1 Introduction and preliminaries
Banach [5], in 1922, proved a contraction principle, this key principle ensures the existence and uniqueness of fixed point theorem for Banach contraction. Later, this famous principle was extended by many authors to more general contractive conditions in different space (see [1, 6‐11]). In 1982, Sessa [12] introduced the notion of weakly commuting maps and derived common fixed point for these maps. The first paper [13] on modular function spaces was published in 1990. After that many authors developed this theory by finding fixed point in modular function spaces. Recently, Chistyakov gave the concept of modular metric spaces in [14, 15].
Definition 1.1
[16]
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Let X be a nonempty set. A modular metric on X is a function \(\omega: (0,\infty)\times X\times X \rightarrow[0, \infty]\) satisfying the following axioms:
(1)
\(u=v\) if and only if \(\omega_{\lambda}(u,v)=0\), for all \(\lambda> 0\);
(2)
\(\omega_{\lambda}(u,v)=\omega_{\lambda}(v,u)\), for all \(\lambda> 0\) and \(u, v \in X\);
(3)
\(\omega_{\lambda+ \nu}(u,v)\leq\omega_{\lambda}(u,w)+\omega_{\nu}(w,v)\), for all \(\lambda, \nu> 0\) and \(u, v, w \in X\).
In the sequel, for a function \(\omega: (0,\infty)\times X\times X \rightarrow[0, \infty]\), we will write for all \(\lambda> 0\) and \(u, v \in X\) and modular metric space as MMS. For related terminologies see [16]. Afterwards many mathematicians studied fixed point properties for modular metric spaces; see [16‐19]. Recently, Azadifar et al. [2] defined compatible mappings in modular metric space and obtained a common fixed point theorem of integral type as an extension of Jungck [20, 21].
$$ \omega_{\lambda}(u,v)=\omega(\lambda, u, v), $$
(1.1)
Definition 1.2
[2]
Let \(X_{\omega}\) be a MMS produced by the metric modular ω. Two mappings \(f,h:X_{\omega}\to X_{\omega}\) on \(X_{\omega}\) are called ω-compatible if \(\omega_{\lambda}(fhx_{n}, hfx_{n})\to0\) as \(n\to\infty\), whenever \(\{x_{n}\}_{n=1}^{\infty}\) is a sequence in \(X_{\omega}\) such that \(hx_{n}\to q\) and \(hx_{n} \to q\) for some point \(q\in X_{\omega}\) and for \(\lambda>0\).
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Further, Mongkolkeha and Kumam [22] obtained a common fixed point theorem for pair of compatible mappings satisfying a generalize weak contraction of integral type in modular spaces. Hussain and Salimi [8] established more general fixed point results for some integral-type contractions in MMS. The main intent of this paper is to establish certain common fixed point theorems for ω-weakly compatible maps under different weak contractive conditions which are more general than the corresponding contractive condition of integral type. Our results are more general and are an extension of [2, 4, 11, 22, 23] in the setting of modular metric spaces.
2 Common fixed point theorems for quasi-type weak contractions of integral type
Here, we define weakly compatible mappings for modular metric space and find of a common fixed point for quasi-type weak contractions of integral type satisfying the condition of weakly compatible in MMS.
Definition 2.1
Let \(X_{\omega}\) be a MMS produced by the metric modular ω, f and h be two self-mappings of \(X_{\omega}\). A point \(x\in X_{\omega}\) is called a coincidence point of f and h if and only if \(fx=hx\). We will call \(q=fx=hx\) a point of coincidence of f and h.
Denote the set of all coincidence points of f and h by \(C(f,h)\).
Definition 2.2
Two mappings \(f, h:X_{\omega}\to X_{\omega}\) are said to be ω-weakly compatible if and only if \(fhq=hfq\) for \(q\in C(f,h)\).
Note that every ω-compatible map is a ω-weakly compatible, but a ω-weakly compatible map needs to be ω-compatible (see Example 2.2).
Lemma 2.1
[24]
Let
f
and
g
be weakly compatible self-maps of a set
X. If
f
and
g
have a unique point of coincidence
\(q = fx = gx\), then
q
is the unique common fixed point of
f
and
g.
Denote by Φ, Θ, Ψ, and Π the collection of lower semicontinuous functions \(\phi:[0,\infty)\to [0, \infty)\) with \(\phi(r)>0\) for all \(r>0\) and \(\phi(r)=0\) if and only if \(r=0\), the collection of Lebesgue integrable functions \(\varphi:[0,\infty)\to[0, \infty)\) which is nonnegative, summable, and, for all \(\epsilon>0\), \(\int_{0}^{\epsilon}\varphi(r)\,\mathrm {d}r>0\), the collection of lower semicontinuous functions \(\psi:[0,\infty)\to[0, \infty)\) for which \(\psi(r)< r\) for all \(r>0\) and the collection of nondecreasing functions \(\pi:[0,\infty)\to[0, \infty)\) such that \(\sum_{n=1}^{\infty}\pi^{n}(r)<+\infty\) for all \(r>0\), where \(\pi^{n}\) is the nth iterate of π, respectively.
Lemma 2.2
[25]
If
\(\pi\in\Pi\), then the following hold:
(i)
\((\pi^{n}(r))_{n\in\mathbb{N}}\)
converges to 0 as
\(n\to\infty \)
for all
\(r\in(0,+\infty)\);
(ii)
\(\pi(r)< r\)
for all
\(r>0\);
(iii)
\(\pi(r)=0\)
if and only if
\(r=0\).
Lemma 2.3
[10]
Let
\(\varphi\in\Theta\)
and
\(\{s_{n}\}_{n\in\mathbb{N}}\)
be a nonnegative sequence with
\(s_{n}\to a\)
as
\(n\to\infty\). Then
$$\lim_{n\to\infty} \int_{0}^{s_{n}}\varphi(r)\,\mathrm {d}r= \int_{0}^{a}\varphi(r)\,\mathrm {d}r. $$
Now we present the main results of this section.
Theorem 2.1
Let
\(X_{\omega}\)
be a MMS. Suppose
\(a, t\in\mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the following assertions:
where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
(1)
\(S(X_{\omega})\subseteq h(X_{\omega})\), \(h(X_{\omega})\)
is a complete subspace of
\(X_{\omega}\);
(2)
S
and
h
are
ω-weakly compatible;
(3)
\(\int_{0}^{\omega_{\lambda/a}(Sx,Sy)}\varphi(r)\,\mathrm {d}r\leq\int_{0}^{\mathcal {M}(x,y)}\varphi(r)\,\mathrm {d}r-\phi (\int_{0}^{\mathcal{M}(x,y)}\varphi (r)\,\mathrm {d}r )\),
$$\begin{aligned} \begin{aligned} \mathcal{M}(x,y)=&\max \biggl\{ \omega_{\lambda/t}(hx,hy), \omega_{\lambda /t}(hx,Sx), \omega_{\lambda/t}(hy,Sy), \frac{\omega_{\lambda /t}(hx,Sy)+\omega_{\lambda/t}(hy,Sx)}{2}, \\ &\frac{\omega_{\lambda/t}(hx,Sy)\omega_{\lambda/t}(hy,Sx)}{1+\omega _{\lambda/t}(hx,hy)}, \frac{\omega_{\lambda/t}(hx,Sx)\omega_{\lambda /t}(hx,Sy)}{2[1+\omega_{\lambda/t}(hx,hy)]}, \frac{\omega_{\lambda/t}(hy, Sy)\omega_{\lambda/t}(hy,Sx)}{2[1+\omega _{\lambda/t}(hx,hy)]} \biggr\} , \end{aligned} \end{aligned}$$
Proof
Choose \(a>2t\) and let \(x_{0}\in X_{\omega}\) be an arbitrary point. Since \(S(X_{\omega})\subseteq h(X_{\omega})\), there is a point \(x_{1}\in X_{\omega}\) such that \(S(x_{0})=h(x_{1})\). On continuing this, we generate a sequence \(\{hx_{n}\}_{n=1}^{\infty}\) as follows: \(Sx_{n}=hx_{n+1}\) for each n. Suppose for any n, \(hx_{n}\neq hx_{n+1}\), since, otherwise, there exists a point of coincidence of S and h, (3) shows that where Since \(hx_{n}=Sx_{n-1}\), it follows that Moreover, and then Now if \(\omega_{\lambda/a}(hx_{n}, hx_{n+1})> \omega_{\lambda /a}(hx_{n-1}, hx_{n})\), then This is a contradiction. So, \(\mathcal{M}(x_{n},x_{n-1})\leq\omega _{\lambda/a}(hx_{n-1}, hx_{n})\). Therefore it shows that the sequence \(\{\int_{0}^{\omega_{\lambda/a}(hx_{n+1}, hx_{n})}\varphi(r)\}\) is decreasing and bounded below. Hence, there is \(k\geq0\) such that If \(k>0\), then by Lemma 2.3 and (2.1), we have a contradiction. So, we get Suppose \(l< a'<2t\), since \(\omega_{\lambda}\) is a decreasing function, so \(\omega_{\lambda/a'}(hx_{n+1},hx_{n})\leq\omega_{\lambda /a}(hx_{n+1}, hx_{n})\), whenever \(a'<2t\leq a\). On considering the limit as \(n\to\infty\) from both sides of this inequality shows that \(\omega _{\lambda/a'}(hx_{n+1},hx_{n})\to0\) for \(t< a'<2t\) and \(\lambda>0\). Thus we have \(\omega_{\lambda/a}(hx_{n+1},hx_{n})\to0\) as \(n\to\infty\) for any \(a>t\). Next, we show that \(\{hx_{n}\}_{n\in\mathbb{N}}\) is a Cauchy sequence. So, for all \(\varepsilon>0\), there exists \(n_{0}\in \mathbb{N}\) such that \(\omega_{\lambda/a}(hx_{n+1}, hx_{n})<\frac{\varepsilon}{a}\) for all \(n\in\mathbb{N}\) with \(n\geq n_{0}\) and \(\lambda>0\). Suppose \(m, n\in\mathbb{N}\) and \(m > n\). Observe that, for \(\frac{\lambda }{a(m-n)}\), there exists \(n_{\frac{\lambda}{(m-n)}}\in\mathbb{N}\) such that for all \(n\geq n_{\frac{\lambda}{(m-n)}}\). Now, we have for all \(m,n\geq n_{\frac{\lambda}{(m-n)}}\). This shows that \(\{hx_{n}\} _{n\in\mathbb{N}}\) is a Cauchy sequence. From completeness of \(h(X_{\omega})\), it follows that there exists \(x^{*}\in X\) such that \(\omega _{\lambda/t}(hx_{n},x^{*})\to0\) as \(n\to\infty\). Consequently, we can find p in \(X_{\omega}\) such that \(h(p)=x^{*}\). By (3), we get where By taking the limit as \(n\to\infty\), we have This shows \(\omega_{\lambda/a}(Sp,x^{*})= 0\) for \(\lambda>0\). Hence \(Tp=x^{*}\) and S and h have the point of coincidence \(x^{*}\). Suppose that \(q\neq x^{*}\) is another point of coincidence of S and h in \(X_{\omega}\). Then \(Tv=hv=q\) for some v in \(X_{\omega}\). By (3), we get where
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(hx_{n+1}, h_{x_{n}})}\varphi(r)\,\mathrm {d}r&= \int _{0}^{\omega_{\lambda/a}(Sx_{n},Sx_{n-1})}\varphi(r)\,\mathrm {d}r \\ &\leq \int_{0}^{\mathcal{M}(x_{n},x_{n-1})}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\mathcal{M}(x_{n},x_{n-1})}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x_{n},x_{n-1})=&\max \biggl\{ \omega_{\lambda /t}(hx_{n},hx_{n-1}), \omega_{\lambda/t}(hx_{n},Sx_{n}), \omega_{\lambda /t}(hx_{n-1},Sx_{n-1}), \\ &\frac{\omega_{\lambda /t}(hx_{n},Sx_{n-1})+\omega_{\lambda/t}(hx_{n-1},Sx_{n})}{2}, \frac{\omega_{\lambda/t}(hx_{n},Sx_{n-1})\omega_{\lambda /t}(hx_{n-1},Sx_{n})}{1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})}, \\ &\frac {\omega_{\lambda/t}(hx_{n},Sx_{n})\omega_{\lambda/t}(hx_{n}, Sx_{n-1})}{2[1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})]}, \frac{\omega_{\lambda/t}(hx_{n-1}, Sx_{n-1})\omega_{\lambda /t}(hx_{n-1}, Sx_{n})}{2[1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})]} \biggr\} . \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x_{n},x_{n-1})=&\max \biggl\{ \omega_{\lambda /t}(hx_{n-1},hx_{n}), \omega_{\lambda/t}(hx_{n}, hx_{n+1}), \frac{\omega _{\lambda/t}(hx_{n-1},hx_{n+1})}{2}, \\ &\frac{\omega_{\lambda /t}(hx_{n-1}, hx_{n})\omega_{\lambda/t}(hx_{n-1}, hx_{n+1})}{2[1+\omega _{\lambda/t}(hx_{n}, hx_{n-1})]} \biggr\} . \end{aligned}$$
$$\begin{aligned} \omega_{\lambda/t}(hx_{n-1},hx_{n+1}) &\leq \omega_{\lambda /2t}(hx_{n-1}, hx_{n})+\omega_{\lambda/2t}(hx_{n}, hx_{n+1}) \\ &\leq\omega_{\lambda/a}(hx_{n-1}, hx_{n})+ \omega_{\lambda/a}(hx_{n}, hx_{n+1}) \end{aligned}$$
$$\begin{aligned} \frac{\omega_{\lambda/t}(hx_{n-1}, hx_{n})\omega_{\lambda/t}(hx_{n-1}, hx_{n+1})}{2[1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})]} &\leq\frac{\omega _{\lambda/t}(hx_{n-1}, hx_{n+1})}{2} \\ &\leq\frac{\omega_{\lambda/a}(hx_{n-1}, hx_{n})+\omega_{\lambda /a}(hx_{n}, hx_{n+1})}{2} \\ &\leq\max \bigl\{ \omega_{\lambda/a}(hx_{n-1}, hx_{n}), \omega_{\lambda /a}(hx_{n}, hx_{n+1}) \bigr\} , \end{aligned}$$
$$\mathcal{M}(x_{n},x_{n-1})\leq\max \bigl\{ \omega_{\lambda/a}(hx_{n-1}, hx_{n}), \omega_{\lambda/a}(hx_{n}, hx_{n+1}) \bigr\} . $$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(hx_{n+1},hx_{n})}\varphi(r)\,\mathrm {d}r &\leq \int _{0}^{\omega_{\lambda/a}(hx_{n},hx_{n+1})}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\omega_{\lambda/a}(hx_{n},hx_{n+1})}\varphi(r)\,\mathrm {d}r \biggr) \\ &< \int_{0}^{\omega_{\lambda/a}(hx_{n},hx_{n+1})}\varphi(r)\,\mathrm {d}r. \end{aligned}$$
$$\begin{aligned} \begin{aligned}[b] \int_{0}^{\omega_{\lambda/a}(hx_{n+1},hx_{n})}\varphi(r)\,\mathrm {d}r &\leq \int _{0}^{\omega_{\lambda/a}(hx_{n},hx_{n-1})}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\omega_{\lambda/a}(hx_{n},hx_{n-1})}\varphi(r)\,\mathrm {d}r \biggr) \\ &< \int_{0}^{\omega_{\lambda/a}(hx_{n},hx_{n-1})}\varphi(r)\,\mathrm {d}r, \end{aligned} \end{aligned}$$
(2.1)
$$\begin{aligned} \lim_{n\to\infty} \int_{0}^{\omega_{\lambda/a}(hx_{n+1},hx_{n})}\varphi (r)\,\mathrm {d}r=k. \end{aligned}$$
$$\begin{aligned} \lim_{n\to\infty}{\omega_{\lambda/a}(hx_{n+1},hx_{n})}=0. \end{aligned}$$
$$\begin{aligned} \omega_{\frac{\lambda}{a(m-n)}}(hx_{n+1},hx_{n})< \frac{\epsilon}{a(m-n)}, \end{aligned}$$
$$\begin{aligned} \omega_{\lambda/l}(hx_{n}, hx_{m}) \leq& \omega_{\frac{\lambda }{a(m-n)}}(hx_{n}, hx_{n+1})+\omega_{\frac{\lambda }{a(m-n)}}(hx_{n+1},hx_{n+2})+\cdots+ \omega_{\frac{\lambda }{a(m-n)}}(hx_{m-1}, hx_{m}) \\ < & \frac{\varepsilon}{a(m-n)}+\frac{\varepsilon}{a(m-n)}+\cdots+\frac {\varepsilon}{a(m-n)} \\ =&\varepsilon/a, \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(hx_{n}, Sp)}\varphi(r)\,\mathrm {d}r&= \int_{0}^{\omega _{\lambda/a}(Sx_{n-1}, Sp)}\varphi(r)\,\mathrm {d}r \\ &\leq \int_{0}^{\mathcal{M}(x_{n-1}, p)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\mathcal{M}(x_{n-1}, p)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x_{n-1},p) =&\max \biggl\{ \omega_{\lambda/t}(hx_{n-1},hp), \omega_{\lambda/t}(hx_{n-1},Sx_{n-1}), \omega_{\lambda/t}(hp,Sp), \\ & \frac{\omega_{\lambda/t}(hx_{n-1},Sp)+\omega_{\lambda/t}(hp,Sx_{n-1})}{2}, \frac{\omega_{\lambda/t}(hx_{n-1},Sp)\omega_{\lambda /t}(hp,Sx_{n-1})}{1+\omega_{\lambda/t}(hx_{n-1}, hp)} \\ &\frac{\omega_{\lambda/t}(hx_{n-1},Sx_{n-1})\omega_{\lambda /t}(hx_{n-1},Sp)}{2[1+\omega_{\lambda/t}(hx_{n-1}, hp)]}, \frac{\omega_{\lambda/t}(hp,Sp)\omega_{\lambda /t}(hp,Sx_{n-1})}{2[1+\omega_{\lambda/t}(hx_{n-1}, hp)]} \biggr\} . \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(x^{*}, Sp)}\varphi(r)\,\mathrm {d}r &\leq \int _{0}^{\omega_{\lambda/t}(x^{*}, Sp)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\omega_{\lambda/t}(x^{*}, Sp)}\varphi(r)\,\mathrm {d}r \biggr) \\ & < \int_{0}^{\omega_{\lambda/t}(x^{*}, Sp)}\varphi(r)\,\mathrm {d}r \\ & \leq \int_{0}^{\omega_{\lambda/a}(x^{*}, Sp)}\varphi(r)\,\mathrm {d}r. \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(hp,hv)}\varphi(r)\,\mathrm {d}r&= \int_{0}^{\omega _{\lambda/a}(Sp,Sv)}\varphi(r)\,\mathrm {d}r \\ &\leq \int_{0}^{\mathcal{M}(p,v)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\mathcal{M}(p,v)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
$$\begin{aligned} \mathcal{M}(p,v) =&\max \biggl\{ \omega_{\lambda/t}(hp, hv), \omega _{\lambda/t}(hp, Sp), \omega_{\lambda/t}(hv, Sv), \\ & \frac{\omega_{\lambda/t}(hv, Sp)+\omega_{\lambda/t}(hp,Sv)}{2}, \frac{\omega_{\lambda/t}(hp,Sv)\omega_{\lambda/t}(hv,Sp)}{1+\omega _{\lambda/t}(hp, hv)}, \\ &\frac{\omega_{\lambda/t}(hp,Sp)\omega_{\lambda/t}(hp,Sv)}{2[1+\omega _{\lambda/t}(hp, hv)]}, \frac{\omega_{\lambda/t}(hv,Sv)\omega_{\lambda/t}(hv,Sp)}{2[1+\omega _{\lambda/t}(hp, hv)]} \biggr\} \\ =&\omega_{\lambda/t}(hp, hv). \end{aligned}$$
So, From this contradiction, we see that S and h have a unique coincidence point \(x^{*}\). By using Lemma 2.1, we get \(x^{*}\) a unique common fixed point of S and h. □
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(hp,hv)}\varphi(r)\,\mathrm {d}r&\leq \int_{0}^{\omega _{\lambda/t}(hp,hv)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\omega_{\lambda /t}(hp,hv)}\varphi(t)\,\mathrm {d}t \biggr) \\ &< \int_{0}^{\omega_{\lambda/t}(hp,hv)}\varphi(r)\,\mathrm {d}r \leq \int _{0}^{\omega_{\lambda/a}(hp,hv)}\varphi(r)\,\mathrm {d}r. \end{aligned}$$
Here is an example to illustrate Theorem 2.1.
Example 2.1
Let \(X_{\omega}=\{0,1,2,3,4,\ldots\}\) and \(\omega_{\lambda}(x,y)=\frac {d(x,y)}{\lambda}\), where Define \(S,h:X_{\omega}\to X_{\omega}\) and \(\varphi, \phi:[0,\infty)\to[0, \infty)\) as
\(\varphi(r)=2r\) and \(\phi(r)=\sqrt{r}\), respectively. Then \(S(X_{\omega})\subseteq h(X_{\omega})\) and \(h(X_{\omega})\) is a complete subspace of \(X_{\omega}\). Note that \(x=0\) is the coincidence point of S and h and This shows that S and h are ω-weakly compatible maps. Now we verify that S and h satisfy condition (3) of Theorem 2.1. Suppose \(a, t \in\mathbb{R}^{+}\), \(a>t\). Then there arise four cases.
$$d(x,y)=\left \{ \textstyle\begin{array}{l@{\quad}l} x+y, & x\neq y, \\ 0, & x=y. \end{array}\displaystyle \right . $$
$$Sx=0\quad \forall x\in X_{\omega}, \quad \quad hx= \left\{ \textstyle\begin{array}{l@{\quad}l} x-1, & \text{if } x\neq0,\\ 0, & \text{if } x=0, \end{array}\displaystyle \right . $$
$$\begin{aligned} Sh(0)=S(0)=h(0)=hS(0). \end{aligned}$$
Case 1: Assume \(x=y=0\). Then condition (3) holds trivially because \(Sx=Sy=hx= {hy=0}\).
Case 2: Assume \(y=0\) and \(x>0\). Then This implies that Also, so, Therefore, Since \(\frac{t}{\lambda} (x-1)\geq1\), this shows that Thus condition (3) is satisfied in this case.
$$\begin{aligned} \omega_{\lambda/a}(Sx, Sy)=\omega_{\lambda/a}(0,0)=0. \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r=0. \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x,y) =&\max \biggl\{ \omega_{\lambda/t}(x-1, 0), \omega _{\lambda/t}(x-1,0), \omega_{\lambda/t}(0,0), \frac{\omega_{\lambda /t}(x-1, 0)+\omega_{\lambda/t}(0,0)}{2}, \\ & \frac{\omega_{\lambda/t}(x-1,0)\omega_{\lambda/t}(0,0)}{1+\omega _{\lambda/t}(x-1, 0)}, \frac{\omega_{\lambda/t}(x-1, 0)\omega_{\lambda /t}(x-1, 0)}{2[1+\omega_{\lambda/t}(x-1, 0)]}, \frac{\omega_{\lambda/t}(0,0)\omega_{\lambda/t}(0, x-1)}{2[1+\omega _{\lambda/t}(x-1, 0)]} \biggr\} \\ =&\max \biggl\{ \frac{t}{\lambda}(x-1), \frac{t/\lambda(x-1)}{2}, \frac {t^{2}/\lambda^{2}(x-1)^{2}}{2 (1+t/\lambda(x-1) )} \biggr\} \\ =&\max \biggl\{ \frac{t}{\lambda}(x-1), \frac{1}{2 (1+\frac {1}{t/\lambda(x-1)} )} \biggr\} \\ =&\frac{t}{\lambda} (x-1), \end{aligned}$$
$$\begin{aligned} \int_{0}^{\mathcal{M}(x,y)}2r\,\mathrm {d}r=\frac{t^{2}}{\lambda^{2}}(x-1)^{2}. \end{aligned}$$
$$\begin{aligned} \begin{aligned} \int_{0}^{\mathcal{M}(x,y)}2r\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathcal {M}(x,y)}2r\,\mathrm {d}r \biggr)&= \frac{t^{2}}{\lambda^{2}}(x-1)^{2}-\frac{t}{\lambda } (x-1) \\ &=\frac{t}{\lambda} (x-1) \biggl[\frac{t}{\lambda} (x-1)-1 \biggr]. \end{aligned} \end{aligned}$$
$$\begin{aligned} \frac{t}{\lambda} (x-1) \biggl[\frac{t}{\lambda} (x-1)-1 \biggr]\geq0= \int _{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r. \end{aligned}$$
Case 3: Assume \(x>y>0\). Then we need to consider two subcases:
Subcase 1: If \(x=y+1\) or \(y=x-1\), then \(Sx=Sy=0\), \(hx=x-1\) and \(hy=x-2\). This implies that and Since \(2x-3\geq x-1\geq x-2\), \(\mathcal{M}(x,y)=\frac{t}{\lambda }(2x-3)\). Hence
$$\begin{aligned} \omega_{\lambda/a}(Sx, Sy)=0 \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x,y) =&\max \biggl\{ \frac{t}{\lambda}(2x-3), \frac {t}{\lambda}(x-1), \frac{t}{\lambda}(x-2), \frac{t/\lambda(2x-3)}{2}, \frac{t^{2}/\lambda^{2}(x-1)(x-2)}{ (1+t/\lambda(2x-3) )}, \\ & \frac{t^{2}/\lambda^{2}(x-1)^{2}}{2 (1+t/\lambda(2x-3) )}, \frac {t^{2}/\lambda^{2}(x-2)^{2}}{2 (1+t/\lambda(2x-3) )} \biggr\} . \end{aligned}$$
$$\begin{aligned} \begin{aligned} \int_{0}^{\mathcal{M}(x,y)}2r\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathcal {M}(x,y)}2r\,\mathrm {d}r \biggr)&= \frac{t^{2}}{\lambda^{2}}(2x-3)^{2}-\frac{t}{\lambda } (2x-3) \\ &\geq0= \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r. \end{aligned} \end{aligned}$$
Subcase 2: If \(x>y+1\) and \(x=2y\), then \(Sx=Sy=0\), \(hx=2y-1\) and \(hy=y-1\). This implies that and Since \(3y-2\geq2y-1\geq y-1\), \(\mathcal{M}(x,y)=\frac{t}{\lambda }(3y-2)\). Hence Now if \(x>2y\), then \(Sx=Sy=0\), \(hx=x-1\) and \(hy=y-1\). This implies that \(\mathcal{M}(x,y)=\frac{t}{\lambda}(x+y-2)\). Hence Thus condition (3) is satisfied in this case.
$$\begin{aligned} \omega_{\lambda/a}(Sx, Sy)=0 \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x,y) =&\max \biggl\{ \frac{t}{\lambda}(3y-2), \frac {t}{\lambda}(2y-1), \frac{t}{\lambda}(y-1), \frac{t/\lambda(3y-2)}{2}, \frac{t^{2}/\lambda^{2}(2y-1)(y-1)}{ (1+t/\lambda(3y-2) )}, \\ & \frac{t^{2}/\lambda^{2}(2y-1)^{2}}{2 (1+t/\lambda(3y-2) )}, \frac {t^{2}/\lambda^{2}(y-1)^{2}}{2 (1+t/\lambda(3y-2) )} \biggr\} . \end{aligned}$$
$$\begin{aligned} \begin{aligned} \int_{0}^{\mathcal{M}(x,y)}2r\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathcal {M}(x,y)}2r\,\mathrm {d}r \biggr)&= \frac{t^{2}}{\lambda^{2}}(3y-2)^{2}-\frac{t}{\lambda } (3y-2) \\ &\geq0= \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r. \end{aligned} \end{aligned}$$
$$\begin{aligned} \begin{aligned} \int_{0}^{\mathcal{M}(x,y)}2r\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathcal {M}(x,y)}2r\,\mathrm {d}r \biggr)&= \frac{t^{2}}{\lambda^{2}}(x+y-2)^{2}-\frac {t}{\lambda} (x+y-2) \\ &\geq0= \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r. \end{aligned} \end{aligned}$$
Case 4: Assume \(x=y>0\). Then \(Sx=Sy=0\) and \(hx=hy=x-1\). This implies that and Hence So, condition (3) is satisfied in this case. Thus all conditions of Theorem 2.1 hold and 0 is a unique common fixed point of S and h.
$$\begin{aligned} \omega_{\lambda/a}(Sx, Sy)=0 \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x,y) &=\max \biggl\{ 0, \frac{t}{\lambda}(x-1), \frac {t^{2}}{\lambda^{2}}(x-1)^{2}, \frac{t^{2}/\lambda^{2}(x-1)^{2}}{2} \biggr\} \\ & =\frac{t^{2}}{\lambda^{2}}(x-1)^{2}. \end{aligned}$$
$$\begin{aligned} \begin{aligned} \int_{0}^{\mathcal{M}(x,y)}2r\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathcal {M}(x,y)}2r\,\mathrm {d}r \biggr)&= \frac{t^{4}}{\lambda^{4}}(x-1)^{4}-\frac {t^{2}}{\lambda^{2}} (x-1)^{2} \\ &\geq0= \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r. \end{aligned} \end{aligned}$$
From Theorem 2.1, we conclude the following results:
Theorem 2.2
Let
\(X_{\omega}\)
be a MMS. Assume that
\(a, t\in\mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
for all
\(x,y\in X_{\omega}\)
and
\(\lambda>0\), where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
T
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\mathfrak {m}_{1}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {m}_{1}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.2)
$$\mathfrak{m}_{1}(x,y)=\max \biggl\{ \omega_{\lambda/t}(hx,hy), \omega _{\lambda/t}(hx,Sx), \omega_{\lambda/t}(hy,Sy), \frac{\omega_{\lambda /t}(hx,Sy)+\omega_{\lambda/t}(hy,Sx)}{2} \biggr\} , $$
Theorem 2.3
Let
\(X_{\omega}\)
be a MMS. Assume that
\(a, t \in\mathbb{R}^{+}\), \(a>t\)
and
\(S,h:X_{\omega}\to X_{\omega}\)
are two self mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
for all
\(x, y\in X_{\omega}\), where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
T
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\omega _{\lambda/t}(hx,hy)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\omega_{\lambda /t}(hx,hy)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.3)
Now we give an Example 2.2 which shows that Theorem 2.3 extends significantly Theorem 2.2 and Theorem 4.2 of [2].
Example 2.2
Let \(X_{\omega}=[0,1]\) and \(\omega_{\lambda}(x,y)=\frac{\vert x-y\vert }{\lambda}\). Define \(S,h:X_{\omega}\to X_{\omega}\) and \(\varphi, \phi:[0,\infty)\to[0, \infty)\) as
\(\varphi(r)=2r\) and \(\phi(r)=\ln(1+r)\), respectively. First of all we verify that S and h satisfies the inequality (2.3). Suppose \(a, t \in\mathbb{R}^{+}\), \(a>t\). Then there are two cases.
$$Sx= \left\{ \textstyle\begin{array}{l@{\quad}l} 1/2 & \text{if } x\in[0,\frac{1}{2}],\\ 3/4 & \text{if } x\in(\frac{1}{2},1], \end{array}\displaystyle \right .\quad\quad hx=\left\{ \textstyle\begin{array}{l@{\quad}l} 1-x & \text{if } x\in[0,\frac{1}{2}],\\ 0 & \text{if } x\in(\frac{1}{2},1], \end{array}\displaystyle \right . $$
Case 1: Let \(x\in[0,\frac{1}{2}]\). Then This implies that Also, so, Therefore, Since \(\ln(1+x)\leq x\) for all \(x\in[0, 1]\), this shows that Thus (2.3) is satisfied in this case.
$$\begin{aligned} \omega_{\lambda/a}(Sx, Sy)=\omega_{\lambda/a} \biggl(\frac{1}{2}, \frac {1}{2} \biggr)=0. \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r=0. \end{aligned}$$
$$\begin{aligned} \omega_{\lambda/t}(hx, hy)=\omega_{\lambda/t} (1-x, 1-y)= \frac {l\vert x-y\vert }{\lambda}, \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/t}(hx,hy)}2r\,\mathrm {d}r={\frac{l\vert x-y\vert }{\lambda}}^{2}. \end{aligned}$$
$$\begin{aligned} \int_{0}^{\omega_{\lambda/t}(hx,hy)}2r\,\mathrm {d}r-\phi \biggl( \int_{0}^{\omega _{\lambda/t}(hx,hy)}2r\,\mathrm {d}r \biggr) = \frac{t^{2}{\vert x-y\vert }^{2}}{\lambda^{2}}- \ln \biggl(1+\frac{t^{2}{\vert x-y\vert }^{2}}{\lambda^{2}} \biggr). \end{aligned}$$
$$\begin{aligned} \frac{t^{2}{\vert x-y\vert }^{2}}{\lambda^{2}}-\ln \biggl(1+\frac{t^{2}{\vert x-y\vert }^{2}}{\lambda ^{2}} \biggr)\geq0= \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}2r\,\mathrm {d}r. \end{aligned}$$
Case 2: Let \(x\in(\frac{1}{2},1]\). Then Thus (2.3) is satisfied trivially in this case.
$$\begin{aligned} \omega_{\lambda/a}(Sx, Sy)=0=\omega_{\lambda/t}(hx,hy). \end{aligned}$$
Next, since \(x=\frac{1}{2}\) is the coincidence point of S and h and showing that S and h are ω-weakly compatible maps. Thus all conditions of Theorem 2.3 hold and \(\frac{1}{2}\) is a unique common fixed point of S and h.
$$\begin{aligned} Sh \biggl(\frac{1}{2} \biggr)=S \biggl(\frac{1}{2} \biggr)=h \biggl(\frac {1}{2} \biggr)=hS \biggl(\frac{1}{2} \biggr), \end{aligned}$$
Further, consider a sequence \(\{x_{n}\}= \{\frac{1}{2}-\frac {1}{n} \}\), \(n\geq2\), in \(X_{\omega}\). We have But Therefore, S and h are not ω-compatible.
$$\begin{aligned} \omega_{\lambda}(Sx_{n}, hx_{n})&= \omega_{\lambda} \biggl(S \biggl(\frac {1}{2}-\frac{1}{n} \biggr), h \biggl(\frac{1}{2}-\frac{1}{n} \biggr) \biggr) \\ &= \omega_{\lambda} \biggl(\frac{1}{2}+\frac{1}{n}, \frac{1}{2} \biggr)\to 0 \quad \text{as } n\to\infty. \end{aligned}$$
$$\begin{aligned} \omega_{\lambda}(Sh{x_{n}}, hS{x_{n}})\nrightarrow0 \quad \text{as } n\rightarrow\infty. \end{aligned}$$
Like to the arguments of Theorem 2.1, we state the following results and exclude their proofs.
Theorem 2.4
Let
\(X_{\omega}\)
be a MMS. Suppose
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
\(a, t\in\mathbb{R}^{+}\)
with
\(a>t\)
for all
\(x,y\in X_{\omega}\), where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\mathfrak {m}_{2}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {m}_{2}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.4)
$$\begin{aligned} \begin{aligned}[b] \mathfrak{m}_{2}(x,y)=&\max \biggl\{ \omega_{\lambda/t}(hx,hy), \omega _{\lambda/t}(hx,Sx), \omega_{\lambda/t}(hy,Sy), \frac{\omega_{\lambda /t}(hx,Sy)+\omega_{\lambda/t}(hy,Sx)}{2}, \\ &\frac{\omega_{\lambda/t}(hy,Sx)\omega_{\lambda/t}(hx,Sy)}{1+\omega _{\lambda/t}(Sx,Sy)}, \frac{\omega_{\lambda/t}(hy,Sx)\omega_{\lambda/t}(hx,Sx)}{2[1+\omega _{\lambda/t}(Sx,Sy)]}, \\ & \frac{\omega_{\lambda/t}(hx,Sy)\omega_{\lambda/t}(hy, Sy)}{2[1+\omega _{\lambda/t}(Sx,Sy)]} \biggr\} , \end{aligned} \end{aligned}$$
(2.5)
Theorem 2.5
Let
\(X_{\omega}\)
be a MMS. Assume that
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
\(a, t\in\mathbb{R}^{+}\)
with
\(a>t\)
such that
for all
\(x,y\in X_{\omega}\), where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\mathfrak {m}_{3}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {m}_{3}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.6)
$$\begin{aligned} \begin{aligned}[b] \mathfrak{m}_{3}(x,y)=&\max \biggl\{ \omega_{\lambda/t}(hx,hy), \omega _{\lambda/t}(hx,Sx), \omega_{\lambda/t}(hy,Sy), \\ &\frac{\omega_{\lambda /t}(hx,Sy)+\omega_{\lambda/t}(hy,Sx)}{2}, \frac{\omega_{\lambda/t}(hy,Sx)\omega_{\lambda/t}(hx,Sy)}{1+\omega _{\lambda/t}(hx,hy)}, \\ &\min \biggl\{ \frac{\omega_{\lambda/t}(hx,Sx)\omega_{\lambda /t}(hx,Sy)}{1+\omega_{\lambda/t}(hx,hy)}, \frac{\omega_{\lambda/t}(hy,Sy)\omega_{\lambda/t}(hy, Sx)}{1+\omega _{\lambda/t}(hx,hy)} \biggr\} \biggr\} , \end{aligned} \end{aligned}$$
(2.7)
Theorem 2.6
Let
\(X_{\omega}\)
be a MMS. Suppose
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and for all
\(x,y\in X_{\omega}\), there exist
\(a, t\in\mathbb {R}^{+}\)
with
\(a>t\)
where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\omega_{\lambda/a}(Sx,Sy)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\mathfrak {m}_{4}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {m}_{4}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.8)
$$\begin{aligned} \begin{aligned}[b] \mathfrak{m}_{4}(x,y)=&\max \biggl\{ \omega_{\lambda/t}(hx,hy), \omega _{\lambda/t}(hx,Sx), \omega_{\lambda/t}(hy,Sy), \\ &\frac{\omega_{\lambda /t}(hx,Sy)+\omega_{\lambda/t}(hy,Sx)}{2}, \frac{\omega_{\lambda/t}(hy,Sx)\omega_{\lambda/t}(hx,Sy)}{1+\omega _{\lambda/t}(Sx,Sy)}, \\ &\min \biggl\{ \frac{\omega_{\lambda/t}(hy,Sx)\omega_{\lambda /t}(hx,Sx)}{1+\omega_{\lambda/t}(Sx,Sy)}, \frac{\omega_{\lambda/t}(hx,Sy)\omega_{\lambda/t}(hy, Sy)}{1+\omega _{\lambda/t}(Sx,Sy)} \biggr\} \biggr\} , \end{aligned} \end{aligned}$$
(2.9)
Theorem 2.7
Let
\(X_{\omega}\)
be a MMS. Suppose
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
for all
\(x,y\in X_{\omega}\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathcal{M}(x,y)\)
is as in Theorem
2.1, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(Sx,Sy))}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi (\mathcal{M}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathcal {M}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.10)
Proof
Choose \(a>2t\). Let \(x_{0}\in X_{\omega}\) be an arbitrary point. Since \(S(X_{\omega})\subseteq h(X_{\omega})\), there is a point \(x_{1}\) in \(X_{\omega}\) such that \(S(x_{0})=h(x_{1})\). By continuing this, we generate a sequence \(\{hx_{n}\}_{n=1}^{\infty}\) as follows: \(Sx_{n}=hx_{n+1}\) for each n. Suppose for any n, \(hx_{n}\neq hx_{n+1}\), since, otherwise, S and h have a point of coincidence, (2.10) shows that where As in the proof of Theorem 2.1, we get Now if \(\omega_{\lambda/a}(hx_{n}, hx_{n+1})> \omega_{\lambda /a}(hx_{n-1}, hx_{n})\), then This gives a contradiction. So, \(\mathcal{M}(x_{n},x_{n-1})\leq\omega _{\lambda/a}(hx_{n-1}, hx_{n})\). Therefore which implies that there exists \(k\geq0\) such that If \(k>0\), then by Lemma 2.3 and (2.11), we get the contradiction. So, we have Since \(\pi\in\Pi\), Lemma 2.2 gives From this we see that \(\{hx_{n}\}_{n\in\mathbb{N}}\) is a Cauchy sequence. Since \(h(X_{\omega})\) is complete, there exists \(x^{*}\in X\) such that \(\omega_{\lambda/t}(hx_{n},x^{*})\to0\) as \(n\to\infty\). Consequently, we can find p in \(X_{\omega}\) such that \(h(p)=x^{*}\) By (2.10), we get where Taking the limit as \(n\to\infty\) and using Lemma 2.3 yields This contradiction gives \(\pi(\omega_{\lambda/a}(Sp,x^{*}))= 0\), by using Lemma 2.2, we get \(\omega_{\lambda/a}(Sp,x^{*})= 0\) for \(\lambda >0\). Hence \(Sp=x^{*}\). Hence \(x^{*}\) is the point of coincidence of S and h. Assume that there is another point of coincidence q in \(X_{\omega}\) such that \(q\neq x^{*}\). Then there exists u in \(X_{\omega}\) such that \(Su=hu=q\). By (2.10), we get where
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n+1}, h{x_{n}}))}\varphi(r)\,\mathrm {d}r&= \int _{0}^{\pi(\omega_{\lambda/a}(Sx_{n}, S{x_{n-1}}))}\varphi(r)\,\mathrm {d}r \\ &\leq \int_{0}^{\pi(\mathcal{M}(x_{n},x_{n-1}))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathcal{M}(x_{n},x_{n-1}))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x,y)=&\max \biggl\{ \omega_{\lambda/t}(hx_{n},hx_{n-1}), \omega _{\lambda/t}(hx_{n},Sx_{n}), \omega_{\lambda/t}(hx_{n-1},Sx_{n-1}), \\ &\frac {\omega_{\lambda/t}(hx_{n},Sx_{n-1})+\omega_{\lambda/t}(hx_{n-1},Sx_{n})}{2}, \frac{\omega_{\lambda/t}(hx_{n},Sx_{n-1})\omega_{\lambda /t}(hx_{n-1},Sx_{n})}{1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})}, \\ &\frac {\omega_{\lambda/t}(hx_{n},Sx_{n})\omega_{\lambda/t}(hx_{n}, Sx_{n-1})}{2[1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})]}, \frac{\omega_{\lambda/t}(hx_{n-1}, Sx_{n-1})\omega_{\lambda /t}(hx_{n-1}, Sx_{n})}{2[1+\omega_{\lambda/t}(hx_{n}, hx_{n-1})]} \biggr\} . \end{aligned}$$
$$\mathcal{M}(x_{n},x_{n-1})\leq\max \bigl\{ \omega_{\lambda/a}(hx_{n-1}, hx_{n}), \omega_{\lambda/a}(hx_{n}, hx_{n+1}) \bigr\} . $$
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n+1},hx_{n}))}\varphi(r)\,\mathrm {d}r &\leq \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n},hx_{n+1}))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\omega_{\lambda/t}(hx_{n},hx_{n+1}))}\varphi(r)\,\mathrm {d}r \biggr) \\ &< \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n},hx_{n+1}))}\varphi(r)\,\mathrm {d}r. \end{aligned}$$
$$\begin{aligned} \begin{aligned}[b] \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n+1},hx_{n}))}\varphi(r)\,\mathrm {d}r &\leq \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n},hx_{n-1}))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\omega_{\lambda/t}(hx_{n},hx_{n-1}))}\varphi(r)\,\mathrm {d}r \biggr) \\ &< \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n},hx_{n-1}))}\varphi(r)\,\mathrm {d}r, \end{aligned} \end{aligned}$$
(2.11)
$$\begin{aligned} \lim_{n\to\infty} \int_{0}^{\pi(\omega_{\lambda /a}(hx_{n+1},hx_{n}))}\varphi(r)\,\mathrm {d}r=k. \end{aligned}$$
$$\begin{aligned} \lim_{n\to\infty}{\pi\bigl(\omega_{\lambda/a}(hx_{n+1},hx_{n}) \bigr)}=0. \end{aligned}$$
$$\begin{aligned} \lim_{n\to\infty}{\omega_{\lambda/a}(hx_{n+1},hx_{n})}=0. \end{aligned}$$
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(hx_{n}, Sp))}\varphi(r)\,\mathrm {d}r&= \int_{0}^{\pi (\omega_{\lambda/a}(Sx_{n-1}, Sp))}\varphi(r)\,\mathrm {d}r \\ &\leq \int_{0}^{\pi(\mathcal{M}(x_{n-1}, p))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathcal{M}(x_{n-1}, p))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
$$\begin{aligned} \mathcal{M}(x_{n-1},p) =&\max \biggl\{ \omega_{\lambda/t}(hx_{n-1},hp), \omega_{\lambda/t}(hx_{n-1},Sx_{n-1}), \omega_{\lambda/t}(hp,Sp), \\ & \frac{\omega_{\lambda/t}(hx_{n-1},Sp)+\omega_{\lambda/t}(hp,Sx_{n-1})}{2}, \frac{\omega_{\lambda/t}(hx_{n-1},Sp)\omega_{\lambda /t}(hp,Sx_{n-1})}{1+\omega_{\lambda/t}(hx_{n-1}, hp)} \\ &\frac{\omega_{\lambda/t}(hx_{n-1},Sx_{n-1})\omega_{\lambda /t}(hx_{n-1},Sp)}{2[1+\omega_{\lambda/t}(hx_{n-1}, hp)]}, \frac{\omega_{\lambda/t}(hp,Sp)\omega_{\lambda /t}(hp,Sx_{n-1})}{2[1+\omega_{\lambda/t}(hx_{n-1}, hp)]} \biggr\} . \end{aligned}$$
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(x^{*}, Sp))}\varphi(r)\,\mathrm {d}r &\leq \int _{0}^{\pi(\omega_{\lambda/t}(x^{*}, Sp))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\pi(\omega_{\lambda/t}(x^{*}, Sp))}\varphi(r)\,\mathrm {d}r \biggr) \\ & < \int_{0}^{\pi(\omega_{\lambda/t}(x^{*}, Sp))}\varphi(r)\,\mathrm {d}r \\ & \leq \int_{0}^{\pi(\omega_{\lambda/a}(x^{*}, Sp))}\varphi(r)\,\mathrm {d}r. \end{aligned}$$
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(hp,hu))}\varphi(r)\,\mathrm {d}r&= \int_{0}^{\pi (\omega_{\lambda/a}(Sp,Su))}\varphi(r)\,\mathrm {d}r \\ &\leq \int_{0}^{\pi(\mathcal{M}(p,u))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\pi(\mathcal{M}(p,u))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
$$\begin{aligned} \mathcal{M}(p,u) =&\max \biggl\{ \omega_{\lambda/t}(hp, hu), \omega _{\lambda/t}(hp, Sp), \omega_{\lambda/t}(hu, Su), \\ & \frac{\omega_{\lambda/t}(hu, Sp)+\omega_{\lambda/t}(hp,Su)}{2}, \frac{\omega_{\lambda/t}(hp,Su)\omega_{\lambda/t}(hu,Sp)}{1+\omega _{\lambda/t}(hp, hu)}, \\ &\frac{\omega_{\lambda/t}(hp,Sp)\omega_{\lambda/t}(hp,Su)}{2[1+\omega _{\lambda/t}(hp, hu)]}, \frac{\omega_{\lambda/t}(hu,Su)\omega_{\lambda/t}(hu,Sp)}{2[1+\omega _{\lambda/t}(hp, hu)]} \biggr\} \\ =&\omega_{\lambda/t}(hp, hu). \end{aligned}$$
So, which is a contradiction. This proves the uniqueness of the point of coincidence. Thus \(x^{*}\) is a unique coincidence point of S and h. By using Lemma 2.1, we see that S and h have a unique common fixed point. □
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(hp,hu))}\varphi(r)\,\mathrm {d}r&\leq \int_{0}^{\pi (\omega_{\lambda/t}(hp,hu))}\varphi(r)\,\mathrm {d}r- \phi \biggl( \int_{0}^{\pi(\omega_{\lambda/t}(hp,hu))}\varphi(r)\,\mathrm {d}r \biggr) \\ &< \int_{0}^{\pi(\omega_{\lambda/t}(hp,hu))}\varphi(r)\,\mathrm {d}r \leq \int _{0}^{\pi(\omega_{\lambda/a}(hp,hu))}\varphi(r)\,\mathrm {d}r, \end{aligned}$$
From Theorem 2.7, we get the following theorems.
Theorem 2.8
Let
\(X_{\omega}\)
be a modular metric space. Assume
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
where
\(\mathfrak{m}_{1}(x,y)\)
is as in Theorem
2.2, \(\varphi\in\Theta\), \(\phi\in\Phi\), and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(Sx,Sy))}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi (\mathfrak{m}_{1}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathfrak {m}_{1}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.12)
Theorem 2.9
Let
\(X_{\omega}\)
be a modular metric space. Suppose
\(a, t\in\mathbb {R}^{+}\)
with
\(a>t\)
and
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
where
\(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(Sx,Sy))}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi (\omega_{\lambda/t}(hx,hy))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi (\omega_{\lambda/t}(hx,hy))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.13)
Remark 2.1
Remark 2.2
In case \(\phi(t)=(1-r)t\), where \(0< r<1\), and \(\phi(t)=t-\psi(t)\), where \(\psi\in\Psi\), then Theorems 2.1-2.9 reduce to corollaries which elongate and generalize Theorems 2.2-4.3 of [2], Theorem 2.1 of [1], Theorems 2.1 and 2.4 of [4], Theorems 2.1-2.4 of [11], Theorems 2.1 and 3.1 of [22], Theorem 2 of [26] and Theorems 3.1 and 3.4 of [3] in the set-up of modular metric space.
By considering similar argument of Theorem 2.7, we state the following results and exclude their proofs.
Theorem 2.10
Let
\(X_{\omega}\)
be a MMS. Suppose
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
for all
\(x,y\in X_{\omega}\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak{m}_{2}(x,y)\)
is as in Theorem
2.4, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(Sx,Sy))}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi (\mathfrak{m}_{2}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathfrak {m}_{2}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.14)
Theorem 2.11
Let
\(X_{\omega}\)
be a MMS. Suppose
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
for all
\(x,y\in X_{\omega}\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak{m}_{3}(x,y)\)
is as in Theorem
2.5, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(Sx,Sy))}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi (\mathfrak{m}_{3}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathfrak {m}_{3}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.15)
Theorem 2.12
Let
\(X_{\omega}\)
be a MMS. Suppose
\(S,h:X_{\omega}\to X_{\omega}\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
2.1
and
for all
\(x,y\in X_{\omega}\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak{m}_{4}(x,y)\)
is as in Theorem
2.6, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\omega_{\lambda/a}(Sx,Sy))}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi (\mathfrak{m}_{4}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi(\mathfrak {m}_{4}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(2.16)
3 Applications to fuzzy metric spaces
In 1988, Grabiec [27] defined contractive mappings on a fuzzy metric space and extended fixed point theorems of Banach and Edelstein in such spaces. Successively, George and Veeramani [28] slightly modified the notion of a fuzzy metric space introduced by Kramosil and Michálek. For more details see [29‐31] and the references therein. In this section we deduce fixed point results in a triangular fuzzy metric space.
Definition 3.1
[28]
The 3-tuple \((X, M, \ast)\) is said to be a fuzzy metric space if X is an arbitrary nonempty set, ∗ is a continuous t-norm and M is a fuzzy set in \(X^{2}\times(0, \infty)\) satisfying the following conditions: for all \(x, y, z \in X\), \(s, t>0\),
(1)
\(M(x, y, t)>0\),
(2)
\(M(x, y, t)=1\) if and only if \(x=y\),
(3)
\(M(x, y, t)=M(y,x,t)\),
(4)
\(M(x, y, t)\ast M(y, z, s) \leq M(x, z, t+s)\),
(5)
\(M(x,y,\cdot): (0, \infty) \rightarrow[0, 1]\) is continuous.
Definition 3.2
[29]
Let \((X, M, \ast)\) be a fuzzy metric space. The fuzzy metric M is called triangular whenever for all \(x,y,z \in X\) and all \(t>0\).
$$\begin{aligned} \frac{1}{M(x, y, t)}-1 \leq\frac{1}{M(x, z, t)}-1 +\frac{1}{M(z, y, t)}-1 \end{aligned}$$
(3.1)
Lemma 3.2
[8]
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Define
for all
\(x,y,z \in X\)
and all
\(\lambda>0\). Then
\(\omega_{\lambda}\)
is a modular metric on
X.
$$\begin{aligned} \omega_{\lambda}(x,y)=\frac{1}{M(x, y, \lambda)}-1 \end{aligned}$$
(3.2)
Definition 3.3
[32]
Two self-mappings S and h of a fuzzy metric space \((X, M, \ast)\) are called weakly compatible if they commute at their coincidence points.
As an application of Lemma 3.2 and the results proved above, we deduce the following new fixed point theorems in triangular fuzzy metric spaces.
Theorem 3.1
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the following assertions:
where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
(1)
\(S(X)\subseteq h(X)\), \(h(X)\)
is a complete subspace of
X;
(2)
S
and
h
are weakly compatible mappings;
(3)
\(\int_{0}^{\frac{1}{M(Sx, Sy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r\leq\int _{0}^{\mathcal{N}(x,y)}\varphi(r)\,\mathrm {d}r-\phi (\int_{0}^{\mathcal {N}(x,y)}\varphi(r)\,\mathrm {d}r )\),
$$\begin{aligned} \begin{aligned} \mathcal{N}(x,y)=&\max \biggl\{ \frac{1}{M(hx, hy, {\lambda/t})}-1, \frac {1}{M(hx, Sx, {\lambda/t})}-1, \\ &\frac{1}{M(hy, Sy, {\lambda/t})}-1, \frac{1}{2M(hx, Sy, {\lambda /t})}+\frac{1}{2M(hy, Sx, {\lambda/t})}-1, \\ &M(hx,hy,{\lambda/t}) \biggl(\frac{1}{M(hx, Sy, {\lambda/t})}-1 \biggr) \biggl( \frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr), \\ & \frac{M(hx,hy,{\lambda/t})}{2} \biggl(\frac{1}{M(hx, Sx, {\lambda /t})}-1 \biggr) \biggl( \frac{1}{M(hx, Sy, {\lambda/t})}-1 \biggr), \\ &\frac{M(hx,hy,{\lambda/t})}{2} \biggl(\frac{1}{M(hy, Sy, {\lambda /t})}-1 \biggr) \biggl( \frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr) \biggr\} , \end{aligned} \end{aligned}$$
Theorem 3.2
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Assume that
\(a, t\in\mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\)
and
\(\lambda\geq0\), where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\frac{1}{M(Sx, Sy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r\leq \int _{0}^{\mathfrak{n}_{1}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {n}_{1}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.3)
$$\begin{aligned} \begin{aligned} \mathfrak{n}_{1}(x,y)=&\max \biggl\{ \frac{1}{M(hx, hy, {\lambda/t})}-1, \frac{1}{M(hx, Sx, {\lambda/t})}-1, \frac{1}{M(hy, Sy, {\lambda /t})}-1, \\ &\frac{1}{2M(hx, Sy, {\lambda/t})}+\frac{1}{2M(hy, Sx, {\lambda /t})}-1 \biggr\} , \end{aligned} \end{aligned}$$
Theorem 3.3
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Assume that
\(a, t\in\mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\)
and
\(\lambda\geq0\), where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\frac{1}{M(Sx, Sy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r\leq \int _{0}^{\frac{1}{M(hx, hy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int _{0}^{\frac{1}{M(hx, hy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.4)
Theorem 3.4
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\frac{1}{M(Sx, Sy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r\leq \int _{0}^{\mathfrak{n}_{2}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {n}_{2}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.5)
$$\begin{aligned} \begin{aligned} \mathfrak{n}_{2}(x,y)=&\max \biggl\{ \frac{1}{M(hx, hy, {\lambda/t})}-1, \frac{1}{M(hx, Sx, {\lambda/t})}-1, \\ &\frac{1}{M(hy, Sy, {\lambda/t})}-1, \frac{1}{2M(hx, Sy, {\lambda /t})}+\frac{1}{2M(hy, Sx, {\lambda/t})}-1, \\ &M(Sx,Sy,{\lambda/t}) \biggl(\frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr) \biggl( \frac{1}{M(hx, Sy, {\lambda/t})}-1 \biggr), \\ & \frac{M(Sx,Sy,{\lambda/t})}{2} \biggl(\frac{1}{M(hy, Sx, {\lambda /t})}-1 \biggr) \biggl( \frac{1}{M(hx, Sx, {\lambda/t})}-1 \biggr), \\ &\frac{M(Sx,Sy,{\lambda/t})}{2} \biggl(\frac{1}{M(hx, Sy, {\lambda /t})}-1 \biggr) \biggl( \frac{1}{M(hy, Sy, {\lambda/t})}-1 \biggr) \biggr\} , \end{aligned} \end{aligned}$$
Theorem 3.5
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\frac{1}{M(Sx, Sy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r\leq \int _{0}^{\mathfrak{n}_{3}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {n}_{3}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.6)
$$\begin{aligned} \begin{aligned} \mathfrak{n}_{3}(x,y)=&\max \biggl\{ \frac{1}{M(hx, hy, {\lambda/t})}-1, \frac{1}{M(hx, Sx, {\lambda/t})}-1, \\ &\frac{1}{M(hy, Sy, {\lambda/t})}-1, \frac{1}{2M(hx, Sy, {\lambda /t})}+\frac{1}{2M(hy, Sx, {\lambda/t})}-1, \\ &M(hx,hy,{\lambda/t}) \biggl(\frac{1}{M(hx, Sy, {\lambda/t})}-1 \biggr) \biggl( \frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr), \\ & \min \biggl\{ \frac{M(hx,hy,{\lambda/t})}{2} \biggl(\frac{1}{M(hx, Sx, {\lambda/t})}-1 \biggr) \biggl( \frac{1}{M(hx, Sy, {\lambda/t})}-1 \biggr), \\ &\frac{M(hx,hy,{\lambda/t})}{2} \biggl(\frac{1}{M(hy, Sy, {\lambda /t})}-1 \biggr) \biggl( \frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr) \biggr\} \biggr\} , \end{aligned} \end{aligned}$$
Theorem 3.6
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
where
\(\varphi\in\Theta\)
and
\(\phi\in\Phi\). Then
S
and
h
have a unique common fixed point.
$$\begin{aligned} \int_{0}^{\frac{1}{M(Sx, Sy, {\lambda/a})}-1}\varphi(r)\,\mathrm {d}r\leq \int _{0}^{\mathfrak{n}_{4}(x,y)}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\mathfrak {n}_{4}(x,y)}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.7)
$$\begin{aligned} \begin{aligned} \mathfrak{n}_{4}(x,y)=&\max \biggl\{ \frac{1}{M(hx, hy, {\lambda/t})}-1, \frac{1}{M(hx, Sx, {\lambda/t})}-1, \\ & \frac{1}{M(hy, Sy, {\lambda/t})}-1, \frac{1}{2M(hx, Sy, {\lambda /t})}+\frac{1}{2M(hy, Sx, {\lambda/t})}-1, \\ &M(Sx,Sy,{\lambda/t}) \biggl(\frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr) \biggl( \frac{1}{M(hx, Sy, {\lambda/t})}-1 \biggr), \\ & \min \biggl\{ \frac{M(Sx,Sy,{\lambda/t})}{2} \biggl(\frac{1}{M(hy, Sx, {\lambda/t})}-1 \biggr) \biggl( \frac{1}{M(hx, Sx, {\lambda/t})}-1 \biggr), \\ &\frac{M(Sx,Sy,{\lambda/t})}{2} \biggl(\frac{1}{M(hx, Sy, {\lambda /t})}-1 \biggr) \biggl( \frac{1}{M(hy, Sy, {\lambda/t})}-1 \biggr) \biggr\} \biggr\} , \end{aligned} \end{aligned}$$
Theorem 3.7
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathcal {N}(x,y)\)
is as in Theorem
3.1, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\frac{1}{M(Sx, Sy, {\lambda/a})}-1)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi(\mathcal{N}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi (\mathcal{N}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.8)
Theorem 3.8
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak {n}_{1}(x,y)\)
is as in Theorem
3.2, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\frac{1}{M(Sx, Sy, {\lambda/a})}-1)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi(\mathfrak{n}_{1}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi (\mathfrak{n}_{1}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.9)
Theorem 3.9
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak {n}_{2}(x,y)\)
is as in Theorem
3.4, \(\varphi\in\Theta\), \(\phi\in\Phi\), and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\frac{1}{M(Sx, Sy, {\lambda/a})}-1)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi(\mathfrak{n}_{2}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi (\mathfrak{n}_{2}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.10)
Theorem 3.10
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak {n}_{3}(x,y)\)
is as in Theorem
3.5, \(\varphi\in\Theta\), \(\phi\in\Phi\)
and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\frac{1}{M(Sx, Sy, {\lambda/a})}-1)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi(\mathfrak{n}_{3}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi (\mathfrak{n}_{3}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.11)
Theorem 3.11
Let
\((X, M, \ast)\)
be a triangular fuzzy metric space. Suppose
\(a, t\in \mathbb{R}^{+}\)
with
\(a>t\)
and
\(S,h:X\to X\)
are two self-mappings satisfying the conditions (1) and (2) of Theorem
3.1
and
for all
\(x,y\in X\), \(a, t\in\mathbb{R}\)
with
\(a>t\), where
\(\mathfrak {n}_{4}(x,y)\)
is as in Theorem
3.6, \(\varphi\in\Theta\), \(\phi\in\Phi\), and
\(\pi\in\Pi\). Then there exists a unique common fixed point of
S
and
h.
$$\begin{aligned} \int_{0}^{\pi(\frac{1}{M(Sx, Sy, {\lambda/a})}-1)}\varphi(r)\,\mathrm {d}r\leq \int_{0}^{\pi(\mathfrak{n}_{4}(x,y))}\varphi(r)\,\mathrm {d}r-\phi \biggl( \int_{0}^{\pi (\mathfrak{n}_{4}(x,y))}\varphi(r)\,\mathrm {d}r \biggr), \end{aligned}$$
(3.12)
Acknowledgements
This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. Therefore, the authors acknowledge with thanks DSR, KAU, for financial support.
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.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this article. All authors read and approved the final manuscript.