phi(t) = c_1 t^{r-1}$, where $c_1 = 1/[1-r]_q$. So, we also have

Since the conditions of Theorem 2.2 are all satisfied, we may deduce that the $q$-difference equation (2.10) is oscillatory. Note that Theorem 1.2 does not apply to (2.10).

Example 2.2. Consider the equation

where $r < 1$, $k > 1$, and $\lambda > 0$ are real numbers.

In view of (1.7), we see that

and

This means that Theorem 1.2 fails for (2.11). Setting $u(t) = t^m$ with $m\geq k-1$, we obtain from (2.5) and (2.6) that

It follows that the conditions of Theorem 2.1 are all satisfied if

or

Let $\varepsilon, \delta > 0$ with $\delta > \varepsilon$, and put $k = 1+\varepsilon$ and $r = 1-\delta$. It is easily seen that (2.12) is satisfied with $m = (\varepsilon+\delta)/4.$ Therefore, we deduce from Theorem 2.1 that the $q$-difference equation (2.11) is oscillatory. It should be noted that neither (2.12) nor (2.13) holds when $r > 1$ and $k > 1$.

3.   Conclusions

In this study, we have explored Leighton's well-known oscillation theorem under the condition (1.4) of being unsatisfied. Through the application of an equivalence transformation, we have demonstrated the possibility of an alternative theorem. Furthermore, we have provided examples to illustrate the results under critical conditions (1.13) and (1.14). Example 2 also highlights the unresolved nature of the problem when both (1.13) and (1.14) are simultaneously satisfied. As a direction for future research, we encourage the reader to investigate the feasibility of obtaining an oscillation theorem similar to Theorem 2.2 under the condition (1.14). It appears that such research and related ones would further contribute to the advancement of the oscillation theory of $q$-difference equations.

Author contributions

All authors contributed equally. All authors have read and approved the final version of the manuscript for publication.

Use of AI tools declaration

The authors explicitly state that they did not utilize any Artificial Intelligence (AI) tools during the development of this article.

Acknowledgments

The authors are grateful to the editor and referees for their invaluable comments that helped improve the paper.

Conflict of interest

The authors declare that they have no conflicts of interest.