Ulam-type stability for Caputo-type fractional delay differential equations

Snezhana Hristova

doi:10.1515/dema-2025-0112

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Ulam-type stability for Caputo-type fractional delay differential equations

Snezhana Hristova

Published/Copyright: March 17, 2025

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From the journal Demonstratio Mathematica Volume 58 Issue 1

Abstract

This study focuses on differential equations incorporating generalized fractional derivatives of the Caputo type. The concept of Ulam-type stability (US) is analyzed in the context of both initial value problems and boundary value problems (BVPs) for the fractional differential equations under investigation. Particular attention is given to addressing certain misconceptions that arise when applying US to BVPs. To mitigate these issues, we propose incorporating a parameter into the boundary conditions as a potential solution. The dependency of the solution on this parameter is established, and a method is outlined for selecting the parameter appropriately. This approach ensures that the solution of the fractional equation is strongly influenced by the arbitrarily chosen solution of the associated inequality. The theoretical findings are further clarified through illustrative examples.

Keywords: Caputo-type fractional derivative; fractional differential equation; delay; initial value problem; boundary value problem; Ulam-type stability

MSC 2010: 34A08; 34D99

1 Introduction

Ulam-type stability (US), sometimes known as Hyers-Ulam stability, Hyers-Ulam-Rassias stability, etc., has been defined and studied by many authors for various types of differential equations. Initially, it was defined for operator equations, and later, it was applied to differential equations (see the formulation given in [1,2]). The study in this area has grown in the last few years, and nowadays, it is a central subject in the mathematical analysis area.

Note the application of US in differential equations with different types of derivatives (such as ordinary derivatives, partial derivatives, and fractional derivatives) could be divided into two main parts:

US for initial value problems (IVPs);
US for problems with boundary conditions.

Note the first type (IVPs) is well and deeply studied by many authors for various types of equations and derivatives. We could mention (but not all of them): [3] for implicit fractional order differential equation with Caputo fractional derivative, [4] for non-instantaneous impulses and the generalized proportional Caputo fractional derivative, [5] for delay differential equation with Riemann-Liouville fractional derivative, [6] for impulsive delay differential equations [7] for non-instantaneous impulsive differential equations with finite state-dependent delay, [8] for impulsive ordinary differential equations, [9,10] for fractional differential equations with Caputo derivative.

The situation with the second type (with boundary conditions) is more difficult to study, and it leads to some typical misunderstandings. These typical misunderstandings are connected with the appropriate differential inequality, their solutions that might not satisfy the boundary conditions of the studied problem. It does not allow us to use the integral presentation of the solution of the given boundary problem to the solution of the differential inequality.

The primary objective of this study is to explore US in the context of boundary value problems (BVPs). Building on foundational concepts and applications of US in IVPs, we highlight key misconceptions associated with its application to problems involving boundary conditions. These issues are demonstrated through several examples. To address them, we propose a novel strategy that incorporates a parameter into the boundary conditions. This method is illustrated using a general linear BVP. For broader applicability, we examine a differential equation that includes a general delay and a Caputo fractional derivative defined with respect to another function (DwrtF). An integral representation of the solution is derived, and the existence of solutions for the given BVP is established. Finally, we analyze US for the problem under consideration and provide sufficient conditions for Ulam-Hyers stability.

The main contributions of the article to the field of stability theory of differential equations, particularly in fractional differential equations, can be summarized as follows:

In Section 2, we give some known literature results about fractional diffintegrals with respect to another function and give an implicit formula for the solution of a linear BVP.
In Section 3, we provide the main steps to study US, and discuss its applications on both initial conditions and boundary conditions. The main concepts are illustrating on simple scalar differential equations with Caputo fractional derivative with respect to another function. In the case of boundary condition, the main misunderstanding is illustrated on the simple example.
In Section 4, the algorithm for the US is applied to a delay differential equation with Caputo fractional derivative with respect to another function and boundary condition.
In Section 3, we apply the proposed new methodology to study US for a linear BVP for a differential equation with instantaneous impulses and the Caputo fractional derivative with respect to another function.
To summarize and emphasize some possible applications of the suggested approach, we complete this article with a conclusion.

2 Some results from fractional calculus

Let 0 < b ≤ ∞ be a fixed number and ψ : [ 0 , b ] → R be a smooth increasing function with ψ ′ ( t ) > 0 almost everywhere in [ 0 , b ] . Note if b < ∞ , we will consider the closed interval [ 0 , b ] , and if b = ∞ , we will consider the half-open interval [ 0 , b ) .

We now give the definitions for fractional differintegral with respect to a given function.

Definition 1

[11] Let δ > 0 . The Riemann-Liouville fractional integral with respect to the function ψ (RLI) is defined by

(1) ℐ ψ ( t ) δ 0 ς ( t ) = 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ς ( s ) d s , t ∈ ( 0 , b ] .

Definition 2

[11] Let δ ∈ ( 0 , 1 ) . The Caputo fractional derivative with respect to the function ψ ( t ) (DwrtF) is defined by

(2) D ψ ( t ) δ 0 C ς ( t ) = ℐ ψ ( t ) 1 − δ 0 1 ψ ′ ( t ) d d t ς ( t ) = 1 Γ ( 1 − δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) − δ ς ′ ( s ) d s , t ∈ ( 0 , b ] .

Remark 1

Note the RLI given in Definition 1 and the DwrtF given in Definition 2 are also called the ψ -fractional integral and the ψ -Caputo fractional derivative, respectively.

Lemma 1

(Theorem 5 [11]) Given a function ς ∈ C 1 [ 0 , b ] and δ ∈ ( 0,1 ) , we have D ψ ( t ) δ 0 C ( ℐ ψ ( t ) δ 0 ) ς ( t ) = ς ( t ) .

Lemma 2

(Theorem 4 [11]) Given a function ς ∈ C 1 [ 0 , b ] and δ ∈ ( 0,1 ) , we have ℐ ψ ( t ) δ 0 ( D ψ ( t ) δ 0 C ) ς ( t ) = ς ( t ) − ς ( 0 ) .

Lemma 3

[11] The solution of the scalar linear fractional differential equation with DwrtF

(3) ( D ψ ( t ) δ 0 C ς ) ( t ) = f ( t ) , t ∈ ( 0 , b ] ,

an initial condition

(4) ς ( 0 ) = V 0 ,

where V 0 ∈ R , δ ∈ ( 0,1 ) , f ∈ C ( [ 0 , b ] , R ) , is

ς ( t ) = V 0 + ( ℐ δ 0 f ) ( t ) = V 0 + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) f ( s ) d s , t ∈ ( 0 , b ] .

We now present the results for fractional differential equations with DwrtF and a boundary condition on the interval [ 0 , b ] .

Consider the fractional differential equation with DwrtF (3) with the linear boundary condition

(5) A ς ( 0 ) + B ς ( b ) = γ ,

where A , B ∈ R are given constants, γ ∈ R .

Based on Lemma 3, we obtain the explicit form of the solution of BVP (3) and (5).

Lemma 4

Let f ∈ C ( [ 0 , b ] , R ) , A , B ∈ R : A + B ≠ 0 , be given constants, and γ ∈ R . Then, the solution of BVP (3) and (5) is

(6) ς ( t ) = γ A + B − B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) f ( s ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) f ( s ) d s , t ∈ [ 0 , b ] .

3 Brief overview of the concepts of US

Consider the ψ -Caputo fractional differential equation with a general delay (DE)

(7) D ψ ( t ) δ 0 C x ( t ) = F ( t , x , x t ) , t ∈ ( 0 , b ] ,

where F : [ 0 , b ] × R × R → R , x t ( Θ ) = x ( t + Θ ) , Θ ∈ [ − κ , 0 ] , κ > 0 .

We recall the basic algorithm in the study of US. The steps of the algorithm are described as follows:

Statement of the problem: The main problem of study consists of two parts:
1. The differential equation (7);
2. The initial condition (IVP), the boundary condition (BVP), or both IVP and BVP.
Integral presentation: Obtain the equivalent integral presentation (integral equation) of the solution of the problem defined in Step 1.
Existence of solution: Define an operator based on the integral equation obtained in Step 2; prove the equivalence between the solution of the given problem and the fixed point of the operator; prove the existence (and uniqueness) of a fixed point, i.e., prove the existence of a solution of the problem defined in Step 1.
Definition of US: Define in an appropriate way the differential inequality deeply connected with the problem given in Step 1. Based on this inequality, define the US of the problem given in Step 1.
Sufficient conditions for US: Based on Steps 2–4, prove the US of the solutions of the studied problem.

3.1 IVPs

First, we discuss the application of US to IVPs of differential equations.

Consider the differential equation with a delay (DE) (7) with the initial condition

(8) x ( t ) = ϕ ( t ) , t ∈ [ − κ , 0 ] ,

where ϕ ∈ C ( [ − κ , 0 ] , R ) .

To focus on US, we skip both Step 2 and Step 3, assuming that for any initial function ϕ ∈ C ( [ − κ , 0 ] , R ) , the IVP (IVP) (7) and (8) has a solution on [ − κ , b ] .

Remark 2

Note that any solution of the IVP (7) and (8) depends significantly on the initial function ϕ ( ⋅ ) .

Let ε > 0 . Consider the fractional differential inequality (DI)

(9) ∣ ( D ψ ( t ) δ 0 C ς ) ( t ) − F ( t , ς ( t ) , ς t ) ∣ ≤ ε , t ∈ ( 0 , b ] .

Remark 3

Note inequality (9) has many solutions for any given function F and a number ε . Also, the value of the function ς ( ⋅ ) has to be known on the initial time interval [ − κ , 0 ] to be able to satisfy the inequality (9). But this value is not given initially; it is not connected with the initial function ϕ ( ⋅ ) in the initial condition (8).

We now recall the definition for US based on the classical articles [12,13]:

Definition 3

The IVP (7) and (8) is Ulam-Hyers stable if there exists a real number C > 0 such that for each ε > 0 and for each solution ς : [ − κ , b ] → R of the DI (9), there exists a solution x : [ − κ , b ] → R of IVP (7) and (8) with ∣ ς ( t ) − x ( t ) ∣ ≤ C ε for t ∈ [ − κ , b ] .

Remark 4

Note in Definition 3 the main point is that for any solution of the inequality, there exists a solution of the equation, i.e., for different solutions of DI, the solution of the equation could be changeable, not one and the same.

Remark 5

Note in Definition 3 it says that there exists a solution of the IVP (7), and (8), but the initial function ϕ ( ⋅ ) in the initial condition (8) is not fixed.

The main idea in the practical application of US for IVP (7) and (8): For a given solution ς : [ − κ , b ] → R of the differential inequality (9), we consider a solution x : [ − κ , b ] → R of the IVP (7) and (8) with the initial function ϕ ( t ) = ς ( t ) , t ∈ [ − κ , 0 ] . Therefore, for any solution ς ( t ) of the inequality (9), the initial condition (8) is changed and the IVP (7) and (8) as well as its solution is different for different ς ( t ) (Remark 4).

Remark 6

If we fixed the initial function ϕ ( ⋅ ) in the initial condition (8) and add the initial condition

(10) ς ( t ) = ϕ ( t ) , t ∈ [ − κ , 0 ] ,

to (9), then again the fractional differential inequalities (9) and (10) will have many solutions depending on ε . But in this case, the meaning of “existing a solution” will be lost in Definition 3 (see Remark 4). Then, we have to change Definition 3 to the following one:

Definition 4

(Modified US) The IVP (7) and (8) is modified Ulam-Hyers stable if there exists a real number C > 0 such that the solution x : [ − κ , b ] → R n of IVP (7), (8) satisfies the inequality ∣ ς ( t ) − x ( t ) ∣ ≤ C ε , t ∈ [ 0 , b ] for any solution ς : [ − κ , b ] → R n of the DI (9) with initial condition (10) and arbitrary ε > 0 .

Note Definition 4 is very restrictive, but it is because of the fixed initial function. And the idea of US, introduced and used by many authors is changed if the initial function of the IVP is fixed.

We will illustrate the main idea of application of US on an fractional differential equation without any delay, i.e., the initial interval will be reduced to an initial point and we will use a constant initial value instead of an initial function.

Example 1

Consider the simple scalar fractional differential equation

(11) D t 2 0.3 x ( t ) = 1 , t ∈ ( 0 , 2 ] ,

with the initial condition

(12) x ( 0 ) = x 0 .

Its solution of IVP (11) and (12) is x ( t ) = x 0 + t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] .

Let ε > 0 be an arbitrary number, and consider the differential inequality

(13) ∣ D t 2 0.3 ς ( t ) − 1 ∣ < ε , t ∈ [ 0 , 2 ] .

Choose an arbitrary solution of the differential inequality (13), for example, ς ( t ) = ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] . Then, consider the IVP (11) and (12) with initial value x 0 = ς ( 0 ) = 0 . Then, the solution of the corresponding IVP is x ( t ) = t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] and

∣ x ( t ) − ς ( t ) ∣ = t 0.6 0.3 Γ ( 0.3 ) − ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) < ε 0.5 < ε , t ∈ [ 0 , 2 ] .

Choose another solution of differential inequality (13), for example, ς ( t ) = 2 + ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] . Then, consider IVP (11) and (12) with initial value x 0 = ς ( 0 ) = 2 with the solution x ( t ) = 2 + t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] , and

∣ x ( t ) − ς ( t ) ∣ = 2 + t 0.6 0.3 Γ ( 0.3 ) − 2 − ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) < ε 0.5 < ε , t ∈ [ 0 , 2 ] .

Obviously, for any particular solution of the differential inequality (13), there exists a solution of IVP (11), (12), depending significantly on the chosen solution of the inequality.

Now, we will illustrate Remark 6. Let us consider the IVP (11) and (12) with a gived initial value x 0 , for example, x 0 = 4 . Its solution is x ( t ) = 4 + t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] .

Choose the initial value of the solution of the differential inequality (13) to be ς ( 0 ) = x 0 = 4 . Then one solution of the differential inequality (13) with the initial condition ς ( 0 ) = 4 is ς ( t ) = 4 + ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] , and

∣ x ( t ) − ς ( t ) ∣ = 4 + t 0.6 0.3 Γ ( 0.3 ) − 4 − ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) < ε 0.5 < ε , t ∈ [ 0 , 2 ] .

Now, consider another solution of the differential inequality (13) with the same initial value, i.e., consider ς 1 ( t ) = 4 + ( 1 + 0.2 ε ) t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] . Then,

∣ x ( t ) − ς 1 ( t ) ∣ = 4 + t 0.6 0.3 Γ ( 0.3 ) − 4 − ( 1 + 0.2 ε ) t 0.6 0.3 Γ ( 0.3 ) < ε 0.2 < ε , t ∈ [ 0 , 2 ] .

In this case, we are not able to use Definition 3 but the changed one, Definition 4.

3.2 BVPs

Consider the differential equation with a delay (DE) (7) with a boundary condition

(14) x ( t ) = ϕ ( t ) , t ∈ [ − κ , 0 ) , A x ( 0 ) + B x ( b ) = γ ,

where ϕ ∈ C ( [ − κ , 0 ] , R ) , A , B ∈ R : A + B > 0 , are given constants, γ ∈ R .

Let ε > 0 . Consider the differential inequality (9).

We will give the definition for US based on the classical articles [12,13] for IVPs.

Definition 5

The BVP (7) and (14) is US if there exists a real number C > 0 such that for each ε > 0 and for each solution ς : [ 0 , b ] → R of the DI (9), there exists a solution x : [ 0 , b ] → R of BVP (7) and (14) with ∣ ς ( t ) − x ( t ) ∣ ≤ C ε for t ∈ [ 0 , b ] .

Remark 7

Similar to the application of US to IVP (Remark 4), the main point in Definition 5 is that for any solution of the inequality, there exists a solution of the equation, i.e., for different solutions of DI, the solution of the equation could be changeable, not one and the same.

The main idea in the practical application of US for BVP (7) and (14): For a given solution ς : [ 0 , b ] → R n of the differential inequality (9), we consider a solution x : [ 0 , b ] → R n of BVP (7) and (14) with an appropriate initial function ϕ ( ⋅ ) and a parameter γ in (14). Both the initial function ϕ ( ⋅ ) and the parameter γ have to depend on the chosen solution ς : [ 0 , b ] → R n of the differential inequality (9). Therefore, for any solution ς ( t ) of the inequality (9), the initial and boundary conditions (14) are changed and the IVP (7), (14) as well as its solution is different for different ς ( t ) (Remark 7).

Note in many published articles about US for any type of BVP for differential equations, the authors choose an arbitrary solution of the corresponding differential inequality. In this case, the chosen solution could not satisfy the given boundary condition, and the integral presentation obtained in Step 2 for the solution of the studied BVP might not be true for the chosen solution of the differential inequality. It leads to some mistakes in the proofs. To avoid it, in our case, we could fix both the initial function ϕ ( ⋅ ) and the constant γ ∈ R in the boundary condition (14), and add the boundary condition

(15) ς ( t ) = ϕ ( t ) , t ∈ [ − κ , 0 ) , A ϕ ( 0 ) + B ς ( b ) = γ ,

to (9). Then, the fractional differential inequalities (9) will have many solutions depending on ε and satisfying the boundary conditions (15). Then, similar to the Section 3.1, we are not able to apply Definition 5 and we have to change Definition 5 to the following one:

Definition 6

(Modified US) The BVP (7) and (14) is modified US if there exists a real number C > 0 such that the solution x : [ − κ , b ] → R of BVP (7) and (14) satisfies ∣ ς ( t ) − x ( t ) ∣ ≤ C ε , t ∈ [ 0 , b ] for any solution ς : [ − κ , b ] → R of the DI (9) with boundary condition (15) and arbitrary ε > 0 .

Definition 6 is more restrictive, and it changes the classical idea of US. Also, it is more difficult to be obtained a solution of (7) satisfying the boundary condition (15) than to solve only the inequality (7).

We will illustrate the aforementioned on an example. To avoid some difficulties with solving the fractional equations, we will consider the case of equations without any delay, i.e., the initial interval will be reduced to an initial point.

Example 2

Consider the scalar linear BVP

(16) D t 2 0.3 x ( t ) = 1 , t ∈ ( 0 , 2 ] , x ( 0 ) + x ( 2 ) = γ .

According to Lemma 4, BVP (16) has a unique solution

(17) x ( t ) = γ 2 − 1 2 Γ ( 0.3 ) ∫ 0 2 ( 2 2 − s 2 ) 0.3 − 1 2 s d s + 1 Γ ( 0.3 ) ∫ 0 t ( t 2 − s 2 ) 0.3 − 1 2 s d s = 0.5 γ + t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] .

Let ε > 0 be an arbitrary number. Consider the differential inequality

(18) ∣ D t 2 0.3 ς ( t ) − 1 ∣ < ε , t ∈ [ 0,1 ] .

First, let γ be given initially, for example, γ = 1 . Then, the exact solution of BVP (16) is x ( t ) = 0.5 + t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] .

Choose an arbitrary solution of inequality (18), for example,

(19) ς ( t ) = ς 0 + ( 1 + 0.5 ε ) t 0.6 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] ,

where ς 0 ∈ R is an arbitrary constant.

Note the solution ς ( t ) does not satisfy the boundary condition ς ( 0 ) + ς ( 2 ) = γ and ∣ x ( t ) − ς ( t ) ∣ = 0.5 + t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) − ς 0 − ( 1 + 0.5 ε ) t 0.6 Γ ( 1.3 ) = 0.5 − 2 0.6 − 1 Γ ( 1.3 ) − ς 0 − 0.5 ε t 0.6 Γ ( 1.3 ) might not be < ε on [ 0 , 2 ] .

Now, choose a solution of the inequality (18) satisfying the boundary condition ς ( 0 ) + ς ( 2 ) = 1 , i.e., ς ( t ) = 0.5 + ( 1 + 0.5 ε ) t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] . Then, ∣ x ( t ) − ς ( t ) ∣ = 0.5 ε t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) < ε on [ 0 , 2 ] . Note, if we change the solution of the inequality (18) satisfying the boundary condition ς ( 0 ) + ς ( 2 ) = 1 m, then the solution of BVP (16) is x ( t ) = 0.5 + t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] will be not changed, it will be fixed.

Therefore, in this case, we have to apply Definition 6.

Now, to keep the classical ideas of US, we choose an arbitrary solution of the differential inequality (18), for example ς ( t ) = ( 1 + 0.5 ε ) t 0.6 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] .

Then, let γ = ς ( 0 ) + ς ( 2 ) = ( 1 + 0.5 ε ) 2 0.6 Γ ( 1.3 ) .

Consider BVP (16) with γ = ( 1 + 0.5 ε ) 2 0.6 Γ ( 1.3 ) . Then, the solution of corresponding BVP is x ( t ) = 0.5 ( 1 + 0.5 ε ) 2 0.6 Γ ( 1.3 ) + t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] , and

∣ x ( t ) − ς ( t ) ∣ = 0.5 ε t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) < ε , t ∈ [ 0 , 2 ] .

Choose another solution of differential inequality (18), for example, ς ( t ) = 2 + ( 1 + 0.5 ε ) t 0.6 0.3 Γ ( 0.3 ) , t ∈ [ 0 , 2 ] . Now, γ = ς ( 0 ) + ς ( 2 ) = 4 + ( 1 + 0.5 ε ) 2 0.6 Γ ( 1.3 ) .

Consider BVP (16) with γ = 4 + ( 1 + 0.5 ε ) 2 0.6 Γ ( 1.3 ) . Then, the solution of corresponding BVP is x ( t ) = 0.5 4 + ( 1 + 0.5 ε ) 2 0.6 Γ ( 1.3 ) + t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) , t ∈ [ 0 , 2 ] , and for t ∈ [ 0 , 2 ] , we obtain

∣ x ( t ) − ς ( t ) ∣ = 0.5 ε t 0.6 − 2 0.6 − 1 Γ ( 1.3 ) < ε , t ∈ [ 0 , 2 ] .

Therefore, for any solution ς ( ⋅ ) of differential inequality (18), we set up a different boundary condition depending on the chosen ς ( ⋅ ) and we obtain a different solution of the corresponding BVP.

In this case, we use Definition 3, which is a generalization of classical definitions for US.

Remark 8

In our further study, we will keep the classical definition and the classical meaning of US, i.e., for every chosen solution of the differential inequality, we will change the boundary condition depending on the chosen solution.

4 US of BVP for Caputo fractional differential equation with respect to another function

US is studied for BVPs for differential equations with a ψ -Caputo fractional derivative in [14]. Unfortunately, the authors choose an arbitrary solution of the corresponding fractional differential inequality, which might not satisfy the boundary condition, and they incorrectly apply the integral presentation for the solution of BVP (see (18) [14]).

Step 1. Statement of the problem.

Consider the BVP for the nonlinear differential equation with the ψ -Caputo fractional derivative (7) with a boundary condition (14).

Introduce the following assumptions:

The function ψ : [ 0 , b ] → R is smooth and increasing with ψ ′ ( t ) > 0 almost everywhere in [ 0 , b ] .
The function F ∈ C ( [ 0 , b ] × R 2 , R ) .
There exists constants K 1 , K 2 > 0 such that ∣ F ( t , x 1 , x 2 ) − F ( t , y 1 , y 2 ) ∣ ≤ K 1 ∣ x 1 − y 1 ∣ + K 2 ∣ x 2 − y 2 ∣ for t ∈ [ 0 , b ] and x 1 , x 2 , y 1 , y 2 ∈ R .

Step 2. Integral presentation.

Define the set

ℰ = { ς : [ − κ , b ] → R : ς ∈ C ( [ − κ , 0 ] , R ) , and for all t ∈ ( 0 , b ] ∃ D ψ ( t ) δ 0 C ς ( t ) } .

Let ς ∈ ℰ . Define the norm ∣ ∣ ς ∣ ∣ κ = sup t ∈ [ − κ , b ] ∣ ς ( t ) ∣ .

Based on Lemma 4, we define the following operator Ω : ℰ → R by the equalities:

Ω ( ς ( t ) ) = γ A + B − B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s , if t ∈ [ 0 , b ] ϕ ( t ) , if t ∈ [ − κ , 0 ) .

Lemma 5

Let the operator Ω have a fixed point η ∈ ℰ . Then, the function η ( ⋅ ) satisfies the boundary condition (14).

Proof

Let η ∈ ℰ be a fixed point of the operator Ω . Then,

η ( 0 ) = Ω η ( 0 ) = γ A + B − B ( A + B ) Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , η ( s ) , η s ) d s

and

η ( b ) = Ω η ( b ) = γ A + B + 1 Γ ( δ ) A A + B ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , η ( s ) , η s ) d s .

Thus,

□ A η ( 0 ) + B η ( b ) = γ − A B ( A + B ) Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , η ( s ) , η s ) d s + A B ( A + B ) Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , η ( s ) , η s ) d s = γ .

Theorem 1

Let conditions (A1)–(A3) be satisfied, ϕ ∈ C ( [ − κ , 0 ] , R ) , and γ ∈ R be a parameter.

Then, the fixed point of the operator Ω in ℰ is a solution of BVP (7) and (14), and vice versa, for any initial function ϕ ∈ C ( [ − κ , 0 ] , R ) and any value of γ ∈ R .

Proof

Let the function ς ∈ ℰ be a fixed point of the operator Ω .

Note the expression γ A + B − B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s does not depend on the time variable t , and its Caputo fractional derivatives with respect to the function ψ is zero.

Then, for any t ∈ ( 0 , b ] , apply Lemma 1 and obtain

D ψ ( t ) δ 0 C ( ℐ ψ ( t ) δ 0 C F ( t , ς ( t ) , ς t ) ) = F ( t , ς ( t ) , ς t ) ,

i.e., the function ς ( ⋅ ) satisfies (7).

According to Lemma 5, the function ς ( ⋅ ) satisfies the boundary value condition (14).

Therefore, the fixed point ς ( ⋅ ) is a solution of BVP (7) and (14).

Let the function ς ∈ ℰ be a solution of BVP (7) and (14). Taking an integral ( ℐ ϕ ( t ) μ 0 ) on both sides of (7) for t ∈ ( 0 , b ] and using Lemma 2, we obtain

ς ( t ) = ς ( 0 ) + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s , t ∈ ( 0 , b ] .

Apply the boundary condition (14), and we obtain

□ ς ( t ) = γ A + B − A ς ( 0 ) + B ς ( b ) A + B + ς ( 0 ) + B A + B ς ( b ) − B A + B ς ( 0 ) − B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s = γ A + B − B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) F ( s , ς ( s ) , ς s ) d s = Ω ( ς ( t ) ) , t ∈ [ 0 , b ] .

Remark 9

If the conditions of Theorem 2 are satisfied, then any solution ς ( ⋅ ) of BVP (7) and (14) satisfies the integral equation ς ( t ) = Ω ( ς ( t ) ) for t ∈ [ − κ , b ] , and this integral equation is called integral presentation of the solution.

Step 3. Existence of a solution.

A. Finite interval. We will consider the case b < ∞ .

According to Theorem 1, the existence of the solution of BVP (7) and (14) is equivalent to the existence of a fixed point of the fractional integral operator Ω .

Theorem 2

Let conditions (A1)–(A3) be satisfied, and the inequality

(20) ( K 1 + K 2 ) A + 2 B A + B ( ψ ( b ) − ψ ( 0 ) ) δ Γ ( 1 + δ ) < 1

holds.

Then, the operator Ω has a fixed point in ℰ .

Proof

Let ς , ς * ∈ ℰ . Apply ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) d s = ( ψ ( t ) − ψ ( 0 ) ) δ δ , and obtain for t ∈ ( 0 , b ] ,

(21) ∣ ( Ω ς ) ( t ) ∣ − ( Ω ς * ) ( t ) ∣ ≤ B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ∣ F ( s , ς ( s ) , ς s ) − F ( s , ς * ( s ) , ς s * ) ∣ d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ∣ F ( s , ς ( s ) , ς s ) − F ( s , ς * ( s ) , ς s * ) ∣ d s ≤ B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ( K 1 ∣ ς ( s ) − ς * ( s ) ∣ + K 2 ∣ ς s − ς s * ∣ ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ( K 1 ∣ ς ( s ) − ς * ( s ) ∣ + K 2 ∣ ς s − ς s * ∣ ) d s ≤ ( K 1 + K 2 ) ∣ ∣ ς − ς * ∣ ∣ κ B A + B 1 Γ ( δ ) ( ψ ( b ) − ψ ( 0 ) ) δ δ + 1 Γ ( δ ) ( ψ ( t ) − ψ ( 0 ) ) δ δ .

Therefore,

∣ ∣ ( Ω ς ) − ( Ω ς * ) ∣ ∣ κ ≤ ( K 1 + K 2 ) A + 2 B A + B ( ψ ( b ) − ψ ( 0 ) ) δ Γ ( 1 + δ ) ∣ ∣ ς − ς * ∣ ∣ κ .

Then, the claim of Theorem 2 follows from the Banach construction principle.□

B. Infinite Interval. We will consider the case b = ∞ and the half-open interval [ − κ , ∞ ) .

Theorem 3

Let conditions (A1)–(A3) be satisfied, there exists a constant M > 0 such that ψ ( t ) ≤ M , t ∈ [ 0 , ∞ ) , and the inequality

(22) ( K 1 + K 2 ) A + 2 B A + B ( M − ψ ( 0 ) ) δ Γ ( 1 + δ ) < 1

holds.

Then, the operator Ω has a fixed point on ℰ with b = ∞ .

The proof of Theorem 3 is similar to the one of Theorem 2, and we omit it.

Step 4. Definition of US.

Let ε > 0 , and consider the ψ -Caputo fractional differential inequalities (9). According to the given examples in the previous section and the discussion about the differential inequality, we will use Definition 5 for Ulam-Hyers stability.

Step 5. Sufficient conditions for Ulam-Hyers stability.

A. Finite interval.

Theorem 4

Let assumptions (A1)–(A3) hold, and inequality (20) holds.

Then, the BVP (7) and (14) is Ulam-Hyers stable.

Proof

Let ε > 0 and ς ∈ ℰ be a solution of the fractional differential inequality (9). Note that the initial values of the function ς ( ⋅ ) on interval [ − κ , 0 ] are arbitrary but continuous. The function ς ( ⋅ ) might not satisfy the initial and boundary conditions (14).

There exists a function g ( ⋅ ) ∈ C ( [ 0 , b ] , R ) : ∣ g ( t ) ∣ < ε , such that

(23) D ψ ( t ) δ 0 C ς ( t ) = F ( t , ς , ς t ) + g ( t ) , t ∈ ( 0 , b ] .

Let ϕ ( t ) = ς ( t ) , t ∈ [ − κ , 0 ) and γ = A ς ( 0 ) + B ς ( b ) . Then, conditions (A2) and (A3) are satisfied for the function G ( t , u , v ) = F ( t , u , v ) + g ( t ) , and according to Lemma 4, we obtain

ς ( t ) = A ς ( 0 ) + B ς ( b ) A + B − B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ( F ( s , ς ( s ) , ς s ) + g ( s ) ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ( F ( s , ς ( s ) , ς s ) + g ( s ) ) d s .

Consider the differential equation (7) with boundary condition (14), where ϕ ( t ) = ς ( t ) , t ∈ [ − κ , 0 ] and γ = A ς ( 0 ) + B ς ( b ) . All conditions of Theorem 2 are satisfied, and therefore, there exists a solution x ∈ ℰ of (7) and (14) for the chosen function ϕ and parameter γ .

Then,

(24) ∣ x ( t ) − ς ( t ) ∣ ≤ B A + B 1 Γ ( δ ) ∫ 0 b ( ψ ( b ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ( ∣ F ( s , ς ( s ) , ς s ) − F ( s , x ( s ) , x s ) ∣ + ∣ g ( s ) ∣ ) d s + 1 Γ ( δ ) ∫ 0 t ( ψ ( t ) − ψ ( s ) ) δ − 1 ψ ′ ( s ) ( ∣ F ( s , ς ( s ) , ς s ) − F ( s , x ( s ) , x s ) ∣ + ε ) d s ≤ B A + B ( ψ ( b ) − ψ ( 0 ) ) δ Γ ( 1 + δ ) ( ( K 1 + K 2 ) ∣ ∣ ς − x ∣ ∣ κ + ε ) + ( ψ ( t ) − ψ ( 0 ) ) δ Γ ( 1 + δ ) ( ( K 1 + K 2 ) ∣ ∣ ς − x ∣ ∣ κ + ε ) ≤ ( ψ ( b ) − ψ ( 0 ) ) δ Γ ( 1 + δ ) A + 2 B A + B ( ( K 1 + K 2 ) ∣ ∣ ς − x ∣ ∣ κ + ε ) ,

∣ ∣ x − ς ∣ ∣ κ ≤ ( ψ ( b ) − ψ ( 0 ) ) δ Γ ( 1 + δ ) A + 2 B A + B ( K 1 + K 2 ) ∣ ∣ ς − x ∣ ∣ κ + ε .

Thus,

( ( A + B ) Γ ( 1 + δ ) − ( A + 2 B ) ( ψ ( b ) − ψ ( 0 ) ) δ ( K 1 + K 2 ) ) ∣ ∣ ς − x ∣ ∣ κ ≤ ( ψ ( b ) − ψ ( 0 ) ) δ ( A + 2 B ) ε ,

which proves the claim of theorem with C = ( ψ ( b ) − ψ ( 0 ) ) δ ( A + 2 B ) ( ( A + B ) Γ ( 1 + δ ) − ( A + 2 B ) ( ψ ( b ) − ψ ( 0 ) ) δ ( K 1 + K 2 ) ) .□

B. Infinite Interval. Let b < ∞ .

Theorem 5

Let assumptions (A1)–(A3) hold, and there exists a constant M > 0 such that ψ ( t ) ≤ M , t ∈ [ 0 , ∞ ) , and inequality (22) holds.

Then, the BVP (7) and (14) is Ulam-Hyers stable.

5 Partial cases

The above obtained results are partial case of some known in the literature results. We will point out some of them:

– IVP for delay equations with DwrtF.
- – Integral presentation.
- In the case DwrtF is applied to the differential equation and A = 1 , B = 0 , the integral presentation obtained in Theorem 2 is reduced to the result of Lemma 1.1 [15] (with slight corrections of typos).
- – US.
- In the case DwrtF is applied to the differential equation and A = a , B = 0 , the US is studied in Theorem 3.1. [15].
– IVP for delay equations with Caputo fractional derivative.
In the case of variable delay, the US of the IVP is studied in Develi and Duman [16], and the obtained results are partial case of our study with A = 1 , B = 0 , K 1 = K 2 , and ϕ ( t ) ≡ t .
– IVP for delay equations with ordinary derivative.
In the case of variable delay, the integral presentation (with slight typo corrections) and US of the IVP is studied in Otrocol and Ilea [17] and the obtained results are partial case of our study with A = 1 , B = 0 , K 1 = K 2 , ϕ ( t ) ≡ t , and δ = 1 .

From the obtained result concerning US, we can obtain as partial cases several results for various problems.

5.1 Application

Consider the following scalar delay fractional differential equation with DwrtF:

(25) ( D ψ ( t ) 0.3 0 C ς ) ( t ) = t 4 t + 2 e − ∣ ς ( t ) ∣ + 0.1 sin ( ς ( t − 1 ) ) , t ∈ ( 0 , 3 ] ,

with boundary condition

(26) ς ( t ) = ϕ ( t ) , t ∈ [ − 1 , 0 ) , ς ( 0 ) + 2 ς ( 3 ) = γ ,

where γ ∈ R is a parameter, ϕ ∈ C ( [ − 1 , 0 ] , R ) .

Case 1. Let ϕ ( t ) = t 2 , t ∈ [ 0 , 3 ] . Therefore, condition (A1) is satisfied.

In this case, F ( t , u , v ) = t 4 t + 2 e − ∣ u ∣ + 0.1 sin ( v ) , i.e., conditions (A2) and (A3) are satisfied with K 1 = 0.2 , K 2 = 0.1 and

( K 1 + K 2 ) A + 2 B A + B 2 0.6 Γ ( 1.3 ) = 0.3 ⁎ 3 ⁎ 3 0.6 2 Γ ( 1.3 ) ≈ 0.969314 < 1 .

According to Theorem 1, BVP (25) and (26) has a solution for t ∈ [ − 1 , 3 ] for any γ ∈ R and ψ ∈ C ( [ − 1 , 0 ] , R ) .

Note if we increase the interval, i.e., consider [ 0 , 4 ] , then the aforementioned inequality is not satisfied and Theorem 1 could not be applied.

According to Theorem 3, BVP (25) and (26) is Ulam-Hyers stable on [ − 1 , 2 ] .

Case 2. Let ψ ( t ) = ∫ 0 t e − s 2 d s . Then, M = 0.8865 and ψ ( t ) ≤ M , t ≥ 0 , and

( K 1 + K 2 ) A + 2 B A + B ( M − ψ ( 0 ) ) 0.3 Γ ( 1.3 ) ≈ 0.48361 < 1 ,

and according to Theorem 2, the BVP (25) and (26) has a solution for t ≥ − 1 for any γ ∈ R and ψ ∈ C ( [ − 1 , 0 ] , R ) .

According to Theorem 4 BVP (25) and (26) is Ulam-Hyers stable on [ − 1 , ∞ ) .

The type of the function ψ ( ⋅ ) has a huge influence on the behavior of the solution. In Case 1 with an increasing function ψ ( ⋅ ) , we could conclude the existence and US only on an appropriately small interval. In the case of a bounded function ψ ( ⋅ ) , we could conclude the existence and US on an infinite interval.

6 Conclusions

In this article, we have concentrated on US for a general linear BVP involving a delay differential equation with a Caputo fractional derivative defined with respect to another function. A novel approach has been introduced to study this form of stability, aiming to eliminate common misunderstandings. We have proposed a new framework that establishes a connection between the solutions of the given problem and those of the associated differential inequality. This approach equips researchers with a reliable method for correctly applying US to a wide range of BVPs. These include ordinary differential equations, fractional differential equations with various types of Caputo fractional derivatives, integrodifferential fractional equations, and partial differential equations involving time-fractional derivatives.

Funding information: The work was partially supported by the Bulgarian National Science Fund under Project KP-06-N62/1 and Fund of FNI Plovdiv University “P. Hilendarski” under Project FP23-FMI-002.
Author contributions: The author confirms the sole responsibility for the conception of the study, presented results, and prepared manuscript.
Conflict of interest: The author states no conflict of interest.

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Received: 2024-09-30

Revised: 2025-01-20

Accepted: 2025-02-05

Published Online: 2025-03-17

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/dema-2025-0112

Keywords for this article

Caputo-type fractional derivative; fractional differential equation; delay; initial value problem; boundary value problem; Ulam-type stability

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