On homogeneous second order linear general quantum difference equations

R E S E A R C H

Open Access

On homogeneous second order linear

general quantum difference equations

Nashat Faried

1

_{, Enas M Shehata}

2*

_{and Rasha M El Zafarani}

1

*_{Correspondence:}

[email protected]

2_{Department of Mathematics,}

Faculty of Science, Menoufia University, Shibin El-koom, Egypt Full list of author information is available at the end of the article

Abstract

In this paper, we prove the existence and uniqueness of solutions of the

β

-Cauchy problem of second order

β

-difference equations

a0(t)D2βy(t) +a1(t)Dβy(t) +a2(t)y(t) =b(t), t∈I,

a0(t)= 0, in a neighborhood of the unique fixed points0of the strictly increasing

continuous function

β

, defined on an intervalI⊆R. These equations are based on the general quantum difference operatorDβ, which is defined by

Dβf(t) = (f(

β

(t)) –f(t))/(

β

(t) –t),

β

(t)=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.

MSC: 39A10; 39A13; 39A70; 47B39

Keywords: a general quantum difference operator; general quantum difference

equations; Euler-Cauchy general quantum difference equation

1 Introduction

Quantum calculus allows us to deal with sets of non-differentiable functions by substi-tuting the classical derivative by a difference operator. Non-differentiable functions are used to describe many important physical phenomena. Quantum calculus has a lot of ap-plications in different mathematical areas such as the calculus of variations, orthogonal polynomials, basic hyper-geometric functions, economical problems with a dynamic na-ture, quantum mechanics and the theory of scale relativity; see,e.g., [–]. The general quantum difference operatorDβis defined, in [, p.], by

Dβf(t) =

_{f(β(t))–f(t)}

β(t)–t , t=s,

f(s), t=s,

wheref :I→Xis a function defined on an intervalI⊆R,Xis a Banach space andβ:I→I

is a strictly increasing continuous function defined onI, which has only one fixed point

s∈Iand satisfies the inequality: (t–s)(β(t) –t)≤ for allt∈I. The functionf is said to

beβ-differentiable onI, if the ordinary derivativefexists ats. Theβ-difference operator

yields the Hahn difference operator whenβ(t) =qt+ω,ω> ,q∈(, ), and the Jackson

q-difference operator whenβ(t) =qt,q∈(, ); see [–]. In [], [, Chapter ], the definition of theβ-derivative, theβ-integral, the fundamental theorem ofβ-calculus, the chain rule, Leibniz’s formula and the mean value theorem were introduced. In [], the

β-exponential,β-trigonometric andβ-hyperbolic functions were presented. In [], the existence and uniqueness of solutions of theβ-initial value problem of the first order were established. In addition, an expansion form of theβ-exponential function was deduced.

This paper is devoted for deducing some results of the solutions of the homogeneous second order linearβ-difference equations which are based onDβ. In Section , we

in-troduce the needed preliminaries of the β-calculus from [, –]. In Section , we prove the existence and uniqueness of solutions of theβ-Cauchy problem of second or-der β-difference equations in a neighborhood of s. We also construct a fundamental

set of solutions for the second order linear homogeneousβ-difference equations when the coefficients are constants and study the different cases of the roots of their charac-teristic equations. Finally, we drive the Euler-Cauchy β-difference equation. Through-out this paper,Jis a neighborhood of the unique fixed pointsofβ andXis a Banach

space. Iff isβ-differentiable two times overI, then the second order derivative off is de-noted byD

βf =Dβ(Dβf). Furthermore,S(y,b) ={y∈X:y–y ≤b}and the rectangle R={(t,y)∈I×X:|t–s| ≤a,y–y ≤b}, wherea,bare fixed positive real numbers.

2 Preliminaries

In this section, we present some needed results associated with theβ-calculus from [, –].

Lemma . The following statements are true:

(i) The sequence of functions{βk_(t)}∞

k=converges uniformly to the constant function ˆ

β(t) :=son every compact intervalV⊆Icontainings.

(ii) The series∞_k=|βk(t) –βk+(t)|is uniformly convergent to|t–s|on every compact intervalV⊆Icontainings.

Lemma . If f :I→Xis a continuous function at s,then the sequence{f(βk(t))}∞k= converges uniformly to f(s)on every compact interval V⊆I containing s.

Theorem . If f :I→Xis continuous at s,then the series

∞

k=(βk(t) –βk+(t))× f(βk(t))is uniformly convergent on every compact interval V⊆I containing s.

Lemma . Let f :I→Xbeβ-differentiable and Dβf(t) = for all t∈I.Then f(t) =f(s) for all t∈I.

Theorem . Assume that f :I→Xand g:I→Rareβ-differentiable functions on I.

Then:

(i) the productfg:I→Xisβ-differentiable onIand

Dβ(fg)(t) =

Dβf(t)

g(t) +fβ(t)Dβg(t)

=Dβf(t)

(ii) f/gisβ-differentiable attand

Dβ(f/g)(t) =

(Dβf(t))g(t) –f(t)Dβg(t) g(t)g(β(t)) ,

provided thatg(t)g(β(t))= .

Theorem . Assume f :I→Xis continuous at s.The function F defined by

F(t) =

∞

k=

βk(t) –βk+(t)fβk(t), t∈I (.)

is aβ-antiderivative of f with F(s) = .Conversely,aβ-antiderivative F of f vanishing at sis given by(.).

Definition . Letf :I→Xanda,b∈I. Theβ-integral off fromatobis

b a

f(t)dβt=

b s

f(t)dβt–

a s

f(t)dβt,

where

x s

f(t)dβt= ∞

k=

βk(x) –βk+(x)fβk(x), x∈I,

provided that the series converges atx=aandx=b.f is calledβ-integrable onIif the series converges ataandbfor alla,b∈I. Clearly, iff is continuous ats∈I, thenf is

β-integrable onI.

Definition . Theβ-exponential functionsep,β(t) andEp,β(t) are defined by

ep,β(t) =



∞

k=[ –p(βk(t))(βk(t) –βk+(t))]

(.)

and

Ep,β(t) = ∞

k=

 +pβk(t)βk(t) –βk+(t), (.)

wherep:I→Cis a continuous function atsand both infinite products are convergent

to a non-zero number for everyt∈Iandep,β(t) = _Ep,

β(t).

It is worth mentioning that both products in (.) and (.) are convergent since

∞

k=|p(βk(t))(βk(t) –βk+(t))|is uniformly convergent. See [, Definition .].

Theorem . Theβ-exponential functions ep,β(t)and E–p,β(t)are,respectively,the unique solutions of theβ-initial value problems:

Dβy(t) =p(t)y(t), y(s) = ,

Dβy(t) = –p(t)y

Theorem . Assume that p,q:I→Care continuous functions at s∈I.The following properties are true:

(i) _e 

p,β(t)=e–p/[+(β(t)–t)p](t),

(ii) ep,β(t)eq,β(t) =ep+q+(β(t)–t)pq(t),

(iii) ep,β(t)/eq,β(t) =e(p–q)/[+(β(t)–t)q](t).

Definition . Theβ-trigonometric functions are defined by

cosp,β(t) =

eip,β(t) +e–ip,β(t)

 ,

sinp,β(t) =

eip,β(t) –e–ip,β(t)

i .

Theorem . For all t∈I.The following relation holds true:

eip,β(t) =cosp,β(t) +isinp,β(t).

Theorem . Assume that the function f:R→Xis continuous at(s,y)∈R and satis-fies the Lipschtiz condition(with respect to y)

f(t,y) –f(t,y)≤Ly–y, for all(t,y), (t,y)∈R.

Then the β-initial value problem Dβy(t) =f(t,y), y(s) =y, t∈I has a unique solu-tion on[s–δ,s+δ],where L is a positive constant andδ=min{a,_Lbb_+M,ρ_L}with M=

sup_(t,y)∈Rf(t,y)<∞,ρ∈(, ).

3 Main results

In this section, we prove the existence and uniqueness of solutions of theβ-Cauchy prob-lem of second orderβ-difference equations in a neighborhood ofs. 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-Cauchyβ-difference equation.

3.1 Existence and uniqueness of solutions

Theorem . Let fi(t,y,y) :I×



i=Si(xi,bi)→X,s∈I,such that the following condi-tions are satisfied:

(i) foryi∈Si(xi,bi),i= , ,fi(t,y,y)are continuous att=s,

(ii) there is a positive constantAsuch that,fort∈I,yi,y˜i∈Si(xi,bi),i= , ,the following

Lipschitz condition is satisfied:

fi(t,y,y) –fi(t,y˜,y˜)≤A 

i=

yi–y˜i.

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

Dβyi(t) =fi

t,y(t),y(t)

Proof Lety= (x,x)Tandb= (b,b)T, where (·,·)T stands for vector transpose. Define

the functionf :I×_i=Si(xi,bi)→X×Xbyf(t,y,y) = (f(t,y,y),f(t,y,y))T. It is easy

to show that system (.) is equivalent to theβ-IVP

Dβy(t) =f

t,y(t), y(s) =y. (.)

Since eachfiis continuous att=s,f is continuous att=s. The functionf satisfies the

Lipschitz condition because fory,y˜∈_i=Si(xi,bi),

f(t,y) –f(t,y˜)=f(t,y,y) –f(t,y˜,y˜)

=



i=

fi(t,y,y) –fi(t,˜y,˜y, )

≤A 

i=

yi–y˜i=Ay–y˜.

Applying Theorem ., see the proof in [], there existsδ>  such that (.) has a unique solution on [s,s+δ]. Hence, theβ-IVP (.) has a unique solution on [s,s+δ].

Corollary . Let f(t,y,y)be a function defined on I×



i=Si(xi,bi)such that the

fol-lowing conditions are satisfied:

(i) for any values ofyi∈Si(xi,bi),i= , ,f is continuous att=s, (ii) f satisfies the Lipschitz condition

f(t,y,y) –f(t,y˜,y˜)≤A 

i=

yi–y˜i,

whereA> ,yi,˜yi∈Si(xi,bi),i= , andt∈I.Then

Dβy(t) =f

t,y(t),Dβy(t)

, Diβ–y(s) =xi,i= ,  (.)

has a unique solution on[s,s+δ].

Proof Consider equation (.). It is equivalent to (.), where{φi(t)}i=is a solution of (.)

if and only ifφ(t) is a solution of (.). Here,

fi(t,y,y) =

y, i= ,

f(t,y,y), i= .

Hence, by Theorem ., there existsδ>  such that system (.) has a unique solution on

[s,s+δ].

The following corollary gives us the sufficient conditions for the existence and unique-ness of the solutions of theβ-Cauchy problem (.).

Corollary . Assume the functions aj(t) :I→C,j= , , ,and b(t) :I→Xsatisfy the

(i) aj(t),j= , , andb(t)are continuous atswitha(t)= for allt∈I, (ii) aj(t)/a(t)is bounded onI,j= , .Then

a(t)Dβy(t) +a(t)Dβy(t) +a(t)y(t) =b(t),

Diβ–y(s) =xi, xi∈X,i= , ,

(.)

has a unique solution on subintervalJ⊆I,s∈J.

Proof Dividing bya(t), we get

Dβy(t) =A(t)Dβy(t) +A(t)y(t) +B(t), (.)

whereAj(t) = –aj(t)/a(t) andB(t) =b(t)/a(t). SinceAj(t) andB(t) are continuous att=s,

the functionf(t,y,y), defined by

f(t,y,y) =A(t)y+A(t)y+B(t),

is continuous att=s. Furthermore,Aj(t) is bounded onI. Consequently, there isA> 

such that|Aj(t)| ≤Afor allt∈I. We can see thatf satisfies the Lipschitz condition with

Lipschitz constantA. Thus,f(t,y,y) satisfies the conditions of Corollary .. Hence, there

exists a unique solution of (.) onJ.

3.2 Fundamental solutions of linear homogeneous

β

-difference equations The second order homogeneous linearβ-difference equation has the form

a(t)Dβy(t) +a(t)Dβy(t) +a(t)y(t) = , t∈I, (.)

where the coefficientsa(t)= ,aj(t),j= ,  are assumed to satisfy the conditions of

Corol-lary ..

Lemma . If the function y is a solution of the homogeneous equation(.),such that y(s) = and Dβy(s) = ,s∈I,then y(t) = ,for all t∈J.

Proof By Corollary ., ifxi= ,i= ,  in theβ-IVP (.), which has a unique solution

onJ, thenysuch thaty(t) =  for allt∈Jis a unique solution of theβ-difference equation (.), which satisfies the given initial conditionsy(s) = ,Dβy(s) = . Hence we have the

desired result.

Theorem . The linear combination cy+cy of any two solutions y and y of the homogeneous linearβ-difference equation(.)is also a solution of it in J,where cand c are arbitrary constants.

Proof The proof is straightforward.

Proof Let

φ=

y Dβy

, φ=

y Dβy

, φ=

y Dβy

,

be the solutions of the linear systemDβyi(t) =ai(t)yi(t),i= , , corresponding,

respec-tively, to the solutionsy,yof homogeneous linearβ-difference equation (.). Sincey,y

are linearly independent inJ, thenφ,φ are linearly independent inJ. Then there exist

two constantsc,csuch thatφ=cφ+cφ. The first component of this isy=cy+cy.

Thus the results hold.

Definition . A set of two linearly independent solutions of the second order homoge-neous linearβ-difference equation (.) is called a fundamental set of it.

Theorem . There exists a fundamental set of solutions of the second order homogeneous linearβ-difference equation(.).

Proof By Corollary ., there exist unique solutionsyandyof equation (.), such that y(s) = ,Dβy(s) =  andy(s) = ,Dβy(s) = .

Suppose thatyandyare linear dependent, so there exist constantscandcnot both

zero, such that

cy(t) +cy(t) = , for allt∈J,

cDβy(t) +cDβy(t) = , for allt∈J.

We havec=c=  att=s, which is a contradiction. Thus the solutionsy andy are

linearly independent inJ. Then there exists a fundamental set of the two solutionsyand

yof equation (.).

Definition . Lety,ybeβ-differentiable functions. Then we define theβ-Wronskian

of the functionsy,y, defined onI, by

Wβ(y,y)(t) =

y(t) y(t) Dβy(t) Dβy(t)

, t∈I.

Lemma . Let y(t),y(t)be functions defined on I.Then,for any t∈I,t=s,

DβWβ(y,y)(t) =

y(β(t)) y(β(t)) Dβy(t) Dβy(t).

. (.)

Proof SinceWβ(y,y)(t) =y(t)Dβy(t) –y(t)Dβy(t), then

DβWβ(y,y)(t) =y

β(t)D_βy(t) –y

β(t)D_βy(t),

Theorem . Assume that y(t)and y(t)are two solutions of equation(.).Then their

β-Wronskian,Wβ,

Wβ(y,y)(t) =e–r(t)+r(t)(β(t)–t),βWβ(y,y)(s), t∈I,

where r(t) = a_a(₍t_t)₎and r(t) = a_a_(₍t_t)₎satisfy the conditions of Corollary..

Proof Sinceyandyare solutions of equation (.), from (.) we have

DβWβ(y,y)(t) =

y(β(t)) y(β(t))

–a(t)

a(t)Dβy(t) – a(t) a(t)Dβy(t)

+

y(β(t)) y(β(t))

–a(t)

a(t)y(t) – a(t) a(t)y(t)

= –a(t)

a(t)

y(t) y(t) Dβy(t) Dβy(t)

+

a(t) a(t)

β(t) –t

y(t) y(t) Dβy(t) Dβy(t)

=–r(t) +r(t)

β(t) –tWβ(y,y)(t),

which has the solution

Wβ(y,y)(t) =Wβ(y,y)(s)e–r(t)+r(t)(β(t)–t),β, t∈I.

Using Theorem . and Lemma ., we can prove the following corollaries.

Corollary . Two solutions yand yofβ-difference equation(.)are linearly depen-dent in J if and only if Wβ(y,y)(t) = ,for all t∈J.

Corollary . The value of Wβ(y,y)(t)ofβ-difference equation(.)either is zero or unequal to zero for all t∈J.

3.3 Homogeneous equations with constant coefficients Equation (.) can be written as

Ly(t) =aDβy(t) +bDβy(t) +cy(t) = , (.)

wherea,b, andcare constants. The characteristic polynomial of equation (.) is

P(λ) =aλ+bλ+c= , (.)

wherey(t) =eλ,β(t) is a solution of equation (.). Since equation (.) is a quadratic

equa-tion with real coefficients, it has two roots, which may be real and different, real but re-peated, or complex conjugates.

Case : real and different roots of the characteristic equation(.).

Letλ andλbe real roots withλ= λ, theny(t) =eλ,β(t) andy(t) =eλ,β(t) are two solutions of equation (.). Therefore,

is a general solution of equation (.), with

c=

Dβy–yλ

λ–λ

e–λ,β(s) and c=

yλ–Dβy

λ–λ

e–λ,β(s).

Example . Find the solution of theβ-initial value problem

Dβy(t) + Dβy(t) + y(t) = , y(s) = ,Dβy(s) = .

By assuming thaty(t) =eλ,β(t), we obtain the solution

y(t) = e–,β(t) – e–,β(t).

Case : complex roots of the characteristic equation(.).

Let λ =ν +iμ and λ =ν –iμ, where ν and μ are real numbers. Then y(t) = e(ν+iμ),β(t) andy(t) =e(ν–iμ),β(t) are two solutions of equation (.). By Theorems ., ., e(ν+iμ),β(t) =eν,β(t)e iμ

+ν(β(t)–t),β(t). So,

e(ν+iμ),β(t) =eν,β(t)

cos μ

+ν(β(t)–t),β(t) +isin μ +ν(β(t)–t),β(t)

.

We have

y(t) +y(t) = eν,β(t)cos μ

+ν(β(t)–t),β(t)

and

y(t) –y(t) = ieν,β(t)sin μ +ν(β(t)–t),β(t).

Therefore,

u(t) =eν,β(t)cos μ

+ν(β(t)–t),β(t) and v(t) =eν,β(t)sin μ +ν(β(t)–t),β(t)

are two solutions of equation (.). If theβ-Wronskian ofuandvis not zero, thenuand

vform a fundamental set of solutions. The general solution of equation (.) is

y(t) =ceν,β(t)cos μ

+ν(β(t)–t),β(t) +ceν,β(t)sin μ +ν(β(t)–t),β(t),

wherecandcare arbitrary constants.

Example . Find the general solution of

Dβy(t) +Dβy(t) +y(t) = . (.)

The characteristic equation isλ_{+λ+  = , and its roots are}

λ,=

–  ±i

√

Thus, the general solution of equation (.) is

y(t) =ce–/,β(t)cos √/ –/(β(t)–t),β

(t) +ce–/,β(t)sin √/ –/(β(t)–t),β

(t).

Case : repeated roots.

Consider the case that the two rootsλandλare equal, so

λ=λ= –b/a.

Therefore, the solutiony(t) =e–b/a,β(t) is one solution of theβ-difference equation (.),

and we give the second solution by the following example:

Example . Solve theβ-difference equation

Dβy(t) + Dβy(t) + y(t) = . (.)

The characteristic equation is (λ+ )_{= , soλ}

=λ= –. Therefore,y(t) =e–,β(t) is a

so-lution of equation (.). To find the second soso-lution, lety(t) =v(t)e–,β(t). ThenDβv(t) = .

Therefore,v(t) =ct+c, wherecandcare arbitrary constants. Then the general solution

is

y(t) =cte–,β(t) +ce–,β(t),

where the two solutionsy(t) =e–,β(t) andy(t) =te–,β(t) form a fundamental set of

so-lutions of equation (.).

3.4 Euler-Cauchy

β

-difference equation

The Euler-Cauchyβ-difference equation takes the form

tβ(t)Dβy(t) +atDβy(t) +by(t) = , t∈I,t=s, (.)

wherea,bare constants. The characteristic equation of (.) is given by

λ+ (a– )λ+b= . (.)

Theorem . If the characteristic equation(.)has two distinct rootsλandλ,then a fundamental set of solutions of (.)is given by eλ/t,β(t)and eλ/t,β(t).

Proof Lety(t) =eλ/t,β(t), whereλis a root of equation (.). It follows that

Dβy(t) =

λ

ty(t), D

 βy(t) =

λ–λ

tβ(t)y(t).

Consequently, we have

tβ(t)Dβy(t) +atDβy(t) +by(t) =

Assume thatλ andλ are distinct roots of the characteristic equation (.). Then, we

have

λ+λ=  –a, λλ=b.

Moreover,Wβ(eλ/t,β,eλ/t,β)(t)= , sinceλ=λ. Hence,eλ/t,β(t) andeλ/t,β(t) form a

fun-damental set of solutions of (.).

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

Theorem . Assume that/β(t)is bounded on I and /∈I.Then the general solution of

the Euler-Cauchyβ-difference equation

tβ(t)D_βy(t) + ( – γ)tDβy(t) +γy(t) = , t∈I, (.)

is given by

y(t) =ceγ_t,β(t) +ceγ_t,β(t)

t s

e– β(τ),β

 +γ_τ(β(τ) –τ)dβτ.

Proof The characteristic equation of (.) is

λ– γ λ+γ= .

Then the characteristic roots areλ=λ=γ. Hence one linearly independent solution of

equation (.) isy(t) =eγ_t,β(t). To obtain the second linearly independent solution, we

can rewrite equation (.) in the form

Dβy(t) +r(t)Dβy(t) +r(t)y(t) = , (.)

wherer(t) =–β(tγ) andr(t) = γ

tβ(t). Consequently,

–r(t) +r(t)

β(t) –t=γ



t –

(γ – )

β(t) .

Letube a solution of equation (.) such thatu(s) = ,Dβu(s) = . Then

Wβ(eγ

t,β,u)(t) =e–r(t)+r(t)(β(t)–t),β(t) =eγ_t–(γ_β–)_(t) ,β(t).

By Theorem ., we find thatusatisfies the followingβ-difference equation:

Dβ

u eγ

t,β

(t) = Wβ(e γ

t,β,u)(t) eγ

t,β(t)eβγ(t),β(β(t))

=

eγ

t –

(γ–) β(t) ,β

(t)

e

γ

t,β(t)( + γ

t(β(t) –t))

Then

u(t) =eγ

t(t)

t s

eα τ –

(γ–) β(τ) ,β

(τ)

e

γ τ,β(

τ)( +γ_τ(β(τ) –τ))dβτ.

Also,

eγ

t –

(γ–) β(t) ,β

(t)

eγ

t,β(t)

=e– β(t),β(t).

Therefore,

y(t) =ceγ_t,β(t) +ceγ_t,β(t)

t s

e– β(τ),β(τ)  +γ_τ(β(τ) –τ)dβτ

is the general solution of equation (.).

4 Conclusion

In this paper, the existence and uniqueness of solutions of theβ-Cauchy problem of sec-ond orderβ-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 equa-tions of these equaequa-tions were studied. Finally, the Euler-Cauchyβ-difference equation was derived.

Acknowledgements

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

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.

Author details

1_{Department of Mathematics, Faculty of Science, Ain Shams University, Cairo, Egypt.}2_{Department of Mathematics,}

Faculty of Science, Menoufia University, Shibin El-koom, Egypt.

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Received: 8 June 2017 Accepted: 8 August 2017 References

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