# Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study

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**—**Theory and Applications)

## Abstract

**:**

## 1. Introduction

## 2. Ordinary Differential Equations and Ulam-Hyers Stability

**Case 1.**Initial value problem (IVP) for ODE (1). Consider the initial condition:

**Definition 1**

**Remark 1.**

**Definition 2.**

**Remark 2.**

**Case 2.**Boundary value problem (BVP) for ODE (1). Consider the boundary condition

**Definition 3.**

**Remark 3.**

**Definition 4.**

**Example 1.**

**Case 1.**(IVP) Consider the initial condition

**Case 1.1.**We will look at Definition 1.

**Case 1.2.**We will look at Definition 2.

**Case 2.**(BVP) Consider the boundary condition:

**Case 2.1.**We will look at Definition 3.

**Case 2.2.**We will look at Definition 4.

**Remark 4.**

## 3. Generalized Proportional Caputo Fractional Differential Equations and Ulam-Hyers Stability

#### 3.1. Preliminary Results from Fractional Calculus

**Remark 5.**

**Remark 6.**

**Lemma 1**

**Corollary 1**

**Lemma 2**

**Corollary 2**

**Lemma 3**

**Lemma 4.**

**Corollary 3.**

#### 3.2. Ulam Type Stability for Initial Value Problems

**A**). The function $f\in C([0,T]\times \mathbb{R}):\phantom{\rule{4pt}{0ex}}f(.,\nu (.))\in {I}^{\alpha ,\rho}[0,T]$ for any function $\nu \in C[0,T]$.

**Definition 5.**

**Theorem 1.**

**Proof.**

**Theorem 2.**

**Definition 6.**

**Remark 7.**

**Definition 7.**

**Remark 8.**

#### 3.3. Existence and Uniqueness of the Solution of Boundary Value Problem

**B**). The condition $a+b{e}^{\frac{\rho -1}{\rho}T}\ne 0$ holds with $\alpha \in (0,1)$, $\rho \in (0,1]$.

**Lemma 5.**

**Definition 8.**

**Remark 9.**

**Theorem 3.**

**Proof.**

**Theorem 4.**

**Proof.**

**Corollary 4.**

**Theorem 5.**

- 1.
- The condition (B) is satisfied.
- 2.
- The condition (A) is satisfied and there exists a constant $L>0$, such that, for $t\in [0,T],{x}_{j}\in \mathbb{R},\phantom{\rule{4pt}{0ex}}j=1,2,$ the inequality $|f(t,{x}_{1})-f(t,{x}_{2})|\le L|{x}_{1}-{x}_{2}|$ holds.
- 3.
- The inequality $L{T}^{\alpha}\left(1+\frac{\left|b\right|}{\left|a+b{e}^{\frac{\rho -1}{\rho}T}\right|}\right)<{\rho}^{\alpha}\mathsf{\Gamma}(1+\alpha )$ holds.

**Proof.**

**Remark 10.**

**Lemma 6.**

**Remark 11.**

#### 3.4. Ulam Type Stability for Boundary Value Problems

**Definition 9.**

**Remark 12.**

**Definition 10.**

**Theorem 6.**

- 1.
- The conditions of Theorem 5 are satisfied.
- 2.
- For any $\epsilon >0$, the inequality (22) has at least one solution from ${C}^{\alpha ,\rho}[0,T]$.

**Proof.**

**Theorem 7.**

- 1.
- The condition (B) is satisfied, and the value of μ is given.
- 2.
- 3.
- The function $f\in C\left(\right[0,T]\times \mathbb{R})$ is such that $f(.,x(.))\in {I}^{\alpha ,\rho}[0,T]$, where $x\left(t\right)$ is the solution from condition 2, and there exists a constant $L>0$, such that, for $t\in [0,T],{x}_{j}\in \mathbb{R},\phantom{\rule{4pt}{0ex}}j=1,2,$, the inequality $|f(t,{x}_{1})-f(t,{x}_{2})|\le L|{x}_{1}-{x}_{2}|$ holds.
- 4.
- The inequality $L{T}^{\alpha}\left(1+\frac{\left|b\right|}{\left|a+b{e}^{\frac{\rho -1}{\rho}T}\right|}\right)<{\rho}^{\alpha}\mathsf{\Gamma}(1+\alpha )$ holds.
- 5.
- For any $\epsilon >0$, the inequality (22) has at least one solution from ${C}^{\alpha ,\rho}[0,T]$.

#### 3.5. Ulam Type Stability for BVP for Caputo Fractional Differential Equations

**Theorem 8.**

- 1.
- The condition (A) is satisfied with $\rho =1$, and there exists a constant $L>0$, such that, for $t\in [0,T],{x}_{j}\in \mathbb{R},\phantom{\rule{4pt}{0ex}}j=1,2,$ the inequality $|f(t,{x}_{1})-f(t,{x}_{2})|\le L|{x}_{1}-{x}_{2}|$ holds.
- 2.
- The inequalities $a+b\ne 0$ and $L{T}^{\alpha}\left(1+\frac{\left|b\right|}{\left|a+b\right|}\right)<\mathsf{\Gamma}(1+\alpha )$ hold.
- 3.
- For any $\epsilon >0$, the inequality (38) has at least one solution.

**Theorem 9.**

- 1.
- The inequality $a+b\ne 0$ holds, and $\alpha \in (0,1)$ and rgb]0,0,0 μ are given.
- 2.
- 3.
- The function $f\in C\left(\right[0,T]\times \mathbb{R})$ is such that $f(.,x(.))\in {I}^{\alpha ,1}[0,T]$, where $x\left(t\right)$ is the solution from condition 2, and there exists a constant $L>0$, such that, for $t\in [0,T],{x}_{j}\in \mathbb{R},\phantom{\rule{4pt}{0ex}}j=1,2,$, the inequality $|f(t,{x}_{1})-f(t,{x}_{2})|\le L|{x}_{1}-{x}_{2}|$ holds.
- 4.
- The inequality $L{T}^{\alpha}\left(1+\frac{\left|b\right|}{\left|a+b\right|}\right)<\mathsf{\Gamma}(1+\alpha )$ holds.
- 5.
- For any $\epsilon >0$, the inequality (38) has at least one solution.

#### 3.6. Examples

**Example 2.**

**Case 1.**(IVP). Consider the initial condition

**Case 1.1.**We will look at Definition 6 with $\rho =1$.

**Case 1.2.**We will look at Definition 7 with $\rho =1$.

**Case 2.**We will consider the linear Caputo fractional differential equation ($\alpha \in (0,1)$) (39) with a boundary condition.

**Case 2.1.**We will look at Definition 9 with $\rho =1$.

**Case 2.2.**We will look at Definition 10 with $\rho =1$.

**Example 3.**

**Remark 13.**

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Agarwal, R.P.; Hristova, S.; O’Regan, D.
Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study. *Axioms* **2023**, *12*, 226.
https://doi.org/10.3390/axioms12030226

**AMA Style**

Agarwal RP, Hristova S, O’Regan D.
Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study. *Axioms*. 2023; 12(3):226.
https://doi.org/10.3390/axioms12030226

**Chicago/Turabian Style**

Agarwal, Ravi P., Snezhana Hristova, and Donal O’Regan.
2023. "Boundary Value Problems for Fractional Differential Equations of Caputo Type and Ulam Type Stability: Basic Concepts and Study" *Axioms* 12, no. 3: 226.
https://doi.org/10.3390/axioms12030226