Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

Changpin Li; Zhiqiang Li

doi:10.1515/ijnsns-2021-0189

Article

Stability and ψ-algebraic decay of the solution to ψ-fractional differential system

Changpin Li and Zhiqiang Li

Published/Copyright: December 2, 2021

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From the journal International Journal of Nonlinear Sciences and Numerical Simulation Volume 24 Issue 2

Abstract

In this article, we focus on stability and ψ-algebraic decay (algebraic decay in the sense of ψ-function) of the equilibrium to the nonlinear ψ-fractional ordinary differential system. Before studying the nonlinear case, we show the stability and decay for linear system in more detail. Then we establish the linearization theorem for the nonlinear system near the equilibrium and further determine the stability and decay rate of the equilibrium. Such discussions include two cases, one with ψ-Caputo fractional derivative, another with ψ-Riemann–Liouville derivative, where the latter is a bit more complex than the former. Besides, the integral transforms are also provided for future studies.

Keywords: ψ-algebraic decay; ψ-fractional derivative; linearization theorem; Mittag–Leffler function; stability

MSC: 34A08; 34D20; 34D30

Corresponding author: Changpin Li, Department of Mathematics, Shanghai University, Shanghai 200444, China, E-mail: lcp@shu.edu.cn

Acknowledgments

The authors wish to thank EiC Prof Björn Birnir and the corresponding editor for their kind handling this paper. They especially thank two anonymous reviewers for their careful reading and giving invaluable correction suggestions. One of referees makes [14], [15], [16] (which cannot be got by us) and [23] available.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.
Research funding: None declared.
Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

Although, there exist some integral transforms for ψ-fractional calculus, they are partial and scattered in different papers [14], [15], [16, 18, 23]. Here we summarize and/or generalize them and then present them here for the convenience of future research.

A.1 The generalized Laplace transform

In the appendix section, we introduce a generalized Laplace transform and a generalized Mellin transform, which can be applied to fractional integrals and derivatives for Definitions 2.1–2.3. Then we investigate their properties.

Definition A.1

Suppose that a function f(t) is defined on ( − ∞ , b ] ( b ∈ R ) . The right generalized Laplace transform of f(t) is defined by

(A.1) F ( s ) = L r ψ { f ( t ) } = ∫ − ∞ b e − s ( ψ ( b ) − ψ ( t ) ) f ( t ) ψ ′ ( t ) d t , s ∈ C .

The inverse right generalized Laplace transform is given by

(A.2) f ( t ) = L r ψ − 1 { F ( s ) } = 1 2 π i ∫ c − i ∞ c + i ∞ e s ( ψ ( b ) − ψ ( t ) ) F ( s ) d s , c > 0 , i 2 = − 1 .

We next present the conditions for the existence of the right generalized Laplace transform.

Theorem A.1

Suppose that a function f(t) is defined on ( − ∞ , b ] ( b ∈ R ) . If

f(t) is continuous or piecewise continuous on every finite subinterval of (−∞, b],
there exist positive constants M > 0 and σ > 0 such that for the given T′ < b,
| f ( t ) | ≤ M e σ ( ψ ( b ) − ψ ( t ) ) , for t < T ′ ,

then the right generalized Laplace transform of f(t) exists when Re(s) > σ.

Proof

Using Definition A.8 and conditions (1) and (2) of theorem gives that

| F ( s ) | ≤ ∫ − ∞ b e − | s | ( ψ ( b ) − ψ ( t ) ) | f ( t ) | ψ ′ ( t ) d t ≤ ∫ − ∞ b e − | s | ( ψ ( b ) − ψ ( t ) ) M e σ ( ψ ( b ) − ψ ( t ) ) ψ ′ ( t ) d t = M ∫ − ∞ b e − ( | s | − σ ) ( ψ ( b ) − ψ ( t ) ) ψ ′ ( t ) d t = M | s | − σ .

Thus the result is true. □

Definition A.2

Assume that functions f(t) and g(t) defined on ( − ∞ , b ] ( b ∈ R ) . We say that the integral ∫ t b f ( ψ − 1 ( ψ ( b ) + ψ ( t ) − ψ ( τ ) ) ) g ( τ ) ψ ′ ( τ ) d τ is the right generalized convolution of f(t) and g(t), that is,

f ( t ) * r ψ g ( t ) = ( f * r ψ g ) ( t ) = ∫ t b f ( ψ − 1 ( ψ ( b ) + ψ ( t ) − ψ ( τ ) ) ) g ( τ ) ψ ′ ( τ ) d τ .

Theorem A.2

(Right convolution theorem). If L r ψ { f ( t ) } = F ( s ) and L r ψ { g ( t ) } = G ( s ) , then

L r ψ { f ( t ) * r ψ g ( t ) } = L r ψ { f ( t ) } L r ψ { g ( t ) } = F ( s ) G ( s ) ;

Or equivalently,

L r ψ − 1 { F ( s ) G ( s ) } = f ( t ) * r ψ g ( t ) .

Proof

According to Definitions A.8 and A.2, and interchanging the order of integration and making the substitution ψ ⁻¹(ψ(b) + ψ(t) − ψ(τ)) = w, we obtain

L r ψ { f ( t ) * r ψ g ( t ) } = ∫ − ∞ b e − s ( ψ ( b ) − ψ ( t ) ) ∫ t b f ( ψ − 1 ( ψ ( b ) + ψ ( t ) − ψ ( τ ) ) ) g ( τ ) ψ ′ ( τ ) d τ ψ ′ ( t ) d t = ∫ − ∞ b ∫ − ∞ b e − s ( 2 ψ ( b ) − ψ ( τ ) − ψ ( w ) ) f ( w ) g ( τ ) ψ ′ ( w ) d w ψ ′ ( τ ) d τ = L r ψ { f ( t ) } L r ψ { g ( t ) } = F ( s ) G ( s ) .

Using repeatedly integration by parts yields the following differential property.□

Lemma A.1

If L r ψ { f ( t ) } = F ( s ) , then

(A.3) L r m { δ ψ n f ( t ) } = ∑ k = 0 n − 1 s n − k − 1 δ ψ k f ( b ) − s n F ( s ) , t < b , n ∈ Z + .

We are ready to present the right generalized Laplace transform for fractional integrals and derivatives.

Theorem A.3

Let n − 1 < α < n ∈ Z + and b ∈ R . Then we have:

(A.4) L r ψ { D t , b − α ψ f ( t ) } = s − α L r ψ { f ( t ) } ,

(A.5) L r ψ { D t , b α ψ f ( t ) } = ( − 1 ) n ∑ k = 0 n − 1 s n − k − 1 D t , b α + k − n ψ f ( t ) t = b − s α L r ψ { f ( t ) } ,

(A.6) L r ψ { D t , b α C ψ f ( t ) } = ( − 1 ) n ∑ k = 0 n − 1 s α − k − 1 δ ψ k f ( b ) − s α L r ψ { f ( t ) } .

Proof

By Definitions 2.1 and A.2, we can rewrite equality (2.2) as the following convolution form

D t , b − α ψ f ( t ) = 1 Γ ( α ) ∫ t b ( ψ ( τ ) − ψ ( t ) ) − ( 1 − α ) f ( τ ) ψ ′ ( τ ) d τ = ( ψ ( b ) − ψ ( t ) ) α − 1 Γ ( α ) * r ψ f ( t ) .

From Theorem A.2, one concludes

L r ψ { D t , b − α ψ f ( t ) } = L r ψ ( ψ ( b ) − ψ ( t ) ) α − 1 Γ ( α ) * r ψ f ( t ) = s − α L r ψ { f ( t ) } .

This shows (A.4).

Next, we prove (A.5). Using equality (2.4) and differential property (A.3) implies that

L r ψ { D t , b α ψ f ( t ) } = ( − 1 ) n L r ψ δ ψ n D t , b − ( n − α ) ψ f ( t ) = ( − 1 ) n ∑ k = 0 n − 1 s n − k − 1 δ ψ k D t , b − ( n − α ) ψ f ( t ) t = b − s n L r ψ { D t , b − ( n − α ) ψ f ( t ) } = ( − 1 ) n ∑ k = 0 n − 1 s n − k − 1 D t , b α + k − n ψ f ( t ) t = b − s α L r ψ { f ( t ) } .

Finally, by means of equalities (2.6), (A.4), and differential property (A.3), one gets

L r ψ { D t , b α C ψ f ( t ) } = ( − 1 ) n L r ψ D t , b − ( n − α ) ψ δ ψ n f ( t ) = ( − 1 ) n s α − n L r ψ δ ψ n f ( t ) = ( − 1 ) n s α − n ∑ k = 0 n − 1 s n − k − 1 δ ψ k f ( b ) − s n L r ψ { f ( t ) } = ( − 1 ) n ∑ k = 0 n − 1 s α − k − 1 δ ψ k f ( b ) − s α L r ψ { f ( t ) } ,

which is (A.6). So the proof is complete. □

Finally, it is not difficult to obtain the right generalized Laplace transform of Mittag–Leffler function

(A.7) ∫ − ∞ b e − s ( ψ ( b ) − ψ ( t ) ) ( ψ ( b ) − ψ ( t ) ) α k + β − 1 E α , β ( k ) ( ± λ ( ψ ( b ) − ψ ( t ) ) α ) ψ ′ ( t ) d t = k ! s α − β ( s α ∓ λ ) k + 1 , R e ( s ) > | λ | 1 α .

A.2 The generalized Mellin transform

In this part, we define generalized Mellin transforms and discuss the corresponding properties.

Definition A.3

Suppose that a function f(t) is defined on [ a , ∞ ) ( a ∈ R ) . The left generalized Mellin transform of f(t) is defined by

(A.8) F ( ξ ) = M l ψ { f ( t ) } = ∫ a ∞ ( ψ ( t ) − ψ ( a ) ) ξ − 1 f ( t ) ψ ′ ( t ) d t , γ 1 < R e ( ξ ) < γ 2 .

The inverse left generalized Mellin transform is given by

(A.9) f ( t ) = M l ψ − 1 { F ( ξ ) } = 1 2 π i ∫ c − i ∞ c + i ∞ ( ψ ( t ) − ψ ( a ) ) − ξ F ( ξ ) d ξ , t > a , c = R e ( ξ ) .

Definition A.4

Suppose that a function f(t) is defined on [ − ∞ , b ) ( b ∈ R ) The right generalized Mellin transform f(t) is defined by

(A.10) F ( ξ ) = M r ψ { f ( t ) } = ∫ − ∞ b ( ψ ( b ) − ψ ( t ) ) ξ − 1 f ( t ) ψ ′ ( t ) d t , b ∈ R , γ 3 < R e ( ξ ) < γ 4 .

The inverse right generalized Mellin transform is given by

(A.11) f ( t ) = M r ψ − 1 { F ( ξ ) } = 1 2 π i ∫ c − i ∞ c + i ∞ ( ψ ( b ) − ψ ( t ) ) − ξ F ( ξ ) d ξ , t < b , c = R e ( ξ ) .

In the sense of generalized Mellin transform, we can define the generalized convolutions and present convolution theorems.

Definition A.5

Suppose that functions f(t) and g(t) are defined on [ a , + ∞ ) ( a ∈ R ) . We call the integral ∫ a ∞ f ψ − 1 ψ ( a ) + ψ ( t ) − ψ ( a ) ψ ( τ ) − ψ ( a ) g ( τ ) ψ ′ ( τ ) d τ ψ ( τ ) − ψ ( a ) the left generalized convolution of f(t) and g(t), i.e.,

f ( t ) * l ψ g ( t ) = ( f * l ψ g ) ( t ) = ∫ a ∞ f ψ − 1 ψ ( a ) + ψ ( t ) − ψ ( a ) ψ ( τ ) − ψ ( a ) g ( τ ) ψ ′ ( τ ) d τ ψ ( τ ) − ψ ( a ) .

Theorem A.4

(Left convolution theorem). If M l ψ { f ( t ) } = F ( ξ ) and M l ψ { g ( t ) } = G ( ξ ) , then

M l ψ { f ( t ) * l ψ g ( t ) } = M l ψ { f ( t ) } M l ψ { g ( t ) } = F ( ξ ) G ( ξ ) ;

Or equivalently,

M l ψ − 1 { F ( ξ ) G ( ξ ) } = f ( t ) * l ψ g ( t ) .

Proof

According to Definitions A.3 and A.5, and changing the order of integration and making the substitution ψ − 1 ψ ( a ) + ψ ( t ) − ψ ( a ) ψ ( τ ) − ψ ( a ) = w , one deduces that

M l ψ { f ( t ) * l ψ g ( t ) } = ∫ a ∞ ( ψ ( t ) − ψ ( a ) ) ξ − 1 ∫ a ∞ f ψ − 1 ψ ( a ) + ψ ( t ) − ψ ( a ) ψ ( τ ) − ψ ( a ) g ( τ ) ψ ′ ( τ ) d τ ψ ( τ ) − ψ ( a ) ψ ′ ( t ) d t = ∫ a ∞ ∫ a ∞ ( ψ ( t ) − ψ ( a ) ) ξ − 1 f ψ − 1 ψ ( a ) + ψ ( t ) − ψ ( a ) ψ ( τ ) − ψ ( a ) g ( τ ) ψ ′ ( t ) d t ψ ′ ( τ ) d τ ψ ( τ ) − ψ ( a ) = ∫ a ∞ ∫ a ∞ ( ψ ( w ) − ψ ( a ) ) ξ − 1 ( ψ ( τ ) − ψ ( a ) ) ξ − 1 f ( w ) g ( τ ) ψ ′ ( w ) d w ψ ′ ( τ ) d τ = M l ψ { f ( t ) } M l ψ { g ( t ) } = F ( ξ ) G ( ξ ) .

All this ends the proof. □

Definition A.6

Suppose that functions f(t) and g(t) are defined on ( − ∞ , b ] ( b ∈ R ) . We call the integral ∫ − ∞ b f ψ − 1 ψ ( b ) − ψ ( b ) − ψ ( t ) ψ ( b ) − ψ ( τ ) g ( τ ) ψ ′ ( τ ) d τ ψ ( b ) − ψ ( τ ) the right generalized convolution of f(t) and g(t), i.e.,

f ( t ) * r ψ g ( t ) = ( f * r ψ g ) ( t ) = ∫ − ∞ b f ψ − 1 ψ ( b ) − ψ ( b ) − ψ ( t ) ψ ( b ) − ψ ( τ ) g ( τ ) ψ ′ ( τ ) d τ ψ ( b ) − ψ ( τ ) .

Theorem A.5

(Right convolution theorem). If M r ψ { f ( t ) } = F ( ξ ) and M r ψ { g ( t ) } = G ( ξ ) , then

M r ψ { f ( t ) * r ψ g ( t ) } = M r ψ { f ( t ) } M r ψ { g ( t ) } = F ( ξ ) G ( ξ ) ;

Or equivalently,

M r ψ − 1 { F ( ξ ) G ( ξ ) } = f ( t ) * r ψ g ( t ) .

Proof

The proof can be done by similar argument of the proof of Theorem A.4. □

The following lemma is the differential property of generalized Mellin transforms.

Lemma A.2

If M l ψ { f ( t ) } = F ( ξ ) and the limits

lim t → a + [ ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 δ ψ n − k − 1 f ( t ) ] , lim t → ∞ [ ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 δ ψ n − k − 1 f ( t ) ] ( k = 0,1 , … , n − 1 )

exist, then

(A.12) M l ψ { δ ψ n f ( t ) } = ∑ k = 0 n − 1 Γ ( 1 + k − ξ ) Γ ( 1 − ξ ) ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 δ ψ n − k − 1 f ( t ) a ∞ + Γ ( 1 + n − ξ ) Γ ( 1 − ξ ) F ( ξ − n ) , t > a , n ∈ Z + .

If M r ψ { f ( t ) } = F ( ξ ) and the limits

lim t → b − [ ( ψ ( b ) − ψ ( t ) ) ξ − k − 1 δ ψ n − k − 1 f ( t ) ] , lim t → − ∞ [ ( ψ ( b ) − ψ ( t ) ) ξ − k − 1 δ ψ n − k − 1 f ( t ) ] ( k = 0,1 , … , n − 1 )

exist, then

(A.13) M r ψ { δ ψ n f ( t ) } = ∑ k = 0 n − 1 ( − 1 ) k Γ ( 1 + k − ξ ) Γ ( 1 − ξ ) ( ψ ( b ) − ψ ( t ) ) ξ − k − 1 δ ψ n − k − 1 f ( t ) − ∞ b + ( − 1 ) n Γ ( 1 + n − ξ ) Γ ( 1 − ξ ) F ( ξ − n ) , t < b , n ∈ Z + .

Proof

A simple application of integration by parts concludes the result so is omitted. □

Next, we present the generalized Mellin transforms corresponding to Definitions 2.1–2.3.

Theorem A.6

Let n − 1 < α < n ∈ Z + and a ∈ R . If M l ψ { f ( t ) } = F ( ξ ) , α < Re(1 − ξ) and the limits, for k = 0, 1, …, n − 1,

lim t → a + [ ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 D a , t α − k − 1 ψ f ( t ) ] , lim t → ∞ [ ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 D a , t α − k − 1 ψ f ( t ) ] ,

lim t → a + [ ( ψ ( t ) − ψ ( a ) ) ξ − α + k δ ψ k f ( t ) ] , lim t → ∞ [ ( ψ ( t ) − ψ ( a ) ) ξ − α + k δ ψ k f ( t ) ]

exist, then:

(A.14) M l ψ { D a , t − α ψ f ( t ) } = Γ ( 1 − ξ − α ) Γ ( 1 − ξ ) F ( ξ + α ) ,

(A.15) M l ψ { D a , t α ψ f ( t ) } = Γ ( 1 − ξ + α ) Γ ( 1 − ξ ) F ( ξ − α ) + ∑ k = 0 n − 1 Γ ( 1 + k − ξ ) Γ ( 1 − ξ ) ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 D a , t α − k − 1 ψ f ( t ) a ∞ ,

(A.16) M l ψ { D a , t α C ψ f ( t ) } = Γ ( 1 − ξ + α ) Γ ( 1 − ξ ) F ( ξ − α ) + ∑ k = 0 n − 1 Γ ( α − k − ξ ) Γ ( 1 − ξ ) ( ψ ( t ) − ψ ( a ) ) ξ − α + k δ ψ k f ( t ) a ∞ .

Proof

Let us begin with the proof of (A.14). Noting that Definitions 2.1 and A.3, interchanging order of integration and making the substitution ψ(t) − ψ(τ) = (ψ(τ) − ψ(a))w, it holds that

M l ψ { D a , t − α ψ f ( t ) } = ∫ a ∞ ( ψ ( t ) − ψ ( a ) ) ξ − 1 1 Γ ( α ) ∫ a t ( ψ ( t ) − ψ ( τ ) ) α − 1 f ( τ ) ψ ′ ( τ ) d τ ψ ′ ( t ) d t = 1 Γ ( α ) ∫ a ∞ ∫ 0 ∞ ( 1 + w ) ξ − 1 w α − 1 d w ( ψ ( τ ) − ψ ( a ) ) ξ + α − 1 f ( τ ) ψ ′ ( τ ) d τ = Γ ( 1 − ξ − α ) Γ ( 1 − ξ ) F ( ξ + α ) ,

in which the integral equality ∫ 0 ∞ ( 1 + w ) ξ − 1 w α − 1 d w = B ( α , 1 − ξ − α ) for α < Re(1 − ξ) is used.

Now we show (A.15). Letting D a , t − ( n − α ) ψ f ( t ) = g ( t ) and using differential property (A.12), one gets

M l ψ { D a , t α ψ f ( t ) } = M l ψ 1 ψ ′ ( t ) d d t n D a , t − ( n − α ) ψ f ( t ) = M l ψ { δ ψ n g ( t ) } = Γ ( 1 − ξ + α ) Γ ( 1 − ξ ) F ( ξ − α ) + ∑ k = 0 n − 1 Γ ( 1 + k − ξ ) Γ ( 1 − ξ ) ( ψ ( t ) − ψ ( a ) ) ξ − k − 1 D a , t α − k − 1 ψ f ( t ) a ∞ .

Finally, we prove (A.16). Letting f ψ ( n ) ( t ) = h ( t ) , and exploiting (A.14) and differential property (A.12) yield

M l ψ { D a , t α C ψ f ( t ) } = M l ψ { D a , t − ( n − α ) ψ δ ψ n f ( t ) } = M l ψ { D a , t − ( n − α ) ψ h ( t ) } = Γ ( 1 − ξ − n + α ) Γ ( 1 − ξ ) H ( ξ + n − α ) = Γ ( 1 − ξ + α ) Γ ( 1 − ξ ) F ( ξ − α ) + ∑ k = 0 n − 1 Γ ( α − k − ξ ) Γ ( 1 − ξ ) ( ψ ( t ) − ψ ( a ) ) ξ − α + k δ ψ k f ( t ) a ∞ .

The proof is thus completed. □

Theorem A.7

Let n − 1 < α < n ∈ Z + and b ∈ R . If M r ψ { f ( t ) } = F ( ξ ) , α < Re(1 − ξ) and the limits, for k = 0, 1, …, n − 1,

lim t → b − [ ( ψ ( b ) − ψ ( t ) ) ξ − k − 1 D t , b α − k − 1 ψ f ( t ) ] , lim t → − ∞ [ ( ψ ( b ) − ψ ( t ) ) ξ − k − 1 D t , b α − k − 1 ψ f ( t ) ] ,

lim t → b − [ ( ψ ( b ) − ψ ( t ) ) ξ − α + k δ ψ k f ( t ) ] , lim t → − ∞ [ ( ψ ( b ) − ψ ( t ) ) ξ − α + k δ ψ k f ( t ) ]

exist, then:

(A.17) M r ψ { D t , b − α ψ f ( t ) } = Γ ( 1 − ξ − α ) Γ ( 1 − ξ ) F ( ξ + α ) ,

(A.18) M r ψ { D t , b α ψ f ( t ) } = Γ ( 1 − ξ + α ) Γ ( 1 − ξ ) F ( ξ − α ) + ∑ k = 0 n − 1 ( − 1 ) k + n Γ ( 1 + k − ξ ) Γ ( 1 − ξ ) ( ψ ( b ) − ψ ( t ) ) ξ − k − 1 D t , b α − k − 1 ψ f ( t ) − ∞ b ,

(A.19) M r ψ { D t , b α C ψ f ( t ) } = Γ ( 1 − ξ + α ) Γ ( 1 − ξ ) F ( ξ − α ) + ∑ k = 0 n − 1 ( − 1 ) k + n Γ ( α − k − ξ ) Γ ( 1 − ξ ) ( ψ ( b ) − ψ ( t ) ) ξ − α + k δ ψ k f ( t ) − ∞ b .

Proof

The proof of this theorem is similar to that of the above theorem and we omit the details. □

By the way, integral transforms have differential definitions. In our experience, the integral transforms defined above are necessary for solving linear ψ-fractional differential equations. But this does not mean that there is no other integral transforms for above equations. For example, we can define the following modified integral transforms.

Definition A.7

Suppose that the given function f(t) defined on [ a , + ∞ ) ( a ∈ R ) . The left modified generalized Laplace transform of f(t) is defined by

(A.20) F ( s ) = L l m ψ { f ( t ) } = ∫ a ∞ e − ( ψ ( s ) − ψ ( a ) ) ( ψ ( t ) − ψ ( a ) ) f ( t ) ψ ′ ( t ) d t , s ∈ C .

Definition A.8

Suppose that a function f(t) is defined on ( − ∞ , b ] ( b ∈ R ) . The right modified generalized Laplace transform of f(t) is defined by

(A.21) F ( s ) = L r m ψ { f ( t ) } = ∫ − ∞ b e − ( ψ ( b ) − ψ ( s ) ) ( ψ ( b ) − ψ ( t ) ) f ( t ) ψ ′ ( t ) d t , s ∈ C .

Remark A.1

Since s is a complex variable, then the function ψ(s) may be a multivalued function. We select the principle value branch of the function ψ(s) under this situation.

It is easy to verify that the following equalities

(A.22) M l ψ [ f ( t ) , ξ ] = 1 Γ ( 1 − ξ ) M l ψ [ L l m ψ [ f ( t ) , s ] , 1 − ξ ] ,

and

(A.23) M r ψ [ f ( t ) , ξ ] = 1 Γ ( 1 − ξ ) M r ψ [ L r m ψ [ f ( t ) , s ] , 1 − ξ ] ,

are valid, where M l ψ [ f ( t ) , ξ ] = M l ψ { f ( t ) } , M r ψ [ f ( t ) , ξ ] = M r ψ { f ( t ) } , L l m ψ [ f ( t ) , s ] = L l m ψ { f ( t ) } , and L r m ψ [ f ( t ) , s ] = L r m ψ { f ( t ) } .

All these integral transforms are powerful tools for studying ψ-type differential systems.

References

[1] T. M. Atanacković, S. Pilipović, B. Stanković, and D. Zorica, Fractional Calculus with Applications in Mechanics, Great Britain, ISTE Ltd and John Wiley & Sons, Inc., 2014.10.1002/9781118577530Search in Google Scholar

[2] R. Almeida, A. B. Malinowska, and M. T. T. Monteiro, “Fractional differential equations with a Caputo derivative with respect to a kernel function and their applications,” Math. Methods Appl. Sci., vol. 41, no. 1, pp. 336–352, 2018. https://doi.org/10.1002/mma.4617.Search in Google Scholar

[3] R. Hilfer, Ed. Applications of Fractional Calculus in Physics, Singapore, World Scientific, 2000.10.1142/3779Search in Google Scholar

[4] X. Liu and L. Ma, “Chaotic vibration, bifurcation, stabilization and synchronization control for fractional discrete–time systems,” Appl. Math. Comput., vol. 385, 2020, Art no. 125423. https://doi.org/10.1016/j.amc.2020.125423.Search in Google Scholar

[5] J. Sabatier, O. P. Agrawal, and J. A. Tenreiro Machado, Eds. Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, The Netherlands, Springer, 2007.10.1007/978-1-4020-6042-7Search in Google Scholar

[6] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Amsterdam, Gordon and Breach Science Publishers, 1993.Search in Google Scholar

[7] V. E. Tarasov and V. V. Tarasova, Economic Dynamics with Memory: Fractional Calculus Approach, Berlin, De Gruyter, 2021.10.1515/9783110627459Search in Google Scholar

[8] D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proc. of the IMACS-SMC, vol. 2, 1996, pp. 963–968.Search in Google Scholar

[9] D. L. Qian, C. P. Li, R. P. Agarwal, and P. J. Y. Wong, “Stability analysis of fractional differential system with Riemann-Liouville derivative,” Math. Comput. Model., vol. 52, no. 5, pp. 862–874, 2010. https://doi.org/10.1016/j.mcm.2010.05.016.Search in Google Scholar

[10] C. P. Li and Y. T. Ma, “Fractional dynamical system and its linearization theorem,” Nonlinear Dynam., vol. 71, pp. 621–633, 2013. https://doi.org/10.1007/s11071-012-0601-1.Search in Google Scholar

[11] C. P. Li and Z. Q. Li, “Stability and logarithmic decay of the solution to Hadamard–type fractional differential equation,” J. Nonlinear Sci., vol. 31, no. 2, 2021, Art no. 31. https://doi.org/10.1007/s00332-021-09691-8.Search in Google Scholar

[12] J. Liouville, “Mémoire sur le changement de la variable indépendante dans le calcul des differentielles indices quelconques,” J. l’Ecole Roy. Polytéchn., vol. 24, pp. 17–54, 1835.Search in Google Scholar

[13] H. Holmgren, “Om differentialkalkylen med indices af hvad natur som helst,” Kongl. Svenska Vetenskaps-Akad. Handl. Stockholm, vol. 5, no. 11, pp. 1–83, 1865.Search in Google Scholar

[14] A. I. Botashev, “On a generalization of Mikusiński’s operational calculus,” in Materials of the First Conf. Young Scientists Acad, Sci. Kirghiz SSR, (1970) [in Russian], Frunze, Ilim, 1970, pp. 109–113.Search in Google Scholar

[15] I. L. Rapoport, “On the structure of an operational calculus for the operator Dψ,” in Research on Integrodifferential Equations in Kirghiz [in Russian], No. 7, Frunze, Ilim, 1970, pp. 200–209.Search in Google Scholar

[16] Yu. A. Brychkov, A. P. Prudnikov, and V. S. Shishov, “Operational calculus,” J. Math. Sci., vol. 15, no. 6, pp. 733–765, 1981. https://doi.org/10.1007/bf01377044.Search in Google Scholar

[17] R. Almeida, “A Caputo fractional derivative of a function with respect to another function,” Commun. Nonlinear Sci. Numer. Simulat., vol. 44, pp. 460–481, 2017. https://doi.org/10.1016/j.cnsns.2016.09.006.Search in Google Scholar

[18] F. Jarad and T. Abdeljawad, “Generalized fractional derivatives and Laplace transform,” Discrete Contin. Dyn. Syst. - S, vol. 13, no. 3, pp. 709–722, 2020. https://doi.org/10.3934/dcdss.2020039.Search in Google Scholar

[19] C. P. Li and F. H. Zeng, Numerical Methods for Fractional Calculus, Boca Raton, USA, Chapman and Hall/CRC, 2015.Search in Google Scholar

[20] C. P. Li and M. Cai, Theory and Numerical Approximations of Fractional Integrals and Derivatives, Philadelphia, SIAM, 2019.Search in Google Scholar

[21] J. Hadamard, “Essai sur létude des fonctions données par leur développement de Taylor,” J. Math. Pure Appl., vol. 8, pp. 101–186, 1892.Search in Google Scholar

[22] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Amsterdam, Elsevier Science, 2006.Search in Google Scholar

[23] H. M. Fahad, M. ur Rehman, and A. Fernandez, “On Laplace transforms with respect to functions and their applications to fractional differential equations,” Math. Method Appl. Sci., https://doi.org/10.1002/mma.7772.Search in Google Scholar

[24] I. Podlubny, Fractional Differential Equations, New York, Academic Press, 1999.Search in Google Scholar

[25] P. Hartman, Ordinary Differential Equations, 2nd ed., Boston, Birkhauser, 1982.Search in Google Scholar

Article note

The current job was partially supported by the National Natural Science Foundation of China under grant no. 11872234.

Received: 2021-04-27

Revised: 2021-09-19

Accepted: 2021-11-04

Published Online: 2021-12-02

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Articles in the same Issue

https://doi.org/10.1515/ijnsns-2021-0189

Keywords for this article

ψ-algebraic decay; ψ-fractional derivative; linearization theorem; Mittag–Leffler function; stability