# Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Material

**Nonlinear operators and the KKM lemma.**Let X be a reflexive Banach space and $T:X\to {2}^{{X}^{*}}$ be a multivalued mapping. A mapping $T:X\to {2}^{{X}^{*}}$ is called bounded if the image of each bounded set in X remains in a bounded subset of ${X}^{*}$. A mapping $T:X\to {2}^{{X}^{*}}$ is called pseudomonotone, provided the following conditions are satisfied

- (i)
- T has nonempty, bounded, closed and convex values.
- (ii)
- T is upper semicontinuous (u.s.c.) from each finite dimensional subspace of X into ${X}^{*}$ equipped with its weak topology.
- (iii)
- if ${u}_{n}\in X$, ${u}_{n}\to u$ weakly in X, ${u}_{n}^{*}\in T{u}_{n}$ and $lim\; sup\phantom{\rule{0.166667em}{0ex}}{\langle {u}_{n}^{*},{u}_{n}-u\rangle}_{{X}^{*}\times X}\le 0$, then to each $y\in X$, there exists ${u}^{*}\left(y\right)\in Tu$ such that $\langle {u}^{*}\left(y\right),u-y\rangle}_{{X}^{*}\times X}\le lim\; inf\phantom{\rule{0.166667em}{0ex}}{\langle {u}_{n}^{*},{u}_{n}-y\rangle}_{{X}^{*}\times X$.

**Lemma**

**1.**

- (a)
- for every $x\in X$, $F\left(x\right)$ is a closed set in Y,
- (b)
- convex hull of any finite set $\{{x}_{1},\dots ,{x}_{r}\}$ of X is contained in ${\bigcup}_{i=1}^{r}F\left({x}_{i}\right)$,
- (c)
- $F\left(x\right)$ is a compact set at least for one $x\in X$.

**The Clarke generalized subgradient and tangent cones.**Let X be a Banach space, $h:X\to \mathbb{R}$ be a locally Lipschitz function, and x, $v\in X$. The Clarke generalized directional derivative of h at x in the direction v is given by

**Lemma**

**2.**

## 3. Formulation of the Problem

**Problem**

**1.**

- H(A):
- $A:V\to {V}^{*}$ is a mapping such that
- (a)
- A is pseudomonotone,
- (b)
- A is strongly monotone with constant ${m}_{A}>0$, i.e.,

$$\langle A{v}_{1}-A{v}_{2},{v}_{1}-{v}_{2}\rangle \ge {m}_{A}{\parallel {v}_{1}-{v}_{2}\parallel}_{V}^{2}\phantom{\rule{4pt}{0ex}}\mathrm{for}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{all}\phantom{\rule{4pt}{0ex}}{v}_{1},{v}_{2}\in V.$$ - H(j):
- $j:V\to \mathbb{R}$ is a mapping such that
- (a)
- j is locally Lipschitz,
- (b)
- ${\parallel \partial j\left(v\right)\parallel}_{{V}^{*}}\le {c}_{0}+{c}_{1}\phantom{\rule{0.166667em}{0ex}}{\parallel v\parallel}_{V}\phantom{\rule{4pt}{0ex}}\mathrm{for}\mathrm{all}\phantom{\rule{4pt}{0ex}}v\in V\phantom{\rule{4pt}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}{c}_{0},{c}_{1}\ge 0$,
- (c)
- $\mathrm{there}\mathrm{exist}\phantom{\rule{4pt}{0ex}}{\alpha}_{j}\ge 0\phantom{\rule{4pt}{0ex}}\mathrm{such}\mathrm{that}$

$${j}^{0}({v}_{1};{v}_{2}-{v}_{1})+{j}^{0}({v}_{2};{v}_{1}-{v}_{2})\le {\alpha}_{j}\phantom{\rule{0.166667em}{0ex}}{\parallel {v}_{1}-{v}_{2}\parallel}_{V}^{2}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}\mathrm{for}\mathrm{all}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4pt}{0ex}}{v}_{1},{v}_{2}\in V.$$ - H(φ):
- $\phi :V\to \mathbb{R}$ is a convex and lower semicontinuous function.
- H(K):
- K is a nonempty, closed and star-shaped in V.
- H(f):
- $f\in {V}^{*}$.

**Theorem**

**1.**

**Remark**

**1.**

## 4. Proof of Theorem 1

**Problem**

**2.**

**Lemma**

**3.**

**Proof.**

- (a)
- for every $v\in V$, the set $F\left(v\right)$ is closed in V,
- (b)
- for any finite set $\{{v}_{1},\dots ,{v}_{r}\}\subset V$, we have $\mathrm{co}\{{v}_{1},\dots ,{v}_{r}\}\subset {\bigcup}_{i=1}^{r}F\left({v}_{i}\right),$
- (c)
- there is ${v}_{0}\in V$ such that $F\left({v}_{0}\right)$ is compact in V.

## 5. Second Existence Result

- H(A)
_{1}: - $A:V\to {V}^{*}$ is a mapping such that
- (a)
- for all $v\in V$, $V\ni u\mapsto \langle Au,v-u\rangle \in \mathbb{R}$ is weakly upper semicontinuous,
- (b)
- for any $v\in V$, there exists ${m}_{v}>0$ such that $\langle Au,u-v\rangle \ge {\alpha}_{A}{\parallel u\parallel}^{2}$ for

all $\parallel u\parallel \ge {m}_{v}$, where ${\alpha}_{A}>0$. - H(j)
_{1}: - $j:V\to \mathbb{R}$ is a mapping such that
- (a)
- j is locally Lipschitz,
- (b)
- ${\parallel \partial j\left(v\right)\parallel}_{{V}^{*}}\le {c}_{0}+{c}_{1}\phantom{\rule{0.166667em}{0ex}}{\parallel v\parallel}_{V}\phantom{\rule{4pt}{0ex}}\mathrm{for}\mathrm{all}\phantom{\rule{4pt}{0ex}}v\in V\phantom{\rule{4pt}{0ex}}\mathrm{with}\phantom{\rule{4pt}{0ex}}{c}_{0},{c}_{1}\ge 0$,
- (c)
- $lim\; sup{j}^{0}({u}_{n};v-{u}_{n})\le {j}^{0}(u;v-u)$ for all $v\in V$ and ${u}_{n}\to u$ weakly in V.

**Theorem**

**2.**

**Proof.**

**Corollary**

**1.**

## 6. Semipermeability Model

**Problem**

**3.**

- H(A)
_{2}: - $A:V\to {V}^{*}$ is a mapping such that $A=-{\sum}_{i,j=1}^{d}{D}_{i}\left({a}_{ij}\left(x\right){D}_{j}\right)$, and
- (i)
- ${a}_{ij}\in {L}^{\infty}(\Omega )$ for i, $j=1,\dots ,d$.
- (ii)
- ${\sum}_{i,j=1}^{d}{a}_{ij}\left(x\right){\xi}_{i}{\xi}_{j}\ge {\alpha}_{0}{\parallel \xi \parallel}^{2}$ for all $\xi \in {\mathbb{R}}^{d}$, a.e. $x\in \Omega $ with ${\alpha}_{0}>0$.

- H(j
_{1}): - ${j}_{1}:\mathbb{R}\to \mathbb{R}$ is such that
- (i)
- ${j}_{1}$ is locally Lipschitz.
- (ii)
- $|\partial {j}_{1}\left(r\right)|\le {c}_{0j}+{c}_{1j}\left|r\right|$ for all $r\in \mathbb{R}$ with ${c}_{0j}$, ${c}_{1j}\ge 0$.
- (iii)
- $(\partial {j}_{1}\left({r}_{1}\right)-\partial {j}_{1}\left({r}_{2}\right))({r}_{1}-{r}_{2})\ge -{\beta}_{1j}{|{r}_{1}-{r}_{2}|}^{2}$ all ${r}_{1}$, ${r}_{2}\in \mathbb{R}$ with ${\beta}_{1j}\ge 0$.

- H(j
_{2}): - ${j}_{2}:\mathbb{R}\to \mathbb{R}$ is such that
- (i)
- ${j}_{2}$ is locally Lipschitz.
- (ii)
- $|\partial {j}_{2}\left(r\right)|\le {c}_{2j}+{c}_{3j}\left|r\right|$ for all $r\in \mathbb{R}$ with ${c}_{2j}$, ${c}_{3j}\ge 0$.
- (iii)
- $(\partial {j}_{2}\left({r}_{1}\right)-\partial {j}_{2}\left({r}_{2}\right))({r}_{1}-{r}_{2})\ge -{\beta}_{2j}{|{r}_{1}-{r}_{2}|}^{2}$ all ${r}_{1}$, ${r}_{2}\in \mathbb{R}$ with ${\beta}_{2j}\ge 0$.

- H(g
_{1}): - ${g}_{1}:\mathbb{R}\to \mathbb{R}$ is such that
- (i)
- ${g}_{1}$ is convex and l.s.c.
- (ii)
- $|\partial {g}_{1}\left(r\right)|\le {c}_{0g}+{c}_{1g}\left|r\right|$ for all $r\in \mathbb{R}$ with ${c}_{0g}$, ${c}_{1g}\ge 0$.

- H(g
_{2}): - ${g}_{2}:\times \mathbb{R}\to \mathbb{R}$ is such that
- (i)
- ${g}_{2}$ is convex and l.s.c.
- (ii)
- $|\partial {g}_{2}\left(r\right)|\le {c}_{2g}+{c}_{3g}\left|r\right|$ for all $r\in \mathbb{R}$ with ${c}_{2g}$, ${c}_{3g}\ge 0$.

- H(f):
- ${f}_{1}\in {L}^{2}(\Omega ),\phantom{\rule{4pt}{0ex}}{f}_{2}\in {L}^{2}\left({\Gamma}_{2}\right),\phantom{\rule{4pt}{0ex}}{f}_{b}\in {L}^{2}\left({\Gamma}_{3}\right).$
- (H
_{0}): - ${\alpha}_{0}>{\beta}_{1j}{\parallel i\parallel}^{2}+{\beta}_{2j}{\parallel \gamma \parallel}^{2}.$

**Problem**

**4.**

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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Migórski, S.; Fengzhen, L. Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets. *Mathematics* **2020**, *8*, 1824.
https://doi.org/10.3390/math8101824

**AMA Style**

Migórski S, Fengzhen L. Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets. *Mathematics*. 2020; 8(10):1824.
https://doi.org/10.3390/math8101824

**Chicago/Turabian Style**

Migórski, Stanisław, and Long Fengzhen. 2020. "Constrained Variational-Hemivariational Inequalities on Nonconvex Star-Shaped Sets" *Mathematics* 8, no. 10: 1824.
https://doi.org/10.3390/math8101824