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])
Anzeige
Assume that \(T\neq \emptyset \). A function \(F:T^{3} \rightarrow [0, \infty ) \) is called quasi-F-metric (q-F-m) if The pair \((T, F)\) is called q-F-m space.
(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 \(p,q,r,s \in T\).
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])
Anzeige
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
Then F is a q-F-m.
$$ F(p,q,r) = \frac{1}{2}\bigl( \vert r - p \vert + \vert p - q \vert \bigr). $$
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
then
□
$$ \varGamma (p,s,s) = \frac{1}{2} \bigl( \vert r-s \vert + \vert s-q \vert \bigr) < \frac{\epsilon }{2}, $$
$$ F(p,q,r)= \frac{1}{2}\bigl( \vert r-p \vert + \vert p-q \vert \bigr)\leqslant \frac{1}{2}\bigl( \vert r-s \vert + \vert s-p \vert + \vert p-s \vert + \vert s-q \vert \bigr)< \epsilon . $$
Example 1.5
Let \(T = \mathbb{R} \) and \(F : T^{3}\rightarrow [0,\infty )\) be a q-F-m defined as
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.
$$ F(p,q,r)= \textstyle\begin{cases} 0, & p=q=r, \\ \vert r-p \vert , & \text{otherwise}. \end{cases} $$
Proof
The proofs of (Γ1) and (Γ2) are obvious. For (Γ3), let \(\epsilon > 0\), and put \(\delta = \frac{\epsilon }{2}\). If
and
then
□
$$ \varGamma (p,s,s) = \vert s-p \vert < \frac{\epsilon }{2} $$
$$ \varGamma (s,q,r) = \vert r-s \vert < \frac{\epsilon }{2}, $$
$$ F(p,q,r)= \vert r-p \vert \leqslant \vert r-s \vert + \vert s-p \vert < \epsilon . $$
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
and
which imply that
□
$$ \varGamma (p,s,s) < \frac{a}{2} $$
$$ \varGamma (s,q,r) < \frac{a}{2}, $$
$$ F(p,q,r)\le \epsilon . $$
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
We say that \(Q(p)\) is ≼-complete when every decreasing (with respect to ≼) F-Cauchy sequence of elements from \(Q(p)\) converges in \(Q(p)\).
$$ Q(p):= \{q \in T : q \preccurlyeq p \}. $$
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
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
$$ q \preccurlyeq p\quad \textit{if and only if}\quad p=q\quad \textit{or}\quad \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) $$
(2.1)
(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
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
Thus \(r\preccurlyeq p\), that is, ≼ is quasi-order on T.
$$\begin{aligned}& r=q\quad \text{or}\quad \varGamma (q,r,r) \leqslant \omega \bigl(g(q)\bigr) \bigl(g(p) - g(r)\bigr), \end{aligned}$$
(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)
$$\begin{aligned} \varGamma (p,r,r) &\leqslant \varGamma (p,q,q) + \varGamma (q,r,r) \\ &\leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) + \omega \bigl(g(q) \bigr) \bigl(g(q)-g(r)\bigr) \\ &\leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) + \omega \bigl(g(p) \bigr) \bigl(g(q)-g(r)\bigr) \\ &=\omega \bigl(g(p)\bigr) \bigl(g(p)\bigr)-g(r)). \end{aligned}$$
Now, a sequence \(\{u_{n}\}\) in \(Q(\hat{p})\) is constructed as follows. Let
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
Since \(u_{n} \in Q(u_{n-1})\), we have \(u_{n} \preccurlyeq u_{n-1}\), and \(\{u_{n}\}\) is decreasing. Also
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
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 \(n\in \mathbb{N}\).
$$\begin{aligned} Q(u_{n}) &= \bigl\{ q \in Q(\hat{p}): q= u_{n} \text{ or } \varGamma (u_{n},q,q) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(q)\bigr)\bigr\} \\ &=\bigl\{ q \in Q(\hat{p}): q\preccurlyeq u_{n}\bigr\} . \end{aligned}$$
$$ g(u_{n}) \leqslant \inf_{p \in Q(u_{n-1})} g(p) + \frac{1}{n}. $$
(2.4)
$$ \varGamma (u_{n-1},u_{n},u_{n}) \leqslant \omega (g(u_{n-1}) \bigl(g(u_{n-1})- g(u_{n})\bigr). $$
$$\begin{aligned} \varGamma (u_{n},u_{m},u_{m})\leqslant{}& \varGamma (u_{n},u_{n+1},u_{n+1}) + \varGamma (u_{n+1},u_{m},u_{m}) \\ \leqslant{}& \varGamma (u_{n},u_{n+1},u_{n+1}) + \varGamma (u_{n},u_{n+2},u _{n+2}) +\cdots + \varGamma (u_{n+1},u_{m},u_{m}) \\ \leqslant{}& \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})- g(u_{n+1})\bigr) + \omega \bigl(g(u_{n+1})\bigr) \bigl(g(u _{n+1})- g(u_{n+2})\bigr) \\ & {}+\cdots + \omega \bigl(g(u_{m-1})\bigr) \bigl(g(u_{m-1})- g(u_{m})\bigr) \\ \leqslant{}& \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})- g(u_{n+1})\bigr) + \omega \bigl(g(u_{n})\bigr) \bigl(g(u _{n+1})- g(u_{n+2})\bigr) \\ &{} +\cdots +\omega \bigl(g(u_{n})\bigr) \bigl(g(u_{m-1})- g(u_{m})\bigr) \\ \leqslant{}& \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})- g(u_{m})\bigr) \leqslant \omega \bigl(g(u _{n})\bigr) \bigl(g(u_{n})- w\bigr). \end{aligned}$$
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
Also, \(\bar{p} \preccurlyeq u_{n}\) for all \(n\in \mathbb{N}\). Thus we have
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 \(p \neq \bar{p}\). □
$$ \varGamma (u_{n},p,p) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(p)\bigr)= \omega (g(u_{n}) \bigl(g(u_{n})-w\bigr)=\rho _{n}. $$
(2.5)
$$ \varGamma (u_{n},\bar{p},\bar{p}) \leqslant \omega (g(u_{n}) \bigl(g(u_{n})-g( \bar{p})\bigr)= \omega (g(u_{n}) \bigl(g(u_{n})-w\bigr)=\rho _{n} $$
(2.6)
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
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
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 \(g(p) \leqslant w \leqslant g(u_{n})\).
$$ q \preccurlyeq p \quad \text{if and only if}\quad p=q\quad \text{or}\quad \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr). $$
(2.7)
$$ \varGamma (u_{n},u_{n+1},u_{n+1}) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(u _{n+1})\bigr). $$
(2.8)
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
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:
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. □
$$ \varGamma (u_{n},u_{m},u_{m}) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(u _{m})\bigr) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(u) \bigr). $$
$$\begin{aligned} Q(u_{n}) &= \bigl\{ q \in T: q=u_{n} \text{ or }\varGamma (u_{n},q,q) \leqslant \omega \bigl(g(u_{n})\bigr) \bigl(g(u_{n})-g(q)\bigr)\bigr\} \\ & =\{q \in T: q\preccurlyeq u_{n}\}. \end{aligned}$$
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
\(p \in T\), \(p \neq \bar{p}\).
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
is satisfied, then we can find
\(\bar{p}\in T\)
such that
\(\{\bar{p}\}= P(\bar{p})\). If the following condition
is satisfied, then we can find
\(\bar{p} \in T\)
such that
\(\bar{p} \in P(\bar{p})\).
$$ \textit{for each } q \in P(p) ,\quad \varGamma (p,q,q)\leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(p)\bigr) $$
(3.1)
$$ \textit{there is }q \in P(p)\textit{ such that } \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)-g(q)\bigr) $$
(3.2)
(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
Then we can find
\(\bar{p} \in T\)
such that
\(g(\bar{p})= \inf_{q \in T}g(q)\).
$$ p \neq \hat{p}\quad \textit{and}\quad \varGamma (\hat{p},p,p) \leqslant \omega \bigl(g(\hat{p})\bigr) \bigl(g(\hat{p})- g(p)\bigr). $$
(3.3)
(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
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, \(\bar{p}\in P(\bar{p})\)).
$$ \varGamma (\bar{p},p,p)> \omega \bigl(g(\bar{p})\bigr) \bigl(g( \bar{p}) - g(p)\bigr)\quad \text{for all } p \in T, p\neq {\bar{p}}. $$
(3.4)
(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
By \((E_{1})\), we get \(G(\hat{p},q)\leqslant G(\hat{p},p) + G(p,q)\), i.e., \(G(\hat{p},q)- G(\hat{p},p) \leqslant G(p,q)\).
$$ q \neq p\quad \text{and} \quad \omega \bigl(g(p)\bigr)G(x,y) + \varGamma (p,q,q) \leqslant 0. $$
(3.5)
Then by (3.5) we get
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, \(\varGamma (p,q,q) \leqslant \omega (g(p))(g(q)-g(p))\).
$$ \omega \bigl(g(p)\bigr) \bigl(G(\hat{p},q)-G(\hat{p},p)\bigr) + \varGamma (p,q,q) \leqslant \omega \bigl(g(p)\bigr)G(p,q) + \varGamma (p,q,q) \leqslant 0. $$
(3.6)
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,
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.
$$ \omega \bigl(g(\bar{p})\bigr) \bigl(g(q)-g(\bar{p})\bigr)+ \varGamma (\bar{p},q,q)\leqslant 0 \quad \text{or}\quad \varGamma (\bar{p},q,q)\leqslant \omega \bigl(g(\bar{p})\bigr) \bigl(g( \bar{p})-g(q)\bigr). $$
(3.7)
(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
Then
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
for all \(p \in T\), \(p \neq \bar{p}\). □
$$ \omega \bigl(g(\hat{p})\bigr)G(\hat{p},\bar{p})+ \varGamma (\hat{p},\bar{p}, \bar{p})\leqslant 0\quad \Rightarrow \quad \omega \bigl(g(\hat{p})\bigr) \bigl(g(\bar{p})-g( \hat{p})\bigr)+ \varGamma (\hat{p},\bar{p},\bar{p})\leqslant 0. $$
$$ \varGamma (\hat{p},\bar{p},\bar{p})\leqslant \omega \bigl(g(\hat{p})\bigr) \bigl(g( \hat{p})-g(\bar{p})\bigr). $$
$$ \omega \bigl(g(\bar{p})\bigr) \bigl(g(p)-g(\bar{p})\bigr) + \varGamma ( \bar{p},p,p)>0\quad \Rightarrow\quad \varGamma (\bar{p},p,p)> \omega \bigl(g(\bar{p})\bigr) \bigl(g(\bar{p})-g(p)\bigr) $$
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
then
P
has a fixed point in
T.
$$ \xi \bigl(\varGamma (p,q,q)\bigr) \leqslant \omega \bigl(g(p)\bigr) \bigl(g(p)- g(q)\bigr), $$
Proof
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
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
\(q \in T\).
$$ \varGamma (p,q,q)) \leqslant \omega \bigl(G(p,t)\bigr) \bigl(G(p,t)- G(q,t) \bigr) $$
Proof
By Theorem 2.2(b), for all \(r \in T\), there exists \(y(r) \in T\) such that
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
which is a contradiction. □
$$ \varGamma \bigl(y(r),q,q\bigr)) > \omega \bigl(G\bigl(y(r),r\bigr)\bigr) \bigl(G \bigl(y(r),r\bigr)- G(q,r)\bigr) $$
$$ \varGamma \bigl(y(r),q,q\bigr)) \leqslant \omega \bigl(G\bigl(y(r),r\bigr)\bigr) \bigl(G\bigl(y(r),r\bigr)- G(q,r)\bigr) , $$
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
for all \(p\geqslant q\). By Corollary 3.4, there exists \(y\in T\) such that \(G(y,p)\geqslant 0\) for all \(p\in T\).
$$ F(p,q,q)) \leqslant \omega \bigl(G(p,x)\bigr) \bigl(G(p,x)- G(q,x)\bigr) $$
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
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.
$$ G(\bar{p},q)+\delta \varGamma (\bar{p},q,q)\geq 0 \quad \text{for all } q\in S, $$
(4.1)
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
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
Hence p̄ is a solution to EP. □
$$ G(u_{n},q)+\frac{1}{n}\varGamma (u_{n},q,q)\geq 0 \quad \text{for each } q \in S. $$
$$ G(\bar{p},q)\geq \limsup_{S\rightarrow \infty }\biggl(G(u_{n_{k}},q)+ \frac{1}{n _{k}}\varGamma (u_{n_{k}},q,q)\biggr)\geq 0\quad \text{for all } q \in S. $$
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:
and
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. □
$$ G(u_{0},u_{n})+\frac{1}{n}\varGamma (u_{0},u_{n},u_{n})\leqslant 0 $$
(4.2)
$$ G(u_{n},p)+\frac{1}{n}\varGamma (u_{n},p,p)>0\quad \text{for all }p\in T. $$
(4.3)
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
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
$$ \varGamma (x,y,y)\leq w\varGamma (x,q,q). $$
$$ \varGamma \bigl(g(p),g(q),g(q)\bigr)\leq w\varGamma (p,q,q)\quad \text{for all } p,q\in T. $$
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
In the same way, we make the sequence \(\lbrace u_{n}\rbrace \) such that \(u_{n+1}\in P(u_{n})\) and
We have
Then, for all \(m,n\in \mathbb{N}\) with \(m>n\), we have
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
for all \(m>n\).
$$ \varGamma (u_{1},u_{2},u_{2})\leq w\varGamma (u_{0},u_{1},u_{1}). $$
$$ \varGamma (u_{n},u_{n+1},u_{n+1})\leq w\varGamma (u_{n-1},u_{n},u_{n})\quad \text{for all }n\in \mathbb{N}. $$
$$\begin{aligned} \varGamma (u_{n},u_{n+1},u_{n+1}) \leq& w\varGamma (u_{n-1},u_{n},u_{n}) \\ \leq& w^{2}\varGamma (u_{n-2},u_{n-1},u_{n-1}) \\ \vdots& \\ \leq& w^{n}\varGamma (u_{0},u_{1},u_{1}). \end{aligned}$$
$$\begin{aligned} \varGamma (u_{n},u_{m},u_{m}) \leq& \varGamma (u_{n},u_{n+1},u_{n+1})+ \varGamma (u_{n+1},u_{m},u_{m}) \\ \leq& \varGamma (u_{n},u_{n+1},u_{n+1})+\varGamma (u_{n+1},u_{n+2},u_{n+2}) \\ &{} +\cdots +\varGamma (u_{m-1},u_{m},u_{m}) \\ \leq& w^{n}\varGamma (u_{0},u_{1},u_{1})+w^{n+1} \varGamma (u_{0},u_{1},u _{1}) \\ &{} +\cdots +w^{m-1}\varGamma (u_{m-1},u_{m},u_{m}) \\ =& w^{n}\bigl(1+w+w^{2}+\cdots +w^{m-n-1}\bigr) \varGamma (u_{0},u_{1},u_{1}) \\ \leq& \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1}). \end{aligned}$$
$$ \varGamma (u_{n},u_{m},u_{m})\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u _{1}) $$
(5.1)
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
By the assumption, there is \(w_{n}\in P(\bar{p})\) such that
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 \(\bar{u}\in P(\bar{u})\).
$$ \varGamma (u_{n},\bar{p},\bar{p})\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u _{1},u_{1}) \quad \text{for all }n\in \mathbb{N}. $$
(5.2)
$$\begin{aligned} \varGamma (u_{n},w_{n},w_{n}) &\leq w\varGamma (u_{n-1},\bar{p},\bar{p}) \\ &\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1}). \end{aligned}$$
(5.3)
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
Now, we construct a sequence \(\lbrace v_{n}\rbrace \) as follows: \(v_{n+1}\in P(v_{n})\) and
Therefore, for all \(n\in \mathbb{N}\), we get
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).
$$ \varGamma (\bar{p},v_{1},v_{1})\leq w\varGamma (\bar{p}, \bar{p},\bar{p}). $$
$$ \varGamma (\bar{p},v_{n+1},v_{n+1})\leq w\varGamma ( \bar{p},v_{n},v_{n})\quad \text{for all }n\in N. $$
$$ \varGamma (\bar{p},v_{n},v_{n})\leq \varGamma (\bar{p},v_{n-1},v_{n-1}) \leq \cdots \leq w^{n} \varGamma (\bar{p},\bar{p},\bar{p}). $$
(5.4)
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
So \(\varGamma (\bar{p},\bar{q},\bar{q})\leq 0\) and \(\varGamma (\bar{p}, \bar{q},\bar{q})=0\). Moreover, we have
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 \(\varGamma ( \bar{p},\bar{p}, \bar{p})=0\). □
$$ \varGamma (\bar{p},v_{n},v_{n})\leq S \quad \Longrightarrow\quad \varGamma (\bar{p}, \bar{q},\bar{q})\leq S=\sup_{n\in \mathbb{N}}w^{n} \varGamma ( \bar{p},\bar{p},\bar{p}). $$
$$\begin{aligned} \varGamma (u_{n},\bar{q},\bar{q}) &\leq \varGamma (u_{n},\bar{p},\bar{p})+ \varGamma (\bar{p},\bar{q},\bar{q}) \\ &\leq \frac{w^{n}}{1-w}\varGamma (u_{0},u_{1},u_{1}) \end{aligned}$$
(5.5)
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.
Availability of data and materials
Not applicable.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.