# On homogeneous second order linear general quantum difference equations

## Abstract

In this paper, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations

$$a_{0}(t)D_{\beta}^{2}y(t)+a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t),\quad t \in I,$$

$$a_{0}(t)\neq0$$, in a neighborhood of the unique fixed point $$s_{0}$$ of the strictly increasing continuous function β, defined on an interval $$I\subseteq{\mathbb{R}}$$. These equations are based on the general quantum difference operator $$D_{\beta}$$, which is defined by $$D_{\beta}{f(t)}= (f(\beta(t))-f(t) )/ (\beta(t)-t )$$, $$\beta(t)\neq t$$. We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation.

## 1 Introduction

Quantum calculus allows us to deal with sets of non-differentiable functions by substituting the classical derivative by a difference operator. Non-differentiable functions are used to describe many important physical phenomena. Quantum calculus has a lot of applications in different mathematical areas such as the calculus of variations, orthogonal polynomials, basic hyper-geometric functions, economical problems with a dynamic nature, quantum mechanics and the theory of scale relativity; see, e.g., [19]. The general quantum difference operator $$D_{\beta}$$ is defined, in [10, p.6], by

$${D}_{\beta}f(t)=\left \{ \textstyle\begin{array}{l@{\quad}l} \frac{f(\beta(t))-f(t)}{\beta(t)-t},&{t}\neq{s_{0}},\\ {{f'}(s_{0})}, &{t}={s_{0}}, \end{array}\displaystyle \right .$$

where $$f:I\rightarrow\mathbb{X}$$ is a function defined on an interval $$I\subseteq{\mathbb{R}}$$, $$\mathbb{X}$$ is a Banach space and $$\beta :I\rightarrow I$$ is a strictly increasing continuous function defined on I, which has only one fixed point $$s_{0}\in{I}$$ and satisfies the inequality: $$(t-s_{0})(\beta(t)-t)\leq0$$ for all $$t\in I$$. The function f is said to be β-differentiable on I, if the ordinary derivative $${f'}$$ exists at $$s_{0}$$. The β-difference operator yields the Hahn difference operator when $$\beta(t)=qt+\omega$$, $$\omega>0$$, $$q \in(0,1)$$, and the Jackson q-difference operator when $$\beta(t)=qt$$, $$q \in(0,1)$$; see [1116]. In [10], [17, Chapter 2], the definition of the β-derivative, the β-integral, the fundamental theorem of β-calculus, the chain rule, Leibniz’s formula and the mean value theorem were introduced. In [18], the β-exponential, β-trigonometric and β-hyperbolic functions were presented. In [19], the existence and uniqueness of solutions of the β-initial value problem of the first order were established. In addition, an expansion form of the β-exponential function was deduced.

This paper is devoted for deducing some results of the solutions of the homogeneous second order linear β-difference equations which are based on $$D_{\beta}$$. In Section 2, we introduce the needed preliminaries of the β-calculus from [10, 1719]. In Section 3, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations in a neighborhood of $$s_{0}$$. We also construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we drive the Euler-Cauchy β-difference equation. Throughout this paper, J is a neighborhood of the unique fixed point $$s_{0}$$ of β and $$\mathbb{X}$$ is a Banach space. If f is β-differentiable two times over I, then the second order derivative of f is denoted by $$D_{\beta}^{2}f=D_{\beta}(D_{\beta}f)$$. Furthermore, $$S(y_{0}, b)=\{y\in \mathbb{X}:\|y-y_{0}\|\leq b\}$$ and the rectangle $$R=\{(t,y)\in{{I}\times \mathbb{X}}:|t-s_{0}|\leq{a},\|y-y_{0}\|\leq{b}\}$$, where a, b are fixed positive real numbers.

## 2 Preliminaries

In this section, we present some needed results associated with the β-calculus from [10, 1719].

### Lemma 2.1

The following statements are true:

1. (i)

The sequence of functions $$\{\beta^{k}(t)\}_{k=0}^{\infty}$$ converges uniformly to the constant function $$\hat{\beta}(t):=s_{0}$$ on every compact interval $$V \subseteq I$$ containing $$s_{0}$$.

2. (ii)

The series $$\sum_{k=0}^{\infty}|\beta^{k}(t)-\beta^{k+1}(t)|$$ is uniformly convergent to $$|t-s_{0}|$$ on every compact interval $$V \subseteq I$$ containing $$s_{0}$$.

### Lemma 2.2

If $$f:I\rightarrow\mathbb{X}$$ is a continuous function at $$s_{0}$$, then the sequence $$\{f(\beta^{k}(t))\}_{k=0}^{\infty}$$ converges uniformly to $$f(s_{0})$$ on every compact interval $$V\subseteq I$$ containing $$s_{0}$$.

### Theorem 2.3

If $$f:I\rightarrow\mathbb{X}$$ is continuous at $$s_{0}$$, then the series $$\sum_{k=0}^{\infty}\| (\beta^{k}(t)-\beta^{k+1}(t) ) f(\beta ^{k}(t))\|$$ is uniformly convergent on every compact interval $$V \subseteq I$$ containing $$s_{0}$$.

### Lemma 2.4

Let $$f:{I}\rightarrow\mathbb{X}$$ be β-differentiable and $${D}_{\beta}f(t)=0$$ for all $$t\in{I}$$. Then $$f(t)=f(s_{0})$$ for all $$t\in{I}$$.

### Theorem 2.5

Assume that $$f:{I}\rightarrow\mathbb {X}$$ and $$g:{I}\rightarrow\mathbb{R}$$ are β-differentiable functions on I. Then:

1. (i)

the product $$fg:I\rightarrow\mathbb{X}$$ is β-differentiable on I and

\begin{aligned} {D}_{\beta}(fg) (t) &=\bigl({D}_{\beta}f(t) \bigr)g(t)+f\bigl(\beta(t)\bigr){D}_{\beta}g(t) \\ & =\bigl({D}_{\beta}f(t)\bigr)g\bigl(\beta(t)\bigr)+f(t){D}_{\beta}g(t), \end{aligned}
2. (ii)

$$f/g$$ is β-differentiable at t and

$${D}_{\beta} ({f}/{g} ) (t)=\frac{({D}_{\beta }f(t))g(t)-f(t){D}_{\beta}g(t)}{g(t)g(\beta(t))},$$

provided that $$g(t)g(\beta(t))\neq{0}$$.

### Theorem 2.6

Assume $$f:{I}\to\mathbb{X}$$ is continuous at $$s_{0}$$. The function F defined by

$$F(t)=\sum_{k=0}^{\infty} \bigl( \beta^{k}(t)-\beta^{k+1}(t) \bigr)f\bigl(\beta^{k}(t) \bigr),\quad t\in{I}$$
(2.1)

is a β-antiderivative of f with $$F(s_{0})=0$$. Conversely, a β-antiderivative F of f vanishing at $$s_{0}$$ is given by (2.1).

### Definition 2.7

Let $$f:{I}\rightarrow{\mathbb{X}}$$ and $$a,b\in{I}$$. The β-integral of f from a to b is

$$\int^{b}_{a}f(t)\,d_{\beta}{t}= \int^{b}_{s_{0}}f(t)\,d_{\beta}{t}- \int ^{a}_{s_{0}}f(t)\,d_{\beta}{t},$$

where

$$\int^{x}_{s_{0}}f(t)\,d_{\beta}{t}=\sum ^{\infty}_{k=0} \bigl(\beta^{k}(x)- \beta^{k+1}(x) \bigr)f\bigl(\beta^{k}(x)\bigr),\quad x\in{I},$$

provided that the series converges at $$x=a$$ and $$x=b$$. f is called β-integrable on I if the series converges at a and b for all $$a,b\in{I}$$. Clearly, if f is continuous at $$s_{0}\in{I}$$, then f is β-integrable on I.

### Definition 2.8

The β-exponential functions $$e_{p,\beta}(t)$$ and $$E_{p,\beta}(t)$$ are defined by

\begin{aligned} e_{p,\beta}(t)=\frac{1}{\prod_{k=0}^{\infty }[1-p(\beta^{k} (t))(\beta^{k}(t)-\beta^{k+1}(t))]} \end{aligned}
(2.2)

and

\begin{aligned} E_{p,\beta}(t)=\prod_{k=0}^{\infty} \bigl[1+ p\bigl(\beta^{k}(t)\bigr) \bigl(\beta^{k} (t) - \beta^{k+1}(t) \bigr) \bigr], \end{aligned}
(2.3)

where $$p:I \rightarrow\mathbb{C}$$ is a continuous function at $$s_{0}$$ and both infinite products are convergent to a non-zero number for every $$t\in I$$ and $$e_{p,\beta}(t)=\frac {1}{E_{p,\beta}(t)}$$.

It is worth mentioning that both products in (2.2) and (2.3) are convergent since $$\sum_{k=0}^{\infty} | p(\beta^{k}(t)) (\beta^{k}(t)-\beta ^{k+1}(t) ) |$$ is uniformly convergent. See [18, Definition 2.1].

### Theorem 2.9

The β-exponential functions $$e_{p,\beta}(t)$$ and $$E_{-p,\beta }(t)$$ are, respectively, the unique solutions of the β-initial value problems:

$$\begin{gathered} D_{\beta}y(t)= p(t)y(t),\quad y(s_{0})=1, \\ D_{\beta}y(t)=-p(t)y\bigl(\beta(t)\bigr), \quad y(s_{0})=1.\end{gathered}$$

### Theorem 2.10

Assume that $$p,q:I\rightarrow\mathbb {C}$$ are continuous functions at $$s_{0}\in I$$. The following properties are true:

1. (i)

$$\frac{1}{e_{p,\beta}(t)}=e_{-p/[1+(\beta(t)-t)p]}(t)$$,

2. (ii)

$$e_{p,\beta}(t)e_{q,\beta}(t)=e_{p+q+(\beta(t)-t)pq}(t)$$,

3. (iii)

$$e_{p,\beta}(t)/e_{q,\beta}(t)=e_{(p-q)/[1+(\beta(t)-t)q]}(t)$$.

### Definition 2.11

The β-trigonometric functions are defined by

$$\begin{gathered} \cos_{p,\beta}(t)=\frac{ e_{ip,\beta}(t)+e_{-ip,\beta }(t)}{2}, \\ \sin_{p,\beta}(t)=\frac{e_{ip,\beta} (t)-e_{-ip,\beta}(t)}{2i}. \end{gathered}$$

### Theorem 2.12

For all $$t\in I$$. The following relation holds true:

\begin{aligned} e_{ip,\beta}(t)=\cos_{p,\beta}(t)+i\sin_{p,\beta}(t). \end{aligned}

### Theorem 2.13

Assume that the function $$f:R\rightarrow {\mathbb{X}}$$ is continuous at $$(s_{0},y_{0})\in{R}$$ and satisfies the Lipschtiz condition (with respect to y)

$$\big\| f(t,y_{1})-f(t,y_{2})\big\| \leq{L}\|y_{1}-y_{2} \|, \quad\textit{for all } (t,y_{1}), (t,y_{2})\in{R}.$$

Then the β-initial value problem $$D_{\beta}{y(t)}=f(t,y)$$, $$y(s_{0})=y_{0}$$, $$t\in{I}$$ has a unique solution on $$[s_{0}-\delta,s_{0}+\delta]$$, where L is a positive constant and $$\delta=\min\{a,\frac{b}{Lb+M},\frac{\rho}{L}\}$$ with $$M=\sup_{(t,y)\in{R}}\|f(t,y)\|<\infty$$, $$\rho\in(0,1)$$.

## 3 Main results

In this section, we prove the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations in a neighborhood of $$s_{0}$$. Furthermore, we construct a fundamental set of solutions for the second order linear homogeneous β-difference equations when the coefficients are constants and study the different cases of the roots of their characteristic equations. Finally, we derive the Euler-Cauchy β-difference equation.

### Theorem 3.1

Let $$f_{i}(t,y_{1},y_{2}):I \times\prod_{i=1}^{2} S_{i}(x_{i}, b_{i})\rightarrow{\mathbb{X}}$$, $$s_{0}\in I$$, such that the following conditions are satisfied:

1. (i)

for $$y_{i}\in S_{i}(x_{i},b_{i})$$, $$i=1,2$$, $$f_{i}(t,y_{1},y_{2})$$ are continuous at $$t=s_{0}$$,

2. (ii)

there is a positive constant A such that, for $$t\in I$$, $$y_{i}, \tilde{y}_{i}\in S_{i}(x_{i},b_{i})$$, $$i=1,2$$, the following Lipschitz condition is satisfied:

$$\big\| f_{i}(t,y_{1},y_{2})-f_{i}(t, \tilde{y}_{1},\tilde{y}_{2})\big\| \leq A \sum _{i=1}^{2}\|y_{i}-\tilde{y}_{i} \|.$$

Then there exists a unique solution of the β-initial value problem, β-IVP,

$$D_{\beta}y_{i}(t)=f_{i}\bigl(t,y_{1}(t),y_{2}(t) \bigr),\quad y_{i}(s_{0})=x_{i}\in {\mathbb{X}}, i =1,2, t \in I.$$
(3.1)

### Proof

Let $$y_{0}=(x_{1},x_{2})^{T}$$ and $$b=(b_{1},b_{2})^{T}$$, where $$(\cdot ,\cdot)^{T}$$ stands for vector transpose. Define the function $$f:I\times \prod_{i=1}^{2}S_{i}(x_{i},b_{i})\rightarrow{\mathbb{X}}\times{\mathbb{X}}$$ by $$f(t,y_{1},y_{2})= (f_{1}(t,y_{1},y_{2}),f_{2}(t, y_{1}, y_{2}) )^{T}$$. It is easy to show that system (3.1) is equivalent to the β-IVP

$$D_{\beta}y(t)=f\bigl(t,y(t)\bigr),\quad y(s_{0})=y_{0}.$$
(3.2)

Since each $$f_{i}$$ is continuous at $$t=s_{0}$$, f is continuous at $$t=s_{0}$$. The function f satisfies the Lipschitz condition because for $$y,\tilde {y}\in\prod_{i=1}^{2}S_{i}(x_{i},b_{i})$$,

\begin{aligned} \big\| f(t,y)-f(t,\tilde{y}) \big\| &= \big\| f(t,y_{1},y_{2})-f(t, \tilde{y}_{1}, \tilde{y}_{2}) \big\| \\ &=\sum_{i=1}^{2} \big\| f_{i}(t,y_{1},y_{2})-f_{i}(t, \tilde{y}_{1},\tilde {y}_{2},) \big\| \\ &\leq A\sum_{i=1}^{2}\|y_{i}- \tilde{y}_{i}\|= A\|y-\tilde{y}\|.\end{aligned}

Applying Theorem 2.13, see the proof in [19], there exists $$\delta>0$$ such that (3.2) has a unique solution on $$[s_{0},s_{0}+\delta]$$. Hence, the β-IVP (3.1) has a unique solution on $$[s_{0},s_{0}+\delta]$$. □

### Corollary 3.2

Let $$f(t,y_{1},y_{2})$$ be a function defined on $$I\times\prod_{i=1}^{2} S_{i}(x_{i},b_{i})$$ such that the following conditions are satisfied:

1. (i)

for any values of $$y_{i}\in S_{i}(x_{i},b_{i})$$, $$i=1,2$$, f is continuous at $$t=s_{0}$$,

2. (ii)

f satisfies the Lipschitz condition

$$\big\| f(t,y_{1},y_{2})-f(t,\tilde{y}_{1}, \tilde{y}_{2}) \big\| \leq A\sum_{i=1}^{2} \|y_{i} -\tilde{y}_{i}\|,$$

where $$A>0$$, $$y_{i},\tilde{y}_{i}\in S_{i}(x_{i},b_{i})$$, $$i=1,2$$ and $$t \in I$$. Then

\begin{aligned} D_{\beta}^{2}y(t)=f\bigl(t,y(t),D_{\beta}y(t)\bigr),\quad D_{\beta}^{i-1}y(s_{0})=x_{i}, i=1,2 \end{aligned}
(3.3)

has a unique solution on $$[s_{0},s_{0} +\delta]$$.

### Proof

Consider equation (3.3). It is equivalent to (3.1), where $$\{\phi_{i}(t)\}_{i=1}^{2}$$ is a solution of (3.1) if and only if $$\phi_{1}(t)$$ is a solution of (3.3). Here,

\begin{aligned} f_{i}(t,y_{1},y_{2})=\left \{ \textstyle\begin{array}{l@{\quad}l} y_{2},& i=1, \\ f (t,y_{1},y_{2}),& i=2. \end{array}\displaystyle \right . \end{aligned}

Hence, by Theorem 3.1, there exists $$\delta>0$$ such that system (3.1) has a unique solution on $$[s_{0},s_{0}+\delta]$$. □

The following corollary gives us the sufficient conditions for the existence and uniqueness of the solutions of the β-Cauchy problem (3.3).

### Corollary 3.3

Assume the functions $$a_{j}(t):I\rightarrow \mathbb{C}$$, $$j=0,1,2$$, and $$b(t):I\rightarrow{\mathbb{X}}$$ satisfy the following conditions:

1. (i)

$$a_{j}(t), j=0,1,2$$ and $$b(t)$$ are continuous at $$s_{0}$$ with $$a_{0}(t)\neq0$$ for all $$t \in I$$,

2. (ii)

$$a_{j}(t)/a_{0}(t)$$ is bounded on I, $$j=1,2$$. Then

$$\begin{gathered} a_{0}(t)D_{\beta}^{2}y(t)+ a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t), \\ D_{\beta}^{i-1}y(s_{0})= x_{i},\quad x_{i} \in{\mathbb{X}}, i=1,2, \end{gathered}$$
(3.4)

has a unique solution on subinterval $$J\subseteq I$$, $$s_{0}\in J$$.

### Proof

Dividing by $$a_{0}(t)$$, we get

$$D_{\beta}^{2}y(t)=A_{1}(t)D_{\beta}y(t)+A_{2}(t)y(t)+B(t),$$
(3.5)

where $$A_{j}(t)=-a_{j}(t)/a_{0}(t)$$ and $$B(t)=b(t)/a_{0}(t)$$. Since $$A_{j}(t)$$ and $$B(t)$$ are continuous at $$t=s_{0}$$, the function $$f(t,y_{1},y_{2})$$, defined by

$$f(t,y_{1},y_{2})=A_{1}(t)y_{2}+A_{2}(t)y_{1}+B(t),$$

is continuous at $$t=s_{0}$$. Furthermore, $$A_{j}(t)$$ is bounded on I. Consequently, there is $$A>0$$ such that $$|A_{j}(t)|\leq A$$ for all $$t \in I$$. We can see that f satisfies the Lipschitz condition with Lipschitz constant A. Thus, $$f(t,y_{1},y_{2})$$ satisfies the conditions of Corollary 3.2. Hence, there exists a unique solution of (3.5) on J. □

### 3.2 Fundamental solutions of linear homogeneous β-difference equations

The second order homogeneous linear β-difference equation has the form

$$a_{0}(t)D_{\beta}^{2}y(t)+a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=0,\quad t \in I,$$
(3.6)

where the coefficients $$a_{0}(t)\neq0$$, $$a_{j}(t)$$, $$j=1,2$$ are assumed to satisfy the conditions of Corollary 3.3.

### Lemma 3.4

If the function y is a solution of the homogeneous equation (3.6), such that $$y(s_{0})=0$$ and $$D_{\beta}y(s_{0})=0$$, $$s_{0}\in I$$, then $$y(t)=0$$, for all $$t\in J$$.

### Proof

By Corollary 3.3, if $$x_{i}=0$$, $$i=1,2$$ in the β-IVP (3.4), which has a unique solution on J, then y such that $$y(t)=0$$ for all $$t \in J$$ is a unique solution of the β-difference equation (3.6), which satisfies the given initial conditions $$y(s_{0})=0$$, $$D_{\beta}y(s_{0})= 0$$. Hence we have the desired result. □

### Theorem 3.5

The linear combination $$c_{1}y_{1}+c_{2}y_{2}$$ of any two solutions $$y_{1}$$ and $$y_{2}$$ of the homogeneous linear β-difference equation (3.6) is also a solution of it in J, where $$c_{1}$$ and $$c_{2}$$ are arbitrary constants.

### Proof

The proof is straightforward. □

### Theorem 3.6

Let $$y_{1}$$ and $$y_{2}$$ be any two linearly independent solutions of the β-difference equation (3.6) in J. Then every solution y of (3.6) can be expressed as a linear combination $$y=c_{1}y_{1}+c_{2}y_{2}$$.

### Proof

Let

$$\phi=\left ( \textstyle\begin{array}{c} y\\ D_{\beta}y \end{array}\displaystyle \right ),\qquad \phi_{1}= \left ( \textstyle\begin{array}{c} y_{1}\\ D_{\beta}y_{1} \end{array}\displaystyle \right ),\qquad \phi_{2}=\left ( \textstyle\begin{array}{c} y_{2}\\ D_{\beta}y_{2} \end{array}\displaystyle \right ),$$

be the solutions of the linear system $$D_{\beta}y_{i}(t)=a_{i}(t)y_{i}(t)$$, $$i=1,2$$, corresponding, respectively, to the solutions $$y_{1}$$, $$y_{2}$$ of homogeneous linear β-difference equation (3.6). Since $$y_{1},y_{2}$$ are linearly independent in J, then $$\phi_{1}$$, $$\phi_{2}$$ are linearly independent in J. Then there exist two constants $$c_{1}$$, $$c_{2}$$ such that $$\phi=c_{1}\phi _{1}+c_{2}\phi_{2}$$. The first component of this is $$y=c_{1}y_{1}+c_{2}y_{2}$$. Thus the results hold. □

### Definition 3.7

A set of two linearly independent solutions of the second order homogeneous linear β-difference equation (3.6) is called a fundamental set of it.

### Theorem 3.8

There exists a fundamental set of solutions of the second order homogeneous linear β-difference equation (3.6).

### Proof

By Corollary 3.3, there exist unique solutions $$y_{1}$$ and $$y_{2}$$ of equation (3.6), such that $$y_{1}(s_{0})=1$$, $$D_{\beta}y_{1}(s_{0})=0$$ and $$y_{2}(s_{0})=0$$, $$D_{\beta}y_{2}(s_{0})=1$$.

Suppose that $$y_{1}$$ and $$y_{2}$$ are linear dependent, so there exist constants $$c_{1}$$ and $$c_{2}$$ not both zero, such that

$$\begin{gathered} c_{1}y_{1}(t)+c_{2}y_{2}(t)= 0, \quad\text{for all } t\in J, \\ c_{1}D_{\beta}y_{1}(t)+c_{2}D_{\beta}y_{2}(t)= 0, \quad\text{for all } t\in J. \end{gathered}$$

We have $$c_{1}=c_{2}=0$$ at $$t=s_{0}$$, which is a contradiction. Thus the solutions $$y_{1}$$ and $$y_{2}$$ are linearly independent in J. Then there exists a fundamental set of the two solutions $$y_{1}$$ and $$y_{2}$$ of equation (3.6). □

### Definition 3.9

Let $$y_{1}$$, $$y_{2}$$ be β-differentiable functions. Then we define the β-Wronskian of the functions $$y_{1}$$, $$y_{2}$$, defined on I, by

\begin{aligned} W_{\beta}(y_{1},y_{2}) (t)= \left \vert \textstyle\begin{array}{c@{\quad}c} y_{1}(t)& y_{2}(t)\\ D_{\beta}y_{1}(t)& D_{\beta}y_{2}(t) \end{array}\displaystyle \right \vert ,\quad t\in I. \end{aligned}

### Lemma 3.10

Let $$y_{1}(t)$$, $$y_{2}(t)$$ be functions defined on I. Then, for any $$t\in I$$, $$t \neq s_{0}$$,

$$D_{\beta}W_{\beta}(y_{1},y_{2}) (t)= \left \vert \textstyle\begin{array}{c@{\quad}c} y_{1}(\beta(t)) & y_{2}(\beta(t)) \\ D_{\beta}^{2}y_{1}(t) & D_{\beta}^{2}y_{2}(t). \end{array}\displaystyle \right \vert .$$
(3.7)

### Proof

Since $$W_{\beta}(y_{1},y_{2})(t)=y_{1}(t)D_{\beta}y_{2}(t)-y_{2}(t)D_{\beta}y_{1}(t)$$, then

$$D_{\beta}W_{\beta}(y_{1},y_{2}) (t)=y_{1}\bigl(\beta(t)\bigr)D_{\beta }^{2}y_{2}(t)-y_{2} \bigl(\beta(t)\bigr)D_{\beta}^{2}y_{1}(t),$$

which is the desired result. □

### Theorem 3.11

Assume that $$y_{1}(t)$$ and $$y_{2}(t)$$ are two solutions of equation (3.6). Then their β-Wronskian, $$W_{\beta}$$,

$$W_{\beta}(y_{1},y_{2}) (t)=e_{-r_{1}(t)+r_{2}(t)(\beta(t)-t),\beta} W_{\beta}(y_{1},y_{2} ) (s_{0}),\quad t\in I ,$$

where $$r_{1}(t)=\frac{a_{1}(t)}{a_{0}(t)}$$ and $$r_{2}(t)=\frac{a_{2} (t)}{a_{0}(t)}$$ satisfy the conditions of Corollary 3.3.

### Proof

Since $$y_{1}$$ and $$y_{2}$$ are solutions of equation (3.6), from (3.7) we have

\begin{aligned} D_{\beta}W_{\beta}(y_{1},y_{2}) (t)&= \left \vert \textstyle\begin{array}{c@{\quad}c} y_{1}(\beta(t))&y_{2}(\beta(t)) \\ - \frac{a_{1}(t)}{a_{0}(t)} D_{\beta}y_{1}(t)&- \frac{a_{1} (t)}{a_{0}(t)} D_{\beta}y_{2}(t) \end{array}\displaystyle \right \vert + \left \vert \textstyle\begin{array}{c@{\quad}c} y_{1}(\beta(t))&y_{2}(\beta(t)) \\ - \frac{a_{2} (t)}{a_{0}(t)}y_{1}(t) &-\frac{a_{2}(t)}{a_{0}(t)}y_{2}(t) \end{array}\displaystyle \right \vert \\ &= - \frac{a_{1}(t)}{a_{0}(t)} \left \vert \textstyle\begin{array}{c@{\quad}c} y_{1}(t)&y_{2}(t) \\ D_{\beta}y_{1} (t) &D_{\beta}y_{2}(t) \end{array}\displaystyle \right \vert + \frac{a_{2}(t)}{a_{0}(t)}\bigl(\beta(t)-t\bigr) \left \vert \textstyle\begin{array}{c@{\quad}c} y_{1}(t)&y_{2}(t) \\ D_{\beta}y_{1}(t) &D_{\beta}y_{2}(t) \end{array}\displaystyle \right \vert \\ &= \bigl[-r_{1}(t)+r_{2}(t) \bigl(\beta(t)-t\bigr) \bigr]W_{\beta}(y_{1},y_{2}) (t),\end{aligned}

which has the solution

$$W_{\beta}(y_{1},y_{2}) (t)= W_{\beta}(y_{1},y_{2}) (s_{0})e_{-r_{1}(t)+r_{2}(t)(\beta(t)-t),\beta} ,\quad t \in I.$$

□

Using Theorem 3.11 and Lemma 3.4, we can prove the following corollaries.

### Corollary 3.12

Two solutions $$y_{1}$$ and $$y_{2}$$ of β-difference equation (3.6) are linearly dependent in J if and only if $$W_{\beta}(y_{1},y_{2})(t)=0$$, for all $$t\in J$$.

### Corollary 3.13

The value of $$W_{\beta}(y_{1},y_{2})(t)$$ of β-difference equation (3.6) either is zero or unequal to zero for all $$t \in J$$.

### 3.3 Homogeneous equations with constant coefficients

Equation (3.6) can be written as

$$Ly(t)=aD_{\beta}^{2}y(t)+bD_{\beta}y(t)+cy(t)=0,$$
(3.8)

where a, b, and c are constants. The characteristic polynomial of equation (3.8) is

$$P(\lambda)=a\lambda^{2}+b \lambda+c=0,$$
(3.9)

where $$y(t)=e_{\lambda,\beta}(t)$$ is a solution of equation (3.8). Since equation (3.9) is a quadratic equation with real coefficients, it has two roots, which may be real and different, real but repeated, or complex conjugates.

Case 1: real and different roots of the characteristic equation ( 3.9 ).

Let $$\lambda_{1}$$ and $$\lambda_{2}$$ be real roots with $$\lambda_{1}\neq \lambda_{2}$$, then $$y_{1}(t)=e_{\lambda_{1},\beta}(t)$$ and $$y_{2}(t)=e_{\lambda_{2},\beta }(t)$$ are two solutions of equation (3.8). Therefore,

$$y(t)=c_{1}e_{\lambda_{1},\beta}(t)+c_{2}e_{\lambda_{2},\beta}(t)$$

is a general solution of equation (3.8), with

$$c_{1}=\frac{D_{\beta}y_{0}-y_{0}\lambda_{2}}{\lambda_{1}-\lambda_{2}}e_{-\lambda _{1},\beta}(s_{0})\quad \text{and}\quad c_{2}=\frac{y_{0}\lambda_{1}-D_{\beta}y_{0}}{\lambda _{1}-\lambda_{2}} e_{-\lambda_{2},\beta}(s_{0}).$$

### Example 3.14

Find the solution of the β-initial value problem

\begin{aligned} D_{\beta}^{2}y(t)+5D_{\beta}y(t)+6y(t)=0,\quad y(s_{0})=2, D_{\beta}y(s_{0})=3. \end{aligned}

By assuming that $$y(t)=e_{\lambda,\beta}(t)$$, we obtain the solution

$$y(t)=9e_{-2,\beta}(t)-7e_{-3,\beta}(t).$$

Case 2: complex roots of the characteristic equation ( 3.9 ).

Let $$\lambda_{1}=\nu+i\mu$$ and $$\lambda_{2}=\nu-i\mu$$, where ν and μ are real numbers. Then $$y_{1}(t)=e_{(\nu+i\mu),\beta}(t)$$ and $$y_{2}(t)=e_{(\nu-i\mu),\beta}(t)$$ are two solutions of equation (3.8). By Theorems 2.102.12, $$e_{(\nu+i\mu),\beta }(t)=e_{\nu,\beta}(t) e_{\frac{i\mu}{1+\nu(\beta(t)-t)},\beta}(t)$$. So,

$$e_{(\nu+i\mu),\beta}(t)=e_{\nu,\beta}(t) \bigl(\cos _{\frac{\mu}{1+\nu(\beta(t)-t)},\beta}(t)+i \sin_{\frac{\mu}{1+\nu(\beta (t)-t)},\beta}(t) \bigr).$$

We have

$$y_{1}(t)+y_{2}(t)=2 e_{\nu,\beta}(t) \cos_{\frac{\mu}{1+\nu (\beta(t)-t)},\beta}(t)$$

and

$$y_{1}(t)-y_{2}(t)=2i e_{\nu,\beta}(t) \sin_{\frac{\mu}{1+\nu(\beta (t)-t)},\beta}(t).$$

Therefore,

$$u(t)=e_{\nu,\beta}(t)\cos_{\frac{\mu}{1+\nu(\beta(t)-t)},\beta}(t) \quad\text{and} \quad v(t)= e_{\nu,\beta}(t)\sin_{\frac{\mu}{1+\nu(\beta(t)-t)},\beta} (t)$$

are two solutions of equation (3.8). If the β-Wronskian of u and v is not zero, then u and v form a fundamental set of solutions. The general solution of equation (3.8) is

$$y(t)=c_{1}e_{\nu,\beta}(t)\cos_{\frac{\mu}{1+\nu(\beta(t)-t)},\beta }(t)+c_{2}e_{\nu,\beta}(t) \sin_{\frac{\mu}{1+\nu(\beta(t)-t)},\beta}(t),$$

where $$c_{1}$$ and $$c_{2}$$ are arbitrary constants.

### Example 3.15

Find the general solution of

$$D_{\beta}^{2}y(t)+D_{\beta}y(t)+y(t)=0.$$
(3.10)

The characteristic equation is $$\lambda^{2}+\lambda+1=0$$, and its roots are

$$\lambda_{1,2}=\frac{-1}{2}\pm i\frac{\sqrt{3}}{2}.$$

Thus, the general solution of equation (3.10) is

$$y(t)=c_{1} e_{-1/2,\beta}(t)\cos_{\frac{\sqrt{3}/2}{1-1/2(\beta(t)-t)},\beta }(t)+c_{2}e_{-1/2,\beta}(t) \sin_{\frac{\sqrt{3}/2}{1-1/2(\beta(t)-t)},\beta}(t).$$

Case 3: repeated roots.

Consider the case that the two roots $$\lambda_{1}$$ and $$\lambda_{2}$$ are equal, so

$$\lambda_{1}=\lambda_{2}=-b/2a.$$

Therefore, the solution $$y_{1}(t)=e_{-b/2a,\beta} (t)$$ is one solution of the β-difference equation (3.8), and we give the second solution by the following example:

### Example 3.16

Solve the β-difference equation

$$D_{\beta}^{2}y(t)+4D_{\beta}y(t)+4y(t)=0.$$
(3.11)

The characteristic equation is $$(\lambda+2)^{2}=0$$, so $$\lambda_{1}=\lambda _{2}=-2$$. Therefore, $$y_{1}(t)= e_{-2,\beta}(t)$$ is a solution of equation (3.11). To find the second solution, let $$y(t)=v(t)e_{-2,\beta}(t)$$. Then $${D_{\beta}^{2}v(t)=0}$$. Therefore, $$v(t)=c_{1}t+c_{2}$$, where $$c_{1}$$ and $$c_{2}$$ are arbitrary constants. Then the general solution is

$$y(t)=c_{1}te_{-2,\beta}(t)+c_{2}e_{-2,\beta}(t),$$

where the two solutions $$y_{1}(t)=e_{-2,\beta}(t)$$ and $$y_{2}(t)=te_{-2,\beta}(t)$$ form a fundamental set of solutions of equation (3.11).

### 3.4 Euler-Cauchy β-difference equation

The Euler-Cauchy β-difference equation takes the form

$$t\beta(t)D_{\beta}^{2}y(t)+atD_{\beta}y(t)+by(t)=0,\quad t\in I, t\neq s_{0},$$
(3.12)

where a, b are constants. The characteristic equation of (3.12) is given by

$$\lambda^{2}+(a-1)\lambda+b=0.$$
(3.13)

### Theorem 3.17

If the characteristic equation (3.13) has two distinct roots $$\lambda_{1}$$ and $$\lambda_{2}$$, then a fundamental set of solutions of (3.12) is given by $$e_{\lambda_{1}/t,\beta}(t)$$ and $$e_{\lambda _{2}/t,\beta}(t)$$.

### Proof

Let $$y(t)=e_{\lambda/t,\beta}(t)$$, where λ is a root of equation (3.13). It follows that

$$D_{\beta}y(t)=\frac{\lambda}{t}y(t),\qquad D_{\beta}^{2}y(t)= \frac{\lambda ^{2}-\lambda}{t\beta(t)}y(t).$$

Consequently, we have

$$t\beta(t)D_{\beta}^{2}y(t)+atD_{\beta}y(t)+by(t)=\bigl( \lambda^{2}+(a-1)\lambda +b\bigr)y(t)=0.$$

Assume that $$\lambda_{1}$$ and $$\lambda_{2}$$ are distinct roots of the characteristic equation (3.13). Then, we have

$$\lambda_{1}+\lambda_{2}=1-a,\qquad \lambda_{1} \lambda_{2}=b.$$

Moreover, $$W_{\beta}(e_{\lambda_{1}/t,\beta},e_{\lambda_{2}/t,\beta})(t)\neq 0$$, since $$\lambda_{1}\neq\lambda_{2}$$. Hence, $$e_{\lambda_{1}/t,\beta}(t)$$ and $$e_{\lambda_{2}/t,\beta}(t)$$ form a fundamental set of solutions of (3.12). □

The following theorem gives us the general solution of the Euler-Cauchy β-difference equation in the double root case.

### Theorem 3.18

Assume that $$1/\beta(t)$$ is bounded on I and $$0\notin I$$. Then the general solution of the Euler-Cauchy β-difference equation

$$t\beta(t)D_{\beta}^{2}y(t)+(1-2\gamma)tD_{\beta}y(t)+ \gamma^{2}y(t)=0,\quad t\in I,$$
(3.14)

is given by

$$y(t)=c_{1}e_{\frac{\gamma}{t},\beta}(t)+c_{2} e_{\frac {\gamma}{t},\beta}(t) \int_{s_{0}}^{t}\frac{ e_{\frac{-1}{\beta(\tau )},\beta}}{1+\frac{\gamma}{\tau}(\beta(\tau)-\tau)}\,d_{\beta}\tau.$$

### Proof

The characteristic equation of (3.14) is

$$\lambda^{2}-2\gamma\lambda+\gamma^{2}=0.$$

Then the characteristic roots are $$\lambda_{1}=\lambda_{2}=\gamma$$. Hence one linearly independent solution of equation (3.14) is $$y_{1}(t)=e_{\frac{\gamma}{t},\beta}(t)$$. To obtain the second linearly independent solution, we can rewrite equation (3.14) in the form

$$D_{\beta}^{2}y(t)+r_{1}(t)D_{\beta}y(t)+r_{2}(t)y(t)=0,$$
(3.15)

where $$r_{1}(t)=\frac{1-2\gamma}{\beta(t)}$$ and $$r_{2}(t)=\frac{\gamma ^{2}}{t\beta(t)}$$. Consequently,

$$-r_{1}(t)+r_{2}(t) \bigl(\beta(t)-t\bigr)= \frac{\gamma^{2}}{t}-\frac{(\gamma-1)^{2}}{\beta(t)}.$$

Let u be a solution of equation (3.15) such that $$u(s_{0})=0$$, $$D_{\beta}u(s_{0})=1$$. Then

$$W_{\beta}( e_{\frac{\gamma}{t},\beta},u) (t)=e_{-r_{1}(t)+r_{2}(t)(\beta (t)-t),\beta}(t)= e_{\frac{\gamma^{2}}{t}-\frac{(\gamma-1)^{2}}{\beta (t)},\beta}(t).$$

By Theorem 2.5, we find that u satisfies the following β-difference equation:

\begin{aligned} D_{\beta}\biggl(\frac{u}{e_{\frac{\gamma}{t},\beta}}\biggr) (t)&=\frac{W_{\beta}( e_{\frac{\gamma}{t},\beta},u)(t)}{ e_{\frac{\gamma}{t},\beta}(t)e_{\frac {\gamma}{\beta(t)},\beta}(\beta(t))} \\ &=\frac{ e_{\frac{\gamma^{2}}{t}-\frac{(\gamma-1)^{2}}{\beta(t)},\beta}(t)}{ e_{\frac{\gamma}{t},\beta}^{2}(t)(1+\frac{\gamma}{t}(\beta(t)-t))}.\end{aligned}

Then

$$u(t)=e_{\frac{\gamma}{t}}(t) \int_{s_{0}}^{t}\frac{ e_{\frac {\alpha^{2}}{\tau}-\frac{(\gamma-1)^{2}}{\beta(\tau)},\beta}(\tau)}{ e_{\frac{\gamma}{\tau},\beta}^{2}(\tau)(1+\frac{\gamma}{\tau}(\beta (\tau)-\tau))}\,d_{\beta}\tau.$$

Also,

$$\frac{e_{\frac{\gamma^{2}}{t}-\frac{(\gamma-1)^{2}}{\beta (t)},\beta}(t)}{ e_{\frac{\gamma}{t},\beta}^{2}(t)}=e_{\frac{-1}{\beta (t)},\beta}(t).$$

Therefore,

$$y(t)=c_{1} e_{\frac{\gamma}{t},\beta}(t)+c_{2}e_{\frac {\gamma}{t},\beta}(t) \int_{s_{0}}^{t}\frac{e_{\frac{-1}{\beta(\tau)},\beta}(\tau)}{1+\frac {\gamma}{\tau}(\beta(\tau)-\tau)}\,d_{\beta}\tau$$

is the general solution of equation (3.14). □

## 4 Conclusion

In this paper, the existence and uniqueness of solutions of the β-Cauchy problem of second order β-difference equations were proved. Moreover, a fundamental set of solutions for second order linear homogeneous β-difference equations when the coefficients are constants was constructed. Also, the different cases of the roots of the characteristic equations of these equations were studied. Finally, the Euler-Cauchy β-difference equation was derived.

## References

1. Agarwal, P, Choi, J: Fractional calculus operators and their image formulas. J. Korean Math. Soc. 53(5), 1183-1210 (2016)

2. Askey, R, Wilson, J: Some basic hypergeometric orthogonal polynomials that generalize the Jacobi polynomials. Mem. Am. Math. Soc. 54, 1-55 (1985)

3. Cresson, J: Non-differentiable variational principles. J. Math. Anal. Appl. 307(1), 48-64 (2005)

4. Cresson, J, Frederico, G, Torres, DFM: Constants of motion for non-differentiable quantum variational problems. Topol. Methods Nonlinear Anal. 33(2), 217-231 (2009)

5. Gasper, G, Rahman, M: Basic Hypergeometric Series. Cambridge University Press, Cambridge (1990)

6. Malinowska, AB, Torres, DFM: Quantum Variational Calculus. SpingerBriefs in Electrical and Computer Engineering - Control, Automation and Robotics. Springer, Berlin (2014)

7. Nottale, L: Fractal Space-Time and Microphysics: Towards a Theory of Scale Relativity. World Scientific, Singapore (1993)

8. Salahshour, S, Ahmadian, A, Senu, N, Baleanu, D, Agarwal, P: On analytical solutions of the fractional differential equation with uncertainty: application to the Basset problem. Entropy 2(17), 885-902 (2015)

9. Tariboon, J, Ntouyas, SK, Agarwal, P: New concepts of fractional quantum calculus and applications to impulsive fractional q-difference equations. Adv. Differ. Equ. 2015, Article ID 18 (2015). doi:10.1186/s13662-014-0348-8

10. Hamza, AE, Sarhan, AM, Shehata, EM, Aldowah, KA: A general quantum difference calculus. Adv. Differ. Equ., 2015, Article ID 182 (2015). doi:10.1186/s13660-015-0518-3

11. Annaby, MH, Hamza, AE, Aldowah, KA: Hahn difference operator and associated Jackson-Nörlund integerals. J. Optim. Theory Appl. 154, 133-153 (2012)

12. Annaby, MH, Mansour, ZS: q-Fractional Calculus and Equations. Springer, Berlin (2012)

13. Bangerezako, G: An introduction to q-difference equations. Preprint, University of Burundi (2008)

14. Hamza, AE, Ahmed, SM: Existence and uniqueness of solutions of Hahn difference equations. Adv. Differ. Equ. 2013, Article ID 316 (2013)

15. Hamza, AE, Ahmed, SM: Theory of linear Hahn difference equations. J. Adv. Math. 4(2), 441-461 (2013)

16. Kac, V, Cheung, P: Quantum Calculus. Springer, New York (2002)

17. Shehata, EM: Generalization of Hahn difference operator and the associated calculus. Ph.D. thesis, Menoufia University (2016)

18. Hamza, AE, Sarhan, AM, Shehata, EM: Exponential, trigonometric and hyperbolic functions associated with a general quantum difference operator. Adv. Dyn. Syst. Appl. (2017, accepted)

19. Hamza, AE, Shehata, EM: Existence and uniqueness of solutions of general quantum difference equations. Adv. Dyn. Syst. Appl. 11(1), 45-58 (2016)

## Acknowledgements

The authors sincerely thank the referees for their valuable suggestions and comments.

## Author information

Authors

### Corresponding author

Correspondence to Enas M Shehata.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.

### Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and permissions

Faried, N., Shehata, E.M. & El Zafarani, R.M. On homogeneous second order linear general quantum difference equations. J Inequal Appl 2017, 198 (2017). https://doi.org/10.1186/s13660-017-1471-3