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 JAMP  Vol.8 No.7 , July 2020
Existence and Stability Results for Impulsive Fractional q-Difference Equation
Abstract: 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.

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 , 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 . 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 ( , ) , f C ( J × , ) . 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 { 0 } , n . If ν , 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 + 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 , ) = { x : J , 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 , ) 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 , ) . 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 , ) P C ( J , ) 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 × and φ k : , 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 and k = 1 , 2 , , m .

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

| f ( t , x ) f ( t , y ) | M ( t ) | x y | , t J , x , y .

(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 , ) .

For any functions x , y P C ( J , ) , 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 ] 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 × satisfy assumption (H2), φ i : , 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).

Cite this paper: Jiang, M. and Huang, R. (2020) Existence and Stability Results for Impulsive Fractional q-Difference Equation. Journal of Applied Mathematics and Physics, 8, 1413-1423. doi: 10.4236/jamp.2020.87107.
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