On homogeneous second order linear general quantum difference equations

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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., [-]. The general quantum difference operator D β is defined, in [, p.], by where f : I → X is a function defined on an interval I ⊆ R, X is a Banach space and β : I → I is a strictly increasing continuous function defined on I, which has only one fixed point s  ∈ I and satisfies the inequality: (ts  )(β(t)t) ≤  for all t ∈ I. The function f is said to be β-differentiable on I, if the ordinary derivative f exists at s  . The β-difference operator yields the Hahn difference operator when β(t) = qt + ω, ω > , q ∈ (, ), and the Jackson q-difference operator when β(t) = qt, q ∈ (, ); see [-]. In [], [, Chapter ], 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 [], the β-exponential, β-trigonometric and β-hyperbolic functions were presented. In [], 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 β . In Section , we introduce the needed preliminaries of the β-calculus from [, -]. In 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  . 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  of β and X is a Banach space. If f is β-differentiable two times over I, then the second order derivative of f is de- where a, b are fixed positive real numbers.

Preliminaries
In this section, we present some needed results associated with the β-calculus from [, -].
Lemma . The following statements are true: (i) The sequence of functions {β k (t)} ∞ k= converges uniformly to the constant function Lemma . If f : I → X is a continuous function at s  , then the sequence {f (β k (t))} ∞ k= converges uniformly to f (s  ) on every compact interval V ⊆ I containing s  . (i) the product fg : I → X is β-differentiable on I and , provided that g(t)g(β(t)) = .
Theorem . Assume f : I → X is continuous at s  . The function F defined by is a β-antiderivative of f with F(s  ) = . Conversely, a β-antiderivative F of f vanishing at s  is given by (.).
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 ∈ I. Clearly, if f is continuous at s  ∈ I, then f is β-integrable on I.
Definition . The β-exponential functions e p,β (t) and E p,β (t) are defined by where p : I → C is a continuous function at s  and both infinite products are convergent to a non-zero number for every t ∈ I and e p, It is worth mentioning that both products in (.) and (.) are convergent since Theorem . The β-exponential functions e p,β (t) and E -p,β (t) are, respectively, the unique solutions of the β-initial value problems: Theorem . Assume that p, q : I → C are continuous functions at s  ∈ I. The following properties are true: Definition . The β-trigonometric functions are defined by Theorem . For all t ∈ I. The following relation holds true: Theorem . Assume that the function f : R → X is continuous at (s  , y  ) ∈ R and satisfies the Lipschtiz condition (with respect to y)

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  . Furthermore, we construct a fundamental set of solutions for the second order linear homogeneous βdifference 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.

Existence and uniqueness of solutions
Lipschitz condition is satisfied: Then there exists a unique solution of the β-initial value problem, β-IVP, has a unique solution on [s  , s  + δ].
Proof Consider equation (.). It is equivalent to (.), where {φ i (t)}  i= is a solution of (.) if and only if φ  (t) is a solution of (.). Here, Hence, by Theorem ., there exists δ >  such that system (.) has a unique solution on [s  , s  + δ].
The following corollary gives us the sufficient conditions for the existence and uniqueness of the solutions of the β-Cauchy problem (.).
Corollary . Assume the functions a j (t) : I → C, j = , , , and b(t) : I → X satisfy the following conditions: (i) a j (t), j = , ,  and b(t) are continuous at s  with a  (t) =  for all t ∈ I, (ii) a j (t)/a  (t) is bounded on I, j = , . Then has a unique solution on subinterval J ⊆ I, s  ∈ J.
Proof Dividing by a  (t), we get where A j (t) = -a j (t)/a  (t) and B(t) = b(t)/a  (t). Since A j (t) and B(t) are continuous at t = s  , the function f (t, y  , y  ), defined by We can see that f satisfies the Lipschitz condition with Lipschitz constant A. Thus, f (t, y  , y  ) satisfies the conditions of Corollary .. Hence, there exists a unique solution of (.) on J.

Fundamental solutions of linear homogeneous β-difference equations
The second order homogeneous linear β-difference equation has the form a  (t)D  β y(t) + a  (t)D β y(t) + a  (t)y(t) = , t ∈ I, (  .  ) where the coefficients a  (t) = , a j (t), j = ,  are assumed to satisfy the conditions of Corollary ..

Lemma . If the function y is a solution of the homogeneous equation
be the solutions of the linear system D β y i (t) = a i (t)y i (t), i = , , corresponding, respectively, to the solutions y  , y  of homogeneous linear β-difference equation (.). Since y  , y  are linearly independent in J, then φ  , φ  are linearly independent in J. Then there exist two constants c  , Thus the results hold. We have c  = c  =  at t = s  , which is a contradiction. Thus the solutions y  and y  are linearly independent in J. Then there exists a fundamental set of the two solutions y  and y  of equation (.).
Definition . Let y  , y  be β-differentiable functions. Then we define the β-Wronskian of the functions y  , y  , defined on I, by , t ∈ I.
Lemma . Let y  (t), y  (t) be functions defined on I. Then, for any t ∈ I, t = s  , .
Theorem . Assume that y  (t) and y  (t) are two solutions of equation (.). Then their β-Wronskian, W β , where r  (t) = a  (t) a  (t) and r  (t) = a  (t) a  (t) satisfy the conditions of Corollary ..
Proof Since y  and y  are solutions of equation (.), from (.) we have which has the solution Using Theorem . and Lemma ., we can prove the following corollaries.

Homogeneous equations with constant coefficients
Equation (.) can be written as where a, b, and c are constants. The characteristic polynomial of equation (.) is where y(t) = e λ,β (t) is a solution of equation (.). Since equation (.) is a quadratic equation with real coefficients, it has two roots, which may be real and different, real but repeated, or complex conjugates. Case : real and different roots of the characteristic equation (.). Let λ  and λ  be real roots with λ  = λ  , then y  (t) = e λ  ,β (t) and y  (t) = e λ  ,β (t) are two solutions of equation (.). Therefore, is a general solution of equation (.), with Example . Find the solution of the β-initial value problem D  β y(t) + D β y(t) + y(t) = , y(s  ) = , D β y(s  ) = .

Example . Find the general solution of
The characteristic equation is λ  + λ +  = , and its roots are Thus, the general solution of equation (.) is -/(β(t)-t) ,β (t).
Case : repeated roots. Consider the case that the two roots λ  and λ  are equal, so λ  = λ  = -b/a. Therefore, the solution y  (t) = e -b/a,β (t) is one solution of the β-difference equation (.), and we give the second solution by the following example: Example . Solve the β-difference equation The characteristic equation is (λ +)  = , so λ  = λ  = -. Therefore, y  (t) = e -,β (t) is a solution of equation (.). To find the second solution, let y(t) = v(t)e -,β (t). Then D  β v(t) = . Therefore, v(t) = c  t + c  , where c  and c  are arbitrary constants. Then the general solution is where the two solutions y  (t) = e -,β (t) and y  (t) = te -,β (t) form a fundamental set of solutions of equation (.).

Euler-Cauchy β-difference equation
The Euler-Cauchy β-difference equation takes the form tβ(t)D  β y(t) + atD β y(t) + by(t) = , t ∈ I, t = s  , (  .   ) where a, b are constants. The characteristic equation of (.) is given by Theorem . If the characteristic equation (.) has two distinct roots λ  and λ  , then a fundamental set of solutions of (.) is given by e λ  /t,β (t) and e λ  /t,β (t).