# Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property

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## Abstract

**:**

## 1. Introduction

**Theorem**

**1**(Ekeland Variational Principle—weak form (wEkVP)).

**Theorem**

**2**(Ekeland Variational Principle—EkVP).

**Remark**

**1.**

**Corollary**

**1.**

**Theorem**

**3.**

- The metric space $(X,d)$ is complete.
- If for every Lipschitz function $f:X\to [0,\infty )$ there exists a point $z\in X$ such that

**Theorem**

**4**(Caristi fixed point theorem).

- If $T:X\to X$ is a mapping satisfying the condition$$\forall x\in X,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}d\left(T\right(x),x)\le \phi \left(x\right)-\phi \left(T\right(x\left)\right)\phantom{\rule{0.166667em}{0ex}},$$
- If $T:X\rightrightarrows X$ is a set-valued mapping satisfying the condition$$\forall x\in X,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\exists y\in T\left(x\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}d(y,x)\le \phi \left(x\right)-\phi \left(y\right)\phantom{\rule{0.166667em}{0ex}},$$
- If $T:X\rightrightarrows X$ is a set-valued mapping with nonempty values satisfying the condition$$\forall x\in X,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall y\in T\left(x\right),\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}d(y,x)\le \phi \left(x\right)-\phi \left(y\right)\phantom{\rule{0.166667em}{0ex}},$$

**Theorem**

**5**(Takahashi minimization principle).

**Remark**

**2.**

## 2. Preliminaries

- reflexive if $(u,u)\in E\left(G\right)$ for all $u\in V\left(G\right)$;
- transitive if $(u,v),\phantom{\rule{0.166667em}{0ex}}(v,w)\in E\left(G\right)$ implies $(u,w)\in E\left(G\right)$.

**Remark**

**3.**

**Definition**

**1.**

- A weighted digraph $G=\left(V\right(G),E(G),d)$, where d is a metric on $V\left(G\right)$, is said to satisfy the OSC property if $(x,{x}_{n})\in E\left(G\right)$ for all $n\in \mathbb{N},$ for every sequence $\left({x}_{n}\right)$ in $V\left(G\right)$ such that ${x}_{n}\stackrel{d}{\to}x$ and $({x}_{n+1},{x}_{n})\in E\left(G\right)$ for all $n\in \mathbb{N}.$
- A partially ordered metric space $(X,d,\u2aaf)$ is said to satisfy the OSC condition if $x\u2aaf{x}_{n}$ for all $n\in \mathbb{N},$ for every sequence $\left({x}_{n}\right)$ in X such that ${x}_{n}\stackrel{d}{\to}x$ and ${x}_{n+1}\u2aaf{x}_{n}$ for all $n\in \mathbb{N}.$

**Remark**

**4.**

**Definition**

**2.**

- A sequence $\left({x}_{n}\right)$ in $V\left(G\right)$ is called a G-sequence if $({x}_{n+1},{x}_{n})\in E\left(G\right)$ for all $n\in \mathbb{N}$.
- A G-Cauchy sequence is a G-sequence that is Cauchy with respect to d.
- A subset Y of $V\left(G\right)$ is called G-closed if $y\in Y$ for every G-sequence $\left({y}_{n}\right)$ in Y, d-convergent to $y\in V\left(G\right)$.
- A function $f:V\left(G\right)\to \mathbb{R}$ is called G-continuous G-LSC) if ${lim}_{n}f\left({x}_{n}\right)=f\left(x\right)$ (resp. $f\left(x\right)\le {lim\; inf}_{n}f\left({x}_{n}\right)$) for every G-sequence $\left({x}_{n}\right)$ in $V\left(G\right)\phantom{\rule{0.277778em}{0ex}}d$-convergent to x.

**Remark**

**5.**

**Proposition**

**1.**

- The relation ${\le}_{\phi}$ is a partial order on X.
- Every ${\le}_{\phi}$-decreasing sequence in X is Cauchy.
- If φ is LSC, then $y{\le}_{\phi}x$ for every sequence $\left({y}_{n}\right)$ in X satisfying ${y}_{n}{\le}_{\phi}x$ and ${y}_{n}\stackrel{d}{\to}y$. Furthermore, the partial order ${\le}_{\phi}$ satisfies the OSC.

**Proof.**

- The proof is a straightforward verification.
- Let $\left({x}_{n}\right)$ be a sequence in X such that ${x}_{n+1}{\le}_{\phi}{x}_{n}$ for all $n\in \mathbb{N}$. The inequality$$0\le d({x}_{n},{x}_{n+1})\le \phi \left({x}_{n}\right)-\phi \left({x}_{n+1}\right)\phantom{\rule{0.166667em}{0ex}}$$$$d({x}_{n},{x}_{n+k})\le \phi \left({x}_{n}\right)-\phi \left({x}_{n+k}\right)\phantom{\rule{0.166667em}{0ex}},$$
- We have$${y}_{n}{\le}_{\phi}x\iff \phi \left({y}_{n}\right)+d({y}_{n},x)\le \phi \left(x\right)\phantom{\rule{0.166667em}{0ex}}.$$

**Remark**

**6.**

## 3. Ekeland Variational Principle in Metric Spaces Endowed with a Graph

**Example**

**1.**

**Theorem**

**6.**

**Proof.**

**Corollary**

**2.**

**Remark**

**7.**

**Theorem**

**7.**

- For every reflexive transitive acyclic weighted digraph $G=\left(V\right(G),E(G),d)$, where d is a G-complete metric on $V\left(G\right)$ such that the OSC property for G-sequences is satisfied, the following property holds.(A${}_{1})$ For any G-closed property $\mathcal{P}$ on $V\left(G\right)$ such that $Dom\left(\mathcal{P}\right)$ is nonempty, every G-lower semi-continuous function $\phi :V\left(G\right)\to [0,\infty )$, any given $\epsilon >0$ and $\lambda >0$ and ${x}_{0}\in Dom\left(\mathcal{P}\right)$ such that$$\phi \left({x}_{0}\right)\le inf\phi (Dom\left(\mathcal{P}\right))+\epsilon \phantom{\rule{0.166667em}{0ex}},$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{i}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}\phi \left(z\right)+{\lambda}^{-1}\epsilon \phantom{\rule{0.166667em}{0ex}}d({x}_{0},z)\le \phi \left({x}_{0}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{ii}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}d(z,{x}_{0})\le \lambda ,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{iii}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}\phi \left(z\right)<\phi \left(x\right)+{\lambda}^{-1}\epsilon \phantom{\rule{0.166667em}{0ex}}d(x,z),\hfill \end{array}$$
- For every partially ordered decreasingly complete metric space $(X,d,\u2aaf)$ with the OSC property for decreasing sequences, the following property holds.(A${}_{2}$) For any decreasingly lower semi-continuous function $\phi :X\to [0,\infty ),$ any $\epsilon >0$, any $\lambda >0$, and any ${x}_{0}\in X$ satisfying$$\phi \left({x}_{0}\right)\le inf\phi \left(X\right)+\epsilon $$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{i}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}\phi \left(z\right)+{\lambda}^{-1}\epsilon \phantom{\rule{0.166667em}{0ex}}d({x}_{0},z)\le \phi \left({x}_{0}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{ii}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}d(z,{x}_{0})\le \lambda ,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{iii}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}\phi \left(z\right)<\phi \left(x\right)+\epsilon {\lambda}^{-1}d(x,z),\hfill \end{array}$$
- For every complete metric space $(X,d)$ the following property holds.(A${}_{3}$) For any LSC function $\phi :X\to [0,\infty ),$ any $\epsilon >0$ and any ${x}_{0}\in X$ satisfying$$\phi \left({x}_{0}\right)\le inf\phi \left(X\right)+\epsilon $$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{i}\right)\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{1.em}{0ex}}\phi \left({x}_{\epsilon}\right)+\epsilon \phantom{\rule{0.166667em}{0ex}}d({x}_{\epsilon},z)\le \phi \left({x}_{0}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{ii}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}d({x}_{\epsilon},{x}_{0})\le 1,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \left(\mathrm{iii}\right)\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{1.em}{0ex}}\phi \left({x}_{\epsilon}\right)<\phi \left(x\right)+\epsilon \phantom{\rule{0.166667em}{0ex}}d(x,{x}_{\epsilon}),\hfill \end{array}$$
- For every complete metric space $(X,d)$ the following property holds.(A${}_{4}$) For any LSC function $\varphi :X\to [0,\infty )$ and for any $\epsilon >0$ there exists ${x}_{\epsilon}\in X$ such that$$\varphi \left({x}_{\epsilon}\right)\le inf\varphi \left(X\right)+\epsilon $$$$\phi \left({x}_{\epsilon}\right)<\phi \left(x\right)+\epsilon \phantom{\rule{0.166667em}{0ex}}d(x,{x}_{\epsilon}),$$

**Proof.**

**Theorem**

**8.**

- Let $G=\left(V\right(G),E(G),d)$ be a reflexive transitive acyclic digraph, where d is a metric on G such that the OSC property holds on G. The following statements are equivalent:
- (i)
- The metric space $\left(V\right(G),d)$ is complete.
- (ii)
- For any lower semi-continuous function $\phi :V\left(G\right)\to [0,\infty )$ and any $\epsilon >0$ there exists ${x}_{\epsilon}\in V\left(G\right)$ such that$$\phi \left({x}_{\epsilon}\right)<\phi \left(x\right)+\epsilon \phantom{\rule{0.166667em}{0ex}}d(x,{x}_{\epsilon}),$$

- Let now $(X,d)$ be a metric space. The following statements are equivalent:
- (i)
- The metric space $(X,d)$ is complete.
- (ii)
- For any partial order ⪯ on X satisfying the OSC property, any continuous function $\phi :X\to [0,\infty )$ and any $\epsilon >0$ there exists ${x}_{\epsilon}\in X$ such that$$\phi \left({x}_{\epsilon}\right)<\phi \left(x\right)+\epsilon \phantom{\rule{0.166667em}{0ex}}d(x,{x}_{\epsilon}),$$

- For a metric space $(X,d)$ the following statements are equivalent:
- (i)
- The metric space $(X,d)$ is complete.
- (ii)
- For any continuous function $\phi :X\to [0,\infty ),$ any $\epsilon >0,\phantom{\rule{0.166667em}{0ex}}\lambda >0$ and any ${x}_{0}\in X$ satisfying$$\phi \left({x}_{0}\right)\le inf\phi \left(X\right)+\epsilon $$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& \phi \left(z\right)+{\lambda}^{-1}\epsilon \le \phi \left({x}_{0}\right),\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& d({x}_{\epsilon},{x}_{0})\le \lambda ,\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}& \phi \left(z\right)<\phi \left(x\right)+{\lambda}^{-1}\epsilon d(x,z),\hfill \end{array}$$
- (iii)
- For any continuous function $\phi :X\to [0,\infty )$ and for any $\epsilon >0$ there exists ${x}_{\epsilon}\in X$ such that$$\phi \left({x}_{\epsilon}\right)<\phi \left(x\right)+\epsilon d(x,{x}_{\epsilon})\phantom{\rule{0.166667em}{0ex}},$$

**Proof.**

## 4. Caristi Fixed Point Theorem and Takahashi Minimization Principle on Weighted Graphs

**Theorem**

**9.**

- 1.
- If $T:X\to X$ is a mapping satisfying the condition$$\forall x\in X,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\left(T\right(x),x)\in E\left(G\right)\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}d\left(T\right(x),x)\le \phi \left(x\right)-\phi \left(T\right(x\left)\right)\phantom{\rule{0.166667em}{0ex}},$$
- 2.
- If $T:X\rightrightarrows X$ is a set-valued mapping such that for every $x\in X$ there exists $y\in T\left(x\right)$ satisfying the condition$$(y,x)\in E\left(G\right)\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}d(y,x)\le \phi \left(x\right)-\phi \left(y\right)\phantom{\rule{0.166667em}{0ex}},$$
- 3.
- Suppose that $T:X\rightrightarrows X$ is a set-valued mapping with nonempty values such that $T\left(X\right)\times X\subseteq E\left(G\right)$ and for every $x\in X$ and all $y\in T\left(x\right)$,$$d(y,x)\le \phi \left(x\right)-\phi \left(y\right)\phantom{\rule{0.166667em}{0ex}}.$$

**Proof.**

**Theorem**

**10.**

**Proof.**

## 5. The Equivalence of Principles

**Theorem**

**11.**

- (wEk)
- The following holds$$\exists z\in V\left(G\right)\phantom{\rule{0.277778em}{0ex}}suchthat\phantom{\rule{0.277778em}{0ex}}S\left(z\right)=\left\{z\right\}\phantom{\rule{0.166667em}{0ex}}.$$
- (Car)
- Any mapping $T:X\to X$ satisfying$$T\left(x\right)\in S\left(x\right)\phantom{\rule{0.277778em}{0ex}}forall\phantom{\rule{0.277778em}{0ex}}x\in V\left(G\right),$$
- (Tak)
- If$$S\left(x\right)\backslash \left\{x\right\}\ne \varnothing \phantom{\rule{0.166667em}{0ex}},$$

**Proof.**

**Corollary**

**3.**

- The metric space $\left(V\right(G),d)$ is G-complete.
- (wEk) For every G-LSC function $\phi :V\left(G\right)\to [0,\infty )$ there exists $z\in V\left(G\right)$ such that$$S\left(z\right)=\left\{z\right\}\phantom{\rule{0.166667em}{0ex}}.$$
- (Car) For every G-LSC function $\phi :V\left(G\right)\to [0,\infty )$ and any mapping $T:V\left(G\right)\to V\left(G\right)$ satisfying$$T\left(x\right)\in S\left(x\right)\phantom{\rule{0.277778em}{0ex}}forall\phantom{\rule{0.277778em}{0ex}}x\in V\left(G\right),$$
- (Tak) For every G-LSC function $\phi :V\left(G\right)\to [0,\infty )$ such that$$S\left(x\right)\backslash \left\{x\right\}\ne \varnothing \phantom{\rule{0.166667em}{0ex}},$$

**Remark**

**8.**

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**MDPI and ACS Style**

Ali, B.; Cobzaş, Ş.; Mabula, M.D.
Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property. *Axioms* **2023**, *12*, 247.
https://doi.org/10.3390/axioms12030247

**AMA Style**

Ali B, Cobzaş Ş, Mabula MD.
Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property. *Axioms*. 2023; 12(3):247.
https://doi.org/10.3390/axioms12030247

**Chicago/Turabian Style**

Ali, Basit, Ştefan Cobzaş, and Mokhwetha Daniel Mabula.
2023. "Ekeland Variational Principle and Some of Its Equivalents on a Weighted Graph, Completeness and the OSC Property" *Axioms* 12, no. 3: 247.
https://doi.org/10.3390/axioms12030247