Ulam Stability for Boundary Value Problems of Differential Equations—Main Misunderstandings and How to Avoid Them

Abstract

Ulam type stability is an important property studied for different types of differential equations. When this type of stability is applied to boundary value problems, there are some misunderstandings in the literature. In connection with this, initially, we give a brief overview of the basic ideas of the application of Ulam type stability to initial value problems. We provide several examples with simulations to illustrate the main points in the application. Then, we focus on some misunderstandings in the application of Ulam stability to boundary value problems. We suggest a new way to avoid these misunderstandings and how to keep the main idea of Ulam type stability when it is applied to boundary value problems of differential equations. We present one possible way to connect both the solutions of the given problem and the solutions of the corresponding inequality. In addition, we provide several examples with simulations to illustrate the ideas for boundary value problems and we also show the necessity of the new way of applying the Ulam type stability. To illustrate the theoretical application of the suggested idea to Ulam type stability, we consider a linear boundary value problem for nonlinear impulsive fractional differential equations with the Caputo fractional derivative with respect to another function and piecewise-constant variable order. We define the Ulam–Hyers stability and obtain sufficient conditions on a finite interval. As partial cases, integral presentations of the solutions of boundary value problems for various types of fractional differential equations are obtained and their Ulam type stability is studied.

1. Introduction

Ulam type stability, sometimes known as Hyers–Ulam stability or Hyers–Ulam–Rassias stability, has been defined and studied by many authors for functional equations (see the formulation given in [1,2] and some recent studies [3,4] and the books [5,6,7]). This type of stability is also applied to various problems for different types of differential equations; it has applications in optimization, biology, economics, etc., and it is a special type of data dependence of solutions. The study in this area has grown and is a central subject in the mathematical analysis area. In this paper, we consider and discuss only its application to differential equations.

The main goal of this paper is to present Ulam type stability for boundary value problems (BVP). We point out the basic misunderstandings in some published paper when this type of stability is considered, and we give one of the possible ways to avoid these misunderstandings and to study Ulam type stability for boundary value problems in a similar way as in the case of initial value problems. A novel approach involving the introduction of a parameter to mitigate these issues is proposed. The presented method is illustrated with specific examples involving a general linear boundary value. To be more precise, we present our ideas on differential equations with a tempered Caputo fractional derivative with respect to another function and with impulses. Note the problem with non-instantaneous impulses is considered in [8] but for the initial value problem and with the generalized proportional Caputo fractional derivative. In this paper, because of the main goal, we study a boundary value problem. Based on some ideas about non-instantaneous impulses in fractional differential equations presented in [8], we consider a differential equation with the more general case of a Caputo type fractional derivative. We present an integral presentation of the solution and prove the existence of the solution of the given BVP. Then, we investigate the Ulam type stability for the studied BVP and sufficient conditions for the Ulam–Hyers stability are derived.

Note the proposed ideas could be applied to study Ulam type stability for various types of boundary value problems for other types of differential equations, such as ordinary differential equations, fractional differential equations with any type of Caputo type fractional derivative, integro-differential fractional equations, and partial differential equations with time-fractional derivatives.

The proposed methodology could be successfully applied to problems in engineering and mathematical physics. For example, in [9], the equilibrium shape (or configuration or state) of the Elastica is described by a boundary value problem for a second-order ordinary differential equation, and since the stability is important in this model, one could investigate the applicability of Ulam type stability. In [10], a generalized two-dimensional space-time reaction–diffusion model is applied to model the calcium dynamics that firmly orchestrates exchanges of calcium flux through intracellular/extracellular sources of calcium to investigate cellular activities and calcium homeostasis. If an appropriate boundary condition is set up in the model, then the suggested ideas could be applied to study Ulam type stability. Therefore, the suggested methodology could broaden the scope of the study and enhance its practical relevance not only to mathematicians but also to numerical analysts, biologists, physicists, and engineers.

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 provide a brief overview of the concepts of Ulam type stability and its applications to various problems such as:

*

Initial value problems to any type of differential equation with various types of derivatives, such as an ordinary derivative or a partial derivative of integer order, any type of fractional derivatives—we briefly make a review and give an algorithm for the application of Ulam type stability;

*

Boundary value problems to any type of differential equation with various types of derivatives—we emphasize the basic misunderstandings of the application of Ulam type stability;

*

A new methodology—we give one of the possible ways to avoid the mentioned misunderstandings by introducing a parameter in the boundary condition;

*

All of the above theoretical explanations are illustrated with appropriate examples.

-

In Section 3, we apply the proposed new methodology to study Ulam type stability for a linear boundary value problem for a differential equation with instantaneous impulses and the Caputo fractional derivative with respect to another function.

-

To summarize and to emphasize some possible applications of the suggested approach, we finish the paper with a conclusion.

2. Brief Overview of the Concepts of Ulam Type Stability

Without loss of generality, we assume the initial time point is zero.

We recall the basic algorithm in the study of Ulam type stability. The steps of the algorithm are described below:

1.

Statement of the problem: Define the main problem of study which consists of two parts:

-

The differential equation $D_{t}x\left(t\right)=F(t,x),t\in [0,b],$ where $x:(0,b]\to {\mathbb{R}}^n$ is the unknown function, and $D_t$ is an applied derivative. Note $b\le \infty$, $n\ge 1$, the applied derivative could be an ordinary derivative or a fractional derivative, and the function F could have more than two arguments (it could contain any type of integral or derivative);

-

The initial condition (IVP), the boundary condition (BVP), or both IVP and BVP.

2.

Integral presentation: Obtain the equivalent integral presentation (integral equation) of the solution of the problem defined in Step 1.

3.

Existence of solution: Define an operator based on the integral equation obtained in Step 2, and prove the existence (and uniqueness) of a fixed point (or points), i.e., prove the existence of a solution of the problem defined in Step 1.

4.

Definition of Ulam type stability: Define in an appropriate way the differential inequality deeply connected with the problem given in Step 1. Based on this inequality, define the Ulam type stability (US) of the problem given in Step 1.

5.

Proof of US: Based on Step 2, Step 3, and Step 4, prove the Ulam type stability of the solutions of the studied problem.

We first discuss the application of Ulam type stability to some types of problems for differential equations.

2.1. Initial Value Problems

First, we discuss the application of Ulam type stability to initial value problems of differential equations.

Consider the differential equation (DE)

$$D_{t}x\left(t\right)=F(t,x),t\in [0,b],$$

(1)

with the initial condition

$$x\left(0\right)=x_0,$$

(2)

where $x_0\in {\mathbb{R}}^n$, and $D_{t}x\left(t\right)$ is an ordinary derivative or any type of Caputo fractional derivative.

To focus on Ulam type stability, we skip both Step 2 and Step 3, assuming that for any initial value $x_0\in {\mathbb{R}}^n$, the initial value problem (IVP) (1), (2) has a solution on $[0,b]$.

Remark 1.

Note that any solution of the IVP (1), (2) depends significantly on the initial value $x_0$, i.e., $x_0\in {\mathbb{R}}^n$ could be considered as a parameter.

Let $\epsilon >0$. Consider the differential inequality (DI)

$$|D_t\nu \left(t\right)-F(t,\nu \left(t\right))|\le \epsilon ,t\in [0,b],$$

(3)

where $|·|$ is an appropriate norm in ${\mathbb{R}}^n$.

We now recall the definition for Ulam type stability based on the classical papers [11,12]:

Definition 1.

The IVP (1), (2) is Ulam–Hyers stable if there exists a real number $C_F>0$ such that for each $\epsilon >0$ and for each solution $\nu :[0,b]\to {\mathbb{R}}^n$ of the DI (3), there exists a solution $x:[0,b]\to {\mathbb{R}}^n$ of IVP (1), (2) with $|\nu \left(t\right)-x\left(t\right)|\le C_F\epsilon$ for $t\in [0,b]$.

The main idea in the practical application of Ulam type stability in this case is that for any solution $\nu :[0,b]\to {\mathbb{R}}^n$ of the differential inequality (3), we consider a solution $x:[0,b]\to {\mathbb{R}}^n$ of the initial value problem (1), (2). As it is mentioned in Remark 1, this solution $x(·)$ depends significantly on the initial value $x_0$. In the application of US, we choose the initial value $x_0=\nu \left(0\right)$ (it exists because of the corresponding Step 3, or the assumption in our case).

Remark 2.

The main point in the application of US to IVPs is the chosen initial value of the given IVP which is equal to the initial value of the chosen arbitrary solution of the corresponding differential inequality.

We illustrate the main idea of the application of Ulam type stability to initial value problems for different types of differential equations.

Example 1

(Ordinary derivative). Consider the simple scalar ordinary differential equation

$$x^{\prime }=x,t\in [0,1],$$

with the initial condition

$$x\left(0\right)=x_0,x_0\in \mathbb{R}.$$

Its solution is $x\left(t\right)=x_0{e}^t$.

Let $\epsilon >0$ be an arbitrary number and consider the differential inequality

$$|\nu \left(t\right)-{\nu }^{\prime }\left(t\right)|<ϵ.$$

We could choose the solution $\nu \left(t\right)=ϵ(t+1),t\in (0,1].$

Now, how do we choose the solution of the differential equation which depends on the initial condition?

Case 1.1. The initial value does not depend on the solution of the inequality. Consider the solution of the given differential equation with the initial value $x_0=2$, i.e., the solution is $x\left(t\right)=2e^t$, and then, the difference $|x\left(t\right)-\nu \left(t\right)|=|2e^t-ϵ(t+1)|$ depends significantly on ϵ, and we cannot find a constant $C_F>0$ independent from ϵ such that $|x\left(t\right)-\nu \left(t\right)|\le C_Fϵ$ (see Figure 1).

Case 1.2. The initial value depends on the solution of the inequality. Consider the solution of the given differential equation with $x_0=\nu \left(0\right)=ϵ$, i.e., $x\left(t\right)=ϵe^t$. Then, the inequality $|x\left(t\right)-\nu \left(t\right)|=ϵ|t+1-e^t|\le ϵ,t\in (0,1]$ holds, i.e., according to Definition 1, the solution of the considered initial value problem is Ulam–Hyers stable with a constant $C_F=1$.

Therefore, the choice of the initial value $x_0=\nu \left(0\right)$ of the solution of the IVP is very important for US.

Example 2

(Caputo fractional derivative). Consider the Caputo fractional derivative

$${\mathcal{D}}_t^{0.3}x\left(t\right)=\frac{1}{\mathsf{\Gamma }(1-0.3)}{\int }_0^t{(t-s)}^{-0.3}{x}^{\prime }\left(s\right)ds,\alpha \in (0,1)$$

(4)

as the derivative $D_{t}x\left(t\right)$ in (1), i.e., consider the scalar fractional differential equation

$${\mathcal{D}}_t^{0.3}x\left(t\right)=x,t\in (0,1],$$

with the initial condition

$$x\left(0\right)=x_0,x_0\in \mathbb{R}.$$

Its solution is $x\left(t\right)=x_0{E}_{0.3}\left(t^{0.3}\right),t\in [0,1],$ where $E_{0.3}(·)$ is the Mittag-Leffler function with one parameter.

Let $\epsilon >0$ be an arbitrary number, and consider the differential inequality

$$|\nu \left(t\right)-{\mathcal{D}}_t^{0.3}\nu \left(t\right)|<ϵ,t\in [0,1].$$

We have the inequality $|t+1-{\mathcal{D}}_t^{0.3}(t+1)|=|t+1-\frac{t^{1-0.3}}{(1-0.3)\mathsf{\Gamma }\left(1.3\right)}|<1,t\in (0,1)$ (see Figure 2). Then, we could choose the solution of the differential inequality $\nu \left(t\right)=ϵ(t+1),t\in [0,1]$.

Case 2.1. The initial value does not depend on the solution of the inequality. Consider the solution of the given differential equation with the initial value $x_0=2$, i.e., the solution is $x\left(t\right)=2e^t$, and then the difference $|x\left(t\right)-\nu \left(t\right)|=|2E_{0.3}\left(t^{0.3}\right)-ϵ(t+1)|$ depends significantly on ϵ, and we cannot find a constant $C_F>0$ independent from ϵ such that $|x\left(t\right)-\nu \left(t\right)\le C_Fϵ$ (see Figure 3).

Case 2.2. The initial value depends on the solution of the inequality. Consider the solution of the given differential equation with $x_0=\nu \left(0\right)=ϵ$, i.e., $x\left(t\right)=ϵE_{0.3}\left(t^{0.3}\right)$. Then, the inequality $|x\left(t\right)-\nu \left(t\right)|=ϵ|t+1-E_{0.3}\left(t^{0.3}\right)|\le C_Fϵ,t\in (0,1]$ holds with the constant $C_F=|2-E_{0.3}\left(1\right)|>0$, i.e., according to Definition 1, the solution of the considered initial value problem is Ulam–Hyers stable (see Figure 4).

Ulam type stability is studied for IVPs for various kinds of differential equations such as ordinary differential equations [11,12,13], ordinary differential equation with impulses [14,15,16,17], and fractional differential equations (FDE) (for example, ref. [18] for the Caputo fractional derivative, ref. [19] for the variable-order Caputo hybrid FDE, ref. [20] for the Caputo type fuzzy FDE with time delays, ref. [21] for the Caputo fractional difference equation, ref. [22] for the Caputo–Fabrizio FDE, ref. [23] for the linear FDE, ref. [24] for the impulsive FDE, ref. [25] for the FDE with the Hilfer–Katugampola operator and impulses, ref. [26] for the Riemann–Liouville FDE with a delay, ref. [27] for the functional FDE, ref. [28] for the tempered Caputo fractional derivative of variable order) (these are only some of the many published papers about initial value problems for various types of differential equations).

2.2. Boundary Value Problems

We discuss the application of Ulam type stability to boundary value problems for various types of differential equations with different types of derivatives. Note this problem has been studied by many authors in many published papers.

2.2.1. Some Misunderstandings in Study Ulam Stability for Boundary Value Problems

Consider the differential Equation (1) with the boundary condition

$$G\left(x\right(0),x(b\left)\right)=0,$$

(5)

where the function G could be a linear function or a nonlinear function, and it could have more than two arguments.

Remark 3.

If we apply Step 2 and Step 3 to the BVP (1), (5), then we prove the existence (uniqueness) of the solution which depends only on the time variable (compare with the case of the IVP and Remark 1).

Let $\epsilon >0$. Consider the differential inequality (3).

We recall the definition for Ulam type stability used by many authors in the literature, which is based on the classical papers [11,12] for initial value problems.

Definition 2.

The BVP (1), (5) is Ulam–Hyers stable if there exists a real number $C_F>0$ such that for each $\epsilon >0$ and for each solution $\nu :[0,b]\to {\mathbb{R}}^n$ of the DI (3), there exists a solution $x:[0,b]\to {\mathbb{R}}^n$ of (1), (5) with $|\nu \left(t\right)-x\left(t\right)|\le C_F\epsilon$ for $t\in [0,b]$.

We focus on two main points in the above definition:

P1.

According to Remark 3, any solution $x(·)$ of BVP (1), (5) depends only on the time variable, and if Step 2 is applied, then this solution exists (and is unique). Thus, Definition 2 means that for any solution $\nu (·)$ of DI (3), the inequality $|\nu \left(t\right)-x\left(t\right)|\le C_F\epsilon ,t\in [0,b]$ holds, where $x(·)$ is the unique solution of BVP (1), (5).

P2.

Any solution $\nu (·)$ of DI (3) might not satisfy the boundary condition (5), and the integral presentation obtained in Step 2 for the solution of BVP (1), (5) might not be true for $\nu (·)$.

We illustrate both above-mentioned points (P1) and (P2) with examples.

Example 3

(Ordinary derivative). Let the derivative $D_t$ in differential Equation (1) be an ordinary derivative, $n=1$, $b=1$, $F(t,x)=1.5x$, and $G(u,v)=u+e^{-1.5}v-2$, i.e., consider the scalar linear BVP

$$x^{\prime }\left(t\right)=1.5x\left(t\right),t\in (0,1],x\left(0\right)+e^{-1.5}x\left(1\right)=2.$$

(6)

The BVP (6) has a unique solution $x\left(t\right)=e^{1.5t},t\in (0,1]$ (see P1).

Let $\epsilon >0$ be an arbitrary number. Consider the differential inequality

$$|{\nu }^{\prime }\left(t\right)-1.5\nu \left(t\right)|<\epsilon ,t\in [0,1].$$

(7)

Choose a solution of DI (7), for example, $\nu \left(t\right)=\epsilon t^2,t\in [0,1]$. Note the solution $\nu \left(t\right)$ does not satisfy the boundary condition $\nu \left(0\right)+e^{-1.5}\nu \left(1\right)=2$ (see P2).

Note the unique solution of BVP (6) does not depend on any solution of DI (7).

The difference $|x\left(t\right)-\nu \left(t\right)|=|e^{1.5t}-\epsilon t^2|$ depends significantly on ϵ. This difference is unbounded with respect to ε. Therefore, there does not exist a constant $C>0$ such that $|\nu \left(t\right)-x\left(t\right)|=|\epsilon t^2-e^{1.5t}|\le C\epsilon$ for any ε and $t\in [0,1]$, i.e., according to Definition 2, the BVP (6) is not Ulam–Hyers stable.

Choose another solution of DI (7), for example, $\nu \left(t\right)=\epsilon e^{1.6t},t\in [0,1]$. This solution does not satisfy the boundary condition $\nu \left(0\right)+e^{-1.5}\nu \left(1\right)=2$ (see P2).

To apply Ulam type stability according to Definition 2, we have to compare the new solution $\nu \left(t\right)=\epsilon e^{1.6t},t\in [0,1]$ of DI (7) with the same unique solution $x\left(t\right)=e^{1.5t},t\in (0,1]$ of BVP (6).

The difference $|x\left(t\right)-\nu \left(t\right)|=|1-\epsilon e^{0.1t}|e^{1.5t}$ depends significantly on ϵ. This difference is unbounded with respect to ε. Therefore, there does not exist a constant $C>0$ such that $|\nu \left(t\right)-x\left(t\right)|=|\epsilon e^{0.1t}-1|e^{1.5t}\le C\epsilon$ for any ε and $t\in [0,1]$, i.e., according to Definition 2, the BVP (6) is not Ulam–Hyers stable.

Example 4

(Fractional derivative). Let the derivative $D_t$ in differential Equation (1) be Caputo fractional derivative (4), $n=1$, $b=1$, $F(t,x)=1.5x$, and $G(u,v)=u+\frac{v}{E_{\alpha }\left(1.5\right)}-2$, i.e., consider the scalar linear BVP

$${\mathcal{D}}_t^{\alpha }x\left(t\right)=1.5x\left(t\right),t\in (0,1],x\left(0\right)+\frac{x\left(1\right)}{E_{\alpha }\left(1.5\right)}=2.$$

(8)

The BVP (8) has a unique solution $x\left(t\right)=E_{\alpha }\left(1.5t^{\alpha }\right),t\in (0,1]$ (see P1).

Let $\epsilon >0$ be an arbitrary number. Consider the differential inequality

$$|{\mathcal{D}}_t^{\alpha }\nu \left(t\right)-1.5\nu \left(t\right)|<\epsilon ,t\in [0,1].$$

(9)

Choose a solution of DI (9), for example, $\nu \left(t\right)=\epsilon t^2,t\in [0,1]$ (see Figure 5).

Note the solution $\nu \left(t\right)$ does not satisfy the boundary condition $\nu \left(0\right)+\frac{\nu \left(1\right)}{E_{\alpha }\left(1.5\right)}=2$ (see P2).

The difference $|x\left(t\right)-\nu \left(t\right)|=|E_{\alpha }\left(1.5t^{\alpha }\right)-\epsilon t^2|$ depends significantly on ϵ (see Figure 6). This difference is unbounded with respect to ε. Therefore, there does not exist a constant $C>0$ such that $|\nu \left(t\right)-x\left(t\right)|=|\epsilon t^2-e^{1.5t}|\le C\epsilon$ for any ε and $t\in [0,1]$, i.e., according to Definition 2, the BVP (8) is not Ulam–Hyers stable.

There are some recently published papers about Ulam type stability with the above-mentioned misunderstandings (P1) and (P2) with proofs assuming that any solution of the corresponding differential inequality satisfies the given boundary condition and applying the integral presentation of the solution of the given BVP to any solution of the differential inequality [29,30,31,32,33,34,35,36,37].

2.2.2. How to Avoid the Misunderstandings

Now, let us recall the situation about the initial value problem. The initial value of the solution of the differential equation has to depend on the initial value of the chosen solution of the differential inequality (see Case 1.2, Example 1 and Case 2.2, Example 2).

To apply the same idea as in the case of the IVP (see Section 2.1), we have to connect the boundary value condition of the given problem (6) with the values of the arbitrary chosen solution of the differential inequality (7). As it is mentioned in Remark 1, we need to have a parameter in the boundary value condition, and later, we have to choose this parameter in an appropriate way to depend on the chosen solution of the differential inequality.

We suggest a possible way to avoid the above misunderstandings and to connect the solution of the given BVP and the arbitrary solution of the corresponding differential inequality.

In the general case, we change the boundary condition (5) to a boundary condition with a parameter

$$g\left(x\right(0),x(b),\theta )=0,$$

(10)

where $\theta \in \mathbb{R}$ is a parameter.

Then, we prove that the BVP (1), (10) has a solution for any parameter $\theta \in \mathbb{R}$.

Remark 4.

Any solution of BVP (1), (10) depends on both the time variable t and the parameter θ.

For an arbitrary $\epsilon >0$, we choose an arbitrary solution $\nu \left(t\right)$ of the differential inequality (3), and we change Definition 2 for the US of BVP (1), (5) to the following definition of the US of BVP (1), (10):

Definition 3.

The BVP (1), (10) is Ulam–Hyers stable if there exists a real number $C_F>0$ such that for each $\epsilon >0$ and for each solution $\nu :[0,b]\to {\mathbb{R}}^n$ of the DI (3), there exists a solution $x:[0,b]\to {\mathbb{R}}^n$ of (1), (10) with $|\nu \left(t\right)-x\left(t\right)|\le C_F\epsilon$ for $t\in [0,b]$.

In a practical application of US, for any $\epsilon$, we choose a solution $\nu (·)$ of the DI (3), choose the parameter $\theta$ to depend on $\nu \left(0\right)$ and $\nu \left(b\right)$, i.e., $\mathsf{\Theta }=\xi \left(\nu \right(0),\nu (b\left)\right)$, and consider the differential Equation (1) with the function $g\left(x\right(0),x(b),\xi (\nu \left(0\right),\nu \left(b\right)\left)\right)$ in the boundary condition (10). Consider the solution $x\left(t\right),t\in [0,b]$ of BVP (1), (10) with the above-defined boundary value condition and study the difference between this solution $x\left(t\right)$ and the chosen solution $\nu \left(t\right)$ of the differential inequality (3). Now, the solution $x\left(t\right)$ depends on the solution $\nu (·)$ of DI(3). Any change in the solution of DI (3) leads to a change in the solution of BVP (1), (10).

Example 5.

Consider again the simple differential equation

$$x^{\prime }\left(t\right)=1.5x\left(t\right),t\in (0,1],$$

(11)

but change the boundary value condition $x\left(0\right)+e^{-1.5}x\left(1\right)=2$ to a boundary value condition with a parameter, i.e., consider the boundary condition

$$x\left(0\right)+e^{-1.5}x\left(1\right)=\mathsf{\Theta },$$

(12)

where $\mathsf{\Theta }\in \mathbb{R}$ is a parameter.

The new BVP (11), (12) has a unique solution $x\left(t\right)=0.5\mathsf{\Theta }{e}^{1.5t},t\in [0,1]$ for any parameter $\mathsf{\Theta }\in \mathbb{R}$.

Then, for an arbitrary solution $\nu \left(t\right)$ of DI (7), we choose the parameter $\mathsf{\Theta }=\nu \left(0\right)+e^{-1.5}\nu \left(1\right)$.

For example, if $\nu \left(t\right)=\epsilon t^2$, then $\mathsf{\Theta }=e^{-1.5}\epsilon$.

Then, the solution of BVP (11), (12) is $x\left(t\right)=0.5\epsilon e^{1.5(t-1)},t\in [0,1]$ and $|\nu \left(t\right)-x\left(t\right)|=\epsilon |t^2-0.5e^{1.5(t-1)}|\le 0.5\epsilon$ for any $\epsilon >0$ and $t\in [0,1]$. Therefore, according to Definition 3, the BVP (11), (12) is Ulam–Hyers stable with a constant $C=0.5$ (compare with Example 3).

Remark 5.

Some authors change Definition 2 concerning the choice of the solution of the differential inequality (3) to a solution of the differential inequality (3) satisfying the boundary condition (5). However, this misses the meaning of Ulam stability (compare with the IVP). In addition, it is more difficult to obtain a solution of the differential inequality (3) and the boundary condition (5) than to obtain any solution of the inequality (3).

Remark 6.

Note the idea of considering a parameter in the boundary condition was suggested and applied to a BVP for a nonlinear generalized proportional Caputo fractional differential equations in [38] and to a BVP for a system of multi-term delay fractional differential equations of Caputo type in [39].

3. Ulam Stability of BVP for Impulsive $\psi$-Caputo Fractional Differential Equation

To illustrate the ideas in Section 2.2, we use the Caputo fractional derivative with respect to another function (called also the $\psi$-Caputo fractional derivative), impulses, and boundary value conditions to the differential equation. To focus on the application of the new idea and the involvement of a parameter in the boundary value condition, we consider a linear boundary value condition.

Ulam type stability is studied for BVPs for differential equations with a $\psi$-Caputo fractional derivative in [40]. Unfortunately, the authors use similar misunderstandings as mentioned in Section 2.2.1. They choose a solution of the corresponding fractional differential inequality and assume this solution satisfies the given boundary condition (see (18) [40]).

In this paper, we suggest a way to apply the US to BVPs for differential equations with a $\psi$-Caputo fractional derivative, and more generally, we consider impulses in the equation and consider the fractional derivative of piecewise-constant order.

3.1. Preliminary Results

3.1.1. Some Results from Fractional Calculus

Let $c<d<\infty$ be fixed numbers and $\psi :[c,d]\to \mathbb{R}$ be a smooth increasing function with ${\psi }^{\prime }\left(t\right)>0$ almost everywhere in $[c,d]$.

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

Definition 4

([41]). Let $\delta >0$. The Riemann–Liouville fractional integral with respect to the function ψ (RLI) is defined by

$${}_c\mathcal{I}_{\psi \left(t\right)}^{\delta }\upsilon \left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\delta \right)}{\int }_c^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\delta -1}{\psi }^{\prime }\left(s\right)\upsilon \left(s\right)ds,t\in (c,d].$$

(13)

Definition 5

([41]). Let $\delta \in (0,1)$. The Caputo fractional derivative with respect to the function $\psi \left(t\right)$ (CFD) is defined by

$$\begin{array}{cc}& {}_c{}^C\mathcal{D}_{\psi \left(t\right)}^{\delta }\upsilon \left(t\right)\hfill \\ & ={}_c\mathcal{I}_{\psi \left(t\right)}^{1-\delta }\left(\frac{1}{{\psi }^{\prime }\left(t\right)}\frac{d}{dt}\right)\upsilon \left(t\right)=\frac{1}{\mathsf{\Gamma }(1-\delta )}{\int }_c^t{(\psi \left(t\right)-\psi \left(s\right))}^{-\delta }{\nu }^{\prime }\left(s\right)ds,t\in (c,d].\hfill \end{array}$$

(14)

Remark 7.

Note the RLI given in Definition 4 and the CFD given in Definition 5 are also called the ψ-fractional integral and the ψ-Caputo fractional derivative, respectively.

Lemma 1

([41]). Let $\delta \in (0,1)$. Then, we have

$$\left({}_c\mathcal{I}^{\delta }\left({}_c{}^C\mathcal{D}^{\delta }\nu \right)\right)\left(t\right)=\nu \left(t\right)-\nu \left(c\right),t\in (c,d],$$

(15)

and

$$\left({}_c{}^C\mathcal{D}^{\delta }\left({}_c\mathcal{I}^{\delta }\nu \right)\right)\left(\tau \right)=\nu \left(\tau \right),\tau \in (c,d].$$

(16)

Lemma 2

([41]). The solution of the IVP for the ψ-Caputo fractional differential equation $\left({}_c{}^C\mathcal{D}^{\delta }\nu \right)\left(t\right)=F\left(t\right)$ for $t\in (c,d]$ and $\nu \left(c\right)=V_0,$ where $\delta \in (0,1)$, $F\in C\left(\right[c,d],\mathbb{R})$, is $\nu \left(t\right)=V_0+\left({}_c\mathcal{I}^{\delta }F\right)\left(t\right),t\in [c,d].$

We now present some results for impulsive $\psi$-Caputo fractional differential equations on the interval $[0,b]$.

3.1.2. Preliminary Results for Scalar Linear Impulsive $\psi$-Caputo Fractional Differential Equations

Let $\psi :[0,b]\to \mathbb{R}$ be a smooth increasing function with ${\psi }^{\prime }\left(t\right)>0$ almost everywhere in $[0,b]$.

Note there are different approaches in the statement of the fractional differential equation with impulses. We use fractional derivatives with changeable lower limit at the impulsive points.

Let the sequences ${\left\{{\tau }_k\right\}}_{k=0}^{p+1},0={\tau }_0<{\tau }_k<{\tau }_{k+1}<{\tau }_{p+1}=b,k=1,2,\dots ,p,$ and ${\left\{{\delta }_k\right\}}_{k=0}^p,{\delta }_k\in (0,1),$ be given.

In the paper, we use the notations $\nu ({\tau }_k-0)={\lim }_{t\to {\tau }_k,t<{\tau }_k}\nu \left(t\right)=\nu \left({\tau }_k\right)$ and $\nu ({\tau }_k+0)={\lim }_{t\to {\tau }_k,t>{\tau }_k}\nu \left(t\right),k=1,2,\dots ,p,$ for a function $\nu :[0,b]\to \mathbb{R}$.

Define

$$\begin{array}{cc}\hfill PC[0,b]& =\{\nu :{\cup }_{j=0}^p({\tau }_k,{\tau }_{k+1}]\to \mathbb{R}:\nu \in C(({\tau }_k,{\tau }_{k+1}],\mathbb{R}),k=0,1,2,\dots ,p,\hfill \\ & \nu \left({\tau }_k\right)<\infty ,\nu ({\tau }_k+0)<\infty ,k=1,2,\dots ,p\}\hfill \end{array}$$

with a norm $\left|\right|\nu \left|\right|={\sup }_{t\in {\cup }_{j=0}^p({\tau }_k,{\tau }_{k+1}]}\left|\nu \left(t\right)\right|.$

Consider the following scalar linear impulsive $\psi$- Caputo fractional differential equation:

$$\begin{array}{cc}& \left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}\nu \right)\left(t\right)=F\left(t\right),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,p,\hfill \\ & \nu ({\tau }_k+0)={\mu }_k\nu \left({\tau }_k\right)+{\gamma }_k,k=1,2,\dots ,p,\hfill \end{array}$$

(17)

with the initial condition

$$\nu \left(0\right)={\nu }_0.$$

(18)

We give an explicit formula for the solution of IVP (17), (18).

Lemma 3.

IVP (17), (18) where $F\in C\left(\right[0,b],\mathbb{R})$, ${\mu }_k,{\gamma }_k\in \mathbb{R},k=1,2,\dots ,p,$ has a solution

$$\begin{array}{cc}& \nu \left(t\right)={\nu }_0\prod _{j=1}^k{\mu }_j+\mathcal{Q}(F,k,t),k=0,1,\dots ,p,\hfill \end{array}$$

(19)

where

$$\mathcal{Q}(F,k,t)=\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}F\right)\left({\tau }_{j+1}\right)+\sum _{j=1}^k\left(\prod _{m=j+1}^k{\mu }_m\right){\gamma }_j+\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}F\right)\left(t\right),$$

(20)

$\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}F\right)\left(t\right)$ is defined by (13) with $c={\tau }_k,\delta ={\delta }_k,d={\tau }_{k+1}$, and

$$\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}F\right)\left({\tau }_{j+1}\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_j\right)}{\int }_{{\tau }_j}^{{\tau }_{j+1}}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left(s\right)\right)}^{{\delta }_j-1}{\psi }^{\prime }\left(s\right)F\left(s\right)ds,j=0,1,2,\dots ,k-1.$$

Proof. 

We use induction.

For $t\in [0,{\tau }_1]$, by direct application of Lemma 2 with $c=0,d={\tau }_1,\delta ={\delta }_0,V_0={\nu }_0,$ we obtain (19).

Let $t\in ({\tau }_1,{\tau }_2]$. Then, from Lemma 2 with $c={\tau }_1,d={\tau }_2,\delta ={\delta }_1$, and $V_0=\nu ({\tau }_1+0)={\mu }_1\nu \left(t_1\right)+{\gamma }_1,$ we obtain

$$\begin{array}{cc}& \nu \left(t\right)={\mu }_1{\nu }_0+{\mu }_1\left({}_{{\tau }_0}\mathcal{I}^{{\delta }_0}F\right)\left({\tau }_1\right)+{\gamma }_1+\left({}_{{\tau }_1}\mathcal{I}^{{\delta }_1}F\right)\left(t\right),t\in ({\tau }_1,{\tau }_2].\hfill \end{array}$$

(21)

Let $t\in ({\tau }_2,{\tau }_3]$. Then, from Lemma 2 with $c={\tau }_2,d={\tau }_3,\delta ={\delta }_2$, and $V_0=\nu ({\tau }_2+0)={\mu }_2\nu \left(t_2\right)+{\gamma }_2,$ we obtain

$$\begin{array}{cc}& \nu \left(t\right)={\mu }_2{\mu }_1{\nu }_0+{\mu }_2{\mu }_1\left({}_{{\tau }_0}\mathcal{I}^{{\delta }_0}F\right)\left({\tau }_1\right)+{\mu }_2\left({}_{{\tau }_1}\mathcal{I}^{{\delta }_1}F\right)\left({\tau }_2\right)+{\mu }_2{\gamma }_1+{\gamma }_2+\left({}_{{\tau }_2}\mathcal{I}^{{\delta }_2}F\right)\left(t\right).\hfill \end{array}$$

(22)

Continuing this process inductively, we obtain (19). □

Consider the impulsive $\psi$-Caputo fractional differential Equation (17) with the linear boundary condition

$$A\nu \left(0\right)+B\nu \left(b\right)=\theta ,$$

(23)

where $A,B\in \mathbb{R}$ are given constants, $\theta \in \mathbb{R}$ is a parameter.

We obtain the explicit form of the solution of BVP (17), (23).

Lemma 4.

Let $F\in C\left(\right[0,b],\mathbb{R})$; ${\mu }_k,{\gamma }_k\in \mathbb{R},k=1,2,\dots ,p$, $A,B\in \mathbb{R}:A+B{\prod }_{j=1}^p{\mu }_j\ne 0,$ are given constants, and $\theta \in \mathbb{R}$ is a parameter. Then, the solution of BVP (17), (23) is

$$\begin{array}{cc}& \nu \left(t\right)=\frac{\theta -B\mathcal{Q}(F,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j+\mathcal{Q}(F,k,t),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p,\hfill \end{array}$$

(24)

where $\mathcal{Q}(F,k,t)$ is defined by (20).

Proof. 

According to Lemma 3, the solution of (17) satisfies (19) for $t=b$, i.e.,

$$\begin{array}{cc}& \nu \left(b\right)=\nu \left(0\right)\prod _{j=1}^p{\mu }_j+\mathcal{Q}(F,p,b).\hfill \end{array}$$

(25)

From the boundary condition (23), we obtain $A\nu \left(0\right)+B\nu \left(b\right)=\nu \left(0\right)\left(A+B{\prod }_{j=1}^p{\mu }_j\right)+B\mathcal{Q}(F,p,b)=\theta$, i.e., $\nu \left(0\right)=\frac{\theta -B\mathcal{Q}(F,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}$.

From (19), we obtain (24). □

3.2. Application of the Algorithm for Ulam Type Stability to BVPs

We apply the algorithm presented in Section 2 for the practical application of Ulam type stability for BVPs.

• Step 1. Statement of the problem

Consider the boundary value problem (BVP) for the nonlinear impulsive differential equation with the $\psi$-Caputo fractional derivative in the form

$$\begin{array}{cc}& \left({}_{{\tau }_k}{}^C\mathcal{D}_{\psi }^{{\delta }_k}\nu \right)\left(t\right)=f(t,\nu \left(t\right))\mathrm{for}t\in (0,b],t\ne t_k,k=1,2,\dots ,p,\hfill \\ & \nu ({\tau }_k+0)={\mu }_k\nu \left({\tau }_k\right)+{\gamma }_k\mathrm{for}k=1,2,\dots ,p,\hfill \end{array}$$

(26)

with the boundary value condition

$$Ax\left(0\right)+Bx\left(b\right))=\theta ,$$

(27)

where $\theta \in \mathbb{R}$ is a parameter. Assume the following assumptions hold:

(A1).

The function $\psi :[0,b]\to \mathbb{R}$ is smooth and increasing with ${\psi }^{\prime }\left(t\right)>0$ almost everywhere in $[0,b]$.

(A2).

The sequence ${\left\{{\tau }_k\right\}}_{k=0}^{p+1},0={\tau }_0<{\tau }_k<{\tau }_{k+1}<{\tau }_{p+1}=b,k=1,2,\dots ,p.$

(A3).

The sequence ${\left\{{\delta }_k\right\}}_{k=0}^p,{\delta }_k\in (0,1).$

(A4).

The constants ${\mu }_k,{\gamma }_k\in \mathbb{R}$, $A,B\in \mathbb{R}$ are such that $A+B{\prod }_{j=1}^p{\mu }_j\ne 0$.

(A5).

The function $f\in C\left(\right[0,b]\times \mathbb{R},\mathbb{R})$.

(A6).

There exists a constant $K>0$ such that $\left|f\right(t,x)-f(t,y\left)\right|\le K|x-y|$ for $t\in [0,b]$ and $x,y\in \mathbb{R}$.

• Step 2. Integral presentation.

Based on Lemma 4, we introduce the following definition:

Definition 6.

The mild solution of BVP (26), (27) is the solution of the integral equation

$$\begin{array}{cc}& \nu \left(t\right)=\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j+\mathbb{L}(f,\nu ,k,t),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p,\hfill \end{array}$$

(28)

where $\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_k\right)}{\int }_{{\tau }_k}^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{{\delta }_k-1}{\psi }^{\prime }\left(s\right)f(s,\nu \left(s\right)))ds$,

$$\begin{array}{cc}& \left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_j\right)}{\int }_{{\tau }_j}^{{\tau }_{j+1}}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left(s\right)\right)}^{{\delta }_j-1}{\psi }^{\prime }\left(s\right)f(s,\nu \left(s\right)))ds,\hfill \\ & forj=0,1,2,\dots ,k-1,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \mathbb{L}(f,\nu ,k,t)& =\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)+\sum _{j=1}^k\left(\prod _{m=j+1}^k{\mu }_m\right){\gamma }_j\hfill \\ & +\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right).\hfill \end{array}$$

(29)

Theorem 1.

Let conditions (A1)–(A5) be satisfied and $\theta \in \mathbb{R}$ be a parameter. The mild solution of BVP (26), (27) is a solution of BVP (26), (27) and vice versa for any value of $\theta \in \mathbb{R}$.

Proof. 

Let the function $\nu \in PC\left(\right[0,b],\mathbb{R})$ be a mild solution of BVP (26), (27) (see Definition 6).

Note the expressions

$$\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right),$$

$$\sum _{j=1}^k\left(\prod _{m=j+1}^k{\mu }_m\right){\gamma }_j,k=1,2,\dots ,p,$$

and

$$\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j$$

do not depend on the time variable t, and their Caputo fractional derivatives with respect to the function $\psi$ are zero.

Then, for any $k=1,2,\dots ,p$, applying Lemma 1 and (16) with $c={\tau }_k,d={\tau }_{k+1},\delta ={\delta }_k,$ we obtain

$$\begin{array}{c}\hfill \left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}\mathbb{L}(f,\nu ,k,·)\right)\left(t\right)=\left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}\left(\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\right)\right)\left(t\right)=f(t,\nu \left(t\right)).\end{array}$$

(30)

Take a fractional derivative $\left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}(·)\right)\left(t\right),t\in ({\tau }_k,{\tau }_{k+1}]$ of both sides of (28), use (30), and obtain $\left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}\nu \right)\left(t\right)=f(t,\nu \left(t\right))$ for $\tau \in (t_k,t_{k+1}],k=0,1,\dots ,p$, i.e., the function $\nu (·)$ satisfies the first equation of (26).

Let $t={\tau }_k,k=1,2,\dots ,p$. From (28), for $t={\tau }_k+0\in ({\tau }_k,{\tau }_{k+1})$ and $t_k-0\in (t_{k-1},t_k]$, applying $\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left({\tau }_k\right)=0$, we have

$$\begin{array}{cc}\hfill \nu (t_k+0)& =\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j+\mathbb{L}(f,\nu ,k,{\tau }_k)\hfill \\ & ={\mu }_k\frac{\theta -B\mathbb{L}(\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^{k-1}{\mu }_j+\sum _{j=0}^{k-1}{\mu }_k\left(\prod _{m=j+1}^{k-1}{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)\hfill \\ & +\sum _{j=1}^k\left(\prod _{m=j+1}^k{\mu }_m\right){\gamma }_j+\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left({\tau }_k\right)\hfill \\ & ={\mu }_k\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^{k-1}{\mu }_j+\sum _{j=0}^{k-1}{\mu }_k\left(\prod _{m=j+1}^{k-1}{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)\hfill \\ & +\sum _{j=1}^{k-1}{\mu }_k\left(\prod _{m=j+1}^{k-1}{\mu }_m\right){\gamma }_j+\left(\prod _{m=k+1}^k{\mu }_m\right){\gamma }_k\hfill \\ & ={\mu }_k\left(\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^{k-1}{\mu }_j+\mathbb{L}(f,\nu ,k-1,{\tau }_k)\right)+{\gamma }_k\hfill \\ & ={\mu }_k\nu \left({\tau }_k\right)+{\gamma }_k.\hfill \end{array}$$

(31)

The equalities in (31) imply that the second equation of (26) is satisfied for all $k=1,2,\dots ,p$.

From (28), using $\mathbb{L}(f,\nu ,0,0)=0$, we obtain

$$\begin{array}{cc}& A\nu \left(0\right)+B\nu \left(b\right)\hfill \\ & =A\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^0{\mu }_j+B\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^p{\mu }_j+B\mathbb{L}(f,\nu ,p,b)\hfill \\ & =\theta -B\mathbb{L}(f,\nu ,p,b)+B\mathbb{L}(f,\nu ,p,b)=\mathsf{\Theta },\hfill \end{array}$$

(32)

i.e., the function $\nu (·)$ satisfies the boundary value condition (27).

Therefore, the mild solution $\nu (·)$ is a solution of BVP (26), (27).

Let the function $\nu \in PC\left(\right[0,b]$ be a solution of BVP (26), (27). Taking an integral $\left({}_{{\tau }_k}\mathcal{I}^{\delta }(·)\right)\left(t\right)$ on both sides of the first equation of (26) for $t\in ({\tau }_k,{\tau }_{k+1}]$ and using Lemma 1, similar to the proof of Lemma 3, we obtain the integral equality (28). □

• Step 3. Existence of a solution.

Define the fractional integral operator $\mathsf{\Omega }:PC[0,b]\to PC[0,b]$ by

$$\begin{array}{cc}\hfill \left(\mathsf{\Omega }\nu \right)\left(\tau \right)& =\frac{\theta -B\mathbb{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j+\mathbb{L}(f,\nu ,k,t),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p.\hfill \end{array}$$

(33)

Theorem 2.

Let conditions (A1)–(A6) be satisfied, $\theta \in \mathbb{R}$, and the inequality

$$KM\mathcal{P}<1$$

(34)

hold, where

$$M=\max \{\frac{1}{\mathsf{\Gamma }({\delta }_j+1)}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_j}:j=0,1,2,\dots ,p\},$$

(35)

and

$$\mathcal{P}=\underset{k=0,1,2,\dots ,p}{\max }\left\{\sum _{j=0}^k\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)\right\}.$$

(36)

Then, the operator Ω has a fixed point.

Proof. 

Let $\nu ,\nu \ast \in PC[0,b]$. Note $\frac{1}{\mathsf{\Gamma }\left({\delta }_j\right)}{\int }_{{\tau }_j}^{\tau }{\left(\psi \left(\tau \right)-\psi \left(s\right)\right)}^{{\delta }_j-1}{\psi }^{\prime }\left(s\right)ds=\frac{{\left(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_j}}{\mathsf{\Gamma }({\delta }_j+1)}$ for $\tau \in ({\tau }_j,{\tau }_{j+1}],j=0,1,2,\dots ,p$, and we obtain for $t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p$ that

$$\begin{array}{cc}\hfill & \left|\right(\mathsf{\Omega }\nu \left)\right(t\left)\right|-(\mathsf{\Omega }\nu \ast )\left(t\right)|\le |\mathbb{L}(f,\nu ,k,t)-\mathbb{L}(f,\nu \ast ,k,t)|\hfill \\ \hfill & \le \sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)\left|\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)-\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu \ast (·))\right)\left({\tau }_{j+1}\right)\right|\hfill \\ \hfill & +\left|\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right)-\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu \ast (·))\right)\left(t\right)\right|\hfill \\ \hfill & \le \parallel \nu -\nu \ast \parallel \sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)K\frac{1}{\mathsf{\Gamma }({\delta }_j+1)}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_j}\hfill \\ \hfill & +\frac{1}{\mathsf{\Gamma }({\delta }_k+1)}{\left(\psi \left({\tau }_{k+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_k}K\parallel \nu -\nu \ast \parallel \hfill \\ \hfill & =K\left|\right|\nu -\nu \ast \left|\right|\sum _{j=0}^k\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)\frac{1}{\mathsf{\Gamma }({\delta }_j+1)}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_j}\hfill \\ \hfill & \le K\parallel \nu -\nu \ast \parallel \sum _{j=0}^p\left(\prod _{m=j+1}^p\left|{\mu }_m\right|\right)\frac{1}{\mathsf{\Gamma }({\delta }_j+1)}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_j}.\hfill \end{array}$$

(37)

Therefore,

$$\left|\right|\left(\mathsf{\Omega }\nu \right)-(\mathsf{\Omega }\nu \ast )\left|\right|\le K\left|\right|\nu -\nu \ast \left|\right|\sum _{j=0}^p\left(\prod _{m=j+1}^p\left|{\mu }_m\right|\right)\frac{1}{\mathsf{\Gamma }({\delta }_j+1)}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right)\right)}^{{\delta }_j}.$$

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

Corollary 1.

Let all the conditions of Theorem 2 be satisfied. Then, the BVP (26), (27) has a solution for any $\theta \in \mathbb{R}$, and it satisfies the integral Equation (28).

The proof follows from Theorem 2 and Theorem 1.

• Step 4. Definition of Ulam type stability

Let $\epsilon >0$ and consider the following impulsive $\psi$-Caputo fractional differential inequalities (FDI):

$$\begin{array}{cc}& \left|\left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}y\right)\left(t\right)-f(t,y\left(t\right))\right|\le \epsilon ,t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,p,\hfill \\ & |y({\tau }_k+0)-{\mu }_{k}y\left({\tau }_k\right)-{\gamma }_k|\le \epsilon ,k=1,2,\dots ,p.\hfill \end{array}$$

(38)

Note if $\nu \in PC[0,b]$ is a solution of (38), then there exist a function $g\in C[0,b]:\left|g\right(t\left)\right|\le \epsilon ,$ $t\in [0,b]$ and constants ${\eta }_k:\left|{\eta }_k\right|\le \epsilon ,k=1,2,\dots ,p,$ such that $\nu \left(t\right)$ is a solution of the following impulsive fractional differential equation

$$\begin{array}{cc}& \left({}_{{\tau }_k}{}^C\mathcal{D}^{{\delta }_k}y\right)\left(t\right)=f(t,y\left(t\right))+g\left(t\right),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,p,\hfill \\ & y({\tau }_k+0)={\mu }_{k}y\left({\tau }_k\right)+{\gamma }_k+{\eta }_k,k=1,2,\dots ,p.\hfill \end{array}$$

(39)

Remark 8.

Note that the solution of the impulsive fractional differential inequalities (38) might not satisfy the boundary condition of (27). A similar comment concerns any solution of the impulsive fractional differential Equation (39).

Definition 7.

The BVP (26), (27) is Ulam–Hyers stable if there exists a real number $C_f>0$ such that for each $\epsilon >0$ and for each solution $y\in PC[0,b]$ of the FDI (38), there exists a solution $\nu \in PC[0,b]$ of (26), (27) with $\theta =Ay\left(0\right)+By\left(b\right)$ such that $\left|\right|\nu -y\left|\right|\le C_f\epsilon$.

• Step 5. Proof of US.

Theorem 3.

Let the assumptions (A1)–(A6) hold and

$$KM\mathcal{P}\mathcal{T}<1,$$

(40)

where $\mathcal{P}$ is defined by (36), M is defined by (35), $\mathcal{Q}={\max }_{k=0,1,2,\dots ,p}\left\{{\prod }_{m=1}^k\left|{\mu }_m\right|\right\}$, and $\mathcal{T}=\left(1+\frac{\left|B\right|\mathcal{Q}}{\left|A+B{\prod }_{j=1}^p{\mu }_j\right|}\right)$.

Then, the BVP (26), (27) is US.

Proof. 

Let $\epsilon >0$ and $y\in PC[0,b]$ be a solution of the fractional differential inequality (38). Note it satisfies the impulsive differential Equation (39). Let $\theta =Ay\left(0\right)+By\left(b\right)$. Then, the conditions (A5) and (A6) are satisfied for the function $G(t,v)=f(t,v)+g\left(t\right)$ and according to Corollary 1, the impulsive differential Equation (39) with a boundary condition $Ay\left(0\right)+By\left(b\right)=\theta$ has a solution satisfying the integral equality

$$\begin{array}{cc}& y\left(t\right)=\frac{\theta -B\mathbb{L}(f+g,y,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j+\mathbb{L}(f+g,y,k,t),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p.\hfill \end{array}$$

(41)

If inequality (40) holds, then the inequality (34) is fulfilled, and all conditions of Theorem 2 are satisfied. According to Corollary 1, there exists a solution $\nu \in PC\left(\right[0,b\left]\right)$ of (26), (27) for the chosen parameter $\theta$, and this solution satisfies the integral equality (28).

Then,

$$\begin{array}{cc}& |y\left(t\right)-\nu \left(t\right)|\le \left|\mathbb{L}(f+g,y,p,b)-\mathbb{L}(f,\nu ,p,b)\right|\left|\frac{B{\prod }_{j=1}^k{\mu }_j}{A+B{\prod }_{j=1}^p{\mu }_j}\right|\hfill \\ & +\left|\mathbb{L}(f+g,y,k,t)-\mathbb{L}(f,\nu ,k,t)\right|,t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p.\hfill \end{array}$$

(42)

For any $k=0,1,2,\dots ,p$ and $t\in ({\tau }_k,{\tau }_{k+1}]$, we apply the inequalities

$$({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}\left|g\right|)\left(t\right)\le \frac{\epsilon }{\mathsf{\Gamma }(1+{\delta }_k)}{(\psi \left(t\right)-\psi \left({\tau }_k\right))}^{{\delta }_k},$$

$$({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}\left|g\right|)\left(t_{j+1}\right)\le \frac{\epsilon }{\mathsf{\Gamma }(1+{\delta }_j)}{(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right))}^{{\delta }_j},$$

$$|f(s,y\left(s\right))-f(s,\nu \left(s\right))|\le K|y\left(s\right)-\nu \left(s\right)|\le K\left|\right|y-\nu \left|\right|,s\in ({\tau }_k,t],$$

and

$$({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}|f(·,y(·))-f(·,\nu (·))|)\left({\tau }_{j+1}\right)\le \frac{K\left|\right|y-\nu \left|\right|}{\mathsf{\Gamma }(1+{\delta }_j)}{(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right))}^{{\delta }_j}$$

and we obtain

$$\begin{array}{cc}& \left|\mathbb{L}(f+g,y,k,t)-\mathbb{L}(f,\nu ,k,t)\right|\hfill \\ & \le \left|\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)\left(\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)-\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,y(·))\right)\left({\tau }_{j+1}\right)\right)\right|\hfill \\ & +\left|\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)(\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}g\right)\left({\tau }_{j+1}\right)\right|+\left|\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}g\right)\left(t\right)\right|\hfill \\ & +\left|\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right)-\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,y(·))\right)\left(t\right)\right|\hfill \\ & \le \sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)\frac{K\left|\right|y-\nu \left|\right|}{\mathsf{\Gamma }(1+{\delta }_j)}{(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right))}^{{\delta }_j}\hfill \\ & +\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)\frac{\epsilon }{\mathsf{\Gamma }(1+{\delta }_j)}{(\psi \left({\tau }_{j+1}\right)-\psi \left({\tau }_j\right))}^{{\delta }_j}+\frac{\epsilon }{\mathsf{\Gamma }(1+{\delta }_k)}{(\psi \left(t\right)-\psi \left({\tau }_k\right))}^{{\delta }_k}\hfill \\ & +\frac{K\left|\right|y-\nu \left|\right|}{\mathsf{\Gamma }(1+{\delta }_k)}{(\psi \left({\tau }_{k+1}\right)-\psi \left({\tau }_k\right))}^{{\delta }_k}\hfill \\ & \le KM\left|\right|y-\nu \left|\right|\sum _{j=0}^k\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)+\epsilon M\sum _{j=0}^k\left(\prod _{m=j+1}^k\left|{\mu }_m\right|\right)\le KM\mathcal{P}\left|\right|y-\nu \left|\right|+\epsilon M\mathcal{P}.\hfill \end{array}$$

(43)

From (42) and (43), we obtain for $t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p,$

$$\begin{array}{cc}\hfill \left|y\right(t)-\nu (t\left)\right|& \le \left(KM\mathcal{P}\left|\right|y-\nu \left|\right|+\epsilon M\mathcal{P}\right)\left(1+\left|\frac{B{\prod }_{j=1}^k{\mu }_j}{A+B{\prod }_{j=1}^p{\mu }_j}\right|\right)\hfill \\ & \le KM\mathcal{P}\mathcal{T}\left|\right|y-\nu \left|\right|+\epsilon M\mathcal{P}\mathcal{T}.\hfill \end{array}$$

(44)

Therefore,

$$\begin{array}{c}\hfill \left|\right|y-\nu \left|\right|\le \epsilon \frac{M\mathcal{P}\mathcal{T}}{1-KM\mathcal{P}\mathcal{T}},\end{array}$$

(45)

or the BVP (26), (27) is Ulam–Hyers stable with a constant $C_f=\frac{M\mathcal{P}\mathcal{T}}{1-KM\mathcal{P}\mathcal{T}}$. □

3.3. Partial Cases

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

3.3.1. Boundary Value Problem for a Fractional Differential Equation without Any Impulses

Consider the fractional differential equation

$$\left({}_0{}^C\mathcal{D}_{\psi }^{\delta }\nu \right)\left(t\right)=f(t,\nu \left(t\right))fort\in (0,b],$$

(46)

with the boundary condition (27), i.e., ${\mu }_k=1,{\gamma }_k=0,k=1,2,\dots ,p$ in (26).

The integral Equation (28) reduces to the following integral equation for the solution of BVP (46), (27):

$$\begin{array}{cc}& \nu \left(t\right)=\frac{\theta -B({}_0\mathcal{I}^{\delta }f(·,\nu (·))\left(b\right)}{A+B}+\left({}_0\mathcal{I}^{\delta }f(·,\nu (·))\right)\left(t\right),t\in (0,b],\hfill \end{array}$$

(47)

where $\left({}_0\mathcal{I}^{\delta }f(·,\nu (·))\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }\left(\delta \right)}{\int }_0^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\delta -1}{\psi }^{\prime }\left(s\right)f(s,\nu \left(s\right))ds$.

Then, Theorem 3 reduces to the following result:

Theorem 4.

Let the assumptions (A1), (A5), (A6) hold, $\delta \in (0,1)$, and

$$K{(\psi \left(b\right)-\psi \left(0\right))}^{\delta }\left(1+\frac{\left|B\right|}{|A+B|}\right)<\mathsf{\Gamma }(1+\delta ).$$

(48)

Then, the BVP (46), (27) is Ulam–Hyers stable.

3.3.2. Boundary Value Problem for an Impulsive Caputo Fractional Differential Equation

Let the function $\psi \left(t\right)\equiv t$, $t\in [0,b]$. Then, the $\psi$-Caputo fractional derivative given by (14) reduces to the Caputo fractional derivative ${}_c{}^C\mathcal{D}^{\delta }\upsilon \left(t\right)$.

Consider the impulsive Caputo fractional differential equation

$$\begin{array}{cc}& \left({}_0{}^C\mathcal{D}^{{\delta }_k}\nu \right)\left(t\right)=f(t,\nu \left(t\right))\mathrm{for}t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,\dots ,p,\hfill \\ & \nu ({\tau }_k+0)={\mu }_k\nu \left({\tau }_k\right)+{\gamma }_k,k=1,2,\dots ,p,\hfill \end{array}$$

(49)

with the boundary condition (27).

The integral Equation (28) reduces to the following integral equation for the solution of BVP (49), (27):

$$\begin{array}{cc}& \nu \left(t\right)=\frac{\theta -B\mathcal{L}(f,\nu ,p,b)}{A+B{\prod }_{j=1}^p{\mu }_j}\prod _{j=1}^k{\mu }_j+\mathcal{L}(f,\nu ,k,t),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p,\hfill \end{array}$$

(50)

where

$$\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_k\right)}{\int }_{{\tau }_k}^t{\left(t-s\right)}^{{\delta }_k-1}f(s,\nu \left(s\right)))ds,$$

$$\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_j\right)}{\int }_{{\tau }_j}^{{\tau }_{j+1}}{\left(t-s\right)}^{{\delta }_j-1}f(s,\nu \left(s\right)))ds,j=0,1,2,\dots ,k-1,$$

and

$$\begin{array}{cc}& \mathcal{L}(f,\nu ,k,t)\hfill \\ & =\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)+\sum _{j=1}^k\left(\prod _{m=j+1}^k{\mu }_m\right){\gamma }_j+\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right).\hfill \end{array}$$

Then, Theorem 3 reduces to the following result:

Theorem 5.

Let the assumptions (A1), (A5), (A6) hold, $\delta \in (0,1)$, and

$$KM\mathcal{P}\mathcal{T},$$

(51)

where $M=\max \{\frac{{\tau }_{j+1}-{\tau }_j}{\mathsf{\Gamma }({\delta }_j+1)}:j=0,1,2,\dots ,p\}$; $\mathcal{P}$ is defined by (36), $\mathcal{T}$ is defined in Theorem 3.

Then, the BVP (46), (27) is Ulam–Hyers stable.

3.3.3. Initial Value Problem for an Impulsive Fractional Differential Equation

Consider the impulsive fractional differential Equation (26) with the initial condition

$$\nu \left(0\right)=\theta ,$$

(52)

i.e., $A=1,B=0$ in (27).

The integral Equation (28) reduces to the following integral equation for the solution of IVP (26), (52):

$$\begin{array}{cc}& \nu \left(t\right)=\theta \prod _{j=1}^k{\mu }_j+\sum _{j=1}^k\left(\prod _{m=j+1}^k{\mu }_m\right){\gamma }_j+\sum _{j=0}^{k-1}\left(\prod _{m=j+1}^k{\mu }_m\right)\left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)\hfill \\ & +\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right),t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,\dots ,p,\hfill \end{array}$$

(53)

where

$$\left({}_{{\tau }_k}\mathcal{I}^{{\delta }_k}f(·,\nu (·))\right)\left(t\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_k\right)}{\int }_{{\tau }_k}^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{{\delta }_k-1}{\psi }^{\prime }\left(s\right)f(s,\nu \left(s\right)))ds,$$

and

$$\begin{array}{cc}& \left({}_{{\tau }_j}\mathcal{I}^{{\delta }_j}f(·,\nu (·))\right)\left({\tau }_{j+1}\right)=\frac{1}{\mathsf{\Gamma }\left({\delta }_j\right)}{\int }_{{\tau }_j}^{{\tau }_{j+1}}{\left(\psi \left({\tau }_{j+1}\right)-\psi \left(s\right)\right)}^{{\delta }_j-1}{\psi }^{\prime }\left(s\right)f(s,\nu \left(s\right)))ds,\hfill \\ & \mathrm{for}j=0,1,\dots ,k-1.\hfill \end{array}$$

Then, Theorem 3 reduces to the following result:

Theorem 6.

Let the assumptions (A1), (A2), (A3), (A5), (A6) hold, ${\mu }_k,{\gamma }_k\in \mathbb{R},k=1,2,\dots ,p$ and

$$KM\mathcal{P}<1,$$

(54)

where $\mathcal{P}$ is defined by (36), M is defined by (35).

Then, the IVP (26), (52) is Ulam–Hyers stable.

Remark 9.

The US for the IVP for an impulsive Caputo fractional differential equation (with $\psi \left(t\right)\equiv t$ in (26)) is studied in [24] but in the case where the lower limit of the fractional derivative is fixed at the initial time.

3.3.4. Initial Value Problem for a Fractional Differential Equation without Any Impulses

Consider the fractional differential Equation (46) with the initial condition (52), i.e., $A=1,B=0$ in (27) and ${\mu }_k=1,\gamma =0,k=1,2,\dots ,p$ in (26).

The integral Equation (28) reduces to the following integral equation for the solution of IVP (46), (52) (compare with [41])

$$\begin{array}{cc}& \nu \left(t\right)=\theta +\frac{1}{\mathsf{\Gamma }\left(\delta \right)}{\int }_0^t{\left(\psi \left(t\right)-\psi \left(s\right)\right)}^{\delta -1}{\psi }^{\prime }\left(s\right)f(s,\nu \left(s\right)))ds,t\in (0,b].\hfill \end{array}$$

(55)

Then, Theorem 3 reduces to the following result:

Theorem 7.

Let the assumptions (A1), (A5), (A6) hold, and $K{(\psi \left(b\right)-\psi \left(0\right))}^{\delta }<\mathsf{\Gamma }(1+\delta )$.

Then, the IVP (46), (52) is Ulam–Hyers stable.

Remark 10.

In the case where $\psi \left(t\right)\equiv t$ in Equation (46), the partial case of the IVP for a Caputo fractional differential equation is studied in [18].

3.4. Application

Example 6.

Let $b=1,p=4,{\tau }_1=0.1,{\tau }_2=0.5,{\tau }_3=0.7,{\tau }_4=0.9,{\mu }_1=2,{\mu }_2=0.5,{\mu }_3=3$, ${\mu }_4=\frac{1}{3},\delta =0.3,\psi \left(t\right)=\frac{t}{t+1}$, ${\delta }_k=0.3$, and $k=0,1,2,3$.

Consider the following scalar impulsive ψ-Caputo fractional differential equation:

$$\begin{array}{cc}& \left({}_{{\tau }_k}{}^C\mathcal{D}_{\psi }^{0.3}\nu \right)\left(t\right)=\frac{t}{8t+1}{e}^{-\left|\nu \right(t\left)\right|},t\in ({\tau }_k,{\tau }_{k+1}],k=0,1,2,3,4,\hfill \\ & \nu ({\tau }_k+0)={\mu }_k\nu \left({\tau }_k\right)+{\gamma }_k,k=1,2,3,4,\hfill \end{array}$$

(56)

with boundary condition

$$\nu \left(0\right)+0.2\nu \left(1\right)=\theta ,$$

(57)

where $\theta \in \mathbb{R}$ is a parameter.

The conditions (A1)–(A3) are satisfied. Also, ${\prod }_{k=1}^4{\mu }_k=1$ and $A+B=1.2\ne 0$, i.e., condition (A4) holds.

In this case, $f(t,y)=\frac{t}{8t+1}{e}^{-\left|y\right|}$, i.e., condition (A6) is satisfied with $K=0.125$.

We have

$$M=\max \{0.5427,0.7284,0.5192,0.4837\}=0.7284,$$

$$\mathcal{P}=\max \{1,1+|{\mu }_1|,1+|{\mu }_2|+|{\mu }_1\left|\right|{\mu }_2|,1+|{\mu }_3|+|{\mu }_3\left|\right|{\mu }_2|+|{\mu }_3\left|\right|{\mu }_2\left|\right|{\mu }_1\left|\right\}=6.5,$$

and $KM\mathcal{P}=0.5966<1$, i.e., all the conditions of Theorem 2 are satisfied; therefore, BVP (56), (57) has a solution.

Also, $\mathcal{Q}=\max \{1,|{\mu }_1|,|{\mu }_1\left|\right|{\mu }_2|,|{\mu }_1\left|\right|{\mu }_2\left|\right|{\mu }_3|,|{\mu }_1\left|\right|{\mu }_2\left|\right|{\mu }_3\left|\right|{\mu }_4\left|\right\}=3$, $|A+B|=1.2$, $\mathcal{T}=1.5$ and $KM\mathcal{P}\mathcal{T}=0.894839<1$, i.e., all the conditions of Theorem 3 are satisfied, and the solution of BVP (56), (57) is Ulam–Hyers stable.

4. Conclusions

The main goal of this paper was to focus on the application of Ulam type stability to boundary value problems for any type of differential equations and to point out the misunderstandings appearing in the literature. We suggested a possible way to avoid these misunderstandings by involving a parameter. We illustrated the suggested idea on a general linear boundary value problem for a nonlinear impulsive differential equation with a Caputo fractional derivative with respect to another function and a piecewise-constant variable order. This involved a parameter in the boundary condition. The existence and uniqueness of the solution depending on this parameter was proved, the Ulam–Hyers stability was defined, and sufficient conditions were obtained. The proposed ideas could be applied to study Ulam type stability for various types of boundary value problems for other types of differential equations, such as ordinary differential equations, fractional differential equation with any type of Caputo type fractional derivative, integro-differential fractional equations, and partial differential equations with time-fractional derivatives.

The suggested new way of the application of Ulam type stability to boundary value problems and the idea of the connection between the solutions of the given problem and the solutions of the corresponding inequality will give researchers a tool for a correct application of Ulam type stability to various types of boundary value problems.