Impulsive quantum (p, q)-difference equations

<jats:p>In this paper we study quantum <jats:inline-formula><jats:alternatives><jats:tex-math>$(p,q)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula>-difference equations with impulse and initial or boundary conditions. We consider first order impulsive <jats:inline-formula><jats:alternatives><jats:tex-math>$(p,q)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula>-difference boundary value problems and second order impulsive <jats:inline-formula><jats:alternatives><jats:tex-math>$(p,q)$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>(</mml:mo><mml:mi>p</mml:mi><mml:mo>,</mml:mo><mml:mi>q</mml:mi><mml:mo>)</mml:mo></mml:math></jats:alternatives></jats:inline-formula>-difference initial value problems. Existence and uniqueness results are proved via Banach’s fixed point theorem.</jats:p>


Introduction and preliminaries
Let p, q be quantum constants satisfying 0 < q < p ≤ 1. The (p, q)-number, [n] p,q , is defined by [n] p,q = p nq n pq .
In [15,16], M. Tunç and E. Göv defined the quantum (p, q)-difference of a function f on the finite interval [a, b] by and a D p,q f (a) = f (a). The (p, q)-difference of a power function f (t) = (ta) α , α ≥ 0, is given by As is customary, we put the following relation: It is obvious that if a = 0, then equations (1.5)-(1.8) are reduced to (1.1)-(1.4), respectively. The domain-shift properties of the (p, q)-difference and (p, q)-integral operators for a function f (t), t ∈ [a, b] are respectively given by Also we remark that if p = 1, then both domains are reduced to [a, b]. For the shifting of the second order (p, q)-difference and integral domains, we consider the following result. respectively.
Proof We have Setting For m = n = 0 and setting 1 we obtain t = p 2 (ba) + a, which implies that Before going to the next result, we would like to recall the operator a Φ r defined by where m, a ∈ R and r ∈ [0, 1]. Some properties of this operator can be found in [17]. ( Substituting into (1.9) yields t a s a f (r) a d p,q r a d p,q s which is completed the proof. The following theorem has been proved in [16].

Theorem 1.4 The fundamental relations of (p, q)-calculus can be stated as
In this paper we study the impulsive (p, q)-difference equations with initial and boundary conditions. We consider four types of problems, two impulsive (p, q)-difference equations of type I and two impulsive (p, q)-difference equations of type II (explained in the next section). Existence and uniqueness results are proved via Banach's contraction mapping principle. Examples illustrating the obtained results are also constructed.

Impulsive (p, q)-difference equations
In this section, we consider the first and second order (p, q)-difference equations with initial or boundary conditions and also prove the existence and uniqueness of solutions for impulsive problems. Firstly, let t k , k = 1, . . . , m, be the impulsive points such that 0 = t 0 < The investigations are based on (p, q)-calculus introduced in the previous section by replacing a point a by t k , quantum numbers p by p k and q by q k , k = 0, 1, . . . , m, and also applying the (p k , q k )-difference and (p k , q k )-integral operators only on a finite subinterval of J. In addition, the consecutive subintervals can be related with jump conditions which provide a meaning of quantum difference equations with impulse effects. There are two types of impulsive problems which will be established in the next two subsections. The consecutive domains of impulsive (p, q)-difference equations of type I are overlapped, while the unknown functions of impulsive equations of type II are defined on disconnected consecutive domains.

Impulsive (p, q)-difference equations of type I
Consider the first-order impulsive (p, q)-difference impulsive boundary value problem of the form where α, β, and γ are real constants with α = -β, the quantum numbers p k , q k satisfy . . , m, are given functions, and t k D p k ,q k is the quantum (p k , q k )-difference operator starting at a point t k , k = 0, 1, . . . , m.
However, by the shifting property of (p, q)-integration applied to the two above equations, we have respectively.

Theorem 2.1
The nonlinear first-order (p, q)-difference boundary value problem (2.1) can be transformed into an integral equation we obtain by using Theorem 1.4 and the shifting property. Next, for , where t 1 is the first impulsive point in J, we also obtain by applying the (p 1 , q 1 )-integration, By the impulsive condition x(t + 1 ) = x(t 1 ) + ϕ 1 (x(t 1 )), it follows, for t ∈ (t 1 , t 2 ], that by (p 2 , q 2 )-integration and due to the impulsive condition x(t + 2 ) = x(t -2 ) + ϕ 2 (x(t 2 )). Repeating this process, we obtain, for t ∈ J k , k = 0, 1, . . . , m, that After that from the boundary condition αx(0) + βx(T) = γ , we have Putting the value of x(0) into (2.3), shows that (2.2) is true and the proof is completed.
Remark 2.2 If α = 0 and β = 0, then the boundary value problem (2.1) can be reduced to the initial value problem with initial condition x(0) = γ /α. Before going to the second-order impulsive problem, we define which are impulsive shifting points of the (p k , q k )-derivative of the unknown function in our system. In addition, we introduce a notation For example, where K i ∈ R, i = 0, 1, 2. Now, we consider the second-order impulsive (p, q)-difference initial value problem of the form where f : [0, ((Tt m )/p 2 m ) + t m ] × R → R, ϕ k : R → R and ϕ * k : R → R, are given functions, λ 1 , λ 2 are given constants. Observe that the distance between the impulsive points t k and τ k in the third equation of (2.4) depends on the value of p k-1 for k = 1, 2, . . . , m. Indeed, which has appeared by the shifting property of (p, q)-calculus as discussed in the previous section.

Theorem 2.3 The impulsive initial value problem of type I given by the (p, q)-difference equation (2.4) can be expressed as an integral equation of the form
pr (s))), r = 0, 1, . . . , k, and b a (·) = 0, when b < a.
Proof By computing the (p 0 , q 0 )-integral of both sides of the first equation of (2.4), we get Applying another (p 0 , q 0 )-integration, we obtain, for t ∈ (0, t 1 ], For t ∈ (t 1 , ((t 2t 1 )/p 2 1 ) + t 1 ], applying the double (p 1 , q 1 )-integration to both sides of the first equation of (2.4), we have where t ∈ (t 1 , t 2 ]. Due to the impulsive conditions and we have Similarly, we deduce the integral equation (2.5), as desired. Now, the existence and uniqueness results for problems (2.1) and (2.4) will be proved by using the Banach's contraction mapping principle. Let us define the space PC(J, R) = {x : J → R: x(t) is continuous everywhere except for some t k in which x(t + k ) and x(tk ) exist and x(tk ) = x(t k ), k = 1, 2, . . . , m}. The set PC(J, R) is a Banach space equipped with the norm x = sup{|x(t)| : t ∈ J}. For convenience, we put , x, y ∈ R and k = 1, 2, . . . , m. If then the boundary value problem (2.1) has a unique solution on J.
Proof In view of Theorem 2.1, we define the operator A : PC(J, R) → PC(J, R) by Define the ball B r 1 = {x ∈ PC(J, R) : x ≤ r 1 } where the positive constant r 1 is defined by The Banach contraction mapping principle is used to claim that there exists a unique fixed point of an operator equation x = Ax in B r 1 . By setting sup t∈J |f (t, 0)| = M 1 , and sup{|ϕ i (0)|, i = 1, 2, . . . , m} = M 2 and using the inequalities which leads to AB r 1 ⊂ B r 1 . To prove that A is a contraction, we let x, y ∈ B r 1 . Then we have for all x, y ∈ R. Proof The proof is similar to that of Theorem 2.4 and is omitted.
In proving our next results, we use the constants: , (ρ * r+1t r ) 2 p r + q r , Applying Theorem 2.9 to define the operator on PC 1 (Λ 1 , R) and following the method of Theorem 2.4, we can easily prove the existence of a unique solution of problem (2.10).

15)
then the boundary value problem of type II (2.10) has a unique solution on Λ 1 .
Proof To show the technique of computation of constants Ω 6 and Ω 7 , we give a short proof. Now we prove that the operator equation x = Bx has a unique fixed point, where the operator B : PC 2 (Λ 2 , R) → PC 2 (Λ 2 , R) is defined, in view of Theorem 2.10, by (s) t k d p k ,q k s, t ∈ (t k , ρ * k+1 ], k = 0, 1, . . . , m. By a similar method as in Theorem 2.4, we can show that the operator B maps a subset of PC 2 (Λ 2 , R) into subset of PC 2 (Λ 2 , R). Next, we will prove that B is a contraction. Let problems. The consecutive domains of impulsive (p, q)-difference equations of type I are overlapped, while the unknown functions of impulsive equations of type II are defined on disjoint consecutive domains. Four types of problems were considered, two impulsive (p, q)-difference equations of type I and two impulsive (p, q)-difference equations of type II. Existence and uniqueness results were proved via Banach's contraction mapping principle. Examples illustrating the obtained results were also presented.