On homogeneous second order linear general quantum difference equations
- Nashat Faried^{1},
- Enas M Shehata^{2}Email author and
- Rasha M El Zafarani^{1}
https://doi.org/10.1186/s13660-017-1471-3
© The Author(s) 2017
Received: 8 June 2017
Accepted: 8 August 2017
Published: 24 August 2017
Abstract
Keywords
a general quantum difference operator general quantum difference equations Euler-Cauchy general quantum difference equationMSC
39A10 39A13 39A70 47B391 Introduction
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, 17–19]. 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, 17–19].
Lemma 2.1
- (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 \(s_{0}\).
- (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
- (i)the product \(fg:I\rightarrow\mathbb{X}\) is β-differentiable on I and$$\begin{aligned} {D}_{\beta}(fg) (t) &=\bigl({D}_{\beta}f(t) \bigr)g(t)+f\bigl(\beta(t)\bigr){D}_{\beta}g(t) \\ & =\bigl({D}_{\beta}f(t)\bigr)g\bigl(\beta(t)\bigr)+f(t){D}_{\beta}g(t), \end{aligned} $$
- (ii)\(f/g\) is β-differentiable at t andprovided that \(g(t)g(\beta(t))\neq{0}\).$${D}_{\beta} ({f}/{g} ) (t)=\frac{({D}_{\beta }f(t))g(t)-f(t){D}_{\beta}g(t)}{g(t)g(\beta(t))}, $$
Theorem 2.6
Definition 2.7
Definition 2.8
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
Theorem 2.10
- (i)
\(\frac{1}{e_{p,\beta}(t)}=e_{-p/[1+(\beta(t)-t)p]}(t)\),
- (ii)
\(e_{p,\beta}(t)e_{q,\beta}(t)=e_{p+q+(\beta(t)-t)pq}(t)\),
- (iii)
\(e_{p,\beta}(t)/e_{q,\beta}(t)=e_{(p-q)/[1+(\beta(t)-t)q]}(t)\).
Definition 2.11
Theorem 2.12
Theorem 2.13
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
- (i)
for \(y_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\), \(f_{i}(t,y_{1},y_{2})\) are continuous at \(t=s_{0}\),
- (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:
Proof
Corollary 3.2
- (i)
for any values of \(y_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\), f is continuous at \(t=s_{0}\),
- (ii)f satisfies the Lipschitz conditionwhere \(A>0\), \(y_{i},\tilde{y}_{i}\in S_{i}(x_{i},b_{i})\), \(i=1,2\) and \(t \in I\). Then$$\big\| f(t,y_{1},y_{2})-f(t,\tilde{y}_{1}, \tilde{y}_{2}) \big\| \leq A\sum_{i=1}^{2} \|y_{i} -\tilde{y}_{i}\|, $$has a unique solution on \([s_{0},s_{0} +\delta]\).$$\begin{aligned} D_{\beta}^{2}y(t)=f\bigl(t,y(t),D_{\beta}y(t)\bigr),\quad D_{\beta}^{i-1}y(s_{0})=x_{i}, i=1,2 \end{aligned}$$(3.3)
Proof
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
- (i)
\(a_{j}(t), j=0,1,2\) and \(b(t)\) are continuous at \(s_{0}\) with \(a_{0}(t)\neq0\) for all \(t \in I\),
- (ii)\(a_{j}(t)/a_{0}(t)\) is bounded on I, \(j=1,2\). Thenhas a unique solution on subinterval \(J\subseteq I\), \(s_{0}\in J\).$$ \begin{gathered} a_{0}(t)D_{\beta}^{2}y(t)+ a_{1}(t)D_{\beta}y(t)+a_{2}(t)y(t)=b(t), \\ D_{\beta}^{i-1}y(s_{0})= x_{i},\quad x_{i} \in{\mathbb{X}}, i=1,2, \end{gathered} $$(3.4)
Proof
3.2 Fundamental solutions of linear homogeneous β-difference equations
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
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 \(y_{2}(s_{0})=0\), \(D_{\beta}y_{2}(s_{0})=1\).
Definition 3.9
Lemma 3.10
Proof
Theorem 3.11
Proof
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
Case 1: real and different roots of the characteristic equation ( 3.9 ).
Example 3.14
Case 2: complex roots of the characteristic equation ( 3.9 ).
Example 3.15
Case 3: repeated roots.
Example 3.16
3.4 Euler-Cauchy β-difference equation
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
The following theorem gives us the general solution of the Euler-Cauchy β-difference equation in the double root case.
Theorem 3.18
Proof
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
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