Equilibrium problems on quasi-weighted graphs

Abstract

We give a new minimization theorem for equilibrium problems on a quasi-weighted graph. This result generalizes the graphical version of the Ekeland’s variational principle for equilibrium problems on weighted graphs (Alfuraidan and Khamsi in Proc Am Math Soc 9:33–40, 2022).

1 Introduction

Ekeland’s variational principle (EVP) [6,7,8,9, 11] is considered as a minimization result for a proper lower semi-continuous bounded from below function on a complete metric space. It is a simple minimization tool that is used to estimate the solution. Motivated by the huge applications of EVP together with the quasi-metric concept that makes sense in reality and inspired by [1, 3], we give a generalized version of the Ekeland’s variational principle for equilibrium problems on a quasi-weighted graph.

We start by recalling the equilibrium problem.

Definition 1.1

[4, 10] Let (T, d) be a metric space and S be a nonempty subset of T. Let $H: S\times S \rightarrow {\mathbb {R}}$ be a bifunction such that $H(s,s)=0$ $\forall$ $s\in S$. The problem of finding ${\overline{s}}\in S$ such that

$$\begin{aligned} H({\overline{s}},s)\ge 0, \; \forall s\in S, \end{aligned}$$

is called an equilibrium problem for $H(\cdot , \cdot )$.

The concept of the equilibrium problem as stated above is not depending on the distance $d(\cdot ,\cdot )$. Hence, we can rephrase the definition in a more abstract form:

Definition 1.2

[1] Let S be a nonempty set. Let $H: S\times S \rightarrow {\mathbb {R}}$ be a bifunction such that $H(s,s)=0$ $\forall$ $s\in S$. Finding ${\overline{s}}\in S$ such that

$$\begin{aligned} H({\overline{s}},s)\ge 0, \; \forall \; s\in S, \end{aligned}$$

is the equilibrium problem for $H(\cdot , \cdot )$.

2 Preliminaries

We start by recalling the definition of the quasi-metric and some needed terminology on the concept of quasi-metric spaces. For more details on these spaces and their relation with EVP, the reader may consult [5].

Definition 2.1

Let T be a set. A function $d: T \times T \rightarrow [0,+\infty )$ is called quasi-metric if the following conditions hold:

• 1. $d(t,s)=0$ if and only if $t=s$,

• 2. $d(t,w)\le d(t,s)+d(s,w),$ $\forall$ $t,s,w \in T$.

The pair (T, d) is called a quasi-metric space.

Remark 2.2

If a quasi-metric d is symmetric, i.e., $d(s,t)=d(s,t)$, $\forall s,t$, then d is a usal metric(distance).

Definition 2.3

[2] Let (T, d) be a quasi-metric space.

• (a) A sequence $\{t_n\} \subset T$ is left-convergent to $t \in T$ if $\lim \nolimits _{n \rightarrow \infty }\ d(t_n,t) = 0$, i.e., $\forall$ $\varepsilon > 0$, there is an $N \in {\mathbb {N}}$ such that

  $$\begin{aligned} d(t_n,t)< \varepsilon , \;\; {\text {for any}}\; \; n\ge N. \end{aligned}$$

  In general, left-limits are not necessarily unique. We denote the set of left-limits by

  $$\begin{aligned} \Omega _l(\{t_n\}):= \left\{ t \in T;\; \lim \limits _{n \rightarrow \infty }\ d(t_n,t) = 0\right\} . \end{aligned}$$

• (b) A sequence $\{t_n\} \subset X$ is right-convergent to $t \in T$ if

  $$\begin{aligned} \lim \limits _{n \rightarrow \infty }\ d(t, t_n) = 0. \end{aligned}$$

  We denote the set of right-limits by

  $$\begin{aligned} \Omega _r(\{t_n\}) = \left\{ t \in T;\; \lim \limits _{n \rightarrow \infty }\ d(t, t_n) = 0\right\} . \end{aligned}$$

• (c) A sequence $\{t_n\} \subset T$ is left-Cauchy (resp. right-Cauchy) if for each $\varepsilon > 0$ there is an $N \in {\mathbb {N}}$ such that

  $$\begin{aligned} d(t_n,t_m)< \varepsilon \ (\mathrm{resp.}\; d(t_m,t_n)< \varepsilon ), \;\; {\text {for any}}\; \; m> n > N. \end{aligned}$$

• (d) (T, d) is a left-complete (resp. right-complete) space if for all left-Cauchy (resp. right-Cauchy) sequences $\{t_n\}$ in T, we have $\Omega _l(\{t_n\}) \ne \emptyset$ (resp. $\Omega _r(\{t_n\}) \ne \emptyset$).

• (e) A function $\Upsilon : T \rightarrow [0, +\infty )$ is a left-lower (resp. right-lower) semicontinuous function if and only if for all sequence $\{t_n\}$ in T, we have

  $$\begin{aligned} \Upsilon (t) \le \ \liminf _{n \rightarrow \infty }\ \Upsilon (t_n), \end{aligned}$$

  $\forall$ $t \in \Omega _l(\{t_n\})$ (resp. $t \in \Omega _r(\{t_n\})$).

The following technical lemma will be needed.

Lemma 2.4

[2] Let (T, d) be a quasi-metric space. The following assertions are true:

• (a) Let $\{t_n\}$ be a sequence in T such that $\Omega _l(\{t_n\})$ and $\Omega _r(\{t_n\})$ are not empty. Then $\Omega _l(\{t_n\})$ and $\Omega _r(\{t_n\})$ are reduced to one point and are equal, i.e., there exists a unique $t \in T$ such that

  $$\begin{aligned} \lim _{n \rightarrow \infty }\ d(t, t_n) = \lim _{n \rightarrow \infty }\ d(t_n, t) = 0. \end{aligned}$$

• (b) Fix $u \in T$. Let $\{t_n\}$ be a sequence in T. We have

  $$\begin{aligned} d(u,t) \le \ \liminf _{n \rightarrow \infty }\ d(u, t_n), \end{aligned}$$

  for all $t \in \Omega _l(\{t_n\})$, i.e., the function $\Upsilon _u: T \rightarrow [0,+\infty )$ defined by $\Upsilon _u(t) = d(u,t)$ is a left-lower semicontinuous function.

Now, we present some basic graph theory definitions that will be used in the sequel. For more detail, the reader can consult any graph theory book.

Definition 2.5

• (1) A directed graph $G=(V,E)$ consists of a finite, nonempty set of vertices V and a set of edges E. Each edge e is an ordered pair (t, s) of vertices.

• (2) If e is an edge of G and $s=t$, then e is called a loop. If E(G) contains all the loops, then G is a reflexive graph.

• (3) A graph G is said to be transitive if $(s,w) \in E(G)$ provided $(s,t) \in E(G)$ and $(t,w) \in E(G)$, $\forall$ $s, t, w \in V(G)$.

• (4) A multi-edge is a set of two or more edges having same ends.

• (5) A graph $G = (V(G), E(G))$ endowed with a quasi-metric $d: V(G) \times V(G) \rightarrow [0,+\infty )$ will be called a quasi-weighted graph.

In this paper, we only consider asymmetric digraphs, i.e., if $(s,t)\in E(G),$ then $(t,s)\notin E(G),$ without multi-edges.

Definition 2.6

Let $(G,d) = (V(G), E(G), d)$ be a quasi-weighted graph.

• (a) A sequence $\{t_n\}$ in V(G) is G-decreasing provided $(t_{n+1},t_n)\in E(G)$, $\forall$ $n\in {\mathbb {N}}$.

• (b) (V(G), d) is a left-G-complete space if, for each left-Cauchy G-decreasing sequence $\{t_n\}$ in V(G), we have $\Omega _l(\{t_n\}) \ne \emptyset$.

• (c) (V(G), d) is a strongly left-G-complete space if, for each left-Cauchy G-decreasing sequence $\{t_n\}$ in V(G), $\Omega _l(\{t_n\})$ is reduced to one element. This element will be called the left G-limit of $\{t_n\}$.

• (d) A function $\Upsilon : V(G) \rightarrow [0, +\infty )$ is a left-lower G-semicontinuous function if and only if for all G-decreasing sequence $\{t_n\}$ in T, we have

  $$\begin{aligned} \Upsilon (t) \le \ \liminf _{n \rightarrow \infty }\ \Upsilon (t_n), \end{aligned}$$

  $\forall t \in \Omega _l(\{t_n\})$.

In the following definition, we adapt the property (OSC) introduced in [3] to the case of quasi-weighted graphs.

Definition 2.7

[2] Let $G = (V(G), E(G),d)$ be a quasi-weighted digraph. We say that G satisfies the property (OSC) if and only if for all G-decreasing sequence $\{t_n\}$ in V(G) and $t \in \Omega _l(\{t_n\})$, we have

• (i) $(t, t_n) \in E(G)$, for all $n \in {\mathbb {N}}$, and

• (ii) $(w, t) \in E(G)$ whenever $(w, t_n) \in E(G)$, for all $n \in {\mathbb {N}}$.

Remark 2.8

In this paper, we study quasi-weighted graphs which make sense in reality. Indeed, we can consider the GPS Manhattan roads as an example of a strongly complete quasi-weighted graph. In order to illustrate this example, consider an oriented square in the plane, see the figure below.

Fig. 1

GPS Manhattan roads

The Manhattan quasi-distance is defined in the following way:

• 1. $d(x,y)=0$ if and only if $x=y$.

• 2. $d(x,y) =$ length of the path following the orientation going from x to y.

Notice that:

• (i) d is not symmetric since, for example,

  $$\begin{aligned} d(u,z)=D(u,v)+D(v,w)+D(w,z) \end{aligned}$$

  while

  $$\begin{aligned} d(z,u)=D(z,u), \end{aligned}$$

  where D is the usual Euclidean distance.

• (ii) If t belong to the path going from x to y, then

  $$\begin{aligned} d(x,t) + d(t,y)= d(x,y). \end{aligned}$$

• (iii) If s does not belong to the path going from x to y, then

  $$\begin{aligned} d(x,y)\le d(x,s)\le d(x,s)+d(s,y). \end{aligned}$$

  Thus d is a quasi-distance.

• (iv) Consider the following sequence: $x_1=v$, $x_2$ is the middle point of [u, v], $x_3$ is the middle point of $[u,x_2]$, $\vdots$, $x_{n+1}$ is the middle point of $[u,x_n]$. Then $d(u,x_1)=D(u,v)$, $d(u,x_2)=\frac{D(u,v)}{2}$, $\vdots$, $d(u,x_n)=\frac{D(u,v)}{2^{n-1}}$. We have $u\in \Omega _l(\{x_n\})=\{u\}$ and

  $$\begin{aligned} d(x_n,u)=D(x_n,v)+D(v,w)+D(w,z)+D(z,u)\nrightarrow 0. \end{aligned}$$

This example captures the case of any left-Cauchy sequence.

3 Main results

We are now ready to state our main result.

Theorem 3.1

Let $G = (V(G), E(G),d)$ be a reflexive transitive quasi-weighted digraph such that (V(G), d) is strongly left G-complete. Assume that G satisfies Property (OSC). Let $H: V(G)\times V(G) \rightarrow {\mathbb {R}}$ be a bifunction. Assume that the following conditions hold:

• (i) $H(s, s) = 0$, $\forall s\in V(G)$;

• (ii) $H(s,t)\le H(s,w)+H(w,t)$, $\forall$ $s,t,w\in V(G)$ such that $(t,s) \in E(G)$ and $(w,t) \in E(G)$;

• (iii) H(s, .) is lower bounded and G-left lower semicontinuous, $\forall s\in V(G)$.

Then, for every $\varepsilon >0$ and $s_0 \in V(G)$, there exists ${\overline{s}} \in V(G)$ such that $({\overline{s}}, s_0) \in E(G)$ and

• (a) $H(s_0,{\overline{s}})+\varepsilon \ d(s_0,{\overline{s}})\le 0$;

• (b) $H({\overline{s}},t)+\varepsilon \ d({\overline{s}},t)>0$, $\forall t \in V(G){\setminus } \{{\overline{s}}\}$ with $(t, {\overline{s}})\in E(G)$.

Proof

If we replace the quasi-metric $d(\cdot ,\cdot )$ by the quasi-metric $\varepsilon \ d(\cdot , \cdot )$, all the assumptions are satisfied. Therefore and without loss of any generality, we may assume $\varepsilon = 1$. Let $s \in V(G)$. Set

$$\begin{aligned} A(s) = \{t \in V(G);\; (t,s) \in E(G)\; \textrm{and}\; H(s,t) + d(s,t) \le 0\}. \end{aligned}$$

Since G is reflexive and the property (i), we get $s \in A(s)$ thus $A(s)\ne$, for any $s \in V(G)$. Let $t \in A(s)$ and $w \in A(t)$. Since G is transitive, we get $(w,s) \in E(G)$. Using (ii), we get

$$\begin{aligned} \begin{array}{llll} H(s,w) + d(s,w) &{}\le &{} H(s,t) + H(t,w) + d(s,t) + d(t,w)\\ &{} = &{} H(s,t) + d(s,t) + H(t,w) + d(t,w)\\ &{} \le &{} 0, \end{array} \end{aligned}$$

which implies $w\in A(s)$, i.e., $A(t) \subset A(s)$. Moreover, for any $s \in V(G)$, set

$$\begin{aligned} r(s) = \sup _{t \in A(s)}\ d(s,t)\;\;\; \textrm{and}\;\; v(s) = \inf _{t \in A(s)} H(s,t). \end{aligned}$$

Note that $v(s) > -\infty$, since $H(s,\cdot )$ is lower bounded for any $s \in V(G)$. For any $s \in V(G)$ and $t \in A(s)$, we have $d(s,t) \le -H(s,t)$ which implies

$$\begin{aligned} r(s) = \sup _{t \in A(s)}\ d(s,t) \le \sup _{t \in A(s)} -H(s,t) = - \inf _{t \in A(s)} H(s,t) = -v(s). \end{aligned}$$

Fix $s_0 \in V(G)$. By definition of $v(s_0)$, there exists $s_1 \in A(s_0)$ such that

$$\begin{aligned} H(s_0,s_1) \le v(s_0) + \frac{1}{2}. \end{aligned}$$

By induction, we construct a sequence $\{s_n\}$ such that $s_{n+1} \in A(s_n)$ and

$$\begin{aligned} H(s_n,s_{n+1}) \le v(s_n) + \frac{1}{2^{n+1}}, \end{aligned}$$

for any $n \in {\mathbb {N}}$. Fix $n \in {\mathbb {N}}$. Since $A(s_{n+1}) \subset A(s_n)$, we get

$$\begin{aligned} \inf _{t \in A(s_n)}\ H(s_{n+1},y) \le \inf _{t \in A(s_{n+1})}\ H(s_{n+1},t) = v(s_{n+1}). \end{aligned}$$

For any $t \in V(G)$, (ii) implies $H(s_n, y) \le H(s_n, s_{n+1}) + H(s_{n+1}, t)$ which gives

$$\begin{aligned} H(s_n, t) - H(s_n, s_{n+1}) \le H(s_{n+1}, t). \end{aligned}$$

Hence

$$\begin{aligned} \begin{array}{lll} v(s_n) - H(s_n, s_{n+1}) &{} = &{} \displaystyle \inf _{t \in A(s_n)}\ H(s_n, t) - H(s_n, s_{n+1})\\ &{}&{}\\ &{}\le &{}\displaystyle \inf _{t \in A(s_n)}\ H(s_{n+1}, t) \le v(s_{n+1}). \end{array} \end{aligned}$$

Since $\displaystyle H(s_n,s_{n+1}) \le v(s_n) + \frac{1}{2^{n+1}}$, we get

$$\begin{aligned} -v(s_n) \le - H(s_n, s_{n+1}) + \displaystyle \frac{1}{2^{n+1}} \le v(s_{n+1}) - v(s_n) + \frac{1}{2^{n+1}}, \end{aligned}$$

which implies

$$\begin{aligned} 0 \le v(s_{n+1}) + \frac{1}{2^{n+1}}. \end{aligned}$$

Therefore, we must have

$$\begin{aligned} r(s_{n+1}) \le - v(s_{n+1}) \le \frac{1}{2^{n+1}}. \end{aligned}$$

In particular, we have

$$\begin{aligned} d(s_n, s_{n+1}) \le r(s_n) \le \frac{1}{2^n}, \end{aligned}$$

which proves that $\{s_n\}$ is left Cauchy. Since this sequence is G-decreasing by construction and V(G) is left G-complete, we conclude that $\Omega _l(\{s_n\})$ is not empty. Let ${\overline{s}}\in \Omega _l(\{s_n\})$. Since G satisfies the property (OSC), we get $({\overline{s}}, s_n) \in E(G)$. Since $H(s,\cdot )$ is left-lower G-semicontinuous, for any $s \in V(G)$, we conclude that

$$\begin{aligned} H(t, {\overline{s}}) \le \liminf _{n \rightarrow \infty }\ H(t, s_n), \end{aligned}$$

for any $t \in V(G)$. By construction of $\{s_n\}$, we know that $s_{n+h} \in A(s_n)$, for any $n, h \in {\mathbb {N}}$, which implies

$$\begin{aligned} \begin{array}{lll} H(s_n, {\overline{s}}) + d(s_n, {\overline{s}}) &{}\le &{}\liminf \limits _{h \rightarrow \infty }\ H(s_n, s_{n+h}) + d(s_n, s_{n+h})\\ &{}\le &{} 0. \end{array} \end{aligned}$$

Hence ${\overline{s}} \in A(s_n)$, which implies $A({\overline{s}}) \subset A(s_n)$, for any $n \in {\mathbb {N}}$. For any $t \in A({\overline{s}})$ and $n \in {\mathbb {N}}$, we have $t \in A(s_n)$, which implies

$$\begin{aligned} d(s_n,t) \le r(s_n) \le \frac{1}{2^n}. \end{aligned}$$

Hence $\lim \limits _{n \rightarrow \infty } \ d(s_n,t) = 0$, i.e., $t \in \Omega _l(\{s_n\})$. Since (V(G), d) is strongly left G-complete, we conclude that $t = {\overline{s}}$, i.e., $A({\overline{s}}) = \{{\overline{s}}\}$. Putting everything together, we get

$$\begin{aligned} H(s_0, {\overline{s}}) + d(s_0, {\overline{s}}) \le 0 \end{aligned}$$

since ${\overline{s}} \in A(s_0)$, and for any $t \in V(G){\setminus } \{{\overline{s}}\}$ with $(t, {\overline{s}})\in E(G)$, we have

$$\begin{aligned} H({\overline{s}},t)+ d({\overline{s}},t)>0, \end{aligned}$$

since $t \not \in A({\overline{s}})$. The proof of Theorem 3.1 is complete. $\square$

The following result is easy to obtain from Theorem 3.1.

Theorem 3.2

Let $G = (V(G), E(G),d)$ be a reflexive transitive quasi-weighted digraph such that (V(G), d) is strongly left G-complete. Assume that G satisfies Property (OSC). Let $H: V(G)\times V(G) \rightarrow {\mathbb {R}}$ be a bifunction. Assume that the following conditions hold:

• (i) $H(s, s) = 0$, $\forall s\in V(G)$;

• (ii) $H(s,t)\le H(s,w)+H(w,t)$, $\forall s,t,w\in V(G)$ such that $(t,s) \in E(G)$ and $(w,t) \in E(G)$;

• (iii) H(s, .) is lower bounded and left G-lower semicontinuous, for every $s\in V(G)$.

Let $K: V(G)\times V(G)\rightarrow {\mathbb {R}}$ be a bifunction such that $H(s,t) \le K(s,t)$, $\forall s, t \in V(G)$. Then for every $\varepsilon > 0$ and $s_0\in V(G)$, $\exists \; {\overline{s}}\in V(G)$ such that $({\overline{s}}, s_0) \in E(G)$ and

• (a) $H(s_0,{\overline{s}})+\varepsilon \ d(s_0,{\overline{s}})\le 0$;

• (b) $K({\overline{s}},t)+\varepsilon \ d({\overline{s}},t)>0$, $\forall t \in V(G){\setminus } \{{\overline{s}}\}$ with $(t, {\overline{s}})\in E(G)$.

As a corollary, we obtain the quasi-metric version of Corollary 3.5 of [1] which is a major improvement to the main result of [3].

Corollary 3.3

Let $G = (V(G), E(G),d)$ be a reflexive transitive quasi-weighted digraph such that (V(G), d) is strongly left G-complete. Assume that G satisfies Property (OSC). Let $H: V(G)\times V(G) \rightarrow {\mathbb {R}}$ be a bifunction. Assume that the following conditions hold:

• (i) $\exists \; \psi : V(G) \rightarrow {\mathbb {R}}$ such that

  $$\begin{aligned} H(s, t) \ge \psi (s) - \psi (t),\; \forall s,t\in V(G); \end{aligned}$$

• (ii) $\psi$ is lower bounded and left G-lower semicontinuous.

Then, for every $\varepsilon >0$ and $s_0 \in V(G)$, $\exists \; {\overline{s}} \in V(G)$ such that $({\overline{s}}, s_0) \in E(G)$ and

• (a) $\psi ({\overline{s}})\le \psi (s_0)-\varepsilon \ d(s_0,{\overline{s}})$;

• (b) $H({\overline{s}},t)+\varepsilon \ d({\overline{s}},t)>0$, $\forall t\in V(G)$ such that $t \ne {\overline{s}}$.