Existence and Stability Results for Impulsive Fractional <i>q</i>-Difference Equation

In this paper, we study the boundary value problem for an impulsive fractional q -difference equation. Based on Banach’s contraction mapping principle, the existence and Hyers-Ulam stability of solutions for the equation which we considered are obtained. At last, an illustrative example is given for the main result.

• Impulsive Fractional <i>q</i>-Difference Equation
• Hyers-Ulam Stability
• Existence
• <i>q</i>-Calculus

1. Introduction

The q-calculus or quantum calculus is an old subject that was initially developed by Jackson [1]; basic definitions and properties of q-calculus can be found in [2]. The fractional q-calculus had its origin in the works by Al-Salam [3] and Agarwal [4]. But the definitions mentioned above about q-calculus can’t be applied to impulse points t k , k ∈ ℤ , such that t k ∈ ( q t , t ) . In [5], the authors defined the concepts of fractional q-calculus by defining a q-shifting operator Φ a q ( m ) = q m + ( 1 − q ) a ,   m , a ∈ ℝ . Using the q-shifting operator, the fractional impulsive q-difference equation was defined. In paper [5] [6] [7], the authors discussed the existence of solutions for the fractional impulsive q-difference equation with Riemann-Liouville and Caputo fractional derivatives respectively. Some other results about q-difference equations can be found in papers [8] - [16] and the references cited therein. Dumitru Baleanu et al. discussed the stability of non-autonomous systems with the q-Caputo fractional derivatives in reference [17]. However, the existence and stability of solutions for the fractional impulsive q-difference have not been yet studied.

Motivated greatly by the above mentioned excellent works, in this paper we investigate the following fractional impulsive q-difference equation with q-integral boundary conditions:

{ t k c D q k α k x ( t ) = f ( t , x ( t ) ) ,   t ∈ J k ⊆ J = [ 0 , T ] ,   t ≠ t k , Δ x ( t k ) = x ( t k + ) − x ( t k ) = φ k ( x ( t k ) ) ,   k = 1 , 2 , ⋯ , m , η 1 x ( 0 ) + η 2 x ( T ) = μ ∑ k = 0 m   t k I q k β k x ( t k + 1 ) . (1)

where t k c D q k α k is the fractional q k -derivative of the Caputo type of order α k on J k , 0 < α k < 1 , 0 < q k < 1 , J 0 = [ 0 , t 1 ] , J 0 = [ 0 , t 1 ] , k = 1 , 2 , ⋯ , m , φ k ∈ C ( ℝ , ℝ ) , f ∈ C ( J × ℝ , ℝ ) . t k I q k β k denotes the Riemann-Liouville q k -fractional integral of order β k > 0 on J k , k = 0 , 1 , 2 , ⋯ , m and η 1 ,   η 2 ,   μ are three constants.

2. Preliminaries on q-Calculus and Lemmas

Here we recall some definitions and fundamental results on fractional q-integral and fractional q-derivative, for the full theory for which one is referred to [5] [6] [7].

For q ∈ ( 0,1 ) , we define a q-shifting operator as a Φ q ( m ) = q m + ( 1 − q ) a . The new power of q-shifting operator is defined as a ( n − m ) q ( 0 ) = 1 ,

a ( n − m ) q ( k ) = ∏ i = 0 k − 1 ( n −   a Φ q i ( m ) ) , k ∈ ℕ ∪ { 0 } , n ∈ ℝ . If ν ∈ ℝ , then   a ( n − m ) q ( ν ) = n ν ∏ i = 0 ∞ 1 − a n Φ q i ( m n ) 1 − a n Φ q i + ν ( m n ) .

The q-derivative of a function f on interval [ a , b ] is defined by

( a D q f ) ( t ) = f ( t ) − f ( a Φ q ( t ) ) ( 1 − q ) ( t − a ) , t ≠ a , ( a D q f ) ( a ) = l i m t → a ( a D q f ) ( t ) .

The q-integral of a function f defined on the interval [ a , b ] is given by

( a I q f ) ( t ) = ∫ a t     f ( s )   a d s = ( 1 − q ) ( t − a ) ∑ i = 0 ∞     q i f ( a Φ q i ( t ) ) , t ∈ [ a , b ] .

Some results about operator a D q and a I q can be found in references [5]. Let us define fractional q-derivative and q-integral on interval [ a , b ] and outline some of their properties [5] [6] [7].

Definition 1 [5] The fractional q-derivative of Riemann-Liouville type of order ν ≥ 0 on interval [ a , b ] is defined by ( a D q 0 f ) ( t ) = f ( t ) and

( a D q ν f ) ( t ) = ( a D q l   a I q l − ν f ) ( t ) , ν > 0,

where l is the smallest integer greater than or equal to ν .

Definition 2 [5] Let α ≥ 0 and f be a function defined on [ a , b ] . The fractional q-integral of Riemann-Liouville type is given by ( a I q 0 f ) ( t ) = f ( t ) and

( a I q α f ) ( t ) = 1 Γ q ( α ) ∫ a t   a ( t − Φ a q ( s ) ) q α − 1 f ( s )   a d q s , α > 0, t ∈ [ a , b ] .

Lemma 1 [5] Let α , β ∈ ℝ + and f be a continuous function on [ a , b ] , a ≥ 0 . The Riemann-Liouville fractional q-integral has the following semi-group property

a I q β   a I q α f ( t ) =   a I q α   a I q β f ( t ) =   a I q α + β f ( t ) .

Lemma 2 [5] Let f be a q-integrable function on [ a , b ] . Then the following equality holds

a D q α   a I q α f ( t ) = f ( t ) ,       for     α > 0, t ∈ [ a , b ] .

Lemma 3 [5] Let α > 0 and p be a positive integer. Then for t ∈ [ a , b ] the following equality holds

a I q α   a D q p f ( t ) =   a D q p   a I q α f ( t ) − ∑ k = 0 p − 1 ( t − a ) α − p + k Γ q ( α + k − p + 1 )   a D q k f ( a ) .

Definition 3 [7] The fractional q-derivative of Caputo type of order α ≥ 0 on interval [ a , b ] is defined by a c D q 0 f ( t ) = f ( t ) and

( a c D q α f ) ( t ) = ( a I q n − α   a D q n f ) ( t ) , α > 0,

where n is the smallest integer greater than or equal to α .

Lemma 4 [7] Let α > 0 and n be the smallest integer great than or equal to α . Then for t ∈ [ a , b ] the following equality holds

a I q α   a c D q α f ( t ) = f ( t ) − ∑ k = 0 n − 1 ( t − a ) k Γ q ( k + 1 )   a D q k f ( a ) .

3. Main Results

In this section, we will give the main results of this paper.

Let P C ( J , ℝ ) = { x : J → ℝ , x ( t ) is continuous everywhere except for some t k at which x ( t k + ) and x ( t k − ) exist, and x ( t k − ) = x ( t k ) , k = 1,2, ⋯ , m } . P C ( J , ℝ ) is a Banach space with the norm

‖ x ‖ = s u p { | x ( t ) | : t ∈ J } .

First, for the sake of convenience, we introduce the following notations:

Λ = η 1 + η 2 − μ ∑ i = 0 m     Ω β i ≠ 0 ,     Ω σ i = t i ( t i + 1 − t i ) q i ( σ i ) Γ q i ( σ i + 1 ) ,

where σ i ∈ { α i , β i , α i + β i } ,   q i ∈ ( 0 , 1 ) ,   i = 0 , 1 , 2 , ⋯ , m .

To obtain our main results, we need the following lemma.

Lemma 5 Let μ ∑ i = 0 m     Ω β i ≠ η 1 + η 2 and h ( t ) ∈ C ( J , ℝ ) . Then for any t ∈ J k , the solution of the following problem

{ t k c D q k α k x ( t ) = h ( t ) ,   t ∈ J k ⊆ J = [ 0 , T ] ,   t ≠ t k , Δ x ( t k ) = x ( t k + ) − x ( t k ) = φ k ( x ( t k ) ) ,   k = 1 , 2 , ⋯ , m , η 1 x ( 0 ) + η 2 x ( T ) = μ ∑ k = 0 m   t k I q k β k x ( t k + 1 ) (2)

is given by

x ( t ) = 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i h ( t i + 1 ) − η 2   t i I q i α i h ( t i + 1 ) )     + ∑ i = 1 m [ μ ( ∑ j = 1 i     φ j ( x ( t j ) ) + ∑ j = 0 i − 1   t j I q j α j h ( t j + 1 ) ) Ω β i − η 2 φ i ( x ( t i ) ) ] }     + ∑ i = 1 k     φ i ( x ( t i ) ) + ∑ i = 0 k − 1   t i I q i α i h ( t i + 1 ) +   t k I q k α k h ( t ) . (3)

Proof. Applying the operator t 0 I q 0 α 0 on both sides of the first equation of (2) for t ∈ J 0 and using Lemma 4, we have

x ( t ) = x ( t 0 ) +   t 0 I q 0 α 0 h ( t ) .

Then we get for t = t 1 that

x ( t 1 ) = x ( t 0 ) +   t 0 I q 0 α 0 h ( t 1 ) . (4)

For t ∈ J 1 , again taking the t 1 I q 1 α 1 to (4) and using the above process, we get

x ( t ) = x ( t 1 + ) +   t 1 I q 1 α 1 h ( t ) .

Applying the impulsive condition x ( t 1 + ) = x ( t 1 ) + φ 1 ( x ( t 1 ) ) , we get

x ( t ) = x ( t 0 ) + φ 1 ( x ( t 1 ) ) +   t 0 I q 0 α 0 h ( t 1 ) +   t 1 I q 1 α 1 h ( t ) .

By the same way, for t ∈ J 2 , we have

x ( t ) = x ( t 0 ) + φ 1 ( x ( t 1 ) ) + φ 2 ( x ( t 2 ) ) +   t 0 I q 0 α 0 h ( t 1 ) +   t 1 I q 1 α 1 h ( t 2 ) +   t 2 I q 2 α 2 h ( t ) .

Repeating the above process for t ∈ J k ⊆ J , k = 0 , 1 , 2 , ⋯ , m , we get

x ( t ) = x ( t 0 ) + ∑ i = 1 k     φ i ( x ( t i ) ) + ∑ i = 0 k − 1   t i I q i α i h ( t i + 1 ) +   t k I q k α k h ( t ) . (5)

From (5), we find that

x ( T ) = x ( t 0 ) + ∑ i = 1 k     φ i ( x ( t i ) ) + ∑ i = 0 k − 1   t i I q i α i h ( t i + 1 ) +   t k I q k α k h ( T ) .

From the boundary condition of (2), we get

x ( t 0 ) = 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i h ( t i + 1 ) − η 2   t i I q i α i h ( t i + 1 ) )     + ∑ i = 1 m [ μ ( ∑ j = 1 i     φ j ( x ( t j ) ) + ∑ j = 0 i − 1   t j I q j α j h ( t j + 1 ) ) Ω β i − η 2 φ i ( x ( t i ) ) ] } . (6)

Substituting (6) to (5), we obtain the solution (3). This completes the proof.

We define an operator G : P C ( J , ℝ ) → P C ( J , ℝ ) as follows:

G x ( t ) = 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i f ( s , x ) ( t i + 1 ) − η 2   t i I q i α i f ( s , x ) ( t i + 1 ) )     + ∑ i = 1 m [ μ ( ∑ j = 1 i     φ j ( x ( t j ) ) + ∑ j = 0 i − 1   t j I q j α j f ( s , x ) ( t j + 1 ) ) Ω β i − η 2 φ i ( x ( t i ) ) ] }     + ∑ i = 1 k     φ i ( x ( t i ) ) + ∑ i = 0 k − 1   t i I q i α i f ( s , x ) ( t i + 1 ) +   t k I q k α k f ( s , x ) ( t ) . (7)

Then, the existence of solutions of system (1) is equivalent to the problem of fixed point of operator G in (7).

Theorem 1 Let f : J × ℝ → ℝ and φ k : ℝ → ℝ , k = 1 , 2 , ⋯ , m be continuous functions. Assume that μ ∑ i = 0 m     Ω β i ≠ η 1 + η 2 and the following conditions are satisfied:

(H₁) There exists a positive constant L such that | φ k ( x ) − φ k ( y ) | ≤ L | x − y | for each x , y ∈ ℝ and k = 1 , 2 , ⋯ , m .

(H₂) There exists a function M ( t ) ∈ C ( J , ℝ + ) such that

| f ( t , x ) − f ( t , y ) | ≤ M ( t ) | x − y | ,     ∀   t ∈ J ,   x , y ∈ ℝ .

(H₃) Δ < 1 .

Then problem (1) has a unique solution on J, where M = s u p t ∈ J | M ( t ) | and

Δ = 1 Λ ∑ i = 1 m ( μ M Ω α i + β i + ( η 2 + M ) Ω α i + μ M ∑ j = 0 i − 1     Ω α j Ω β i + μ L i Ω β i )     + 1 Λ ( μ Ω α 0 + β 0 + η 2 Ω α 0 ) + m L ( 1 Λ η 2 + 1 ) .

Proof. The conclusion will follow once we have shown that the operator G defined (7) is a construction with respect to a suitable norm on P C ( J , ℝ ) .

For any functions x , y ∈ P C ( J , ℝ ) , we have

| ( G x ) ( t ) − ( G y ) ( t ) | ≤ 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i | f ( s , x ) − f ( s , y ) | ( t i + 1 ) + η 2   t i I q i α i | f ( s , x ) − f ( s , y ) | ( t i + 1 ) )     + ∑ i = 1 m [ μ ( ∑ j = 1 i | φ j ( x ( t j ) ) − φ j ( y ( t j ) ) | + ∑ j = 0 i − 1   t j I q j α j | f ( s , x ) − f ( s , y ) | ( t j + 1 ) ) Ω β i       + η 2 | φ i ( x ( t i ) ) − φ i ( y ( t i ) ) | ] } + ∑ i = 1 m | φ i ( x ( t i ) ) − φ i ( y ( t i ) ) |     + ∑ i = 0 m − 1   t i I q i α i | f ( s , x ) − f ( s , y ) | ( t i + 1 ) +   t m I q m α m | f ( s , x ) − f ( s , y ) | ( t ) .

By conditions (H₁) and (H₂), we get

| ( G x ) ( t ) − ( G y ) ( t ) | ≤ 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i ( M ‖ x − y ‖ ) ( t i + 1 ) + η 2   t i I q i α i ( M ‖ x − y ‖ ) ( t i + 1 ) )         + ∑ i = 1 m [ μ ( ∑ j = 1 i     L ‖ x − y ‖ + ∑ j = 0 i − 1   t j I q j α j ( M ‖ x − y ‖ ) ) Ω β i + η 2 L ‖ x − y ‖ ] }

        + ∑ i = 1 m     L ‖ x − y ‖ + ∑ i = 0 m − 1   t i I q i α i ( M ‖ x − y ‖ ) ( t i + 1 ) +   t m I q m α m ( M ‖ x − y ‖ ) ( t m + 1 ) ≤ { 1 Λ ∑ i = 1 m ( μ M Ω α i + β i + η 2 Ω α i + μ L i Ω β i + μ M ∑ j = 0 i − 1     Ω α j Ω β i + M Ω α i )         + 1 Λ ( μ Ω α 0 + β 0 + η 2 Ω α 0 ) + m L ( 1 Λ η 2 + 1 ) } ‖ x − y ‖ ,

which implies that

‖ G x − G y ‖ ≤ Δ ‖ x − y ‖ .

Thus the operator G is a contraction in view of the condition (H₃). By Banach’s contraction mapping principle, the problem (1) has a unique solution on J. This completes the proof.

In the following, we study the Hyers-Ulam stability of impulsive fractional q-difference Equation (1). Let ε > 0 , ϵ > 0 and δ : [ 0, T ] → ℝ be a continuous function. Consider the inequalities:

{ | t k c D q k α k x ¯ ( t ) − f ( t , x ¯ ( t ) ) | ≤ δ ( t ) ε ,   t ∈ J k ⊆ J = [ 0 , T ] ,   t ≠ t k , k = 0 , 1 , ⋯ , m , | Δ x ¯ ( t k ) − ϕ k ( x ¯ ( t k ) ) | ≤ ϵ ε ,   k = 1 , 2 , ⋯ , m , η 1 x ¯ ( 0 ) + η 2 x ¯ ( T ) = μ ∑ k = 0 m   t k I q k β k x ¯ ( t k + 1 ) . (8)

Now, we give out the definition of Hyers-Ulam stability of system (1).

Definition 4 System (1) is Hyers-Ulam stable with respect to system (8), if there exists A f > 0 such that

| x ¯ − x ˜ | ≤ A f ε

for all t ∈ J , where x ¯ is the solution of (8), and x ˜ of the solution for system (1).

Theorem 2 Assume f : J × ℝ → ℝ satisfy assumption (H₂), φ i : ℝ → ℝ , i = 1 , 2 , ⋯ , m are continuous functions and satisfy assumption (H₁) and the condition (H₃) holds,   s u p t ∈ J δ ( t ) ≤ 1 . Then the system (1) is Hyers-Ulam stable with respect to system (8).

Proof. Let t k c D q k α k x ¯ ( t ) = f ( t , x ¯ ( t ) ) + g ( t ) , k = 0 , 1 , ⋯ , m and   Δ x ¯ ( t k ) = φ k ( x ¯ ( t k ) ) + g k , k = 1 , 2 , ⋯ , m . Consider the system

{   t k c D q k α k x ¯ ( t ) = f ( t , x ¯ ( t ) ) + g ( t ) ,   t ∈ J k ⊆ J = [ 0 , T ] ,   t ≠ t k , Δ x ¯ ( t k ) = φ k ( x ¯ ( t k ) ) + g k ,   k = 1 , 2 , ⋯ , m . η 1 x ¯ ( 0 ) + η 2 x ¯ ( T ) = μ ∑ k = 0 m   t k I q k β k x ¯ ( t k + 1 ) . (9)

Similarly to the system in Theorem 1, system (9) is equivalent to the following integral equation in Lemma 5.

x ¯ ( t ) = 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i ( f ( s , x ¯ ) + g ( s ) ) ( t i + 1 ) − η 2   t i I q i α i ( f ( s , x ¯ ) + g ( s ) ) ( t i + 1 ) )     + ∑ i = 1 m [ μ ( ∑ j = 1 i ( φ j ( x ¯ ( t j ) ) + g j ) + ∑ j = 0 i − 1   t j I q j α j ( f ( s , x ¯ ) + g ( s ) ) ( t j + 1 ) ) Ω β i

            − η 2 ( φ i ( x ¯ ( t i ) ) + g i ) ] } + ∑ i = 1 k ( φ i ( x ¯ ( t i ) ) + g i )     + ∑ i = 0 k − 1   t i I q i α i ( f ( s , x ¯ ) + g ( s ) ) ( t i + 1 ) +   t k I q k α k ( f ( t , x ¯ ) + g ( t ) ) (10)

Now, we define the operator G ˜ as following

G ˜ x ( t ) = 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i f ( s , x ) ( t i + 1 ) − η 2   t i I q i α i f ( s , x ) ( t i + 1 ) )     + ∑ i = 1 m [ μ ( ∑ j = 1 i     φ j ( x ( t j ) ) + ∑ j = 0 i − 1   t j I q j α j f ( s , x ) ( t j + 1 ) ) Ω β i − η 2 φ i ( x ( t i ) ) ] }     + ∑ i = 1 k     φ i ( x ( t i ) ) + ∑ i = 0 k − 1   t i I q i α i f ( s , x ) ( t i + 1 ) +   t k I q k α k f ( s , x ) ( t ) + G ( t ) = G x + G ( t ) . (11)

where

G ( t ) = 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i g ( t i + 1 ) − η 2   t i I q i α i g ( t i + 1 ) )       + ∑ i = 1 m [ μ ( ∑ j = 1 i     g j + ∑ j = 0 i − 1   t j I q j α j g ( t j + 1 ) ) Ω β i − η 2 g i ] }     + ∑ i = 1 k     g i + ∑ i = 0 k − 1   t i I q i α i g ( t i + 1 ) +   t k I q k α k g ( t ) . (12)

Note that

‖ G ˜ x − G ˜ y ‖ = ‖ G x − G y ‖ .

Then the existence of a solution of (1) implies the existence of a solution to (9), it follows from Theorem 1 that G ˜ is a contraction. Thus there is a unique fixed point x ¯ of G ˜ , and respectively x ˜ of G .

Since t ∈ [ 0, T ] and   s u p t ∈ J δ ( t ) ≤ 1 , we obtain

‖ G ‖ = max t ∈ J | G ( t ) | = max t ∈ J | 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i g ( t i + 1 ) − η 2   t i I q i α i g ( t i + 1 ) )       + ∑ i = 1 m [ μ ( ∑ j = 1 i     g j + ∑ j = 0 i − 1   t j I q j α j g ( t j + 1 ) ) Ω β i − η 2 g i ] }       + ∑ i = 1 k     g i + ∑ i = 0 k − 1   t i I q i α i g ( t i + 1 ) +   t k I q k α k g ( t ) |

≤ max t ∈ J | 1 Λ { ∑ i = 0 m ( μ   t i I q i α i + β i g ( t i + 1 ) − η 2   t i I q i α i g ( t i + 1 ) )       + ∑ i = 1 m [ μ ( ∑ j = 1 i     g j + ∑ j = 0 i − 1   t j I q j α j g ( t j + 1 ) ) Ω β i − η 2 g i ] }

      + ∑ i = 1 m     g i + ∑ i = 0 m − 1   t i I q i α i g ( t i + 1 ) +   t m I q m α m g ( t ) | ≤ { 1 Λ ∑ i = 1 m ( μ Ω α i + β i + η 2 Ω α i + μ ϵ i Ω β i + μ ∑ j = 0 i − 1     Ω α j Ω β i + Ω α i )     + 1 Λ ( μ Ω α 0 + β 0 + η 2 Ω α 0 ) + m ϵ ( 1 Λ η 2 + 1 ) } ε . (13)

Then, we get

‖ x ¯ − x ˜ ‖ = ‖ G ˜ x ¯ − G x ˜ ‖ = ‖ G x ¯ − G x ˜ + G ( t ) ‖ ≤ ‖ G x ¯ − G x ˜ ‖ + ‖ G ‖ ≤ Δ ‖ x ¯ − x ˜ ‖ + { 1 Λ ∑ i = 0 m ( μ Ω α i + β i + η 2 Ω α i + μ ϵ i Ω β i + μ ∑ j = 0 i − 1     Ω α j Ω β i + Ω α i )     + 1 Λ ( μ Ω α 0 + β 0 + η 2 Ω α 0 ) + m ϵ ( 1 Λ η 2 + 1 ) } ε . (14)

By condition (H₃), we have

‖ x ¯ − x ˜ ‖ ≤ ( 1 − Δ ) − 1 { 1 Λ ∑ i = 0 m ( μ Ω α i + β i + η 2 Ω α i + μ ϵ i Ω β i + μ ∑ j = 0 i − 1     Ω α j Ω β i + Ω α i )     + m ϵ ( 1 Λ η 2 + 1 ) } ε . (15)

Let A f = ( 1 − Δ ) − 1 { 1 Λ ∑ i = 0 m ( μ Ω α i + β i + η 2 Ω α i + μ ϵ i Ω β i + μ ∑ j = 0 i − 1     Ω α j Ω β i + Ω α i )               + m ϵ ( 1 Λ η 2 + 1 ) } , then ‖ x ¯ − x ˜ ‖ ≤ A f ε .

This completes the proof.

Remark 1 Note that (1) has a very general form, as special instances results from (1), when, η 1 = η 2 = 1 , μ = 0 , (1) reduces to the antiperiodic boundary value problem of the impulsive fractional q-difference equation:

{ t k c D q k α k x ( t ) = f ( t , x ( t ) ) ,   t ∈ J k ⊆ J = [ 0 , T ] ,   t ≠ t k , Δ x ( t k ) = x ( t k + ) − x ( t k ) = φ k ( x ( t k ) ) ,   k = 1 , 2 , ⋯ , m , x ( 0 ) + x ( T ) = 0.

4. Example

Consider the following boundary value problem:

{ t k c D 3 k + 1 4 k + 3 k + 1 3 k + 2 x ( t ) = sin 2 t t 2 + 50 2 | x ( t ) | 1 + | x ( t ) | + 3 t 4 ,   t ∈ [ 0 , 3 2 ] \ { t 1 , t 2 } , Δ x ( t k ) = 1 200 k x 2 ( t k ) + 2 | x ( t k ) | 1 + | x ( t k ) | + k 5 , t k = k 2 ,   k = 1 , 2 , 8 3 x ( 0 ) + 1 6 x ( 3 2 ) = 1 2 ∑ k = 0 2   t k I 3 k + 1 4 k + 3 k + 1 k 2 + 2 x ( t k + 1 ) . (16)

Corresponding to boundary value problem (1), one see that α k = k + 1 3 k + 2 , β k = k + 1 k 2 + 2 , q k = 3 k + 1 4 k + 3 , t k = k 2 , f ( t , x ) = sin 2 t t 2 + 50 2 | x ( t ) | 1 + | x ( t ) | + 3 4 , φ k ( x ( t k ) ) = 1 200 k x 2 ( t k ) + 2 | x ( t k ) | 1 + | x ( t k ) | . Through a simple calculation, we get

| f ( t , x ) − f ( t , y ) | ≤ sin 2 t t 2 + 25 | x − y | , M ( t ) = sin 2 t t 2 + 25 ≤ 1 25 = M ,

| φ k ( x ) − φ k ( y ) | ≤ 1 200 k | x − y | ≤ 1 200 | x − y | , L = 1 200 ,

Λ ≐ 1.7875 > 0 ,   Δ ≐ 0.4873 < 1.

From Theorem 1, the problem (16) has a unique solution x on [ 0, 3 2 ] . Furthermore, the solution x is Hyers-Ulam stable with respect to the following system

{ | t k c D 3 k + 1 4 k + 3 k + 1 3 k + 2 x ( t ) − sin 2 t t 2 + 50 2 | x ( t ) | 1 + | x ( t ) | − 3 t 4 | ≤ δ ( t ) ε ,   t ∈ [ 0 , 3 2 ] \ { t 1 , t 2 } , | Δ x ( t k ) − 1 200 k x 2 ( t k ) + 2 | x ( t k ) | 1 + | x ( t k ) | − k 5 | ≤ ϵ ε , t k = k 2 ,   k = 1 , 2 , 8 3 x ( 0 ) + 1 6 x ( 3 2 ) = 1 2 ∑ k = 0 2   t k I 3 k + 1 4 k + 3 k + 1 k 2 + 2 x ( t k + 1 ) , (17)

where ϵ > 0 , ε > 0 , sup t ∈ [ 0 , 3 2 ] δ ( t ) < 1 .

5. Conclusion

In this paper, we study the existence and Hyers-Ulam stability of solutions for impulsive fractional q-difference equation. We obtain some results as following: 1) Using the q-shifting operator, the results of existence of solutions for impulsive fractional q-difference equation with q-integral boundary conditions are obtained. 2) The Hyers-Ulam stability of the nonlinear impulsive fractional q-difference equations was obtained.

Funding

This research was supported by Science and Technology Foundation of Guizhou Province (Grant No. [2016] 7075), by the Project for Young Talents Growth of Guizhou Provincial Department of Education under (Grant No. Ky [2017] 133), and by the project of Guizhou Minzu University under (Grant No.16yjrcxm002).

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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