Open Access

On homogeneous second order linear general quantum difference equations

Journal of Inequalities and Applications20172017:198

https://doi.org/10.1186/s13660-017-1471-3

Received: 8 June 2017

Accepted: 8 August 2017

Published: 24 August 2017

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.

Keywords

a general quantum difference operator general quantum difference equations Euler-Cauchy general quantum difference equation

MSC

39A10 39A13 39A70 47B39

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.

3.1 Existence and uniqueness of solutions

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.

Declarations

Acknowledgements

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

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Faculty of Science, Ain Shams University
(2)
Department of Mathematics, Faculty of Science, Menoufia University

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