# Characterizations of Well-Posedness for Generalized Hemivariational Inequalities Systems with Derived Inclusion Problems Systems in Banach Spaces

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

**Definition**

**1.**

**Definition**

**2.**

**Proposition**

**1.**

**Definition**

**3.**

**Definition**

**4**

**.**Let S be a nonempty set in E. The measure of noncompactness (MNC) μ of the set S is formulated below

**Definition**

**5**

**.**Let $\mathcal{H}(\xb7,\xb7)$ be the Hausdorff metric on the collection $CB\left({E}^{*}\right)$ of all nonempty, closed and bounded subsets of ${E}^{*}$, formulated below

**Theorem**

**1**

**.**Suppose that C is nonempty, closed and convex in E and ${C}^{*}$ is nonempty, closed, convex and bounded in ${E}^{*}$. Let $\phi :E\to \mathbf{R}$ be a proper convex l.s.c. functional and $\upsilon \in C$ be arbitrary. Assume that $\forall u\in C$, $\exists {u}^{*}\left(u\right)\in {C}^{*}$ s.t.

## 3. Metric Characterizations of Well-Posedness for SGHVIs

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

**(HA)**: (a) ${A}_{1}:{V}_{1}\times {V}_{2}\to {2}^{{V}_{1}^{*}}$ is monotone in the first variable, i.e., $\forall {u}_{1},{v}_{1}\in {V}_{1}$ and ${u}_{2}\in {V}_{2}$,

**(HJ)**: (a) $J:{V}_{1}\times {V}_{2}\to \mathbf{R}$ is locally Lipschitz with respect to the first variable and second variable on ${V}_{1}\times {V}_{2}$;

**Lemma**

**1**

**.**Suppose that the functional $J:{V}_{1}\times {V}_{2}\to \mathbf{R}$ satisfies the hypotheses (a), (b) in

**(HJ)**. Then, for any sequence ${\mathbf{u}}^{n}=({u}_{1}^{n},{u}_{2}^{n})\in \mathcal{V}$ strongly converging towards $\mathbf{u}=({u}_{1},{u}_{2})\in \mathcal{V}$ and ${v}_{k}^{n}\in {V}_{k}$ strongly converging towards ${v}_{k}\in {V}_{k}$, one has

**Proposition**

**2.**

**(HA)**and $J:{V}_{1}\times {V}_{2}\to \mathbf{R}$ satisfies the hypothesis

**(HJ)**. Then, ${\Omega}_{\alpha}\left(\u03f5\right)={\Delta}_{\alpha}\left(\u03f5\right)\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\forall \u03f5>0$.

**Proof.**

**Lemma**

**2.**

**(HA)**, and $J:{V}_{1}\times {V}_{2}\to \mathbf{R}$ satisfies the hypothesis

**(HJ)**. Then, for any $\u03f5>0,\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}{\Omega}_{\alpha}\left(\u03f5\right)={\Delta}_{\alpha}\left(\u03f5\right)$ is closed in $\mathcal{V}={V}_{1}\times {V}_{2}$.

**Proof.**

**(HJ)**on the functional J, Lemma 1 ensures that

**Theorem**

**2.**

**(HA)**, ${A}_{2}:{V}_{1}\times {V}_{2}\to {2}^{{V}_{2}^{*}}$ satisfy the hypothesis (e) in

**(HA)**, and $J:{V}_{1}\times {V}_{2}\to \mathbf{R}$ satisfy the hypothesis

**(HJ)**. Then, the SGHVI is strongly α-well-posed if and only if

**Proof.**

**(HJ)**, and ${\alpha}_{1}$ and ${\alpha}_{2}$ are continuous, so we can obtain by similar arguments to those in (7)–(9) that

**Theorem**

**3.**

**(HA)**and $J:{V}_{1}\times {V}_{2}\to \mathbf{R}$ satisfy the hypothesis

**(HJ)**. Then, the SGHVI is strongly α-well-posed in the generalized sense if and only if

**Proof.**

## 4. Equivalence for Well-Posedness of the SGHVI and SDIP

**Definition**

**9.**

**Definition**

**10.**

**Definition**

**11.**

**Lemma**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

**Theorem**

**5.**

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**MDPI and ACS Style**

Ceng, L.-C.; Li, J.-Y.; Wang, C.-S.; Zhang, F.-F.; Hu, H.-Y.; Cui, Y.-L.; He, L.
Characterizations of Well-Posedness for Generalized Hemivariational Inequalities Systems with Derived Inclusion Problems Systems in Banach Spaces. *Symmetry* **2022**, *14*, 1341.
https://doi.org/10.3390/sym14071341

**AMA Style**

Ceng L-C, Li J-Y, Wang C-S, Zhang F-F, Hu H-Y, Cui Y-L, He L.
Characterizations of Well-Posedness for Generalized Hemivariational Inequalities Systems with Derived Inclusion Problems Systems in Banach Spaces. *Symmetry*. 2022; 14(7):1341.
https://doi.org/10.3390/sym14071341

**Chicago/Turabian Style**

Ceng, Lu-Chuan, Jian-Ye Li, Cong-Shan Wang, Fang-Fei Zhang, Hui-Ying Hu, Yun-Ling Cui, and Long He.
2022. "Characterizations of Well-Posedness for Generalized Hemivariational Inequalities Systems with Derived Inclusion Problems Systems in Banach Spaces" *Symmetry* 14, no. 7: 1341.
https://doi.org/10.3390/sym14071341