Common fixed point theorems in modified intuitionistic fuzzy metric spaces with common property (E.A.)

Abstract

In this article, we utilize the notions of the property (E.A.) and common property (E.A.) in the setting of modified intuitionistic fuzzy metric spaces to prove a result interrelating the property (E.A.) with common property (E.A.). Also using the common property (E.A.), we prove some common fixed point theorems in modified intuition-istic fuzzy metric spaces satisfying an implicit relation. Some related results are also derived besides furnishing an illustrative example.

AMS Subject Classification (2000): Primary 54H 25; Secondary 47H 10.

1 Introduction and preliminaries

The concept of fuzzy set was introduced in 1965 by Zadeh [1]. Since then, with a view to utilize this concept in topology and analysis, many authors have extensively developed the theory of fuzzy sets along with their applications (e.g., [2–9], 39). In 1986, with similar endeavor, Atanassov [10] introduced and studied the concept of intuitionistic fuzzy sets. Using the idea of intuitionistic fuzzy set, a generalization of fuzzy metric space was introduced by Park [11] which is now known as modified intuitionistic fuzzy metric space wherein notions of continuous t-norm and continuous t-conorm are employed.

Fixed point theory is one of the most fruitful and effective tools in mathematics which has enormous applications in several branches of science especially in chaos theory, game theory, theory of differential equations, etc. Intuitionistic fuzzy metric notion is also useful in modeling some physical problems wherein it is necessary to study the relationship between two probability functions as noticed in [12]. For instance, it has a concrete physical visualization in the context of two-slit experiment as the foundation of E-infinity theory of high energy physics whose details are available in El Naschie in [13–15]. Since the topology induced by intuitionistic fuzzy metric coincides with the topology induced by fuzzy metric (see [12]), Saadati et al. [16] reframed the idea of intuitionistic fuzzy metric spaces and proposed a new notion under the name of modified intuitionistic fuzzy metric spaces by introducing the idea of continuous t-representable.

In 1986, Jungck [17] introduced the notion of compatible mappings in metric spaces and utilized the same (as a tool) to improve commutativity conditions in common fixed point theorems. This concept has frequently been employed to prove existence theorems on common fixed points. In recent past, several authors (e.g., [18–31]) proved various fixed point theorems employing relatively more general contractive conditions. However, the study of common fixed points of non-compatible mappings is also equally interesting which was initiated by Pant [32]. Recently, Aamri and Moutawakil [33] and Liu et al. [34] respectively, defined the property (E.A.) and common property (E.A.) and utilize the same to prove common fixed point theorems in metric spaces. Most recently, Kubiaczyk and Sharma [35] defined the property (E.A.) in Menger PM spaces and utilize the same to prove results on common fixed points wherein the authors claim their results for strict contractions which are merely valid upto contractions. Similar results are also proved by Imdad et al. [23] via common property (E.A). The aim of this article is to utilize the no tion of the property (E.A.) and common property (E.A) to prove some common fixed point theorems in modified intuitionistic fuzzy metric spaces. Our results generalize several previously known results in various spaces which include results in intuitionistic fuzzy metric spaces and metric spaces. Some related results are also derived besides furnishing an illustrative example.

Lemma 1.1. [36] Consider the set L* and operation ${\le }_{L^{*}}$ defined by

$$L^{*}=\left\{\left(x_1,x_2\right):\left(x_1,x_2\right)\in {\left[0,1\right]}^2\text{and}{x}_1+x_2\le 1\right\}$$

$\left(x_1,x_2\right){\le }_{L^{*}}\left(y_1,y_2\right)\iff x_1\le y_1$ and x₂ ≥ y₂, for every (x₁, x₂), (y₁, y₂) ∈ L*. Then $\left(L^{*},{\le }_{L^{*}}\right)$ is a complete lattice.

Definition 1.1. [10] An intuitionistic fuzzy set ${\mathcal{A}}_{\zeta ,\eta }$ in a universe U is an object ${\mathcal{A}}_{\zeta ,\eta }=\left\{\left({\zeta }_{\mathcal{A}}\left(u\right),{\eta }_{\mathcal{A}}\left(u\right)\left|u\right.\in U\right)\right\}$, where, for all $u\in U,{\zeta }_{\mathcal{A}}\left(u\right)\in \left[0,1\right]$ and ${\eta }_{\mathcal{A}}\left(u\right)\in \left[0,1\right]$ are called the membership degree and the non-membership degree, respectively, of $u\in {\mathcal{A}}_{\zeta ,\eta }$, and furthermore they satisfy ${\zeta }_{\mathcal{A}}\left(u\right)+{\eta }_{\mathcal{A}}\left(u\right)\le 1$.

For every z _i = (x _i , y _i ) ∈ L*, if c _i ∈ [0,1] such that ${\sum }_{j=1}^n{c}_j=1$ then it is easy to see that

$$c_1\left(x_1,y_1\right)+\cdots +c_n\left(x_n,y_n\right)=\sum _{j=1}^n{c}_j\left(x_j,y_j\right)=\left(\sum _{j=1}^n{c}_j{x}_j,\sum _{j=1}^n{c}_j{y}_j\right)\in L^{*}.$$

We denote its units by $0_{L^{*}}=\left(0,1\right)$ and $1_{L^{*}}=\left(1,0\right)$. Classically, a triangular norm * = T on [0,1] is defined as an increasing, commutative, associative mapping T : [0,1]² → [0,1] satisfying T(1, x) = 1 * x = x, for all x ∈ [0,1]. A triangular co-norm S = ⋄ is defined as an increasing, commutative, associative mapping S : [0,1]² → [0,1] satisfying S(0, x) = 0⋄x = x, for all x ∈ [0,1]. Using the lattice $\left(L^{*},{\le }_{L^{*}}\right)$ these definitions can straightforwardly be extended.

Definition 1.2. [37] A triangular norm (t-norm) on L* is a mapping $\mathcal{T}:{\left(L^{*}\right)}^2\to L^{*}$ satisfying the following conditions:

• (I) $\left(\forall x\in L^{*}\right)\left(\mathcal{T}\left(x,1_{L^{*}}\right)=x\right)$ (boundary condition),

• (II) $\left(\forall \left(x,y\right)\in {\left(L^{*}\right)}^2\right)\left(\mathcal{T}\left(x,y\right)=\mathcal{T}\left(y,x\right)\right)$ (commutativity),

• (III) $\left(\forall \left(x,y,z,\right)\in {\left(L^{*}\right)}^3\right)\left(\mathcal{T}\left(x,\mathcal{T}\left(y,z\right)\right)=\mathcal{T}\left(\mathcal{T}\left(x,y\right),z\right)\right)$ (associativity),

• (IV) $\left(\forall \left(x,x^{\prime },y,y^{\prime }\right)\in {\left(L^{*}\right)}^4\right)\left(x{\le }_{L^{*}}{x}^{\prime }\right)$ and $\left(y{\le }_{L^{*}}{y}^{\prime }\to \mathcal{T}\left(x,y\right){\le }_{L^{*}}\mathcal{T}\left(x^{\prime },y^{\prime }\right)\right)$ (monotonic-ity).

Definition 1.3. [36, 37] A continuous t-norm 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 x = (x₁, x₂), y = (y₁, y₂) ∈ L*,

$$\mathcal{T}\left(x,y\right)=\left(x_1*y_1,x_2♢y_2\right).$$

Now, we define a sequence $\left\{{\mathcal{T}}^n\right\}$ recursively by $\left\{{\mathcal{T}}^1=\mathcal{T}\right\}$ and

$${\mathcal{T}}^n\left(x^{\left(1\right)},...,x^{\left(n+1\right)}\right)=\mathcal{T}\left({\mathcal{T}}^{n-1}\left(x^{\left(1\right)},...,x^{\left(n\right)}\right),x^{\left(n+1\right)}\right)$$

for n ≥ 2 and x⁽ⁱ⁾∈ L*.

Definition 1.4. [36, 37] A negator on L* is any decreasing mapping $\mathcal{N}:L^{*}\to L^{*}$ satisfying $\mathcal{N}\left(0_{L^{*}}\right)=1_{L^{*}}$ and $\mathcal{N}\left(1_{L^{*}}\right)=0_{L^{*}}$. If $\mathcal{N}\left(\mathcal{N}\left(x\right)\right)=x$, for all x ∈ L*, then is called an involutive negator. A negator on [0,1] is a decreasing mapping N : [0,1] → [0,1] satisfying N(0) = 1 and N(1) = 0. N _s denotes the standard negator on [0,1] defined as (for all x ∈ [0,1])N _s (x) = 1-x.

Definition 1.5. [16] Let M, N are fuzzy sets from X² × (0, ∞) to [0,1] such that M(x, y, t) + N(x, y, t) ≤ 1 for all x, y ∈ X and t > 0. The 3-tuple $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ is said to be an intuitionistic fuzzy metric space if X is an arbitrary (non-empty) set, is a continuous t-representable and ${ℳ}_{M,N}$ is a mapping X² × (0, ∞) → L* (an intuitionistic fuzzy set, see Definition 1.1) satisfying the following conditions for every x, y ∈ X and t, s > 0:

• (I) ${ℳ}_{M,N}\left(x,y,t\right){>}_{L^{*}}{0}_{L^{*}}$,

• (II) ${ℳ}_{M,N}\left(x,y,t\right)=1_{L^{*}}$ if and only if x = y,

• (III) ${ℳ}_{M,N}\left(x,y,t\right)={ℳ}_{M,N}\left(y,x,t\right)$,

• (IV) ${ℳ}_{M,N}\left(x,y,t+s\right){\ge }_{L^{*}}\mathcal{T}\left({ℳ}_{M,N}\left(x,z,t\right),{ℳ}_{M,N}\left(z,y,s\right)\right)$,

• (V) ${ℳ}_{M,N}\left(x,y,.\right):\left(0,\infty ,\right)\to L^{*}$ is continuous.

In this case ${ℳ}_{M,N}$ is called an intuitionistic fuzzy metric. Here,

$${ℳ}_{M,N}\left(x,y,t\right)=\left(M\left(x,y,t\right),N\left(x,y,t\right)\right).$$

Remark 1.1. [38] In an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right),M\left(x,y,.\right)$ is non-decreasing and N(x, y,.) is non-increasing for all x, y ∈ X. Hence $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ is non-decreasing function for all x, y ∈ X.

Example 1.1. [16] Let (X, d) be a metric space. Denote $\mathcal{T}\left(a,b\right)=\left(a_1{b}_1,\text{min}\left\{a_2+b_2,1\right\}\right)$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L* and let M and N be fuzzy sets on X² × (0, ∞) defined as follows:

$${ℳ}_{M,N}\left(x,y,t\right)=\left(M\left(x,y,t\right),N\left(x,y,t\right)\right)=\left(\frac{h{t}^n}{h{t}^n+md\left(x,y\right)},\frac{md\left(x,y\right)}{h{t}^n+md\left(x,y\right)}\right)$$

for all h, m, n, t ∈ R⁺. Then $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ is an intuitionistic fuzzy metric space.

Example 1.2. [16] Let X = ℕ. Denote $\mathcal{T}\left(a,b\right)=\left(\text{max}\left\{0,a_1+b_1-1\right\},a_2+b_2-a_2{b}_2\right)$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L* and let M and N be fuzzy sets on X² × (0, ∞) defined as follows:

$${ℳ}_{M,N}\left(x,y,t\right)=\left(M\left(x,y,t\right),N\left(x,y,t\right)\right)=\left\{\begin{array}{ccc}\left(\frac{x}{y},\frac{y-x}{y}\right)\hfill & \text{if}\hfill & x\le y\hfill \\ \left(\frac{y}{x},\frac{x-y}{x}\right)\hfill & \text{if}\hfill & y\le x\hfill \end{array}\right.,$$

for all x, y ∈ X and t > 0. Then $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ is an intuitionistic fuzzy metric space.

Definition 1.6. [16] Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be an intuitionistic fuzzy metric space. For t > 0, define the open ball B(x, r, t) with center x ∈ X and radius 0 < r < 1, as

$$B\left(x,r,t\right)=\left\{y\in X:{ℳ}_{M,N}\left(x,y,t\right){>}_{L^{*}}\left(N_s\left(r\right),r\right)\right\}.$$

A subset A ⊂ X is called open if for each x ∈ A, there exist t > 0 and 0 < r < 1 such that B(x, r, t) ⊆ A. Let ${\tau }_{{ℳ}_{M,N}}$ denote the family of all open subsets of X. ${\tau }_{{ℳ}_{M,N}}$ is called the topology induced by intuitionistic fuzzy metric.

Note that this topology is Hausdorff (see Remark 3.3 and Theorem 3.5 of [11]).

Definition 1.7. [16] A sequence {x _n } in an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ is called a Cauchy sequence if for each 0 < ϵ < 1 and t > 0, there exists n₀ ∈ ℕ such that

$${ℳ}_{M,N}\left(x_n,y_m,t\right){>}_{L^{*}}\left(N_s\left(\epsilon \right),\epsilon \right)$$

and for each n, m ≥ n₀ here N _s is the standard negator. The sequence {x _n } is said to be convergent to x ∈ X in the intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ and denoted by $x_n{\to }^{{ℳ}_{M,N}}x$ if ${ℳ}_{M,N}\left(x_n,x,t\right)\to 1_{L^{*}}$ whenever n → ∞ for every t > 0. An intuitionistic fuzzy metric space is said to be complete if and only if every Cauchy sequence is convergent.

Lemma 1.2. [9] Let ${ℳ}_{M,N}$ be an intuitionistic fuzzy metric. Then, for any $t>0,{ℳ}_{M,N}\left(x,y,t\right)$ is non-decreasing with respect to t, in $\left(L^{*},{\le }_{L^{*}}\right)$, for all x, y ∈ X.

Definition 1.8. [16] Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be an intuitionistic fuzzy metric space. ℳ is said to be continuous on X × X × (0, ∞) if

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x_n,y_n,t_n\right)={ℳ}_{M,N}\left(x,y,t\right),$$

whenever a sequence {(x _n , y _n , t _n )} in X × X × (0, ∞) converges to a point (x, y, t) ∈ X × X × (0, ∞), i.e.,

$$\begin{array}{l}\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x_n,x,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(y_n,y,t\right)=1_{L^{*}}\\ \text{and}\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x,y,t_n\right)={ℳ}_{M,N}\left(x,y,t\right).\end{array}$$

Lemma 1.3. [16] Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be an intuitionistic fuzzy metric space. Then ℳ is continuous function on X × X × (0, ∞).

Proof. The proof is similar to that of fuzzy metric space case (see Proposition 1 of [39]).

Definition 1.9. [16] Let f and g be mappings from an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ into itself. Then the pair of these mappings is said to be weakly compatible if they commute at their coincidence point, that is, fx = gx implies that fgx = gfx.

Definition 1.10. [16] Let f and g be mappings from an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ into itself. Then the mappings are said to be compatible if

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(fg{x}_n,gf{x}_n,t\right)=1_{L^{*}},\forall t>0$$

whenever {x _n } is a sequence in X such that

$$\underset{n\to \infty }{\text{lim}}f{x}_n=\underset{n\to \infty }{\text{lim}}g{x}_n=x\in X.$$

Definition 1.11. Let f and g be mappings from an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ into itself. Then the mappings are said to be non-compatible if there exists at least one sequence {x _n } in X such that $\underset{n\to \infty }{\text{lim}}f{x}_n=\underset{n\to \infty }{\text{lim}}g{x}_n=x\in X$ but $\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(fg{x}_n,gf{x}_n,t\right)\ne 1_{L^{*}}$ or non-existent for at least one t > 0.

Proposition 1.1. [16] If self-mappings f and g of an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ are compatible, then they are weakly compatible.

The converse is not true as seen in following example.

Example 1.3. [16] Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be an intuitionistic fuzzy metric space, where X = [0,2] and ${ℳ}_{M,N}\left(x,y,t\right)=\left(\frac{1}{t+d\left(x,y\right)},\frac{d\left(x,y\right)}{t+d\left(x,y\right)}\right)$ for all t > 0 and x, y ∈ X. Denote $\mathcal{T}\left(a,b\right)=\left(a_1{b}_1,\text{min}\left\{a_2+b_2,1\right\}\right)$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L*. Define self-maps f and g on X as follows:

$$f\left(x\right)=\left\{\begin{array}{ccc}2\hfill & \text{if}\hfill & 0\le x\le 1\hfill \\ \frac{x}{2}\hfill & \text{if}\hfill & 1<x\le 2\hfill \end{array},g\left(x\right)=\left\{\begin{array}{ccc}2\hfill & \text{if}\hfill & x=1\hfill \\ \frac{x+3}{5}\hfill & \text{if}\hfill & x\ne 1\hfill \end{array}.\right.\right.$$

Then we have g 1 = f 1 = 2 and g 2 = f 2 = 1. Also gf 1 = fg 1 = 1 and gf 2 = fg 2 = 2. Thus pair (f, g) is weakly compatible. Again, $f{x}_n=1-\frac{1}{4n},g{x}_n=1-\frac{1}{10n}$. Thus fx _n → 1, gx _n → 1. Further $gf{x}_n=\frac{4}{5}-\frac{1}{20n},fg{x}_n=2$. Now

$$\begin{array}{l}\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(fg{x}_n,gf{x}_n,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(2,\frac{4}{5}-\frac{1}{20n},t\right)\\ =\left(\frac{t}{t+\frac{6}{5}},\frac{\frac{6}{5}}{t+\frac{6}{5}}\right){<}_{L^{*}}{1}_{L^{*}}\forall t>0.\end{array}$$

Hence the pair (f, g) is not compatible.

Motivated by Aamri and Moutawakil [33], we have

Definition 1.12. [16] Let f and g be two self-mappings of an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$. We say that f and g satisfy the property (E.A.) if there exists a sequence {x _n } in X such that

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(f{x}_n,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(g{x}_n,u,t\right)=1_{L^{*}}$$

for some u ∈ X and t > 0.

Example 1.4. [16] Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be an intuitionistic fuzzy metric space, where X = ℝ and ${ℳ}_{M,N}\left(x,y,t\right)=\left(\frac{1}{t+\left|x-y\right|},\frac{\left|x-y\right|}{t+\left|x-y\right|}\right)$ for every x, y ∈ X and t > 0. Define self-maps f and g on X as follows:

$$fx=2x+1,gx=x+2.$$

Consider the sequence $\left\{x_n=1+\frac{1}{n},n=1,2,....\right\}$ Thus we have

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(f{x}_n,3,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(g{x}_n,3,t\right)=1_{L^{*}}$$

for every t > 0. Then f and g satisfy the property (E.A.).

In the next example, we show that there do exist pairs of mappings which do not share the property (E.A.).

Example 1.5. [16] Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be an intuitionistic fuzzy metric space, where X = ℝ and ${ℳ}_{M,N}\left(x,y,t\right)=\left(\frac{1}{t+\left|x-y\right|},\frac{\left|x-y\right|}{t+\left|x-y\right|}\right)$ for every x, y ∈ X and t > 0. Define self-maps f and g on X as fx = x + 1, gx = x + 2. In case ∃ a sequence {x _n } such that

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(f{x}_n,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(g{x}_n,u,t\right)=1_{L^{*}}$$

for some u ∈ X, then

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(f{x}_n,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x_n+1,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x_n,u-1,t\right)=1_{L^{*}}$$

and

$$\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(g{x}_n,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x_n+2,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(x_n,u-2,t\right)=1_{L^{*}},$$

so that x _n → u - 1 and x _n → u - 2 which is a contradiction. Hence f and g do not satisfy the property (E.A.).

Motivated by Liu et al. [34] and Imdad et al. [23, 24], we also have

Definition 1.13. Two pairs (f, S) and (g, T) of self-mappings of an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ are said to satisfy the common property (E.A.) if there exist two sequences {x _n } and {y _n } in X such that

$$\begin{array}{ll}\hfill \underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(f{x}_n,u,t\right)& =\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(S{x}_n,u,t\right)=\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(g{y}_n,u,t\right)\\ =\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(T{y}_n,u,t\right)=1_{L^{*}}\end{array}$$

for some u ∈ X and t > 0.

Definition 1.14. [40] Two finite families of self-mappings ${\left\{f_i\right\}}_{i=1}^m$ and ${\left\{g_k\right\}}_{i=1}^n$ of a set X are said to be pairwise commuting if:

• (i) f _i f _j = f _j f _i i, j ∈ {1, 2,...,m},

• (ii) g _k g _l = g _l g _k k, l ∈ {1,2,...,n},

• (iii) f _i g _k = g _k f _i i ∈ {1,2,...,m} and k ∈ {1,2,...,n}.

2 Implicit relations

Let Ψ be the set of all continuous functions F(t₁, t₂,..., t₆) : $L^{{*}^6}\to L^{*}$, satisfying the following conditions (for all u, v, 1 ∈ L*, u = (u₁, u₂), v = (v₁, v₂) and $1=1_{L^{*}}=\left(1,0\right)$):

(F₁) : for all $u,v{>}_{L^{*}}{0}_{L^{*}},F\left(u,v,u,v,v,u\right){\ge }_{L^{*}}{0}_{L^{*}}$, or $F\left(u,v,v,u,u,v\right){\ge }_{L^{*}}{0}_{L^{*}}$, implies that $u{\ge }_{L^{*}}v$.

(F₂) : $F\left(u,u,1,1,u,u\right){\ge }_{L^{*}}{0}_{L^{*}}$ implies that $u{\ge }_{L^{*}}1$.

Example 2.1. Define F(t₁, t₂, t₃, t₄, t₅, t₆) = 15t₁ - 13t₂ + 5t₃ - 7t₄ + t₅ - t₆. Then F ∈ Ψ.

Example 2.2. Define $F\left(t_1,t_2,t_3,t_4,t_5,t_6\right)=t_1-\frac{1}{2}{t}_2-\frac{5}{6}{t}_3+\frac{1}{3}{t}_4+t_5-t_6$. Then F ∈ Ψ.

3 Results

The following lemma is proved to interrelate the property (E.A.) with common property (E.A.) in the setting of modified intuitionistic fuzzy metric spaces:

Lemma 3.1. Let A, B, S and T be four self-mappings of a modified IFMS $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ satisfying the following conditions:

• (I) the pair (A, S) (or (B, T)) satisfies the property (E.A.),

• (II) A(X) ⊂ T(X) (or B(X) ⊂ S(X)),

• (III) B(y _n ) converges for every sequence y _n in X whenever T(y _n ) converges (or A(x _n ) converges for every sequence x _n in X whenever S(x _n ) converges),

• (IV) for all x, y ∈ X, s > 0, F ∈ Ψ,

  $$\begin{array}{l}F\left({ℳ}_{M,N}\left(Ax,By,s\right),{ℳ}_{M,N}\left(Sx,Ty,s\right),{ℳ}_{M,N}\left(Ty,By,s\right)\right.,\\ \left.{ℳ}_{M,N}\left(Sx,Ax,s\right),{ℳ}_{M,N}\left(Ax,Ty,s\right),{ℳ}_{M,N}\left(By,Sx,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.\end{array}$$

  (3.1)

Then the pairs (A, S) and (B, T) share the common property (E.A.).

Proof. Since the pair (A, S) enjoys the property (E.A.), there exists a sequence {x _n } in X such that

$$\underset{n\to \infty }{\text{lim}}A{x}_n=\underset{n\to \infty }{\text{lim}}S{x}_n=z,\text{for}\text{some}z\in X,$$

implying thereby $\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(A{x}_n,S{x}_n,s\right)=1_{L^{*}}$. Since A(X) ⊂ T(X), therefore for each {x _n } there exists {y _n } in X such that Ax _n = Ty _n . Therefore, $\underset{n\to \infty }{\text{lim}}A{x}_n=\underset{n\to \infty }{\text{lim}}T{y}_n=z$. Thus, in all we have Ax _n → z, Sx _n → z and Ty _n → z. Now, we show that $\underset{n\to \infty }{\text{lim}}{ℳ}_{M,N}\left(B{y}_n,z,s\right)=1_{L^{*}}$. On using inequality (3.1), we have

$$\begin{array}{l}F\left({ℳ}_{M,N}\left(A{x}_n,B{y}_n,s\right),{ℳ}_{M,N}\left(S{x}_n,T{y}_n,s\right),{ℳ}_{M,N}\left(T{y}_n,B{y}_n,s\right),\right.\\ \left.{ℳ}_{M,N}\left(S{x}_n,A{x}_n,s\right),{ℳ}_{M,N}\left(A{x}_n,T{y}_n,s\right),{ℳ}_{M,N}\left(B{y}_n,S{x}_n,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

which on making n → ∞, reduces to

$$F\left({ℳ}_{M,N}\left(z,\underset{n\to \infty }{\text{lim}}B{y}_n,s\right),1_{L^{*}},{ℳ}_{M,N}\left(z,\underset{n\to \infty }{\text{lim}}B{y}_n,s\right),1_{L^{*}},{ℳ}_{M,N}\left(\underset{n\to \infty }{\text{lim}}B{y}_n,z,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.$$

Using (F₁), we get ${ℳ}_{M,N}\left(\underset{n\to \infty }{\text{lim}}B{y}_n,z,s\right)\ge 1_{L^{*}}$, for all s > 0 so that ${ℳ}_{M,N}\left(\underset{n\to \infty }{\text{lim}}B{y}_n,z,s\right)=1_{L^{*}}$, i.e., $\underset{n\to \infty }{\text{lim}}B{y}_n=z$ which shows that the pairs (A, S) and (B, T) share the common property (E.A.).

Our next result is a common fixed point theorem via the common property (E.A.).

Theorem 3.1. Let A, B, S and T be four self-mappings of a modified IFMS $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ satisfying the condition (3.1). Suppose that

• (I) the pairs (A, S) and (B, T) share the common property (E.A.) and

• (II) S(X) and T(X) are closed subsets of X.

Then the pair (A, S) as well as (B, T) have a coincidence point. Moreover, A, B, S and T have a unique common fixed point in X provided both the pairs (A, S) and (B, T) are weakly compatible.

Proof. Since the pairs (A, S) and (B, T) share the common property (E.A.), there exist two sequences {x _n } and {y _n } in X such that

$$\underset{n\to \infty }{\text{lim}}A{x}_n=\underset{n\to \infty }{\text{lim}}S{x}_n=\underset{n\to \infty }{\text{lim}}B{y}_n=\underset{n\to \infty }{\text{lim}}T{y}_n=z,\text{for}\text{some}z\in X.$$

Since S(X) is a closed subset of X, therefore $\underset{n\to \infty }{\text{lim}}S{x}_n=z\in S\left(X\right)$. Also, there exists a point u ∈ X such that Su = z. Now, we show that ${ℳ}_{M,N}\left(Au,z,s\right)=1_{L^{*}}$. On using inequality (3.1), we have

$$\begin{array}{l}F\left({ℳ}_{M,N}\left(Au,B{y}_n,s\right),{ℳ}_{M,N}\left(Su,T{y}_n,s\right),{ℳ}_{M,N}\left(T{y}_n,B{y}_n,s\right),\right.\\ \left.{ℳ}_{M,N}\left(Su,Au,s\right),{ℳ}_{M,N}\left(Au,T{y}_n,s\right),{ℳ}_{M,N}\left(B{y}_n,Su,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

which on making n → ∞, reduces to

$$F\left({ℳ}_{M,N}\left(Au,z,s\right),1_{L^{*}},1_{L^{*}},{ℳ}_{M,N}\left(z,Au,s\right),{ℳ}_{M,N}\left(Au,z,s\right),1_{L^{*}}\right){\ge }_{L^{*}}{0}_{L^{*}}.$$

Using (F₁), we get ${ℳ}_{M,N}\left(Au,z,s\right)\ge 1_{L^{*}}$, for all s > 0 so that ${ℳ}_{M,N}\left(Au,z,s\right)=1_{L^{*}}$, that is Au = z = Su. Thus, u is a coincidence point of the pair (A, S).

Since T(X) is a closed subset of X, therefore $\underset{n\to \infty }{\text{lim}}T{y}_n=z\in T\left(X\right)$. Also, there exists a point w ∈ X such that Tw = z. Now, we show that ${ℳ}_{M,N}\left(Bw,z,s\right)=1_{L^{*}}$. On using inequality (3.1), we have

$$\begin{array}{l}F\left({ℳ}_{M,N}\left(A{x}_n,Bw,s\right),{ℳ}_{M,N}\left(S{x}_n,Tw,s\right),{ℳ}_{M,N}\left(Tw,Bw,s\right),\right.\\ \left.{ℳ}_{M,N}\left(S{x}_n,A{x}_n,s\right),{ℳ}_{M,N}\left(A{x}_n,Tw,s\right),{ℳ}_{M,N}\left(Bw,S{x}_n,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

which on making n → ∞, reduces to

$$F\left({ℳ}_{M,N}\left(z,Bw,s\right),1_{L^{*}},{ℳ}_{M,N}\left(z,Bw,s\right),1_{L^{*}},1_{L^{*}},{ℳ}_{M,N}\left(Bw,z,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.$$

Using (F₁), we get ${ℳ}_{M,N}\left(z,Bw,s\right)\ge 1_{L^{*}}$, for all s > 0 so that ${ℳ}_{M,N}\left(Bw,z,s\right)=1_{L^{*}}$, that is Bw = z = Tw. Thus, w is a coincidence point of the pair (B, T).

Since Au = Su and the pair (A, S) is weakly compatible, therefore Az = ASu = SAu = Sz. Now we need to show that z is a common fixed point of the pair (A, S). Now, we show that ${ℳ}_{M,N}\left(Az,z,s\right)=1_{L^{*}}$. On using inequality (3.1), we have

$$\begin{array}{l}F\left({ℳ}_{M,N}\left(Az,Bw,s\right),{ℳ}_{M,N}\left(Sz,Tw,s\right),{ℳ}_{M,N}\left(Tw,Bw,s\right),\right.\\ \left.{ℳ}_{M,N}\left(Sz,Az,s\right),{ℳ}_{M,N}\left(Az,Tw,s\right),{ℳ}_{M,N}\left(Bw,Sz,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

implying thereby

$$F\left({ℳ}_{M,N}\left(Az,z,s\right),{ℳ}_{M,N}\left(Az,z,s\right),1_{L^{*}},1_{L^{*}},{ℳ}_{M,N}\left(Az,z,s\right),{ℳ}_{M,N}\left(Az,z,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.$$

Using (F₂), we get ${ℳ}_{M,N}\left(Az,z,s\right)\ge 1_{L^{*}}$, for all s > 0 so that ${ℳ}_{M,N}\left(Az,z,s\right)=1_{L^{*}}$, that is Az = z which shows that z is a common fixed point of the pair (A, S).

Also Bw = Tw and the pair (B, T) is weakly compatible, therefore Bz = BTw = TBw = Tz. Next, we show that z is a common fixed point of the pair (B, T). To accomplish this, we show that ${ℳ}_{M,N}\left(Bz,z,s\right)=1_{L^{*}}$. On using inequality (3.1), we have

$$\begin{array}{l}F\left({ℳ}_{M,N}\left(Au,Bz,s\right),{ℳ}_{M,N}\left(Su,Tz,s\right),{ℳ}_{M,N}\left(Tz,Bz,s\right),\right.\\ \left.{ℳ}_{M,N}\left(Su,Au,s\right),{ℳ}_{M,N}\left(Au,Tz,s\right),{ℳ}_{M,N}\left(Bz,Su,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

or

$$F\left({ℳ}_{M,N}\left(z,Bz,s\right),{ℳ}_{M,N}\left(z,Bz,s\right),1_{L^{*}},1_{L^{*}},{ℳ}_{M,N}\left(z,Bz,s\right),{ℳ}_{M,N}\left(Bz,z,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.$$

Using (F₂), we get ${ℳ}_{M,N}\left(Bz,z,s\right)\ge 1_{L^{*}}$, for all s > 0 so that ${ℳ}_{M,N}\left(Bz,z,s\right)=1_{L^{*}}$, that is Bz = z which showsthat z is a common fixed point of the pair (B, T). Uniqueness of the common fixed point is an easy consequence of the inequality (3.1) (in view of condition (F₂)).

Theorem 3.2. The conclusions of Theorem 3.1 remain true if the condition (II) of Theorem 3.1 is replaced by the following.

(II') $\overline{A\left(X\right)}\subset T\left(X\right)$ and $\overline{B\left(X\right)}\subset S\left(X\right)$.

As a corollary of Theorem 3.2, we can have the following result which is also a variant of Theorem 3.1.

Corollary 3.1. The conclusions of Theorems 3.1 and 3.2 remain true if the conditions (II) and (II') are replaced by following.

(II") A(X) and B(X) are closed subset of X provided A(X) ⊂ T(X) and B(X) ⊂ S(X).

Theorem 3.3. Let A, B, S and T be four self-mappings of a modified IFMS $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ satisfying the condition (3.1). Suppose that

• (I) the pair (A, S) (or (B, T)) satisfies the property (E.A.),

• (II) A(X) ⊂ T(X) (or B(X) ⊂ S(X)),

• (III) B(y _n ) converges for every sequence y _n in X whenever T(y _n ) converges (or A(x _n ) converges for every sequence x _n in X whenever S(x _n ) converges), and

• (IV) S(X) (or T(X)) be closed subset of X.

Then the pair (A, S) as well as (B, T) have a coincidence point. Moreover, A, B, S and T have a unique common fixed point in X provided that the pairs (A, S) and (B, T) are weakly compatible.

Proof. In view of Lemma 3.1, the pairs (A, S) and (B, T) share the common property (E.A.), i.e., there exist two sequences {x _n } and {y _n } in X such that

$$\underset{n\to \infty }{\text{lim}}A{x}_n=\underset{n\to \infty }{\text{lim}}S{x}_n=\underset{n\to \infty }{\text{lim}}B{y}_n=\underset{n\to \infty }{\text{lim}}T{y}_n=z,\text{for}\text{some}z\in X.$$

As S(X) is a closed subset of X, on the lines of Theorem 3.1, one can show that the pair (A, S) has a point of coincidence, say u, i.e., Au = Su. Since A(X) ⊂ T(X) and Au ∈ T(X), there exists w ∈ X such that Au = Tw. Now, we show that ${ℳ}_{M,N}\left(Bw,z,s\right)=1_{L^{*}}$. On using inequality (3.1), we have

$$\begin{array}{l}F\left({ℳ}_{M,N}\left(A{x}_n,Bw,s\right),{ℳ}_{M,N}\left(S{x}_n,Tw,s\right),{ℳ}_{M,N}\left(Tw,Bw,s\right),\right.\\ \left.{ℳ}_{M,N}\left(S{x}_n,A{x}_n,s\right),M\left(A{x}_n,Tw,s\right),{ℳ}_{M,N}\left(Bw,S{x}_n,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

which on making n → ∞, reduces to

$$F\left({ℳ}_{M,N}\left(z,Bw,s\right),1_{L^{*}},{ℳ}_{M,N}\left(z,Bw,s\right),1_{L^{*}},1_{L^{*}},{ℳ}_{M,N}\left(Bw,z,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.$$

Using (F₁), we get ${ℳ}_{M,N}\left(z,Bw,s\right)\ge 1_{L^{*}}$, for all s > 0, so that ${ℳ}_{M,N}\left(Bw,z,s\right)=1_{L^{*}}$, that is Bw = z. Hence Bw = z = Tw. Therefore, w is a coincidence point of the pair (B, T). The rest of the proof can be completed on the lines of Theorem 3.1.

By choosing A, B, S, and T suitably, one can deduce corollaries for a pair as well as triod of mappings. As a simple we drive the following corollary for a pair of mappings.

Corollary 3.2. Let A and S be two self-mappings of a modified IFMS $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ satisfying the following conditions:

• (I) the pair (A, S) satisfies the property (E.A.) and A(x _n ) converges for every sequence {x _n } in X whenever S(x _n ) converges,

• (II) S(X) is closed subset of X and

• (III) for all x, y ∈ X, s > 0, F ∈ Ψ,

  $$\begin{array}{l}F\left({ℳ}_{M,N}\left(Ax,Ay,s\right),{ℳ}_{M,N}\left(Sx,Sy,s\right),{ℳ}_{M,N}\left(Sy,Ay,s\right),\right.\\ \left.{ℳ}_{M,N}\left(Sx,Ax,s\right),{ℳ}_{M,N}\left(Ax,Sy,s\right),{ℳ}_{M,N}\left(Ay,Sx,s\right)\right){\ge }_{L^{*}}{0}_{L^{*}}.\end{array}$$

Then the pair (A, S) has a coincidence point. Moreover, A and S have a unique common fixed point in X provided that the pair (A, S) is weakly compatible.

As an application of Theorem 3.1, we can have the following result for four finite families of self-mappings. While proving this result, we utilize Definition 1.14 which is a natural extension of commutativity condition to two finite families of mappings.

Theorem 3.4. Let {A₁, A₂,...,A _m },{B₁, B₂,...,B _p },{S₁, S₂,...,S _n } and {T₁, T₂,...,T _q } be four finite families of self-mappings of a modified IFMS $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ with A = A₁A₂... A _m , B = B₁B₂...B _p , S = S₁S₂...S _n and T = T₁T₂...T _q satisfying inequality (3.1) and the pairs (A, S) and (B, T) share the common property (E.A). If S(X) and T(X) are closed subsets of X, then the pairs (A, S) and (B, T) have a coincidence point each.

Moreover, A _i , S _k , B _r and T _t have a unique common fixed point provided the pairs of families ({A _i }, {S _k }) and ({B _r },{T _t }) commute pairwise, where i ∈ {1,..., m }, k ∈ {1,...,n }, r ∈ {1,...,p } and t ∈ {1,...,q }.

Proof. The proof follows on the lines of Theorem 4.1 due to Imdad and Ali [41] and Theorem 3.1 due to Imdad et al. [40].

By setting A₁ = A₂ = ⋯ = A _m = A, B₁ = B₂ = ⋯ = B _p = B, S₁ = S₂ = ⋯ = S _n = S and T₁ = T₂ = ⋯ = T _q = T in Theorem 3.4, we deduce the following:

Corollary 3.3. Let A, B, S and T be four self-mappings of an intuitionistic fuzzy metric space $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ such that the pairs (A^m, Sⁿ) and (B^p, T^q) share the common property (E.A.) and also satisfy the condition (for all x, y ∈ X, s > 0, F ∈ Ψ),

$$\begin{array}{l}F({ℳ}_{M,N}(A^{m}x,B^{p}y,s),{ℳ}_{M,N}(S^{n}x,T^{q}y,s),{ℳ}_{M,N}(B^{p}y,T^{q}y,s),\\ {ℳ}_{M,N}(A^{m}x,S^{m}x,s),{ℳ}_{M,N}(A^{m}x,T^{q}y,s),{ℳ}_{M,N}(S^{m}x,B^{p}y,s)){\ge }_{L^{*}}{0}_{L^{*}}\end{array}$$

where m, n, p and q are positive integers. If Sⁿ(X) and T^q(X) are closed subsets of X, then A, B, S and T have a unique common fixed point provided AS = SA and BT = TB.

Finally, we conclude this article with the following example.

Example 3.1. Let $\left(X,{ℳ}_{M,N},\mathcal{T}\right)$ be a modified IFMS, where X = [0, 1], $\mathcal{T}\left(a,b\right)=\left(a_1{b}_1,\text{min}\left\{a_2+b_2,1\right\}\right)$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L* with

$${ℳ}_{M,N}\left(x,y,t\right)=\left(\frac{t}{t+\left|x-y\right|},\frac{\left|x-y\right|}{t+\left|x-y\right|}\right),t>0.$$

Define A, B, S and T by Ax = Bx = 1,

$$S\left(x\right)=T\left(x\right)=\left\{\begin{array}{cc}1,\hfill & \text{if}x\in \left[0,1\right]\cap Q\hfill \\ \frac{1}{3},\hfill & \text{if}x\notin \left[0,1\right]\cap Q.\hfill \end{array}\right.$$

Also define

$$F\left(t_1,t_2,t_3,t_4,t_5,t_6\right)=15t_1-13t_2+5t_3-7t_4+t_5-t_6.$$

Also, A²(X) = {1} = S²(X) whereas for all x, y ∈ X and s > 0

$$\begin{array}{c}F\left(t_1,t_2,t_3,t_4,t_5,t_6\right)=15\left(1,0\right)-13\left(1,0\right)+5\left(1,0\right)-7\left(1,0\right)+\left(1,0\right)-\left(1,0\right)\\ =\left(0,0\right){\ge }_{L^{*}}{0}_{L^{*}}=\left(0,1\right)\end{array}$$

or

$$0\ge 0\text{and}0\le 1\text{which}\text{is}\text{always}\text{a}\text{reality}.$$

This demonstrates the verification of the esteemed implicit function. The remaining requirements of Corollary 3.3 can be easily verified. Notice that 1 is the unique common fixed point of A, B, S, and T.

However, this implicit function does not hold for the maps A, B, S, and T in respect of Theorem 3.1. Otherwise, with x = 0 and $y=\frac{1}{\surd 2}$, we get

$$F\left(t_1,t_2,t_3,t_4,t_5,t_6\right)=15\left(1,0\right)-13\left(\frac{t}{t+\frac{2}{3}},\frac{\frac{2}{3}}{t+\frac{2}{3}}\right)+5\left(\frac{t}{t+\frac{2}{3}},\frac{\frac{2}{3}}{t+\frac{2}{3}}\right)-7\left(1,0\right)\left(\frac{t}{t+\frac{2}{3}},\frac{\frac{2}{3}}{t+\frac{2}{3}}\right)-\left(1,0\right)$$

which contradicts the definition of L*. Thus Corollary 3.3 is a partial generalization of Theorem 3.1 and can be situationally useful.