# Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

- (a)
- demicontinuous if ${u}_{n}\to u$ in H implies $A{u}_{n}\rightharpoonup Au$ in H;
- (b)
- strongly monotone if there exists constant ${m}_{A}>0$ such that$${(Au-Av,u-v)}_{H}\ge {m}_{A}{\parallel u-v\parallel}_{H}^{2}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in H;$$
- (c)
- Lipschitz continuous if there exists constant ${L}_{A}>0$ such that$${\parallel Au-Av\parallel}_{H}\le {L}_{A}{\parallel u-v\parallel}_{H}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in H.$$

**Definition**

**2.**

- (a)
- nonexpansive on K if there exists a constant ${k}_{A}\in [0,1]$ such that$${\parallel Au-Av\parallel}_{H}\le {k}_{A}{\parallel u-v\parallel}_{H}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u,v\in K;$$
- (b)
- a contraction if it is nenexpansive on K with constant ${k}_{A}\in [0,1)$.

**Theorem**

**1.**

**Definition**

**3.**

- (a)
- monotone if$${({u}_{1}^{*}-{u}_{2}^{*},{u}_{1}-{u}_{2})}_{H}\ge 0\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}({u}_{1},{u}_{1}^{*}),({u}_{2},{u}_{2}^{*})\in Gr(T);$$
- (b)
- relaxed monotone if there exists constant ${\alpha}_{T}>0$ such that$${({u}_{1}^{*}-{u}_{2}^{*},{u}_{1}-{u}_{2})}_{H}\ge -{\alpha}_{T}{\parallel {u}_{1}-{u}_{2}\parallel}_{H}^{2}\phantom{\rule{1.em}{0ex}}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}({u}_{1},{u}_{1}^{*}),({u}_{2},{u}_{2}^{*})\in Gr(T);$$
- (c)
- maximal monotone if it is monotone and, for any $v,\phantom{\rule{0.166667em}{0ex}}{v}^{*}\in H$, the following implication holds:$${({u}^{*}-{v}^{*},u-v)}_{H}\ge 0\phantom{\rule{4pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\forall \phantom{\rule{0.166667em}{0ex}}u\in D(T),\phantom{\rule{4pt}{0ex}}{u}^{*}\in Tu\phantom{\rule{4pt}{0ex}}\u27f9\phantom{\rule{4pt}{0ex}}v\in D(T)\phantom{\rule{4pt}{0ex}}\mathrm{and}\phantom{\rule{4pt}{0ex}}{v}^{*}\in Tv.$$

**Theorem**

**2.**

**Proposition**

**1.**

**Proposition**

**2.**

**Definition**

**4.**

**Proposition**

**3.**

- (a)
- $\partial j(u)$ is a nonempty convex and $weakly$ compact subset of H, for all $u\in H$;
- (b)
- the graph of the Clarke subdifferential $\partial j$ is closed in $H\times {H}_{w}$ topology;
- (c)
- for all $u,v\in H$, one has$${j}^{0}(u;v)=max\{\phantom{\rule{0.166667em}{0ex}}{(\xi ,v)}_{H}\mid \xi \in \partial j(u)\phantom{\rule{0.166667em}{0ex}}\}.$$

**Definition**

**5.**

**Proposition**

**4.**

**Proposition**

**5.**

**Proposition**

**6.**

**Definition**

**6.**

**Theorem**

**3.**

## 3. A Parametric Variational–Hemivariational Inequality

**Remark**

**1.**

**Proposition**

**7.**

**Proof.**

**Proposition**

**8.**

**Proof.**

**Proposition**

**9.**

**Proof.**

**Corollary**

**1.**

**Proposition**

**10.**

**Proof.**

**Theorem**

**4.**

**Proof.**

## 4. An First Elliptic Variational–Hemivariational Inequaliy

**Theorem**

**5.**

**Proof.**

**Example**

**1.**

**Example**

**2.**

## 5. An Second Elliptic Variational–Hemivariational Inequaliy

**Theorem**

**6.**

**Proof.**

**Example**

**3.**

## 6. A History-Dependent Variational–Hemivariational Inequaliy

**Theorem**

**7.**

**Proof.**

## 7. Conclusions

## Funding

## Conflicts of Interest

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**MDPI and ACS Style**

Sofonea, M.
Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces. *Axioms* **2022**, *11*, 136.
https://doi.org/10.3390/axioms11030136

**AMA Style**

Sofonea M.
Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces. *Axioms*. 2022; 11(3):136.
https://doi.org/10.3390/axioms11030136

**Chicago/Turabian Style**

Sofonea, Mircea.
2022. "Monotonicity Arguments for Variational–Hemivariational Inequalities in Hilbert Spaces" *Axioms* 11, no. 3: 136.
https://doi.org/10.3390/axioms11030136