# Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator

^{*}

## Abstract

**:**

## 1. Introduction

- (i)
- If $\lambda >8{sin}^{2}\frac{m\pi}{T}$ with$0\le m\le \left\{\begin{array}{c}(T-1)/2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}if\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}is\phantom{\rule{4pt}{0ex}}odd\hfill \\ (T-2)/2\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}if\phantom{\rule{4pt}{0ex}}T\phantom{\rule{4pt}{0ex}}is\phantom{\rule{4pt}{0ex}}even\hfill \end{array}\right.,$then problem (3) has at least $2m+1$ distinct pairs of nontrivial solutions.
- (ii)
- If T is even and $\lambda >8$, then (3) has at least T distinct pairs of nontrivial solutions.

**Theorem 1.**

## 2. Variational Approach and Preliminaries

**Proposition 1.**

**Proof.**

**Proposition 2.**

**Proof.**

**Remark 1.**

- $\underline{T\mathit{odd}}:$
- $${\lambda}_{0}=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{2.em}{0ex}}{\lambda}_{2k-1}={\lambda}_{2k}=4{sin}^{2}\frac{k\pi}{T},\phantom{\rule{1.em}{0ex}}k=1,\dots ,\frac{T-1}{2};$$
- $\underline{T\mathit{even}}:$
- $${\lambda}_{0}=0,\phantom{\rule{4pt}{0ex}}\phantom{\rule{2.em}{0ex}}{\lambda}_{2k-1}={\lambda}_{2k}=4{sin}^{2}\frac{k\pi}{T},\phantom{\rule{1.em}{0ex}}k=1,\dots ,\frac{T-2}{2},\phantom{\rule{4pt}{0ex}}\phantom{\rule{2.em}{0ex}}{\lambda}_{T-1}=4.$$

## 3. Main Result

**Theorem 2.**

- (i)
- If$$\lambda >8{sin}^{2}\frac{m\pi}{T}\phantom{\rule{4pt}{0ex}}(=2{\lambda}_{2m})\phantom{\rule{4pt}{0ex}}with0\le m\le \left\{\begin{array}{c}(T-1)/2\phantom{\rule{4pt}{0ex}}ifTisodd\hfill \\ (T-2)/2\phantom{\rule{4pt}{0ex}}ifTiseven\hfill \end{array}\right.,$$
- (ii)
- If T is even and$$\lambda >8\phantom{\rule{4pt}{0ex}}(=2{\lambda}_{T-1}),$$

**Proof.**

**Example 1.**

**Corollary 1.**

**Proof.**

**Corollary 2.**

- (i)
- If$$\underset{x\to 0}{lim\; inf}\frac{F\left(x\right)}{{x}^{2}}>4{sin}^{2}\frac{m\pi}{T}\phantom{\rule{4pt}{0ex}}with0\le m\le \left\{\begin{array}{c}(T-1)/2\phantom{\rule{4pt}{0ex}}ifTisodd\hfill \\ (T-2)/2\phantom{\rule{4pt}{0ex}}ifTiseven\hfill \end{array}\right.,$$
- (ii)
- If T is even and$$\underset{x\to 0}{lim\; inf}\frac{F\left(x\right)}{{x}^{2}}>4,$$

**Proof.**

**Example 2.**

**Remark 2.**

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Jebelean, P.; Şerban, C.
Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator. *Mathematics* **2022**, *10*, 1595.
https://doi.org/10.3390/math10091595

**AMA Style**

Jebelean P, Şerban C.
Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator. *Mathematics*. 2022; 10(9):1595.
https://doi.org/10.3390/math10091595

**Chicago/Turabian Style**

Jebelean, Petru, and Călin Şerban.
2022. "Multiple Periodic Solutions for Odd Perturbations of the Discrete Relativistic Operator" *Mathematics* 10, no. 9: 1595.
https://doi.org/10.3390/math10091595