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Generalized Ekeland’s variational principle with applications

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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.

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

  1. (i)

    \(F(p,q,r)=0 \) if and only if \(p=q=r \),

  2. (ii)

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

  3. (iii)

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

  4. (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. (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. (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. (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. (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. (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. (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 .

  1. (i)

    If the relation on T has transitivity and reflexive properties, then it is quasi-order.

  2. (ii)

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

  3. (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 \(g(p) \leqslant \lim_{n \rightarrow \infty } g(u_{n})\).

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

  1. (a)

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

  2. (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

  1. (a)

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

  2. (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

  1. (a)

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

  2. (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. □

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:

  1. (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 \(\bar{p} \in P(\bar{p})\).

  2. (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 \(g(\bar{p})= \inf_{q \in T}g(q)\).

  3. (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

    1. (A)

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

    2. (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 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 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

  1. (C)

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

  2. (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 \(p\in T\).

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

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 . 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.

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Acknowledgements

The authors are thankful to the anonymous referees for giving valuable comments and suggestions which helped to improve the final version of this paper. Also, we wish to thank the area editor for useful comments, especially about Introduction and Definition 1.2, that improved the presentation of the paper.

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All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

Correspondence to Reza Saadati.

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MSC

  • 49J53
  • 47H10
  • 90C33
  • 91B50

Keywords

  • Γ-distance
  • Ekeland’s variational principle
  • Equilibrium problems
  • Quasi-F-metric space