# Existence and uniqueness of solutions for a class of integral equations by common fixed point theorems in IFMT-spaces

## Abstract

In this paper, our aim is to address the existence and uniqueness of solutions for a class of integral equations in IFMT-space. Therefore, we introduce the concept of IFMT-spaces and prove a common fixed point theorem in a complete IFMT-space; next we study an application.

## 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 [48].

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 $$E_{\varsigma,\mathcal{M}_{M,N}}$$.

## 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 [1219].

### 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. □

## 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

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

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## Acknowledgements

The author is grateful to the reviewers for their valuable comments and suggestions.

## Author information

Authors

### Competing interests

The author declares to have no competing interests.

### Author’s contributions

Only the author contributed in writing this paper.

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Reprints and Permissions

Saadati, R. Existence and uniqueness of solutions for a class of integral equations by common fixed point theorems in IFMT-spaces. J Inequal Appl 2016, 205 (2016). https://doi.org/10.1186/s13660-016-1148-3

• Accepted:

• Published:

• DOI: https://doi.org/10.1186/s13660-016-1148-3

• 54E40
• 54E35
• 54H25

### Keywords

• integral equations
• nonlinear IF contractive mapping
• complete IFMT-space
• fixed point theorem