Generalized Ekeland’s variational principle with applications

Abstract

By using the concept of Γ-distance, we prove EVP (Ekeland’s variational principle) on quasi-F-metric (q-F-m) spaces. We apply EVP to get the existence of the solution to EP (equilibrium problem) in complete q-F-m spaces with Γ-distances. Also, we generalize Nadler’s fixed point theorem.

1 Introduction and preliminaries

Ekeland [1] was first to study EVP. EVP is a theorem that shows that for some optimization problems there exist nearly optimal solutions. In this paper, we study the concept of Γ-distances defined on a q-F-m space which generalizes the notion of w-distance. We inaugurate EVP in the setting of q-F-m spaces with Γ-distances but without completeness assumption and then in the setting of complete q-F-m spaces with Γ-distances. The equilibrium version of the EVP in the setting of q-F-m spaces with Γ-distances is also presented. We prove some equivalences of our variational principles with Caristi–Kirk type fixed point theorem for multi-valued maps, Takahashi’s minimization theorem, and some other related results. As applications of our results, we derive existence results for solutions of equilibrium problems and fixed point theorems for multi-valued maps. We also extend Nadler’s fixed point theorem for multi-valued maps to q-F-m spaces with Γ-distances. The results of this paper extend and generalize many results that have appeared recently in Al-Homidan, Ansari, and Yao [2], Lin, Balaj, and Ye [3], Bianchi, Kassay, and Pini [4, 5], Ha [6], and Lin and Du [7].

Definition 1.1

([8])

Assume that $T\neq \emptyset$. A function $F:T^{3} \rightarrow [0, \infty )$ is called quasi-F-metric (q-F-m) if

• (i) $F(p,q,r)=0$ if and only if $p=q=r$,

• (ii) $F(p,p,q)>0$ for all $p,q \in T$, with $p\neq q$,

• (iii) $F(p,p,r)\leq F(p,q,r)$ for all $p,q,r\in T$, with $r\neq q$,

• (iv) $F(p,q,r) \leq F(p,s,s) + F(s,q,r)$ for all $p,q,r,s \in T$.

The pair $(T, F)$ is called q-F-m space.

Let $(T,F)$ be a q-F-m space.

• (1) A sequence $\{u_{n}\}$ in T is an F-Cauchy sequence if, for every $\varepsilon >0$, there exists a positive integer $n_{0}$ such that $F(u_{m},u_{n},u_{\ell }) < \varepsilon$ for all $m,n,\ell \geq n_{0}$.

• (2) A sequence $\{u_{n}\}$ in T is F-convergent to a point $u \in T$ if, for every $\varepsilon > 0$, there exists a positive integer $n_{0}$ such that $F(u_{m},u_{n},u) < \varepsilon$ for all $m , n \geq n_{0}$.

In this paper, T is assumed to be a q-F-m space.

Definition 1.2

([9])

A function $\varGamma :T^{3} \rightarrow [0, \infty )$ is called a Γ-distance if

(Γ1):

$\varGamma (p,q,r) \leq \varGamma (p,s,s)+ \varGamma (s,q,r)$ for all $p,q,r \in T$,

(Γ2):

for each $p\in T$, the functions $\varGamma (p,\cdot , \cdot ) : T\rightarrow [0,\infty )$ are lower semicontinuous,

(Γ3):

for every $\varepsilon > 0$, there exists $\delta > 0$ such that $\varGamma (p,s,s) \leq \delta$ and $\varGamma (s,q,r) \leq \delta$ imply $F(p,q,r) \leq \varepsilon$.

It is easy to see that if the functions $\varGamma (p,\cdot ,\cdot ) : T \rightarrow [0,\infty )$ are lower semicontinuous, then the functions $\varGamma (p,q,\cdot ), \varGamma (p,\cdot ,q) : T \rightarrow [0,\infty )$ are lower semicontinuous, also we conclude that if $q\in T$ and $\{u_{m}\}$ is a sequence in T which converges to a point $p\in T$ (with respect to the quasi-F-metric) and $\varGamma (q,u_{m},u _{m})\le K$ for some $K = K(q) > 0$, then $\varGamma (q,p,p)\le K$.

Example 1.3

Let $T = \mathbb{R}$ and $F : T^{3}\longrightarrow [0,\infty )$. Define

$$ F(p,q,r) = \frac{1}{2}\bigl( \vert r - p \vert + \vert p - q \vert \bigr). $$

Then F is a q-F-m.

Example 1.4

The function $\varGamma :=F$, given in the above example, is a Γ-distance.

Proof

The proofs of (Γ1) and (Γ2) are obvious. For (Γ3), let $\epsilon > 0$, and put $\delta = \frac{\epsilon }{2}$. If

$$ \varGamma (p,s,s) = \frac{1}{2} \bigl( \vert r-s \vert + \vert s-q \vert \bigr) < \frac{\epsilon }{2}, $$

then

$$ F(p,q,r)= \frac{1}{2}\bigl( \vert r-p \vert + \vert p-q \vert \bigr)\leqslant \frac{1}{2}\bigl( \vert r-s \vert + \vert s-p \vert + \vert p-s \vert + \vert s-q \vert \bigr)< \epsilon . $$

 □

Example 1.5

Let $T = \mathbb{R}$ and $F : T^{3}\rightarrow [0,\infty )$ be a q-F-m defined as

$$ F(p,q,r)= \textstyle\begin{cases} 0, & p=q=r, \\ \vert r-p \vert , & \text{otherwise}. \end{cases} $$

Then the function $\varGamma : T^{3} \rightarrow [0,\infty )$ defined by $\varGamma (p,q,r) = |r-p|$ for each $q,r\in T$ is a Γ-distance. But it is not a q-F-m on T.

Proof

The proofs of (Γ1) and (Γ2) are obvious. For (Γ3), let $\epsilon > 0$, and put $\delta = \frac{\epsilon }{2}$. If

$$ \varGamma (p,s,s) = \vert s-p \vert < \frac{\epsilon }{2} $$

and

$$ \varGamma (s,q,r) = \vert r-s \vert < \frac{\epsilon }{2}, $$

then

$$ F(p,q,r)= \vert r-p \vert \leqslant \vert r-s \vert + \vert s-p \vert < \epsilon . $$

 □

Example 1.6

Let $T = \mathbb{R}$ and $F : T^{3}\longrightarrow [0,\infty )$ be a q-F-m defined as in Example 1.3. Then the function $\varGamma : T^{3} \rightarrow [0,\infty )$ defined by $\varGamma (p,q,r) = a$ for each $p,q,r\in T$, in which $a>0$, is a Γ-distance.

Proof

The proofs of (Γ1) and (Γ2) are obvious. For (Γ3), let $\epsilon > 0$, and put $\delta = \frac{a}{2}$. Then we have that

$$ \varGamma (p,s,s) < \frac{a}{2} $$

and

$$ \varGamma (s,q,r) < \frac{a}{2}, $$

which imply that

$$ F(p,q,r)\le \epsilon . $$

 □

Remark 1.7

([10])

Let Γ be a Γ-distance. If ξ from $\mathbb{R_{+}}$ to $\mathbb{R_{+}}$ is a decreasing and sub-additive function with $\xi (0)=0$, then $\xi \circ \varGamma$ is a Γ-distance.

Now, we present some properties of Γ-distance.

Lemma 1.8

([9])

Let $\{u_{n}\}$, $\{v_{n}\}$ be two sequences in T and $\{\rho _{n} \}$, $\{\varphi _{n}\}$ be nonnegative sequences converging to 0, and let $p, q, r, s \in T$. Then we have

• (1) $\varGamma (q,u_{n},u_{n}) \leq \rho _{n}$ and $\varGamma (u _{n},q,r) \leq \varphi _{n}$ for all $n\in \mathbb{N}$ imply that $F(q,q,r) < \varepsilon$ and $q=r$;

• (2) $\varGamma (v_{n},u_{n},u_{n}) \leq \rho _{n }$ and $\varGamma (u_{n},v_{m},r) \leq \rho _{n}$ for any $m > n \in \mathbb{N}$ imply that $F(v_{n},v_{m},r) \rightarrow 0$ and hence $v_{n} \rightarrow r$;

• (3) if $\varGamma (u_{n},u_{m},u_{\ell }) \leq \rho _{n}$ for all $m,n,\ell \in \mathbb{N}$ with $\ell \leq n \leq m$, then $\{u_{n}\}$ is an F-Cauchy sequence;

• (4) if $\varGamma (u_{n},s,s) \leq \rho _{n}$ for all $n\in \mathbb{N}$, then the sequence $\{u_{n} \}$ is an F-Cauchy sequence.

Definition 1.9

([2])

Let T have a binary relation ≼.

• (i) If the relation ≼ on T has transitivity and reflexive properties, then it is quasi-order.

• (ii) A sequence $\{u_{n} \}$ in T is said to be decreasing when $u_{n+1} \preccurlyeq u_{n}$ for all $n\in \mathbb{N}$.

• (iii) The relation ≼ is called lower closed when, for each p in T, $Q(p) = \{q \in T : q \preccurlyeq p\}$ is lower closed; in other words, if $\{u_{n} \} \subset Q(p)$ is decreasing and converges to $\tilde{p} \in T$, then $\tilde{p} \in Q(p)$.

Definition 1.10

Suppose that $(T,F)$ is a q-F-m space quasi-ordered by ≼. Define

$$ Q(p):= \{q \in T : q \preccurlyeq p \}. $$

We say that $Q(p)$ is ≼-complete when every decreasing (with respect to ≼) F-Cauchy sequence of elements from $Q(p)$ converges in $Q(p)$.

Definition 1.11

A function $g: T\rightarrow \mathbb{R}\cup \{+\infty \}$ is lower semicontinuous from above (in short, lsca) if, for every sequence $\{u_{n}\}_{n \in \mathbb{N}} \subset T$ converging to $p\in T$ and satisfying $g(u_{n+1}) \leqslant g(u_{n})$ for all $n\in \mathbb{N}$, we have $g(p) \leqslant \lim_{n \rightarrow \infty } g(u_{n})$.

2 Ekeland’s variational principle (EVP)

Here, we give two generalizations of EVP by using the concept of Γ-distance, both in the incomplete and the complete q-F-m spaces.

Theorem 2.1

Assume that $\varGamma : T\times T \times T \longrightarrow \mathbb{R} _{+}$ is a Γ-distance on a q-F-m space $(T,F)$ (not necessarily complete). Let $\omega : (-\infty ,\infty ] \rightarrow (0, \infty )$ be an increasing function and $g : T \rightarrow \mathbb{R} \cup \{\infty \}$ be lsca, bounded from below and proper. The relation ≼ defined by

$$ q \preccurlyeq p\quad \textit{if and only if}\quad p=q\quad \textit{or}\quad \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) $$

(2.1)

is quasi-order. Further, assume that there exists $\hat{p}\in T$ such that $\inf_{p \in T}g(p)< g(\hat{p})$ and $Q(\hat{p})=\{q \in T:q \preccurlyeq \hat{p} \}$ are ≼-complete. Then we can find $\bar{p} \in T$ such that

• (a) $\varGamma (\hat{p}, \bar{p},\bar{p}) \leqslant \omega (g( \hat{p})(g(\hat{p})-g(\bar{p}))$,

• (b) $\varGamma (\bar{p},p,p) > \omega (g(\bar{p}))(g(\bar{p})-g(p))$, $p \in T$, $p \neq \bar{p}$.

Proof

Reflexivity is obvious. We prove that ≼ is transitive. Let $r\preccurlyeq q$ and $q \preccurlyeq p$. Then we have

$$\begin{aligned}& r=q\quad \text{or}\quad \varGamma (q,r,r) \leqslant \omega \bigl(g(q)\bigr) \bigl(g(p) - g(r)\bigr), \end{aligned}$$

(2.2)

$$\begin{aligned}& q=p\quad \text{or}\quad \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr). \end{aligned}$$

(2.3)

If $r=q$ or $p=q$, then transitivity is confirmed. Let $p \neq q \neq r$. Since $\varGamma (p,q,r)\geqslant 0$ and $\omega (p) > 0$, from (2.2) and (2.3), we get $g(q)\geqslant g(r)$ and $g(p)\geqslant g(q)$, i.e., $g(r)\leqslant g(q)\leqslant g(p)$. Since ω is increasing, we get $\omega (g(q)) \leqslant \omega (g(p))$. By using (Γ1), (2.2), and (2.3), we obtain

$$\begin{aligned} \varGamma (p,r,r) &\leqslant \varGamma (p,q,q) + \varGamma (q,r,r) \\ &\leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) + \omega \bigl(g(q) \bigr) \bigl(g(q)-g(r)\bigr) \\ &\leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) + \omega \bigl(g(p) \bigr) \bigl(g(q)-g(r)\bigr) \\ &=\omega \bigl(g(p)\bigr) \bigl(g(p)\bigr)-g(r)). \end{aligned}$$

Thus $r\preccurlyeq p$, that is, ≼ is quasi-order on T.

Now, a sequence $\{u_{n}\}$ in $Q(\hat{p})$ is constructed as follows. Let

$$\begin{aligned} Q(u_{n}) &= \bigl\{ q \in Q(\hat{p}): q= u_{n} \text{ or } \varGamma (u_{n},q,q) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(q)\bigr)\bigr\} \\ &=\bigl\{ q \in Q(\hat{p}): q\preccurlyeq u_{n}\bigr\} . \end{aligned}$$

Put $\hat{p}= u_{0}$ and choose $u_{2} \in Q(u_{1})$ so that $g(u_{2}) \leqslant \inf_{p \in Q(u_{1})} g(p) + \frac{1}{2}$. Suppose that $u_{n-1} \in T$ is defined and choose $u_{n} \in Q(u_{n-1})$ so that

$$ g(u_{n}) \leqslant \inf_{p \in Q(u_{n-1})} g(p) + \frac{1}{n}. $$

(2.4)

Since $u_{n} \in Q(u_{n-1})$, we have $u_{n} \preccurlyeq u_{n-1}$, and $\{u_{n}\}$ is decreasing. Also

$$ \varGamma (u_{n-1},u_{n},u_{n}) \leqslant \omega (g(u_{n-1}) \bigl(g(u_{n-1})- g(u_{n})\bigr). $$

Hence $g(u_{n}) \leqslant g(u_{n-1})$ for all $n \in \mathbb{N}$, that is, $\{g(u_{n})\}$ is decreasing. Also, g is bounded from below, so $\{g(u_{n})\}$ is convergent. Let $\lim_{n \rightarrow \infty } g(u _{n})= w$. Also, we prove that the sequence $\{u_{n}\}$ is F-Cauchy in $Q(\hat{p})$. Assume that $n < m$. Then we have

$$\begin{aligned} \varGamma (u_{n},u_{m},u_{m})\leqslant{}& \varGamma (u_{n},u_{n+1},u_{n+1}) + \varGamma (u_{n+1},u_{m},u_{m}) \\ \leqslant{}& \varGamma (u_{n},u_{n+1},u_{n+1}) + \varGamma (u_{n},u_{n+2},u _{n+2}) +\cdots + \varGamma (u_{n+1},u_{m},u_{m}) \\ \leqslant{}& \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})- g(u_{n+1})\bigr) + \omega \bigl(g(u_{n+1})\bigr) \bigl(g(u _{n+1})- g(u_{n+2})\bigr) \\ & {}+\cdots + \omega \bigl(g(u_{m-1})\bigr) \bigl(g(u_{m-1})- g(u_{m})\bigr) \\ \leqslant{}& \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})- g(u_{n+1})\bigr) + \omega \bigl(g(u_{n})\bigr) \bigl(g(u _{n+1})- g(u_{n+2})\bigr) \\ &{} +\cdots +\omega \bigl(g(u_{n})\bigr) \bigl(g(u_{m-1})- g(u_{m})\bigr) \\ \leqslant{}& \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})- g(u_{m})\bigr) \leqslant \omega \bigl(g(u _{n})\bigr) \bigl(g(u_{n})- w\bigr). \end{aligned}$$

Put $\rho _{n}= \omega (g(u_{n}))(g(u_{n})- w)$. Then $\lim_{n\rightarrow \infty } \rho _{n} = 0$, and according to Lemma 1.8(3), the sequence $\{u_{n}\}$ is nonincreasing and F-Cauchy in $Q(\hat{p})$. ≼-completeness $Q(\hat{p})$ implies that $\{u_{n}\}$ converges to a point $\bar{p} \in P(\hat{p})$. From transitivity of ≼, we conclude $Q(u_{n}) \subset Q(u _{n-1})$ for all $n\in \mathbb{N}$.

Now we are ready to show that $\{\bar{p}\}= Q(\bar{p})$. Assume that $p \in Q(\bar{p})$, and $p \neq \bar{p}$. Then $\varGamma (\bar{p},p,p) \leqslant \omega (g(\bar{p}))(g(\bar{p})-g(p))$. Since Γ is nonnegative and $\omega \geqslant 0$, we conclude that $g(p) \leqslant g(\bar{p})$.

Since $\bar{p}\in Q(\hat{p}) = Q(u_{0})$, we have $\bar{p} \in Q(u _{n-1})$ for all $n\in \mathbb{N}$. Thus $p \preccurlyeq \bar{p}$ and $\bar{p}\preccurlyeq u_{n-1}$, and so $p \preccurlyeq u_{n-1}$ (transitivity of ≼) for $n \in \mathbb{N}$. Also, we have $g(\bar{p}) \leqslant g(u_{n}) \leqslant g(p)+\frac{1}{n}$ and $\lim_{n \rightarrow \infty } g(u_{n}) = w$. Hence $g(\bar{p}) \leqslant w \leqslant g(p) \leqslant g(\bar{p})$ and so $g(\bar{p}) = w = g(p)$. Since $p \preccurlyeq u_{n}$ for all $n \in \mathbb{N}$, we get

$$ \varGamma (u_{n},p,p) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(p)\bigr)= \omega (g(u_{n}) \bigl(g(u_{n})-w\bigr)=\rho _{n}. $$

(2.5)

Also, $\bar{p} \preccurlyeq u_{n}$ for all $n\in \mathbb{N}$. Thus we have

$$ \varGamma (u_{n},\bar{p},\bar{p}) \leqslant \omega (g(u_{n}) \bigl(g(u_{n})-g( \bar{p})\bigr)= \omega (g(u_{n}) \bigl(g(u_{n})-w\bigr)=\rho _{n} $$

(2.6)

and $\lim_{n\rightarrow \infty }\rho _{n}=0$. By using (2.5), (2.6), and Lemma 1.8(1), we conclude that $p =\bar{p}$, $\{\bar{p}\}=Q(\bar{p})$, and so we have $\varGamma ( \bar{p},p,p) > \omega (g(\bar{p}))(g(\bar{p})-g(p))$ for all $p \in T$ and $p \neq \bar{p}$. □

Theorem 2.2

Assume that $(T,F)$ is a complete q-F-m space and that $\varGamma : T \times T \times T \longrightarrow \mathbb{R}_{+}$ is a Γ-distance on Z. Let $\omega : (-\infty ,\infty ] \rightarrow (0, \infty )$ be an increasing function, and $g : T \rightarrow \mathbb{R} \cup \{\infty \}$ be lsca, bounded from below, and proper. Let $\hat{p} \in T$ in which $\inf_{p \in T}g(p)< g(\hat{p})$. Then we can find $\bar{p} \in T$ such that

• (a) $\varGamma (\hat{p}, \bar{p},\bar{p}) \leqslant \omega (g(\hat{p})(g(\hat{p})-g(\bar{p}))$,

• (b) $\varGamma (\bar{p},p,p) > \omega (g(\bar{p}))(g( \bar{p})-g(p))$, $p \in T$, $p \neq \bar{p}$.

Proof

Define a relation ≼ by

$$ q \preccurlyeq p \quad \text{if and only if}\quad p=q\quad \text{or}\quad \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr). $$

(2.7)

In the proof of the previous theorem, we proved that ≼ is quasi-order. Now we are ready to show that ≼ is lower closed. According to Definition 1.9, assume that the sequence $\{u_{n}\}_{n \in \mathbb{N}}$ is decreasing in T, which converges to p, and $u_{n+1} \preccurlyeq u_{n}$. We have

$$ \varGamma (u_{n},u_{n+1},u_{n+1}) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(u _{n+1})\bigr). $$

(2.8)

Since $\varGamma \geqslant 0$ and $\omega \geqslant 0$, we have $g(u_{n+1}) \leqslant g(u_{n})$, and so $\{g(u_{n})\}$ is a decreasing sequence. Since g is bounded from below, we have that $\lim_{n\rightarrow \infty }g(u_{n})$ is finite. Let $\lim_{n\rightarrow \infty }g(u_{n})= w$. Then $w \leqslant g(u_{n})$ for all $n\in \mathbb{N}$. Since g is lsca, we conclude that $g(p) \leqslant \lim_{n\rightarrow \infty }g(u_{n})$, and so we get $g(p) \leqslant w \leqslant g(u_{n})$.

Assume that $n\in \mathbb{N}$ is fixed. For all $m\in \mathbb{N}$, where $m>n$, similar to the proof of Theorem 2.1, we get

$$ \varGamma (u_{n},u_{m},u_{m}) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(u _{m})\bigr) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(u) \bigr). $$

Therefore, we conclude that $g(p) \leqslant g(u_{n})$ for all $n\in \mathbb{N}$. Let $K = \omega (g(u_{n}))(g(u_{n})-g(u))$. According to (Γ2), we have $\varGamma (u_{n},u_{m},u_{m}) \leqslant K$ and then $\varGamma (u_{n},p,p) \leqslant K$ for $n \in \mathbb{N}$. Then, for all $n\in \mathbb{N}$, we get $\varGamma (u_{n},p,p) \leqslant K = \omega (g(u_{n}))(g(u_{n})-g(u))$. So $p\preccurlyeq u_{n}$ and we conclude that ≼ is lower closed for all $r \in T$. Also $Q(r)=\{q \in T: q \preccurlyeq r\}$ is lower closed. The sequence $\{u_{n}\}$ is constructed as follows:

$$\begin{aligned} Q(u_{n}) &= \bigl\{ q \in T: q=u_{n} \text{ or }\varGamma (u_{n},q,q) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(q)\bigr)\bigr\} \\ & =\{q \in T: q\preccurlyeq u_{n}\}. \end{aligned}$$

Then, for all $n\in \mathbb{N}$, $Q(u_{n})$ is a lower closed subset of a complete q-F-m space and therefore ≼-complete. The assertion concludes from Theorem 2.1. □

Corollary 2.3

Assume that g, Γ, T, and ω are the same as in Theorem 2.2. Let $\xi : \mathbb{R_{+}}\longrightarrow \mathbb{R_{+}}$ be increasing and sub-additive with $\xi (0)=0$. If there is $\hat{p} \in T$, such that $\inf_{p\in Z}g(p) < g(\hat{p})$, then there is $\bar{p} \in T$ such that

• (a) $\xi (\varGamma (\hat{p},\bar{p},\bar{p})) \leqslant \omega (g( \hat{p})(g(\hat{p})-g(\bar{p}))$,

• (b) $\xi (\varGamma (\bar{p},p,p)) > \omega (g(\bar{p})(g(\bar{p})-g(p))$ for all $p \in T$, $p \neq \bar{p}$.

Proof

From Remark 1.7, $\xi \circ \varGamma$ is a Γ-distance on T. So, by Theorem 2.2, we obtain the conclusion. □

3 Equivalences

Theorem 3.1

Assume that $(T,F)$ is a complete q-F-m space. Let $\varGamma : T \times T\times T \rightarrow \mathbb{R_{+}}$ be a Γ-distance on T, $\omega : (-\infty , \infty ]\rightarrow (0, \infty )$ be an increasing function and g be lsca, proper, and bounded from below. Then the following statements are equivalent to Theorem 2.2:

• (i) (Caristi–Kirk fixed point theorem). Let $P : T \rightarrow 2^{T}$ be a multi-valued mapping with nonempty values. If the following condition

  $$ \textit{for each } q \in P(p) ,\quad \varGamma (p,q,q)\leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(p)\bigr) $$

  (3.1)

  is satisfied, then we can find $\bar{p}\in T$ such that $\{\bar{p}\}= P(\bar{p})$. If the following condition

  $$ \textit{there is }q \in P(p)\textit{ such that } \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) $$

  (3.2)

  is satisfied, then we can find $\bar{p} \in T$ such that $\bar{p} \in P(\bar{p})$.

• (ii) (Takahashi’s minimization theorem). Assume that, for all $\hat{p} \in T$ with $\inf_{r \in T} g(r) < g(\hat{p})$, there is $p \in T$ such that

  $$ p \neq \hat{p}\quad \textit{and}\quad \varGamma (\hat{p},p,p) \leqslant \omega \bigl(g(\hat{p})\bigr) \bigl(g(\hat{p})- g(p)\bigr). $$

  (3.3)

  Then we can find $\bar{p} \in T$ such that $g(\bar{p})= \inf_{q \in T}g(q)$.

• (iii) (Equilibrium version of EVP). Let $G :T\times T \rightarrow \mathbb{R} \cup \{\infty \}$ be a function satisfying:

  ($E_{1}$):

  for every $p, q,r \in T$, $G(p,r) \leqslant G(p, q)+G(q,r)$;

  ($E_{2}$):

  for all fixed $p \in T$, the function $G(p,\cdot) : T \rightarrow \mathbb{R}\cup \{\infty \}$ is proper and lsca;

  ($E_{3}$):

  there is $p \in T$ such that $\inf_{p \in T} G( \hat{p},p) > -\infty$.

  Then we can find $\bar{p}\in T$ such that

  • (A) $\omega (g(\hat{p}) G(\hat{p},\bar{p}) + \varGamma (\hat{p}, \bar{p},\bar{p})\leqslant 0$,

  • (B) $\omega (g(\bar{p}) G(\bar{p},p) + \varGamma (\bar{p},p,p) > 0$ for all $p \in T$, $p \neq \bar{p}$.

Proof

Assertion (i) follows from Theorem 2.2. By Theorem 2.2(b), there exists $\bar{p} \in T$ such that

$$ \varGamma (\bar{p},p,p)> \omega \bigl(g(\bar{p})\bigr) \bigl(g( \bar{p}) - g(p)\bigr)\quad \text{for all } p \in T, p\neq {\bar{p}}. $$

(3.4)

We prove that $\{\bar{p}\}= T(\bar{p})$ (respectively, $\bar{p} \in T(\bar{p})$). On the contrary, assume that $q \in P(\bar{p})$ and $q \neq \bar{q}$. Then, by (3.1), $\varGamma (\bar{p},q,q) \leqslant \omega (g(\bar{p}))(g(\bar{p})-g(q))$, and by (3.4), $\varGamma (\bar{p},q,q) > \omega (g(\bar{p}))(g(\bar{p})-g(q))$. Therefore $\{\bar{p}\}= P( \bar{p})$ (respectively, $\bar{p}\in P(\bar{p})$).

(i) ⇒ (ii): Let $P:T\to 2^{T}$. Then we define $P(p)= \{q \in T: \varGamma (p,q,q) \leqslant \omega (g(p))(g(p)-g(q))\}$ for every $p \in T$. Then P has property (3.1). By (i), there exists $\bar{p} \in T$ such that $\{\bar{p}\}= P(\bar{p})$. Moreover, by assumption, there exists $p \in T$ such that $p \neq\hat{p}$ and $\varGamma (\hat{p},p,p) \leqslant \omega (g(\hat{p}))(g(\hat{p})-g(p))$ for all $\hat{p} \in T$ when $\inf_{r \in T} g(r)< g(\hat{p})$. Therefore, $p \in T(\hat{p})$ and $P(\hat{p}) \setminus \{\hat{p}\} \neq \emptyset$. Hence $g(\bar{p})= \inf_{p \in T}g(p)$.

(ii) ⇒ (iii): Let $g: T\to \mathbb{R}\cup \{\infty \}$. Then we define $g(p)=G(\hat{p},p)$, where p̂ is the same as in $(E_{3})$. Then from $(E_{3})$ we get $\inf_{p \in T}g(p)> -\infty$, and so g is bounded from below. Assume that (A) is false. So, for all $p \in T$, we can find $q \in T$ such that

$$ q \neq p\quad \text{and} \quad \omega \bigl(g(p)\bigr)G(x,y) + \varGamma (p,q,q) \leqslant 0. $$

(3.5)

By $(E_{1})$, we get $G(\hat{p},q)\leqslant G(\hat{p},p) + G(p,q)$, i.e., $G(\hat{p},q)- G(\hat{p},p) \leqslant G(p,q)$.

Then by (3.5) we get

$$ \omega \bigl(g(p)\bigr) \bigl(G(\hat{p},q)-G(\hat{p},p)\bigr) + \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr)G(p,q) + \varGamma (p,q,q) \leqslant 0. $$

(3.6)

So, for every $p \in T$, we can find $q \in T$ such that $q \neq p$ and $\omega (g(p))(g(q)-g(p)) + \varGamma (p,q,q) \leqslant 0$. Also, $\varGamma (p,q,q) \leqslant \omega (g(p))(g(q)-g(p))$.

Now, by (ii), $g(\bar{p})=\inf_{q \in T}g(q)\leqslant g(r)$. Replace p by p̄ in the last relation. Then there exists $q \in T$ such that $q \neq p$ and $\omega (g(\bar{p}))(G(\hat{p},q)-G(\hat{p}, \bar{p})) + \varGamma (\bar{p},q,q)\leqslant 0$, that is,

$$ \omega \bigl(g(\bar{p})\bigr) \bigl(g(q)-g(\bar{p})\bigr)+ \varGamma (\bar{p},q,q)\leqslant 0 \quad \text{or}\quad \varGamma (\bar{p},q,q)\leqslant \omega \bigl(g(\bar{p})\bigr) \bigl(g( \bar{p})-g(q)\bigr). $$

(3.7)

Since $q \neq p$, by using Lemma 1.8(1), $\varGamma (\bar{p}, \bar{p},\bar{p}) \neq 0$, and $\varGamma (\bar{p},q,q) \neq 0$, we get $\varGamma (\bar{p},q,q)> 0$, and by (3.7), we obtain $0 < \omega (g(\bar{p}))(g(\bar{p})-g(q)) \Rightarrow g(q)< g(\bar{p})$. That is a contradiction.

(iii) ⇒ Theorem 2.2: Let $G: T\times T \rightarrow \mathbb{R}\cup \{\infty \}$ be a function defined by $G(p,q)= g(q)-h(p)$ for all $p,q \in T$. According to Theorem 2.2, G satisfies all the conditions of (iii). By (A), we get

$$ \omega \bigl(g(\hat{p})\bigr)G(\hat{p},\bar{p})+ \varGamma (\hat{p},\bar{p}, \bar{p})\leqslant 0\quad \Rightarrow \quad \omega \bigl(g(\hat{p})\bigr) \bigl(g(\bar{p})-g( \hat{p})\bigr)+ \varGamma (\hat{p},\bar{p},\bar{p})\leqslant 0. $$

Then

$$ \varGamma (\hat{p},\bar{p},\bar{p})\leqslant \omega \bigl(g(\hat{p})\bigr) \bigl(g( \hat{p})-g(\bar{p})\bigr). $$

Also, by (B), we get $\omega (g(\bar{p}))G(\bar{p},p)+ \varGamma ( \bar{p},p,p)>0$ for all $p \in T$, $p \neq \bar{p}$. Then

$$ \omega \bigl(g(\bar{p})\bigr) \bigl(g(p)-g(\bar{p})\bigr) + \varGamma ( \bar{p},p,p)>0\quad \Rightarrow\quad \varGamma (\bar{p},p,p)> \omega \bigl(g(\bar{p})\bigr) \bigl(g(\bar{p})-g(p)\bigr) $$

for all $p \in T$, $p \neq \bar{p}$. □

Corollary 3.2

Let g, Γ, T, ω be the same as in Theorem 3.1 and suppose that $\xi : \mathbb{R_{+}} \rightarrow \mathbb{R_{+}}$ is a subadditive and increasing function such that $\xi (0)=0$. Assume that $P: T\rightarrow 2^{T}$ is a multi-valued mapping with nonempty values. If, for all $p\in T$, there is $q \in P(p)$ such that

$$ \xi \bigl(\varGamma (p,q,q)\bigr) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)- g(q)\bigr), $$

then P has a fixed point in T.

Proof

Note that $\xi \circ \varGamma$ is a Γ-distance on T by Remark 1.7. Then, by Theorem 3.1(i), P has a fixed point in T. □

Corollary 3.3

Suppose that $(T,F)$ is a complete q-F-m space. Let $\varGamma : T \times T\times T \rightarrow \mathbb{R_{+}}$ be a Γ-distance on T and $G: T\times T \rightarrow \mathbb{R}$ be a function satisfying the conditions:

$(F_{1})$ :

$G(p,r)\leqslant G(p,q)+ G(q,r)$ for all $p,q,r \in T$;

$(F_{2})$ :

for every constant $p \in T$, the function $G(p,\cdot) : T\rightarrow \mathbb{R}$ is lsca and bounded from below. Then, for each $\epsilon > 0$ and every $\hat{p}\in T$, there exists $\bar{p}\in T$ such that

• (C) $G(\hat{p},\bar{p})+ \epsilon \varGamma (\hat{p},\bar{p}, \bar{p})\leqslant 0$;

• (D) $G(\bar{p},p) + \epsilon \varGamma (\bar{p},p,p) >0$ for all $p\in T$, $p\neq\bar{p}$.

Proof

Let $g: T\to \mathbb{R}\cup \{\infty \}$. Then we define $g(\hat{p})= G(p,\hat{p})$ for all $\hat{p}\in T$ and fixed $p\in T$. Then, by Theorem 3.1(iii), (C) and (D) are established. □

Corollary 3.4

Let $G:T\times T\rightarrow (-\infty , \infty )$ be proper, lsca, and bounded from below in the first argument and $\omega : (-\infty , \infty ) \rightarrow (0, \infty )$ be nondecreasing. Assume that, for every $p \in T$ with $\{ x \in T : G(p,x)< 0 \}\neq \emptyset$, there exists $q=q(p) \in T$ with $q \neq p$ such that

$$ \varGamma (p,q,q)) \leqslant \omega \bigl(G(p,t)\bigr) \bigl(G(p,t)- G(q,t) \bigr) $$

for all $t\in \{p \in T: G(x,p)> \mathrm{inf}_{a\in T} G(a,p)\}$. Then there exists $y \in T$ such that $G(y,p)\geqslant 0$ for all $q \in T$.

Proof

By Theorem 2.2(b), for all $r \in T$, there exists $y(r) \in T$ such that

$$ \varGamma \bigl(y(r),q,q\bigr)) > \omega \bigl(G\bigl(y(r),r\bigr)\bigr) \bigl(G \bigl(y(r),r\bigr)- G(q,r)\bigr) $$

for all $q \in T$ and $p\neq y(r)$. We show that there exists $y \in T$ such that $G(y,q)\geqslant 0$ for all $q\in T$. Suppose it is false. Then, for all $p \in T$, there exists $q \in T$ such that $G(p,q)< 0$, and thus $\{x \in T: G(p,x)<0\}\neq$ ∅. Then, according to the assumption, there exists $q=q(y(r))$, $q\neq y(r)$ such that

$$ \varGamma \bigl(y(r),q,q\bigr)) \leqslant \omega \bigl(G\bigl(y(r),r\bigr)\bigr) \bigl(G\bigl(y(r),r\bigr)- G(q,r)\bigr) , $$

which is a contradiction. □

Example 3.5

Let $T=[0,1]$ and $F(p,q,r)=\frac{1}{2} \max \{|p-q|, |p-r|, |q-r|\}$. So $(T,F)$ is a complete q-F-m. Assume that $G: T\times T \rightarrow \mathbb{R}$ is defined by $G(p,q)=3p-2q$. Then the function $x \rightarrow G(p,q)$ is proper, lsca, and bounded from below. Also, for every $q \in T$, $G(1,q)\geqslant 0$ and for all $p \in [ \frac{2}{3},1]$, $G(p,q)\geqslant 0$ for all $q\in T$. On the other hand, when $p\in [0,\frac{2}{3}]$ and $q\in [\frac{3}{2}p,1]$, we have $G(p,q)=3p-2q <0$. Then $\{x\in T, G(p,x) < 0\}\neq \emptyset$. Let $p,q \in T$ and $p\geqslant q$. Then we have $p-q=\frac{1}{3}\{(3p-2x)-(3q-2x) \}$ for all $x\in T$. Suppose that $\omega : [0, \infty ) \rightarrow [0, \infty )$ with $\omega (t)=\frac{1}{3}$. Then

$$ F(p,q,q)) \leqslant \omega \bigl(G(p,x)\bigr) \bigl(G(p,x)- G(q,x)\bigr) $$

for all $p\geqslant q$. By Corollary 3.4, there exists $y\in T$ such that $G(y,p)\geqslant 0$ for all $p\in T$.

4 Equilibrium problem

The EP (equilibrium problem) is a new research subject in nonlinear science and engineering [11].

Definition 4.1

Suppose that S is a nonempty subset of a metric space T, $G:S\times S\rightarrow \mathbb{R}$ is a function on $\mathbb{R}$, and Γ is a Γ-distance on T. Let $\delta >0$. If there is $\bar{p}\in T$ such that

$$ G(\bar{p},q)+\delta \varGamma (\bar{p},q,q)\geq 0 \quad \text{for all } q\in S, $$

(4.1)

then p̄ is a δ-solution to EP. Moreover, if (4.1) is satisfied as strict, then p̄ is called a δ-solution to strict EP.

Theorem 4.2

Suppose that $S\neq \emptyset$ is a compact subset of a complete metric space T and that Γ is a Γ-distance. If a real-valued function $G:S\times S\rightarrow \mathbb{R}$ satisfies the following conditions:

$(E_{1})$ :

$G(p,r)\leq G(p,q)+G(q,r)$ for all $p,q,r\in S$;

$(E_{2})$ :

the function $G(p,\cdot ):S \rightarrow \mathbb{R}$ is $lsca$ and bounded from below for each fixed $p\in T$;

$(E_{3})$ :

the function $G(\cdot ,q):S\rightarrow \mathbb{R}$ is upper semicontinuous for each fixed $q\in S$, then we can find a solution $\bar{p}\in S$ to EP.

Proof

By Corollary 3.3, there is $u_{n}\in S$ such that

$$ G(u_{n},q)+\frac{1}{n}\varGamma (u_{n},q,q)\geq 0 \quad \text{for each } q \in S. $$

In other words, for $\epsilon =\frac{1}{n}$, $u_{n}\in S$ is a δ-solution to EP. Since S is compact, there is a subsequence $\lbrace u_{n_{k}}\rbrace$ of $\lbrace u_{n}\rbrace$ such that $u_{n_{k}}\rightarrow \bar{p}$. Since $G(\cdot ,q)$ is upper semicontinuous, we have

$$ G(\bar{p},q)\geq \limsup_{S\rightarrow \infty }\biggl(G(u_{n_{k}},q)+ \frac{1}{n _{k}}\varGamma (u_{n_{k}},q,q)\biggr)\geq 0\quad \text{for all } q \in S. $$

Hence p̄ is a solution to EP. □

Definition 4.3

Assume that $(T,F)$ is a complete q-F-m space and that Γ is a Γ-distance on T. An element $u_{0}\in T$ satisfies the condition $(\varXi )$ if every sequence $\lbrace u_{n}\rbrace$⊂T, satisfying $G(u_{0},u_{n})\leq \frac{1}{n}$ for all $n\in \mathbb{N}$ and $G(u_{n},p)+\frac{1}{n}\varGamma (u_{n},p,p)\geq 0$ for every $p\in T$ and $n\in \mathbb{N}$, has a convergent subsequence.

Theorem 4.4

Suppose that $(T,F)$ is a complete q-F-m space and that Γ is a Γ-distance on T. Let $G:T\times T\longrightarrow \mathbb{R}$ satisfy conditions $(F_{1})$ and $(F_{2})$ of Corollary 3.3 and G be upper semicontinuous in the first variable. If $u_{0}\in T$ satisfies the condition $(\varXi )$, then we can find a solution $\bar{p}\in T$ to EP.

Proof

If in Corollary 3.3 we put $\epsilon =\frac{1}{n}$, then for every $n\in \mathbb{N}$ and for each $u_{0}\in T$, there is $u_{n}\in T$ satisfying the following conditions:

$$ G(u_{0},u_{n})+\frac{1}{n}\varGamma (u_{0},u_{n},u_{n})\leqslant 0 $$

(4.2)

and

$$ G(u_{n},p)+\frac{1}{n}\varGamma (u_{n},p,p)>0\quad \text{for all }p\in T. $$

(4.3)

Since $\varGamma (u_{0},u_{n},u_{n})\geq 0$, by (4.2), we conclude that $G(u_{0},u_{n})\leq 0$ for all $n\in \mathbb{N}$. From (Ξ), there is a subsequence $\lbrace u_{n}\rbrace$ converging to $\bar{p}\in T$. Since $G(\cdot ,p)$ is upper semicontinuous and by (4.3), we get that p̄ is a solution to EP. □

5 A generalization of Nadler’s fixed point theorem

In this section, we are ready to prove Nadler’s fixed point theorem in q-F-m spaces with Γ-distance.

Definition 5.1

Suppose that $(T,F)$ is a q-F-m space. A mapping $P :T\longrightarrow 2^{T}$ is called Γ-contractive if there are a Γ-distance Γ on T and w in $[0,1]$ such that, for all $p,q\in T$ and $x\in P(p)$, there is $y\in P(q)$ satisfying

$$ \varGamma (x,y,y)\leq w\varGamma (x,q,q). $$

Then w ∈ $\mathbb{R}$ is called a Γ-contractive constant. In particular, $g:T\rightarrow T$ is said to be Γ-contractive if there are a Γ-distance on T and $w\in [0,1]$ such that

$$ \varGamma \bigl(g(p),g(q),g(q)\bigr)\leq w\varGamma (p,q,q)\quad \text{for all } p,q\in T. $$

Theorem 5.2

Suppose that $(T,F)$ is a complete q-F-m space, $P:T\rightarrow 2^{T}$ is a Γ-contractive multi-valued mapping, and Γ is a Γ-distance such that, for each p in T, $P(p)$ is a nonempty closed subset. Then there is $\bar{p}\in T$ such that $\bar{p}\in P( \bar{p})$ and $\varGamma (\bar{p},\bar{p},\bar{p})=0$.

Proof

Suppose that Γ is a Γ-distance on T and $w\in [0,1)$ is a Γ-contractive constant. Assume that $u_{0}\in T$ and $u_{1}\in P(u_{0})$ is fixed. Then, by the definition of Γ-contractivity, there exists $u_{2}\in P(u_{1})$ such that

$$ \varGamma (u_{1},u_{2},u_{2})\leq w\varGamma (u_{0},u_{1},u_{1}). $$

In the same way, we make the sequence $\lbrace u_{n}\rbrace$ such that $u_{n+1}\in P(u_{n})$ and

$$ \varGamma (u_{n},u_{n+1},u_{n+1})\leq w\varGamma (u_{n-1},u_{n},u_{n})\quad \text{for all }n\in \mathbb{N}. $$

We have

$$\begin{aligned} \varGamma (u_{n},u_{n+1},u_{n+1}) \leq& w\varGamma (u_{n-1},u_{n},u_{n}) \\ \leq& w^{2}\varGamma (u_{n-2},u_{n-1},u_{n-1}) \\ \vdots& \\ \leq& w^{n}\varGamma (u_{0},u_{1},u_{1}). \end{aligned}$$

Then, for all $m,n\in \mathbb{N}$ with $m>n$, we have

$$\begin{aligned} \varGamma (u_{n},u_{m},u_{m}) \leq& \varGamma (u_{n},u_{n+1},u_{n+1})+ \varGamma (u_{n+1},u_{m},u_{m}) \\ \leq& \varGamma (u_{n},u_{n+1},u_{n+1})+\varGamma (u_{n+1},u_{n+2},u_{n+2}) \\ &{} +\cdots +\varGamma (u_{m-1},u_{m},u_{m}) \\ \leq& w^{n}\varGamma (u_{0},u_{1},u_{1})+w^{n+1} \varGamma (u_{0},u_{1},u _{1}) \\ &{} +\cdots +w^{m-1}\varGamma (u_{m-1},u_{m},u_{m}) \\ =& w^{n}\bigl(1+w+w^{2}+\cdots +w^{m-n-1}\bigr) \varGamma (u_{0},u_{1},u_{1}) \\ \leq& \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1}). \end{aligned}$$

Then the sequence $\{\rho _{n}\}=\{\frac{w^{n}}{1-w}\}$ is a nonnegative sequence on $\mathbb{R}$ tending to 0 as $n\rightarrow \infty$. By Lemma 1.8(3), $\lbrace u_{n}\rbrace$ is an F-Cauchy sequence in T. The sequence $\lbrace u_{n}\rbrace$ is convergent to a $\bar{p}\in T$ since T is complete. Let $n\in \mathbb{N}$. Then we have

$$ \varGamma (u_{n},u_{m},u_{m})\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u _{1}) $$

(5.1)

for all $m>n$.

Let $S =\frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1})$. Then $S\geq 0$. Now, by $(\varGamma 2)$ and $\varGamma (u_{n},u_{m},u_{m})\leq S$, we have $\varGamma (u_{n},\bar{p},\bar{p})\leq S$ for all $n\in \mathbb{N}$. Since n is an arbitrary constant, we have

$$ \varGamma (u_{n},\bar{p},\bar{p})\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u _{1},u_{1}) \quad \text{for all }n\in \mathbb{N}. $$

(5.2)

By the assumption, there is $w_{n}\in P(\bar{p})$ such that

$$\begin{aligned} \varGamma (u_{n},w_{n},w_{n}) &\leq w\varGamma (u_{n-1},\bar{p},\bar{p}) \\ &\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1}). \end{aligned}$$

(5.3)

By (5.2), (5.3), and Lemma 1.8(2), we have that the sequence $\lbrace w_{n}\rbrace$ converges to p̄. Since $P(\bar{p})$ is closed, we have $\bar{u}\in P(\bar{u})$.

Now, we prove that $\varGamma (\bar{u},\bar{u},\bar{u})=0$. Since P is Γ-contractive, there is $v_{1}\in P(\bar{p})$ such that

$$ \varGamma (\bar{p},v_{1},v_{1})\leq w\varGamma (\bar{p}, \bar{p},\bar{p}). $$

Now, we construct a sequence $\lbrace v_{n}\rbrace$ as follows: $v_{n+1}\in P(v_{n})$ and

$$ \varGamma (\bar{p},v_{n+1},v_{n+1})\leq w\varGamma ( \bar{p},v_{n},v_{n})\quad \text{for all }n\in N. $$

Therefore, for all $n\in \mathbb{N}$, we get

$$ \varGamma (\bar{p},v_{n},v_{n})\leq \varGamma (\bar{p},v_{n-1},v_{n-1}) \leq \cdots \leq w^{n} \varGamma (\bar{p},\bar{p},\bar{p}). $$

(5.4)

Since $\varGamma (\bar{p},\bar{p},\bar{p})\geq 0$ and $w^{n}\geq 0$ for all $n\in \mathbb{N}$ and $w^{n}\rightarrow 0$ as $n\rightarrow \infty$, $\lbrace v_{n}\rbrace$ is an F-Cauchy sequence in T according to Lemma 1.8(4).

On the other hand, since T is complete, $\lbrace v_{n}\rbrace$ converges to $\bar{q}\in T$. Let $S=\sup_{n\in \mathbb{N}}w^{n}\varGamma (\bar{p}, \bar{p},\bar{p})$. Then, from (5.4) and $(\varGamma _{2})$, we have

$$ \varGamma (\bar{p},v_{n},v_{n})\leq S \quad \Longrightarrow\quad \varGamma (\bar{p}, \bar{q},\bar{q})\leq S=\sup_{n\in \mathbb{N}}w^{n} \varGamma ( \bar{p},\bar{p},\bar{p}). $$

So $\varGamma (\bar{p},\bar{q},\bar{q})\leq 0$ and $\varGamma (\bar{p}, \bar{q},\bar{q})=0$. Moreover, we have

$$\begin{aligned} \varGamma (u_{n},\bar{q},\bar{q}) &\leq \varGamma (u_{n},\bar{p},\bar{p})+ \varGamma (\bar{p},\bar{q},\bar{q}) \\ &\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1}) \end{aligned}$$

(5.5)

for all $n\in \mathbb{N}$. By (5.2), (5.5), and by Lemma 1.8(1), we obtain $\bar{p}=\bar{q}$ and so $\varGamma ( \bar{p},\bar{p}, \bar{p})=0$. □

As a new approach, one can generalize the results presented in [12,13,14,15,16,17,18,19,20,21] in q-F-m spaces with Γ-distance.