1 Introduction and preliminaries

First of all, we would like to introduce the concept of IFMT-space, which is a non-trivial generalization of IFM-space introduced by Park [1] and Saadati and Park [2] and Saadati et al. [3]; also we use results from [4–8].

We say the pair \((L^{*},\leq_{L^{*}})\) is a complete lattice whenever \(L^{*}\) is a non-empty set and we have the operation \(\leq_{L^{*}}\) defined by

$$L^{*} = \bigl\{ (a ,b) : (a ,b) \in[0,1]\times[0,1] \mbox{ and } a+b\leq 1\bigr\} , $$

\((a,b)\leq_{L^{*}} (c,d)\Longleftrightarrow a\leq c\), and \(b \geq d\), for each \((a,b), (c,d)\in L^{*}\).

Definition 1.1

([9])

An IF set \(\mathcal{F}_{\alpha,\beta}\) in a universe U is an object \(\mathcal{F}_{\alpha,\beta} = \{(\alpha_{\mathcal{F}}(u),\beta_{\mathcal{F}}(u)) | u\in U\}\), in which, for all \(u \in U\), \(\alpha_{\mathcal{F}}(u) \in[0,1]\), and \(\beta_{\mathcal{F}}(u) \in[0,1]\) are said the membership degree and the non-membership degree, respectively, of u in \(\mathcal{F}_{\alpha,\beta}\), and furthermore they satisfy \(\alpha_{\mathcal{F}}(u)+\beta_{\mathcal{F}}(u) \leq1\).

We consider \(0_{L^{*}} = (0,1)\) and \(1_{L^{*}} = (1,0)\) as its units.

Definition 1.2

([4])

The mapping \(\mathcal{T} : L^{*}\times L^{*} \longrightarrow L^{*}\) satisfying the following conditions:

  • \((\forall a \in L^{*})\) \((\mathcal{T}(a,1_{L^{*}} )=a)\),

  • \((\forall(a,b) \in L^{*}\times L^{*})\) \((\mathcal{T}(a,b) = \mathcal{T}(b,a))\),

  • \((\forall(a,b, c) \in L^{*}\times L^{*} \times L^{*})\) \((\mathcal{T}(a,\mathcal {T}(b, c)) = \mathcal{T}(\mathcal{T}(a,b), c))\),

  • \((\forall(a,a',b,b')\in L^{*}\times L^{*} \times L^{*}\times L^{*})\) (\(a \leq _{L^{*}}a'\) and \(b \leq_{L^{*}} b' \Longrightarrow\mathcal{T}(a,b)\leq_{L^{*}} \mathcal{T}(a',b')\)).

is said to be a triangular norm (t-norm) on \(L^{*}\).

\(\mathcal{T}\) is said to be a continuous t-norm if the triple \((L^{*},\leq_{L^{*}},\mathcal{T})\) is an Abelian topological monoid with unit \(1_{L^{*}}\).

Definition 1.3

([4])

\(\mathcal{T}\) on \(L^{*}\) is called continuous t-representable if and only if there exist a continuous t-norm ∗ and a continuous t-conorm ⋄ on \([0,1]\) such that, for all \(a=(a_{1},a_{2}), b=(b_{1},b_{2}) \in L^{*}\),

$$\mathcal{T}(a,b) = (a_{1}\ast b_{1},a_{2} \diamond b_{2}). $$

For example, \(\mathcal{T}(a,b) = (a_{1} b_{1},\min(a_{2}+ b_{2},1))\) for all \(a=(a_{1},a_{2})\) and \(b=(b_{1},b_{2})\) in \(L^{*}\) is a continuous t-representable.

Definition 1.4

The decreasing mapping \(\mathcal{N}: L^{*} \longrightarrow L^{*}\) satisfying \(\mathcal{N}(0_{L^{*}} ) = 1_{L^{*}}\) and \(\mathcal{N}(1_{L^{*}} ) = 0_{L^{*}}\) is said a negator on \(L^{*}\). We say \(\mathcal{N}\) is an involutive negator if \(\mathcal{N}(\mathcal{N}(a)) = a\), for all \(a \in L^{*}\). The decreasing mapping \(N : [0,1]\longrightarrow[0,1]\) satisfying \(N(0) = 1\) and \(N(1) = 0\) is said to be a negator on \([0,1]\). The standard negator on \([0,1]\) is defined, for all \(a \in[0,1]\), by \(N_{s}(a) = 1-a\), denoted by \(N_{s}\). We show \((N_{s}(a),a)=\mathcal{N}_{s}(a)\).

Definition 1.5

If for given \(\alpha\in(0,1)\) there is \(\beta\in(0,1)\) such that

$$\mathcal{T}^{m}\bigl(\mathcal{N}_{s}(\beta),\ldots, \mathcal{N}_{s}(\beta )\bigr)>_{L^{*}}\mathcal{N}_{s}( \alpha), \quad m\in \mathbf{N}, $$

then \(\mathcal{T}\) is a H-\(type\) t-norm.

A typical example of such t-norms is

$$\wedge(a,b)= \bigl(\operatorname{Min}(a_{1},b_{1}), \operatorname{Max}(a_{2},b_{2}) \bigr), $$

for every \(a=(a_{1},a_{2})\) and \(b=(b_{1},b_{2})\) in \(L^{*}\).

Definition 1.6

The tuble \((X,\mathcal{M}_{M,N},\mathcal{T})\) is said to be an IFMT-space if X is an (non-empty) set, \(\mathcal{T}\) is a continuous t-representable, and \(\mathcal{M}_{M,N}\) is a mapping \(X^{2} \times [0,+\infty) \to L^{*}\) (in which \(M,N\) are fuzzy sets from \(X^{2} \times [0,+\infty)\) to \([0,1]\) such that \(M(x,y,t)+N(x,y,t)\leq1\) for all \(x,y\in X\) and \(t>0\)) satisfying the following conditions for every \(x,y,z\in X\) and \(t,s>0\):

  1. (a)

    \(\mathcal{M}_{M,N}(x,y,t)>_{L} 0_{L^{*}}\);

  2. (b)

    \(\mathcal{M}_{M,N}(x,y,t)=\mathcal{M}_{M,N}(y,x,t)=1_{L^{*}} \) iff \(x=y\);

  3. (c)

    \(\mathcal{M}_{M,N}(x,y,t)=\mathcal{M}_{M,N}(y,x,t)\) for each \(x,y \in X\);

  4. (d)

    \(\mathcal{M}_{M,N}(x,y,K(t+s))\geq_{L^{*}}\mathcal{T}(\mathcal {M}_{M,N}(x,z,t),\mathcal{M}_{M,N}(z,y,s))\) for some constant \(K\geq1\);

  5. (e)

    \(\mathcal{M}_{M,N}(x,y,\cdot):[0,\infty)\longrightarrow L^{*}\) is continuous.

Also \(\mathcal{M}_{M,N}\) is said an IFMT. Note that for an IFMT-space

$$\mathcal{M}_{M,N}(x,y,t)=\bigl(M(x,y,t),N(x,y,t)\bigr). $$

\((X,\mathcal{M}_{M,N},\mathcal{T})\) is called a Menger IFMT-space if

$$\lim_{t\to\infty}\mathcal{M}_{M,N}(x,y,t)=\lim _{t\to\infty}\mathcal {M}_{M,N}(y,x,t)=1_{L^{*}}. $$

Remark 1.7

The space of all real functions \(\alpha(x) \), \(x\in[0,1] \) such that \(\int_{0}^{1} {|\alpha(x)|}^{q} \,dx<\infty\), denoted by \(L_{q} \) (\(0< q<1\)), is a metric type space. Consider

$$d(\alpha,\beta)=\biggl({ \int_{0}^{1} {\bigl|{\alpha(x)-\beta(x)}\bigr|}^{q} \,dx}\biggr)^{\frac{1}{q}} , $$

for each \(\alpha,\beta\in L_{q} \). Then d is a metric type space with \(K=2^{\frac{1}{q}} \).

Example 1.8

We consider the set of Lebesgue measurable functions on \([0,1] \) such that \(\int_{0}^{1} {|\alpha(x)|}^{q} \,dx < \infty\), where \(q>0 \) is a real number denoted by \(\mathfrak{M} \). Consider

$$\mathcal{M}_{M,N}(x,y,t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0_{L^{*}} & \mbox{if }t\leq0,\\ (\frac{t}{t+{(\int_{0}^{1} {|{\alpha(x)-\beta(x)}|}^{q} \,dx})^{\frac{1}{q}}}, \frac{(\int_{0}^{1} {|{\alpha(x)-\beta(x)}| }^{q} \,dx)^{\frac{1}{q}}}{t+{(\int_{0}^{1} {|{\alpha(x)-\beta (x)}|}^{q} \,dx})^{\frac{1}{q}}} ) & \mbox{if }t> 0. \end{array}\displaystyle \right . $$

So from Remark 1.7, we have \((M,\mathcal{M}_{M,N},\wedge) \) is IFMT-space with \(K=2^{\frac{1}{q}} \).

Definition 1.9

Let \((X,\mathcal{M}_{M,N},\mathcal{T})\) be a Menger IFMT-space.

  1. (1)

    A sequence \(\{x_{n}\}_{n}\) in X is said to be convergent to x in X if, for every \(\epsilon>0\) and \(\lambda\in0\), there exists a positive integer N such that \(\mathcal{M}_{M,N}(x_{n},x,\epsilon)>1-\lambda\) whenever \(n\geq N\).

  2. (2)

    A sequence \(\{x_{n}\}_{n}\) in X is called a Cauchy sequence if, for every \(\epsilon>0\) and \(\lambda L^{*}-\{0_{L^{*}}\}\), there exists a positive integer N such that \(\mathcal{M}_{M,N}(x_{n},x_{m},\epsilon)>_{L} \mathcal{N}(\lambda) \) whenever \(n, m\geq N\).

  3. (3)

    A Menger IFMT-space \((X,\mathcal{M}_{M,N},\mathcal{T})\) is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.

Remark 1.10

Khamsi and Kreinovich [10] proved, if \((X,\mathcal{M}_{M,N},\mathcal{T})\) is a IFMT-space and \(\{u_{n}\}\) and \(\{v_{n}\}\) are sequences such that \(u_{n}\to u\) and \(v_{n}\to v\), then

$$\lim_{n\to\infty} \mathcal{M}_{M,N}(u_{n},v_{n},t)= \mathcal{M}_{M,N}(u,v,t). $$

Remark 1.11

Let for each \(\sigma\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) there exists a \(\varsigma\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) (which does not depend on n) with

$$ \mathcal{T}^{n-1}\bigl(\mathcal{N}(\varsigma),\ldots, \mathcal {N}(\varsigma)\bigr)>_{L}\mathcal{N}(\sigma)\quad \mbox{for each } n\in \{1,2,\ldots\}. $$
(1)

Lemma 1.12

([11])

Let \((X,\mathcal{M}_{M,N},\mathcal{T})\) be a Menger IFMT-space. If we define \(E_{\varsigma,\mathcal{M}_{M,N}}: X^{2}\longrightarrow{ \mathbb{R}}^{+}\cup\{0\}\) by

$$E_{\varsigma,\mathcal{M}_{M,N}}(x,y)=\inf\bigl\{ t>0 : \mathcal {M}_{M,N}(x,y,t)>_{L} \mathcal{N}(\varsigma)\bigr\} $$

for each \(\varsigma\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) and \(x,y\in X\), then we have the following:

  1. (1)

    For any \(\sigma\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\), there exists a \(\varsigma\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) such that

    $$E_{\mu,\mathcal{M}_{M,N}}(x_{1},x_{k})\leq KE_{\varsigma,\mathcal{M}_{M,N}}(x_{1},x_{2})+K^{2}E_{\varsigma,\mathcal {M}_{M,N}}(x_{2},x_{3})+ \cdots+K^{n-1}E_{\varsigma,\mathcal{M}_{M,N}}(x_{k-1},x_{k}) $$

    for any \(x_{1},\ldots,x_{k}\in X\).

  2. (2)

    For each sequence \(\{x_{n}\}\) in X, we have \(\mathcal{M}_{M,N}(x_{n},x,t)\longrightarrow1_{L^{*}}\) if and only if \(E_{\varsigma,\mathcal{M}_{M,N}}(x_{n} ,x)\to0\). Also the sequence \(\{x_{n}\}\) is Cauchy w.r.t. \(\mathcal {M}_{M,N}\) if and only if it is Cauchy with \(E_{\varsigma,\mathcal{M}_{M,N}}\).

2 Common fixed point theorems

In this section we study some common fixed point theorems in Menger IFMT-spaces, ones can find similar results in others spaces at [12–19].

Definition 2.1

Let f and g be mappings from a Menger IFMT-space \((X,\mathcal {M}_{M,N},\mathcal{T})\) into itself. The mappings f and g are called weakly commuting if

$$\mathcal{M}_{M,N}(fgx,gfx,t)\geq_{L} \mathcal{M}_{M,N}(fx,gx,t) $$

for each x in X and \(t>0\).

Now we assume that Φ is the set of all functions

$$\phi :[0,\infty)\longrightarrow[0,\infty) $$

which satisfy \(\lim_{n\to \infty} \phi^{n}(t)=0\) for \(t>0\) and are onto and strictly increasing. Also, we denote by \(\phi^{n}(t)\) the nth iterative function of \(\phi(t)\).

Remark 2.2

Note that \(\phi\in\Phi\) implies that \(\phi(t)< t\) for \(t>0\). Consider \(t_{0}>0\) with \(t_{0} \leq\phi(t_{0})\). Since ϕ is a nondecreasing function we get \(t_{0} \leq\phi^{n}(t_{0})\) for every \(n\in \{1,2,\ldots\}\), which is a contradiction. Also \(\phi(0)=0\).

Lemma 2.3

([11])

If a Menger IFMT-space \((X,\mathcal{M}_{M,N},\mathcal{T})\) obeys the condition

$$\mathcal{M}_{M,N}(x, y,t) = C, \quad \textit{for all } t > 0 , $$

then we get \(C = 1_{L^{*}}\) and \(x=y\).

Theorem 2.4

Consider the complete Menger IFMT-space \((X,\mathcal{M}_{M,N},\mathcal {T})\). Assume that f and g are weakly commuting self-mappings of X such that:

  1. (a)

    \(f(X)\subseteq g(X)\);

  2. (b)

    f or g is continuous;

  3. (c)

    \(\mathcal{M}_{M,N}(fx,fy,\phi(t))\geq_{L} \mathcal {M}_{M,N}(gx,gy,t)\) in which \(\phi\in\Phi\).

  1. (i)

    Now let (1) hold and let there exist a \(x_{0}\in X\) with

    $$E_{\mathcal{M}_{M,N}}(g x_{0}, f x_{0})=\sup\bigl\{ E_{\gamma,\mathcal {M}_{M,N}}( g x_{0}, f x_{0}): \gamma \in L^{*}- \{0_{L^{*}},1_{L^{*}}\}\bigr\} < \infty, $$

    therefore f and g have a common fixed point which is unique.

Proof

(i) Select \(x_{0}\in X\) with \(E_{\mathcal{M}_{M,N}}(g x_{0}, f x_{0})<\infty \). Select \(x_{1} \in X\) with \(f x_{0}=g x_{1}\). Now select \(x_{n+1}\) such that \(fx_{n}=gx_{n+1}\). Now \(\mathcal{M}_{M,N}(fx_{n},fx_{n+1},\phi ^{n+1}(t)) \geq_{L} \mathcal{M}_{M,N}(gx_{n},gx_{n+1},\phi^{n}(t))=\mathcal{M}_{M,N}(f x_{n-1},fx_{n},\phi^{n}(t)) \geq_{L} \cdots\geq \mathcal{M}_{M,N}(gx_{0}, gx_{1},t)\).

We have for each \(\lambda \in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) (see Lemma 1.9 of [11])

$$\begin{aligned} E_{\lambda, \mathcal{M}_{M,N}}(fx_{n}, f x_{n+1}) =&\inf \bigl\{ \phi^{n+1}(t)>0: \mathcal{M}_{M,N}\bigl(f x_{n}, f x_{n+1},\phi^{n+1}(t)\bigr) >_{L}\mathcal{N}( \lambda) \bigr\} \\ \leq& \inf\bigl\{ \phi^{n+1}(t)>0: \mathcal{M}_{M,N}(g x_{0}, f x_{0}, t) >_{L} \mathcal{N}(\lambda) \bigr\} \\ \leq& \phi^{n+1} \bigl( \inf\bigl\{ t>0: \mathcal{M}_{M,N}(g x_{0}, f x_{0}, t) >_{L} \mathcal{N}(\lambda) \bigr\} \bigr) \\ = &\phi^{n+1} \bigl(E_{\lambda,\mathcal{M}_{M,N}} (g x_{0}, f x_{0}) \bigr) \\ \leq& \phi^{n+1}\bigl(E_{\mathcal{M}_{M,N}} (g x_{0}, f x_{0}) \bigr). \end{aligned}$$

Thus \(E_{\lambda, \mathcal{M}_{M,N}}(f x_{n}, f x_{n+1}) \leq\phi^{n+1} ( E_{\mathcal{M}_{M,N}} (g x_{0}, f x_{0}) )\) for each \(\lambda\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) and so

$$E_{\mathcal{M}_{M,N}} (f x_{n}, fx_{n+1}) \leq\phi^{n+1} \bigl( E_{\mathcal {M}_{M,N}} (g x_{0}, f x_{0}) \bigr). $$

Let \(\epsilon>0\). Select \(n\in\{1,2,\ldots\}\); therefore \(E_{\mathcal{M}_{M,N}}(f x_{n}, f x_{n+1})< \frac{\epsilon-\phi(\epsilon)}{K}\). For \(\lambda\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) there exists a \(\mu\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) with

$$\begin{aligned} E_{\lambda, \mathcal{M}_{M,N}}(f x_{n}, f x_{n+2}) \leq & KE_{\mu, \mathcal{M}_{M,N}}(f x_{n}, f x_{n+1})+ KE_{\mu, \mathcal{M}_{M,N}}(f x_{n+1}, f x_{n+2}) \\ \leq& KE_{\mu, \mathcal{M}_{M,N}}(f x_{n}, f x_{n+1})+\phi\bigl( KE_{\mu, \mathcal{M}_{M,N}}( f x_{n}, f x_{n+1})\bigr) \\ \leq& KE_{\mathcal{M}_{M,N}}( f x_{n}, f x_{n+1})+ \phi\bigl(K E_{\mathcal {M}_{M,N}}(f x_{n}, f x_{n+1})\bigr) \\ \leq&K \frac{\epsilon-\phi(\epsilon)}{K}+ \phi \biggl(K \frac{\epsilon -\phi(\epsilon)}{K} \biggr) \\ \leq& \epsilon. \end{aligned}$$

We can continue this process for every \(\lambda\in L^{*}-\{ 0_{L^{*}},1_{L^{*}}\}\); then

$$E_{\mathcal{M}_{M,N}}(f x_{n}, f x_{n+2}) \leq\epsilon. $$

For \(\lambda\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) there exists a \(\mu\in L^{*}-\{0_{L^{*}},1_{L^{*}}\}\) with

$$\begin{aligned} E_{\lambda, \mathcal{M}_{M,N}}(f x_{n}, x_{n+3}) \leq& KE_{\mu, \mathcal {M}_{M,N}}(f x_{n}, f x_{n+1})+KE_{\mu, \mathcal{M}_{M,N}}(f x_{n+1}, f x_{n+3}) \\ \leq& KE_{\mu, \mathcal{M}_{M,N}}(f x_{n}, f x_{n+1})+ \phi\bigl(K E_{\mu, \mathcal{M}_{M,N}}(f x_{n}, f x_{n+2})\bigr) \\ \leq& KE_{\mathcal{M}_{M,N}}(f x_{n}, f x_{n+1})+\phi\bigl(K E_{\mathcal {M}_{M,N}}(f x_{n}, f x_{n+2})\bigr) \\ \leq& \epsilon-\phi(\epsilon)+ \phi(\epsilon)=\epsilon, \end{aligned}$$

from \(\mathcal{M}_{M,N}(fx_{n+1},fx_{n+3},\phi(t)) \geq_{L} \mathcal{M}_{M,N}(gx_{n+1},gx_{n+3},t)=\mathcal{M}_{M,N}(fx_{n}, fx_{n+2},t)\) we have \(E_{\lambda,\mathcal{M}_{M,N}}(fx_{n+1}, fx_{n+3}) \leq \phi(E_{\mu,\mathcal{M}_{M,N}}(fx_{n},fx_{n+2}))\), which implies that

$$E_{\mathcal{M}_{M,N}}(f x_{n}, f x_{n+3}) \leq\epsilon. $$

By using induction

$$E_{\mathcal{M}_{M,N}}( f x_{n}, f x_{n+k}) \leq\epsilon\quad \mbox{for } k \in \{1,2,\ldots\}, $$

and we conclude that \(\{f x_{n}\}_{n}\) is a Cauchy sequence and by the completeness of X, \(\{f x_{n}\}_{n}\) converges to a point named z in X. Also \(\{g x_{n}\}_{n}\) converges to z. Now we assume that the mapping f is continuous. Then \(\lim_{n}ffx_{n}= fz\) and \(\lim_{n}fgx_{n}=fz\). Also, since f and g are weakly commuting,

$$\mathcal{M}_{M,N}(fgx_{n},gfx_{n},t) \geq_{L} \mathcal{M}_{M,N}(fx_{n},gx_{n},t). $$

Take \(n \to\infty\) in the above inequality and we get \(\lim_{n}gfx_{n} = fz\), by the continuity of \(\mathcal {M}\). Now, we show that \(z=fz\). Assume that \(z\neq fz\). From (c) for each \(t>0\) we have

$$\mathcal{M}_{M,N}\bigl(fx_{n},ffx_{n}, \phi^{k+1}(t)\bigr)\geq_{L} \mathcal{M}_{M,N} \bigl(gx_{n},gfx_{n},\phi^{k}(t)\bigr),\quad k\in \mathbb{N}. $$

Suppose that \(n\to\infty\) in the above inequality; we get

$$\mathcal{M}_{M,N}\bigl(z,fz,\phi^{k+1}(t)\bigr) \geq_{L} \mathcal{M}_{M,N}\bigl(z,fz,\phi^{k}(t) \bigr). $$

Furthermore we have

$$\mathcal{M}_{M,N}\bigl(z,fz,\phi^{k}(t)\bigr) \geq_{L} \mathcal{M}_{M,N}\bigl(z,fz,\phi^{k-1}(t) \bigr) $$

and

$$\mathcal{M}_{M,N}\bigl(z,fz,\phi(t)\bigr) \geq_{L} \mathcal{M}_{M,N}(z,fz,t). $$

Also

$$\mathcal{M}_{M,N}\bigl(z,fz,\phi^{k+1}(t)\bigr) \geq_{L} \mathcal{M}_{M,N}(z,fz,t). $$

Next, we have (see Remark 2.2)

$$\mathcal{M}_{M,N}\bigl(z,fz,\phi^{k+1}(t)\bigr) \leq_{L} \mathcal{M}_{M,N}(z,fz,t). $$

Then \(\mathcal{M}_{M,N}(z,fz,t)=C\) and from Lemma 2.3, we conclude that \(z=fz\). By assumption we have \(f(X)\subseteq g(X)\); then there exists a \(z_{1}\) in X such that \(z=fz=gz_{1}\). Now,

$$\mathcal{M}_{M,N}(ffx_{n},fz_{1},t) \geq_{L} \mathcal{M}_{M,N}\bigl(gfx_{n},gz_{1}, \phi^{-1}(t)\bigr). $$

Take \(n\to\infty\); we get

$$\mathcal{M}_{M,N}(fz,fz_{1},t)\geq_{L} \mathcal{M}_{M,N}\bigl(fz,gz_{1},\phi^{-1}(t) \bigr)=1_{L^{*}}, $$

then \(fz=fz_{1}\), i.e., \(z=fz=fz_{1}=gz_{1}\). Also for each \(t>0\) we get

$$\mathcal{M}_{M,N}(fz,gz,t)=\mathcal{M}_{M,N}(fgz_{1},gfz_{1},t) \geq_{L} \mathcal{M}_{M,N}(fz_{1},gz_{1},t)= \varepsilon_{0}(t) $$

since f and g are weakly commuting, from which we can conclude that \(fz=gz\). This implies that z is a common fixed point of f and g.

Now we prove the uniqueness. Assume that \(z' \neq z\) is another common fixed point of f and g. Now, for each \(t>0\) and \(n \in \mathbb{N}\), we have

$$\begin{aligned} \mathcal{M}_{M,N}\bigl(z,z',\phi^{n+1}(t) \bigr) =& \mathcal{M}_{M,N}\bigl(fz,fz',\phi ^{n+1}(t)\bigr)\geq_{L} F_{gz,gz'}\bigl( \phi^{n}(t)\bigr)=F_{z,z'}\bigl(\phi^{n}(t)\bigr). \end{aligned}$$

Also of course we have

$$\begin{aligned} \mathcal{M}_{M,N}\bigl(z,z',\phi^{n}(t)\bigr) \geq_{L} \mathcal{M}_{M,N}\bigl(z,z', \phi^{n-1}(t)\bigr) \end{aligned}$$

and

$$\begin{aligned} \mathcal{M}_{M,N}\bigl(z,z',\phi^{n}(t)\bigr) \geq_{L} \mathcal{M}_{M,N}\bigl(z,z',t\bigr). \end{aligned}$$

As a result

$$\begin{aligned} \mathcal{M}_{M,N}\bigl(z,z',\phi^{n+1}(t)\bigr) \geq_{L} & \mathcal{M}_{M,N}\bigl(z,z',t\bigr). \end{aligned}$$

On the other hand we have

$$\begin{aligned} \mathcal{M}_{M,N}\bigl(z,z',t\bigr)\leq_{L} \mathcal{M}_{M,N}\bigl(z,z',\phi^{n+1}(t)\bigr). \end{aligned}$$

Then \(\mathcal{M}_{M,N}(z,z',t)=C\), see Lemma 2.3, implies that \(z=z'\), which is contradiction. Then z is the unique common fixed point of f and g. □

3 The existence and uniqueness of solutions for a class of integral equations

Assume that \(X=C([1,3],(-\infty,2.1443888))\) and

$$\mathcal{M}_{M,N}(x,y,t) = \left \{ \textstyle\begin{array}{@{}l@{\quad}l} 0 & \mbox{if }t\leq0,\\ (\inf_{\ell\in[1,3]}\frac{t}{t+ ( x(\ell)-y(\ell))^{2}}, \sup_{\ell \in[1,3]}\frac{( x(\ell)-y(\ell))^{2}}{t+ ( x(\ell)-y(\ell))^{2}} ) & \mbox{if }t> 0, \end{array}\displaystyle \right . $$

for \(x,y\in X\), then \((M,\mathcal{M}_{M,N},\wedge) \) is a complete IFTM-space with \(K=2\).

We consider the mapping \(T:X\to X\) by

$$T\bigl(x(\ell)\bigr)=4+ \int_{1}^{\ell}\bigl(x(u)-u^{2}\bigr) e^{1-u}\,du. $$

Put \(g(x)=T(x)\) and \(f(x)=T^{2}(x)\). Since \(fg=gf\), f and g are (weakly) commuting. Now, for \(x,y\in X\) and \(t>0\),

$$\begin{aligned}[b] &\mathcal{M}_{M,N}(fx,fy,t) \\ &\quad= \mathcal{M}_{M,N}\bigl(T\bigl(Tx(\ell )\bigr),T\bigl(Ty(\ell) \bigr),t\bigr) \\ &\quad= \biggl(\inf_{\ell\in[1,3]}\frac{t}{t+|\int_{1}^{\ell}(Tx(u)-Ty(u)) e^{1-u}\,du|^{2}}, \sup _{\ell\in[1,3]}\frac{|\int_{1}^{\ell}(Tx(u)-Ty(u)) e^{1-u}\,du|^{2}}{t+|\int_{1}^{\ell}(Tx(u)-Ty(u)) e^{1-u}\,du|^{2}} \biggr) \\ &\quad\ge \biggl(\frac{t}{t+\frac{1}{e^{4}}|\int_{1}^{3} (Tx(u)-Ty(u))\,du|^{2}}, \frac{\frac{1}{e^{4}}|\int_{1}^{3} (Tx(u)-Ty(u))\,du|^{2}}{t+\frac{1}{e^{4}}|\int _{1}^{3} (Tx(u)-Ty(u))\,du|^{2}} \biggr) \\ &\quad= \mathcal{M}_{M,N}(gx,gy,t), \end{aligned} $$

then

$$\mathcal{M}_{M,N}(fx,fy, \biggl(\frac{t}{e^{4}} \biggr) \ge_{L} \mathcal {M}_{M,N}(gx,gy,t). $$

Thus all conditions of Theorem 2.4 are satisfied for \(\phi (t)=\frac{t}{e^{4}}\) and so f and g have a unique common fixed point, which is the unique solution of the integral equations

$$x(\ell)=4+ \int_{1}^{\ell}\bigl(x(u)-u^{2}\bigr) e^{1-u}\,du $$

and

$$x(\ell)=(1-\ell)^{2} e^{1-\ell}+ \int_{1}^{\ell}\int_{1}^{u} \bigl(x(v)-v^{2}\bigr) e^{2-(u+v)}\,dv\,du. $$