# Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with p-Laplacian Operator

^{1}

^{2}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition 1.**

**Definition 2.**

**Lemma 1.**

**Lemma 2.**

**Lemma 3**

- (i)
- $\left\{u\in P(\theta ,b,d)|\theta (u)>b\right\}\ne \xd8$ and $\theta (\Psi u)>b$ for $u\in P(\theta ,b,d)$, where $P(\theta ,b,d)=\left\{u\in P|\theta (u)\ge b\phantom{\rule{4pt}{0ex}}and\phantom{\rule{4pt}{0ex}}\u2225u\u2225\le d\right\}$.
- (ii)
- $\u2225\Psi u\u2225<a$ for $u\in {P}_{a}$.
- (iii)
- $\theta (\Psi u)>b$ for $u\in P(\theta ,b,c)$ with $\u2225\Psi u\u2225>d$.

**H1.**

- A priori bound

**Lemma 4.**

**Proof.**

## 3. Main Results

**Lemma 5.**

**Proof.**

**H2.**

**H3.**

**Lemma 6.**

**Proof.**

**Theorem 1.**

**Proof.**

**Remark 1.**

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Shen, T.
Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with *p*-Laplacian Operator. *Fractal Fract.* **2023**, *7*, 427.
https://doi.org/10.3390/fractalfract7060427

**AMA Style**

Shen T.
Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with *p*-Laplacian Operator. *Fractal and Fractional*. 2023; 7(6):427.
https://doi.org/10.3390/fractalfract7060427

**Chicago/Turabian Style**

Shen, Tengfei.
2023. "Multiplicity of Positive Solutions to Hadamard-Type Fractional Relativistic Oscillator Equation with *p*-Laplacian Operator" *Fractal and Fractional* 7, no. 6: 427.
https://doi.org/10.3390/fractalfract7060427