# On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

**maximal element**of T, if $T\left(x\right)=\varnothing $.

**H-space**is a topological space X, together with a family {$\Gamma $${}_{D}$} of some nonempty contractible subsets of X indexed by $D\in \langle X\rangle $ such that ${\Gamma}_{D}$ ⊂ ${\Gamma}_{{D}^{\prime}}$ whenever $D\subset {D}^{\prime}$. The notion of H-space was introduced in 1988 by Bardaro and Ceppitelli [22]. Since then, there have appeared numerous applications and generalizations in the literature [8,16,18,19,23,25,26]. Given an H-space (X, {$\Gamma $${}_{D}$}), a nonempty subset C of X is said to be

**H-convex**if $\Gamma $${}_{D}$ ⊂ C for all $D\in \langle C\rangle $. For a nonempty subset C of X, we define the

**H-convex hull**of C as $H\text{-}coC:=\bigcap \left\{W\right|\phantom{\rule{4pt}{0ex}}W\phantom{\rule{4.pt}{0ex}}\mathrm{is}\phantom{\rule{4.pt}{0ex}}H\text{-}\mathit{convex}\phantom{\rule{4.pt}{0ex}}\mathrm{in}\phantom{\rule{4.pt}{0ex}}X\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}C\subset W\}.$ Moreover, for any $D\in \langle X\rangle $, $H\text{-}coD$ is called a

**polytope**. We say that X is an

**H-space with covering polytopes**, if for any subset C of X, and $y\in H\text{-}coC$, there is a $D\in \langle C\rangle $ such that the polytope $H\text{-}coD$ contains y. For example, a locally convex topological vector space X is an H-space with covering polytopes, by setting $\Gamma $${}_{D}$ =$coD$ for all $D\in \langle X\rangle $.

- (1)
- T is said to be
**of class${\mathcal{L}}_{\phi}$**, if- (a)
- for each $x\in X$, $\phi \left(x\right)\notin H$-$coT\left(x\right)$,
- (b)
- for each $y\in Y$, ${T}^{-1}\left(y\right)$ is open in X.

- (2)
- A set-valued mapping ${T}_{x}:X\u27f6{2}^{Y}$ is an
**${\mathcal{L}}_{\phi}$-majorant of T at x**, if there exists an open neighborhood ${N}_{x}$ of x in X such that- (a)
- for each $z\in X$, $T\left(z\right)\subset {T}_{x}\left(z\right)$,
- (b)
- for each $z\in {N}_{x}$, $\phi \left(z\right)\notin H$-$co{T}_{x}\left(z\right)$,
- (c)
- for each $y\in Y$, ${T}_{x}^{-1}\left(y\right)$ is open in X.

- (3)
- T is said to be
**${\mathcal{L}}_{\phi}$-majorized**if, for each $x\in X$ with $T\left(x\right)\ne \varnothing $, there exists an ${\mathcal{L}}_{\phi}$-majorant of T at x.

**Remark**

**1.**

## 3. Results on Maximal Elements

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

**Theorem**

**1.**

**Proof.**

**Remark**

**2.**

**Theorem**

**2.**

**Proof.**

- (a)
- $T\left(z\right)\subset {T}_{x}\left(z\right)$ for all $z\in X$;
- (b)
- $z\notin H$-$co{T}_{x}\left(z\right)$ for all $z\in {N}_{x}$;
- (c)
- ${T}_{x}^{-1}\left(y\right)$ is open in C for each $y\in C$.

## 4. Results on Abstract Economies

**abstract economy**is defined as a family of order quadrauples $\mathrm{\Omega}:={({C}_{\alpha},{A}_{\alpha},{B}_{\alpha},{P}_{\alpha})}_{\alpha \in I}$ such that, for each $\alpha \in I$, ${A}_{\alpha},{B}_{\alpha}:C\u27f6{2}^{{C}_{\alpha}}$ are constraint correspondences, and ${P}_{\alpha}:C\u27f6{2}^{{C}_{\alpha}}$ is a preference correspondence. An

**equilibrium**of $\mathrm{\Omega}$ is a point $\widehat{x}\in C$ such that, for each $\alpha \in I$, ${\widehat{x}}_{\alpha}={\pi}_{\alpha}\left(\widehat{x}\right)\in cl{B}_{\alpha}\left(\widehat{x}\right)$ and ${A}_{\alpha}\left(\widehat{x}\right)\cap {P}_{\alpha}\left(\widehat{x}\right)=\varnothing $, where ${\pi}_{\alpha}:C\u27f6{C}_{\alpha}$ denotes the projection mapping from C onto ${C}_{\alpha}$. For more details on this, see Gale and Mas-Colell [5], and Shafer and Sonnenschein [3]. It is known that, if (${X}_{\alpha}$,$\Gamma $${}^{\alpha}$${}_{{D}_{\alpha}}$)${}_{\alpha \in I}$ is a family of H-spaces, Tarafder [8,26] has shown that the product space $X={\prod}_{\alpha \in I}{X}_{\alpha}$ with product topology is also an H-space, together with the family $\left\{{\Gamma}_{D}\phantom{\rule{4pt}{0ex}}\right|\phantom{\rule{4pt}{0ex}}D\in \langle X\rangle \}$, where $\Gamma {}_{D}$ is defined by $\Gamma {}_{D}={\prod}_{\alpha \in I}{\Gamma}_{{\pi}_{\alpha}\left(D\right)}^{\alpha}$. In addition, the product of H-convex subsets is also H-convex. For two correspondences $S,T:C\u27f6{2}^{{C}_{\alpha}}$, the correspondence $S\cap T:C\u27f6{2}^{{C}_{\alpha}}$ is defined by $(S\cap T)\left(x\right)=S\left(x\right)\cap T\left(x\right)$ for each $x\in C$.

**Lemma**

**3.**

- (1)
- ${C}_{\alpha}$ is an H-convex subset of ${X}_{\alpha}$, and ${K}_{\alpha}$ is a nonempty compact subset of ${C}_{\alpha}$;
- (2)
- ${P}_{\alpha}:C\u27f6{2}^{{C}_{\alpha}}$ is ${\mathcal{L}}_{{\pi}_{\alpha}}$-majorized, where $C:={\prod}_{\alpha \in I}{C}_{\alpha}$;
- (3)
- the set ${D}_{\alpha}:=\{x\in C|\phantom{\rule{4pt}{0ex}}{P}_{\alpha}\left(x\right)\ne \varnothing \}$ is open in C;
- (4)
- there exists ${y}_{\alpha}\in {K}_{\alpha}$ such that ${y}_{\alpha}\in H\text{-}co{P}_{\alpha}\left(x\right)$ for all $x\in C\backslash K$, where $K:={\prod}_{\alpha \in I}{K}_{\alpha}$.

**Proof.**

- (a)
- ${P}_{\alpha}\left(z\right)\subset {T}_{\alpha}\left(z\right)$ for each $z\in C$;
- (b)
- ${\pi}_{\alpha}\left(z\right)\notin H$-$co{T}_{\alpha}\left(z\right)$ for each $z\in {N}_{\alpha}$;
- (c)
- ${T}_{\alpha}^{-1}\left(y\right)$ is open in C for all $y\in {C}_{\alpha}$.

**Theorem**

**3.**

- (1)
- ${A}_{\alpha}\left(x\right)\ne \varnothing $ and H-$co{A}_{\alpha}\left(x\right)\subset {B}_{\alpha}\left(x\right)$ for each $x\in C$, where $C:={\prod}_{\alpha \in I}{C}_{\alpha}$;
- (2)
- the mapping ${A}_{\alpha}:C\u27f6{2}^{{C}_{\alpha}}$ has open lower sections;
- (3)
- the mapping $cl{B}_{\alpha}:C\u27f6{2}^{{C}_{\alpha}}$ is upper semicontinuous;
- (4)
- the mapping ${A}_{\alpha}\cap {P}_{\alpha}:C\u27f6{2}^{{C}_{\alpha}}$ is ${\mathcal{L}}_{{\pi}_{\alpha}}$-majorized;
- (5)
- the set ${D}_{\alpha}:=\{x\in C\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}{A}_{\alpha}\left(x\right)\cap {P}_{\alpha}\left(x\right)\ne \varnothing \}$ is open in C;
- (6)
- there exists ${y}_{\alpha}\in {K}_{\alpha}$ such that ${y}_{\alpha}\in H\text{-}co({A}_{\alpha}\cap {P}_{\alpha})\left(x\right)$ for all $x\in C\backslash K$, where $K:={\prod}_{\alpha \in I}{K}_{\alpha}$.

**Ω**.

**Proof.**

- (a)
- $({A}_{\alpha}\cap {P}_{\alpha})\left(z\right)\subset {S}_{\alpha}\left(z\right)$ for each $z\in C$;
- (b)
- ${\pi}_{\alpha}\left(z\right)\notin H$-$co{S}_{\alpha}\left(z\right)$ for each $z\in {N}_{\alpha}$;
- (c)
- ${S}_{\alpha}^{-1}\left(y\right)$ is open in C for all $y\in {C}_{\alpha}$.

**Ω**. ☐

**Remark**

**3.**

- (1)
- We focus on the setting of general H-spaces without any linear or convex structure;
- (2)
- The set I of agents can be any infinite set;
- (3)
- The strategy set ${C}_{\alpha}$ need not be compact or metrizable;
- (3)

**Ω**still has an equilibrium point, even though the correspondences are not lower semicontinuous and the strategy sets are not compact.

**Example**

**1.**

**l.c.-space**, if X is an uniform space with uniformity $\mathcal{U}$ having a base $\mathcal{B}$ of symmetric entourages such that, for each $V\in \mathcal{B}$, the set $V\left(E\right):=\{y\in X\mid (x,y)\in V\mathrm{for}\mathrm{some}x\in E\}$ is H-convex whenever E is H-convex. In the setting of $l.c.$-spaces, we now establish a new fixed point theorem as follows:

**Theorem**

**4.**

- (1)
- the mapping $T\cap P:C\u27f6{2}^{C}$ is $\mathcal{L}$-majorized;
- (2)
- the set $D:=\{x\in C\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}T\left(x\right)\cap P\left(x\right)\ne \varnothing \}$ is open in C;
- (3)
- there exists $y\in K$ such that $C\backslash K\subset {\left(H\text{-}co(T\cap P)\right)}^{-1}\left(y\right)$.

**Proof.**

**Corollary**

**1.**

**Proof.**

**Example**

**2.**

## 5. Results on Eigenvector Problems

**kernel**of f is the set $\mathrm{ker}f$ denoted by

**eigenvalue**and the corresponding

**eigenvector**of f, respectively, if $f\left(v\right)=\lambda v$. In this section, we study the following

**eigenvector problem**(EIVP, for short):

- (EIVP)
- Find $v\in C$ with $v\ne \theta $ and $\lambda \in \mathcal{K}$ such that $f\left(v\right)=\lambda v$.

**Lemma**

**4.**

**Lemma**

**5.**

**Proof.**

**Theorem**

**5.**

- (H)
- there exists a nonempty compact subset K of C such that for each $x\in C\backslash K$ there exists $y\in K$ such that $d\left(f\right(y),\Re (x\left)\right)<d\left(f\right(x),\Re (x\left)\right)$.

**Proof.**

**Remark**

**4.**

**Corollary**

**2.**

**Example**

**3.**

## 6. Conclusions

- (1)
- In order to prove our results, we introduce a new concept of H-spaces with covering polytopes, and develop some technical tools. It is known that the mapping H-$coT$ of a set-valued mapping T in any locally convex topological vector space preserves the open lower sections. Lemma 1 indicates that such a property still holds in any H-space with covering polytopes. The reader might study further some general topological space, such as G-spaces [27] and $FC$-spaces [33], and offers some interesting property to avoid such an additional assumption.
- (2)
- Many existence theorems of maximal elements do not need any compactness on the strategy sets. In the literature, there are many ways to control the noncompact case. One classical way is to consider various set-valued mappings, such as compact mapping or condensing mapping. The other way is to offer a proper and novel coercive condition on the strategy set, such as [25] (Theorem 4), [24] (Theorem 3), [34] (Theorem 3.3 and its corollary), and [35] (Theorem 3 and its corollary), which are closely related to our coercive conditions.
- (3)
- KKM theory and fixed point theorem are often related, and play crucial roles in proving the existence of maximal elements. They are logically equivalent with various optimization problems, such as minimax inequality, variational inequality, coincidence theorem, complementarity problems, and equilibrium problems. As we know, most of the earlier works present existence theorems of maximal elements by using fixed point theorems. However, we have an interesting tour in this paper. In fact, we apply a general KKM theorem (see Lemma 2) of Chang and Ma [13] (Theorem 1) to establish our existence theorems of maximal elements, and then obtain a general fixed point theorem. More results on KKM theory can be found in [27,28,30,31,33].

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Chu, L.-J.; Du, W.
On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems. *Symmetry* **2019**, *11*, 789.
https://doi.org/10.3390/sym11060789

**AMA Style**

Chu L-J, Du W.
On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems. *Symmetry*. 2019; 11(6):789.
https://doi.org/10.3390/sym11060789

**Chicago/Turabian Style**

Chu, Liang-Ju, and Wei–Shih Du.
2019. "On Maximal Elements with Applications to Abstract Economies, Fixed Point Theory and Eigenvector Problems" *Symmetry* 11, no. 6: 789.
https://doi.org/10.3390/sym11060789