# On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1**

**Definition**

**2**

**Lemma**

**1**

**Lemma**

**2**

**Definition**

**3**

- (i)
- $ImL$ is a closed subset of Z;
- (ii)
- $dimkerL=codimImL<+\infty $.

**Theorem**

**1**

- (1)
- $Lx\ne \lambda Nx$ for each $(x,\lambda )\in \left(\right)open="["\; close="]">(domL\setminus kerL)\cap \partial \Omega $;
- (2)
- $Nx\notin ImL$ for each $x\in kerL\cap \partial \Omega $; and
- (3)
- $deg(JQN{|}_{kerL},\phantom{\rule{0.166667em}{0ex}}\Omega \cap kerL,\phantom{\rule{0.166667em}{0ex}}0)\ne 0$, where $Q:Z\to Z$ is a continuous projection as above with $ImL=kerQ$ and $J:ImQ\to kerL$ is any isomorphism.

## 3. Main Results

**Lemma**

**3.**

**Proof.**

**Lemma**

**4.**

**Proof.**

**Lemma**

**5.**

**Lemma**

**6.**

**Proof.**

**Case**

**1.**

**Case**

**2.**

**Case**

**3.**

**Case**

**4.**

**Lemma**

**7.**

**Proof.**

**Lemma**

**8.**

**Proof.**

**Theorem**

**2.**

**Proof.**

- (1)
- $L(u,v)\ne \lambda N(u,v)$, for every $(u,v,\lambda )\in \left[\right(\mathrm{dom}L\setminus \mathrm{Ker}L)\cap \partial \Omega ]\times (0,1)$.
- (2)
- $N(u,v)\notin \mathrm{Im}L$ for every $u\in \mathrm{Ker}L\cap \partial \Omega $.
- (3)
- Let $H\left(\right)open="("\; close=")">(u,v),\lambda $. Here, we let I and the isomorphism $J:ImQ\to kerL$, which are both identical operators. Via the homotopy property of degree, we obtain that$$\begin{array}{c}\hfill deg\left(\right)open="("\; close=")">{JQN|}_{kerL},\Omega \cap kerL,0\phantom{\rule{1.em}{0ex}}\\ =deg\left(\right)open="("\; close=")">H(\xb7,0),\Omega \cap kerL,0\hfill \end{array}\hfill \phantom{\rule{1.em}{0ex}}& =deg\left(\right)open="("\; close=")">H(\xb7,1),\Omega \cap kerL,0\hfill $$

## 4. Example

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Hu, L.; Han, Y.; Zhang, S.
On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions. *Axioms* **2021**, *10*, 103.
https://doi.org/10.3390/axioms10020103

**AMA Style**

Hu L, Han Y, Zhang S.
On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions. *Axioms*. 2021; 10(2):103.
https://doi.org/10.3390/axioms10020103

**Chicago/Turabian Style**

Hu, Lei, Yaozhen Han, and Shuqin Zhang.
2021. "On the Existence of Coupled Fractional Jerk Equations with Multi-Point Boundary Conditions" *Axioms* 10, no. 2: 103.
https://doi.org/10.3390/axioms10020103