#
Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with ϕ_{c}-Laplacian

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

**Lemma**

**1.**

- (${a}_{1}$)
- $\frac{\underset{\Phi \left(x\right)\le r}{sup}\Psi \left(x\right)}{r}<\frac{\Psi \left(\overline{x}\right)}{\Phi \left(\overline{x}\right)};$
- (${a}_{2}$)
- for each $\lambda \in {\Lambda}_{r}:=\left(\right)open="("\; close=")">\frac{\Phi \left(\overline{x}\right)}{\Psi \left(\overline{x}\right)},\frac{r}{\underset{\Phi \left(x\right)\le r}{sup}\Psi \left(x\right)}$ the functional $\Phi -\lambda \Psi $ is coercive.

**Lemma**

**2.**

## 2. Preliminaries

**Lemma**

**3.**

**Lemma**

**4.**

**Lemma**

**5.**

## 3. Main Results

**Theorem**

**1.**

- (i)
- $f(k,\xi )>0$ for each $k\in \mathbb{Z}(1,N)$ and $\xi \in [-c,c]$;
- (ii)
- $\frac{{F}_{d}}{2(\sqrt{1+{d}^{2}}-1)}>\frac{{F}_{c}}{-1+\sqrt{1+\frac{4{c}^{2}}{N+1}}};$
- (iii)
- $\underset{\left|\xi \right|\to +\infty}{lim\; sup}\frac{F(k,\xi )}{\left|\xi \right|}<\frac{2{F}_{c}}{N(\sqrt{4{c}^{2}+N+1}-\sqrt{N+1})}$.

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

- (i)
- $f(k,\xi )>0$ for each $k\in \mathbb{Z}(1,N)$ and $\xi \in [0,c]$;
- (ii)
- $\frac{{F}_{d}}{2(\sqrt{1+{d}^{2}}-1)}>\frac{{F}_{c}}{-1+\sqrt{1+\frac{4{c}^{2}}{N+1}}};$
- (iii)
- $\underset{\xi \to +\infty}{lim\; sup}\frac{F(k,\xi )}{\xi}<\frac{2{F}_{c}}{N(\sqrt{4{c}^{2}+N+1}-\sqrt{N+1})}$.

**Proof.**

**Theorem**

**2.**

- (${T}_{1}$)
- $f(k,\xi )>0$ for each $k\in \mathbb{Z}(1,N)$ and $\xi \in [-c,c]$;
- (${T}_{2}$)
- $max\left(\right)open="\{"\; close="\}">\frac{2(\sqrt{1+{d}^{2}}-1)}{{F}_{d}},\frac{2\sqrt{N(N+1)}sin\frac{N\pi}{2(N+1)}}{\beta}$
- (${T}_{3}$)
- there exist a positive constant β such that$$\underset{\left|t\right|\to +\infty}{lim\; inf}\frac{F(k,t)}{\left|t\right|}>\beta $$for each $k\in \mathbb{Z}(1,N).$

**Proof.**

## 4. Examples

**Example**

**1.**

**Example**

**2.**

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Chen, Y.; Zhou, Z.
Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with *ϕ _{c}*-Laplacian.

*Symmetry*

**2020**,

*12*, 1839. https://doi.org/10.3390/sym12111839

**AMA Style**

Chen Y, Zhou Z.
Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with *ϕ _{c}*-Laplacian.

*Symmetry*. 2020; 12(11):1839. https://doi.org/10.3390/sym12111839

**Chicago/Turabian Style**

Chen, Yanshan, and Zhan Zhou.
2020. "Existence of Three Solutions for a Nonlinear Discrete Boundary Value Problem with *ϕ _{c}*-Laplacian"

*Symmetry*12, no. 11: 1839. https://doi.org/10.3390/sym12111839