Existence of common coincidence point of intuitionistic fuzzy maps

Abstract

The aim of this paper is mainly to find the existence of a common coincidence point for three intuitionistic fuzzy set-valued maps in the context of (α, β) - level sets of an intuitionistic fuzzy set. The main result has been applied to derive the solution of a system of nonlinear integral equations. Moreover, an interesting example is put forth to demonstrate the applicability of the method.

Keywords

Common coincidence point intuitionistic fuzzy mapping fuzzy mapping complete metric space

1 Introduction

With the beginning of modern science, uncertainty was mostly regarded as objectionable in science and the indication was to escape it. This approach progressively changed with the development of probability theory in the field of statistical mechanics. Probability theory effectively defined the phenomenon of uncertainty and was thought to be appropriate for dealing with all types of ambiguities. With the advent of fuzzy set theory [30] in 1965, a tremendous modification was observed in the classical ideas of probability theory. Many authors have developed fuzzy set theory by studying and extending its various aspects with more specific aims and objectives. Consequently, Heilpern [12] generalized the Banach contraction principle and achieved fixed point result for fuzzy contractions in a complete metric linear space as an extension of the fixed point theorem for multivalued mappings of Nadler [21]. Subsequently, several researchers (see, [1, 7, 8, 15, 22]) contributed valuably to fixed point theory through fuzzy mappings and highlighted its applications through practical examples. On the parallel, Tripathy and Baruah [27], Das [11], Tripathy et al. [29], Tripathy and Das [28], have applied this theory for studying properties of sequences of fuzzy real numbers.

Out of numerous fuzzy set generalizations in different directions, the concept of intuitionistic fuzzy set (IF-set) theory proposed by Atanassov [4 –6] is of interest to us. IF-set theory is superior to fuzzy set theory in handling physical formulations because it can define the degrees of both membership and non-membership with a degree of hesitancy and have been extensively used in a variety of scientific research areas with different objectives. In addition, a lot of work has been done regarding IF-sets. For instance, Bustince and Burillo [10] analyzed that the concept of vague sets and IF-sets are same. Liu and Wang [16] reduced the degree of uncertainty of the elements in a universe using the intuitionistic fuzzy point operators. In this way, some optimization problems with the induction of IF-sets are described by Plamen Angelov [3], Valentina Radeva [24] and Hristo Aladjov [2]. Recently, Azam et al. [9] studied common and coincidence point theorems on a complete metric space with applications using the idea of intuitionistic fuzzy mappings defined by Shen et al. [25].

This article is primarily focused to find the common coincidence point theorems for three intuitionistic fuzzy mappings on a complete metric space in association with (α, β) - level and (M, m) - level sets of an IF-set [33]. Further, a non trivial example and an application are given to illustrate the usability of our theory. Therefore, these results may help us to understand ϵ^∞-theory of high energy physics, because the activities in this case are frequently IF-sets, (see, [19, 20]). Moreover, there is a use of fuzziness in Physics, an original limit through which we can know about the actions of quantum particles and the negligible scales of nature. These scales assist us to determine the possibilities regarding existence and behaviour of substances. In these circumstances, our research may protect a level of fuzziness in the quantum theory.

2 Preliminaries

In this section, we recall the concept of fuzzy sets and IF-sets.

Let (Z, d) be a metric space and (V, d_V) a metric linear space. Let CB (2^Z) be the family of all nonempty closed and bounded subset of Z and C (2^Z) be the class of all nonempty compact subset of Z.

For A, B ∈ C (2^Z) we define $d (x, A) = inf_{y \in A} d (x, y),$ $d (A, B) = inf_{x \in A, y \in B} d (x, y) .$

Then the Hausdorff metric d_H on C (2^Z) induced by d is given by $d_{H} (A, B) = max {sup_{a \in A} d (a, B), sup_{b \in B} d (A, b)} .$

Definition 2.1. [12, 26] A fuzzy set A in the universe of discourse W is defined by the set of ordered pairs A = { (w, μ_A (w)) : w ∈ W }, where μ_A (w) is the grade of membership of element w in set A.

Definition 2.2. [12, 26] Let A be a fuzzy set. Then α - level set of A is a crisp set denoted by [A] _α and is defined by ${[A]}_{α} = {w \in W : A (w) \geq α} if α \in (0, 1],$ ${[A]}_{0} = cl {w \in W : A (w) > 0} .$

Definition 2.3. [12] A fuzzy set A in a metric linear space V is said to be an approximate quantity if and only if [A] _α is compact and convex in V for each α ∈ [0, 1] with $_{υ \in V}^{\sup} A (υ) = 1.$

Definition 2.4. [12, 26] Let W be an arbitrary set and Z be a metric space. A mapping T is called fuzzy mapping if T is a mapping from W into I^Z (the collection of all fuzzy sets on Z).

Definition 2.5. [7] An element z^∗ ∈ Z is called a fuzzy fixed point of a fuzzy mapping T : Z → I^Z if there exists α ∈ (0, 1] such that z^∗ ∈ [T (z^∗)] _α.

For α ∈ [0, 1] such that [A] _α, [B] _α ∈ C (2^Z), the following notations have been recorded from [7]. $D_{α} (A, B) = d_{H} ({[A]}_{α}, {[B]}_{α}),$ $d_{\infty} (A, B) = sup_{α} D_{α} (A, B) .$ $C (Z) = {A \in I^{Z} : {[A]}_{α} \in C (2^{Z}) for each α \in [0, 1]},$ $F (Z) = {A \in I^{Z} : {[A]}_{α} \in C (2^{Z}) for some α \in [0, 1]} .$

Lemma 2.6. [13] Let (Z, d) be a metric space and A, B ∈ CB (2^Z) with d_H (A, B) < δ, then for each a ∈ A, there exists an element b ∈ B such that d (a, b) < δ.

Lemma 2.7. [21] Let (Z, d) be a metric space and A, B ∈ CB (2^Z). If a ∈ A, then d (a, B) ≤ d_H (A, B).

Definition 2.8. [4] Let W be a universal set. An IF-set $\tilde{A}$ in W is an object of the form $\tilde{A} = {〈 w, μ_{\tilde{A}} (w), υ_{\tilde{A}} (w) 〉 : w \in W}$ where $μ_{\tilde{A}} : W \to [0, 1]$ and $υ_{\tilde{A}} : W \to [0, 1]$ such that $0 \leq μ_{\tilde{A}} (w) + υ_{\tilde{A}} (w) \leq 1$ for every w ∈ W. $μ_{\tilde{A}} (w)$ and $υ_{\tilde{A}} (w)$ denote the membership and non-membership values of w in $\tilde{A}$ . As an example it is shown in Figs. 1 and 2.

Fig. 1

General representation of membership function.

Fig. 2

General representation of non-membership function.

Example 2.9. Let W be the set of all positive real numbers, representing the aggregate amount of possible income in thousands dollars earned by an individual and IF-set $\tilde{A}$ denotes “high income”. If we survey a fair sample of individuals and find out that none of them consider the income below $20, 000 be the higher income and the proportion p of the selected sample consider the income w ranging $20, 000 to $75, 000 as the higher income which is approximately $p = \frac{w - 20}{55} .$ Given the above survey outcomes, we conclude that every individual of the selected sample consider the income over $75, 000 as the higher income. Then the graph (see Fig. 3) and IF-set $\tilde{A}$ is given as;

$μ_{\tilde{A}} (w) = {\begin{matrix} 1, & if & w < 20 \\ \frac{515 - w}{550}, & if & 20 \leq w \leq 75 \\ 0, & if & 75 < w, \end{matrix}$

Fig. 3

Intuitionistic fuzzy set

$υ_{\tilde{A}} (w) = {\begin{matrix} 0, & if & w < 20 \\ \frac{w - 20}{55}, & if & 20 \leq w \leq 75 \\ 1, & if & 75 < w . \end{matrix}$

Definition 2.10. [4] Let $\tilde{A}$ be IF-set, then α - level set of $\tilde{A}$ is a crisp set denoted by ${[\tilde{A}]}_{α}$ and is defined by ${[\tilde{A}]}_{α} = {w \in W : μ_{\tilde{A}} (w) \geq α$ and $υ_{\tilde{A}} (w) \leq 1 - α}$ if α ∈ [0, 1].

Definition 2.11. [33] Let $\tilde{A}$ IF-set, then (α, β) - level set of $\tilde{A}$ is a crisp set denoted by ${[\tilde{A}]}_{(α, β)}$ and is defined by ${[\tilde{A}]}_{(α, β)} = {w \in W : μ_{\tilde{A}} (w) \geq α$ and $υ_{\tilde{A}} (w) \leq β}$ if (α, β) ∈ (0, 1] × [0, 1).

Example 2.12. Let W = {w₁, w₂, w₃, w₄, w₅} and $\tilde{A}$ be IF-set defined as: $\tilde{A}$ = {(w₁, 0.5, 0.4), (w₂, 0.7, 0.2), (w₃, 0.1, 0.8), (w₄, 0.2, 0.7), (w₅, 0.3, 0.6)}.

The (α, β) -level sets are defined by $\begin{matrix} [\tilde{A}]_{(0.1, 0.8)} & = {w_{1}, w_{2}, w_{3}, w_{4}, w_{5}} \\ [\tilde{A}]_{(0.3, 0.6)} & = {w_{1}, w_{2}, w_{5}} \\ [\tilde{A}]_{(0.5, 0.4)} & = {w_{1}, w_{2}} \\ [\tilde{A}]_{(0.7, 0.2)} & = {w_{2}} . \end{matrix}$

Definition 2.13. [14] Any crisp set A can be represented as an IF-set by its intuitionistic fuzzy characteristic function $\ddot{A} = (Γ_{A}, Φ_{A})$ defined as: $Γ_{A} (w) = {\begin{array}{l} 1 & i f w \in A \\ 0 & i f w \notin A, \end{array}$ and $Φ_{A} (w) = {\begin{array}{l} 0 & i f w \in A \\ 1 & i f w \notin A, \end{array}$ for all w ∈ W.

Definition 2.14. [25] Let W be an arbitrary set, Z be a metric space. A mapping T is called intuitionistic fuzzy mapping if T is a mapping from W into (IFS) ^Z (class of intuitionistic fuzzy subsets of Z).

Definition 2.15. [9] A pointz^∗ ∈ Z is said to be an intuitionistic fuzzy fixed point of an intuitionistic fuzzy mapping T : Z → (IFS) ^Z if there exists (α, β) ∈ (0, 1] × [0, 1) such that z^∗ ∈ [T (z^∗)] _(α,β).

Definition 2.16. [9] An IF-set $\tilde{A}$ in a metric linear space V is said to be an approximate quantity if and only if ${[\tilde{A}]}_{(α, β)}$ is compact and convex in V for each (α, β) ∈ (0, 1] × [0, 1) with $sup_{v \in V} μ_{\tilde{A}} (v) = 1 and inf_{v \in V} υ_{\tilde{A}} (v) = 0 .$ For our convenience, we can define ${[\tilde{A}]}_{(M, m)}$ by ${[\tilde{A}]}_{(M, m)} = {\begin{matrix} w \in W : μ_{\tilde{A}} (w) = M \\ and υ_{\tilde{A}} (w) = m \end{matrix}},$ where $\begin{matrix} M & = & max_{w \in W} μ_{\tilde{A}} (w), \\ m & = & min_{w \in W} υ_{\tilde{A}} (w) . \end{matrix}$ It is noted that in Example 2.12, ${[\tilde{A}]}_{(M, m)}$ $= {[\tilde{A}]}_{(0.7, 0.2)} = {w_{2}} .$

Some subcollections of (IFS) ^Z are defined as follows: $R (Z) = {\begin{matrix} A \in {(IFS)}^{Z} : {[A]}_{(α, β)} \in C (2^{Z}), \\ for some (α, β) \in (0, 1] \times [0, 1) \end{matrix}},$ $M (Z) = {\begin{matrix} A \in {(IFS)}^{Z} : {[A]}_{(α, β)} \in C (2^{Z}), \\ for each (α, β) \in (0, 1] \times [0, 1) \end{matrix}} .$ For (α, β) ∈ (0, 1] × [0, 1) such that [A] _(α,β), [B] _(α,β) ∈ C (2^Z), the following notions are given by $D_{(α, β)} (A, B) = d_{H} ({[A]}_{(α, β)}, {[B]}_{(α, β)}),$ $d_{(\infty, \infty)} (A, B) = sup_{(α, β)} D_{(α, β)} (A, B) .$

3 Common coincidence theorems for intuitionistic fuzzy mappings

In this section, common coincidence theorems for three intuitionistic fuzzy mappings using notions of Hausdorff metric and d_∞ metric for IF-sets are presented.

Theorem 3.1. Let W be a nonempty set, (Z, d) be a metric space and F, G, S : W → (IFS) ^Z be intuitionistic fuzzy mappings. Assume that for each ξ ∈ W, there exists (α, β) _F(ξ), (α, β) _G(ξ), (α, β) _S(ξ) ∈ [0, 1) × (0, 1] such that ${[F (ξ)]}^{{(α, β)}_{F} (ξ)},$ ${[G (ξ)]}^{{(α, β)}_{G} (ξ)},$ ${[S (ξ)]}^{{(α, β)}_{S} (ξ)} \in C (2^{Z})$ . If $\begin{matrix} \underset{ξ \in W}{\cup} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)} \cup^{} {[G (ξ)]}^{{(α, β)}_{G} (ξ)}) \\ \subseteq \underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)}, \end{matrix}$ (3.1) and either $\begin{matrix} \underset{ξ \in W}{\cup^{}} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)} \cup^{} {[G (ξ)]}^{{(α, β)}_{F} (ξ)}) \\ or \underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{F} (ξ)} \end{matrix}$ is complete. If k ∈ [0, 1) such that for all ξ, η ∈ W, $\begin{matrix} d_{H} ({[F (ξ)]}_{{(α, β)}_{F} (ξ)}, {[G (η)]}_{{(α, β)}_{G} (η)}) \\ \leq & k d ({[S (ξ)]}_{{(α, β)}_{S} (ξ)}, {[S (η)]}_{{(α, β)}_{S} (η)}) \end{matrix}$ (3.2) Thus, there exists ω ∈ W such that $\begin{matrix} {[F (ω)]}^{{(α, β)}_{G} (ω)} \\ \cap^{} {[G (ω)]}^{{(α, β)}_{G} (ω)} \cap^{} {[S (ω)]}^{{(α, β)}_{G} (ω)} \neq φ . \end{matrix}$

Proof. Assume that ξ₀ is an arbitrary and fixed element of W, then by hypothesis ${[F (ξ_{0})]}^{{(α, β)}_{F} (ξ_{0})} \in C (2^{Z}) .$ Let $η_{1} \in {[F (ξ_{0})]}^{{(α, β)}_{F} (ξ_{0})} \neq φ .$ Then by using expression 3.1, there exists some ξ₁ ∈ W, such that $η_{1} \in {[S (ξ_{1})]}^{{(α, β)}_{S} (ξ_{1})} .$ Thus, $η_{1} \in {[F (ξ_{0})]}^{{(α, β)}_{F} (ξ_{0})} \cap^{} {[S (ξ_{1})]}^{{(α, β)}_{S} (ξ_{1})} .$ Now inequality 3.2 yields $\begin{matrix} d_{H} ({[F (ξ_{0})]}^{{(α, β)}_{F} (ξ_{0})}, {[G (ξ_{1})]}_{{(α, β)}_{G} (ξ_{1})}) \\ < & \sqrt{k} d ({[S (ξ_{0})]}^{{(α, β)}_{S} (ξ_{0})}, {[S (ξ_{1})]}^{{(α, β)}_{S} (ξ_{1})}) . \end{matrix}$ Then, by Lemma 2.6 there exists η₂ ∈ [G (ξ₁)] _{(α,β)_G(ξ₁)} such that $d (η_{1}, η_{2}) < \sqrt{k} d ( {[S (ξ_{0})]}^{{(α, β)}_{S} (ξ_{0})}, {[S (ξ_{1})]}^{{(α, β)}_{S} (ξ_{1})}) .$ Next from the expression 3.1 and for η₂ ∈ [G (ξ₁)] _{(α,β)_G(ξ₁)}, we may choose ξ₂ ∈ W, such that $η_{2} \in {[S (ξ_{2})]}^{{(α, β)}_{S} (ξ_{2})} .$ Therefore, $η_{2} \in {[G (ξ_{1})]}_{{(α, β)}_{G} (ξ_{1})} \cap^{} {[S (ξ_{2})]}^{{(α, β)}_{S} (ξ_{2})} .$ Furthermore, inequality 3.2 implies $\begin{matrix} d_{H} ({[G (ξ_{1})]}_{{(α, β)}_{G} (ξ_{1})}, {[F (ξ_{2})]}^{{(α, β)}_{F} (ξ_{2})}) \\ < & \sqrt{k} d ({[S (ξ_{1})]}^{{(α, β)}_{S} (ξ_{1})}, {[S (ξ_{2})]}^{{(α, β)}_{S} (ξ_{2})}) . \end{matrix}$ Then, by Lemma 2.6 there exists η₃ ∈ [F (ξ₂)] _{(α,β)_F(ξ₂)} such that $d (η_{2}, η_{3}) < \sqrt{k} d ( {[S (ξ_{1})]}^{{(α, β)}_{S} (ξ_{1})}, {[S (ξ_{2})]}^{{(α, β)}_{S} (ξ_{2})}) .$ By induction, the sequences {ξ_n} and {η_n} can be constructed as follows: $η_{2 n + 1} \in {[F (ξ_{2 n})]}^{{(α, β)}_{F} (ξ_{2 n})} \cap^{} {[S (ξ_{2 n + 1})]}^{{(α, β)}_{S} (ξ_{2 n + 1})},$ $\begin{matrix} η_{2 n + 2} & \in & {[G (ξ_{2 n + 1})]}^{{(α, β)}_{G} (ξ_{2 n + 1})} \\ \cap^{} & {[S (ξ_{2 n + 2})]}^{{(α, β)}_{S} (ξ_{2 n + 2})}, \end{matrix}$ where n is any non-negative integer. Consequently, $\begin{matrix} d (η_{2 n + 1}, η_{2 n + 2}) \\ < \sqrt{k} d ({[S (ξ_{2 n})]}^{{(α, β)}_{S} (ξ_{2 n})}, {[S (ξ_{2 n + 1})]}^{{(α, β)}_{S} (ξ_{2 n + 1})}) \\ < \sqrt{k} d (η_{2 n}, η_{2 n + 1}) \\ < k d (η_{2 n - 1}, η_{2 n}) \\ < k^{\frac{3}{2}} d (η_{2 n - 2}, η_{2 n - 1}) \dots < k^{\frac{2 n + 1}{2}} d (η_{0}, η_{1}) . \end{matrix}$ Now we show that {η_q} is a Cauchy sequence. For this, let p > q and p, q are of opposite parity that is p = 2b and q = 2a + 1 for a > 0 and b > 0. Then $\begin{matrix} d (η_{2 a + 1}, η_{2 b}) \\ \leq & d (η_{2 a + 1}, η_{2 a + 2}) + d (η_{2 a + 2}, η_{2 a + 3}) + \dots \\ + d (η_{2 b - 1}, η_{2 b}) \\ < & (\begin{matrix} k^{\frac{2 a + 1}{2}} + k^{\frac{2 a + 2}{2}} + \\ \dots + k^{\frac{2 b - 1}{2}} \end{matrix}) d (η_{0}, η_{1}) \\ < & (\frac{k^{\frac{2 a + 1}{2}}}{1 - k^{\frac{1}{2}}}) d (η_{0}, η_{1}) . \end{matrix}$ (3.3) Similarly, if p, q are of like parity, then we obtain $\begin{matrix} d (η_{2 a}, η_{2 b}) \\ \leq & d (η_{2 a}, η_{2 a + 1}) + d (η_{2 a + 1}, η_{2 a + 2}) + \dots \\ + d (η_{2 b - 1}, η_{2 b}) \\ < & (\begin{matrix} k^{a} + k^{\frac{2 a + 1}{2}} + \dots \\ + k^{\frac{2 b - 1}{2}} \end{matrix}) d (η_{0}, η_{1}) \\ < & (\frac{k^{a}}{1 - k^{\frac{1}{2}}}) d (η_{0}, η_{1}) . \end{matrix}$ (3.4) and $\begin{matrix} d (η_{2 a + 1}, η_{2 b + 1}) \\ \leq & d (η_{2 a + 1}, η_{2 a + 2}) + d (η_{2 a + 2}, η_{2 a + 3}) + \\ \dots + d (η_{2 b}, η_{2 b + 1}) \\ < & (k^{\frac{2 a + 1}{2}} + k^{\frac{2 a + 2}{2}} + \dots + k^{b}) d (η_{0}, η_{1}) \\ < & (\frac{k^{\frac{2 a + 1}{2}}}{1 - k^{\frac{1}{2}}}) d (η_{0}, η_{1}) . \end{matrix}$ (3.5) Thus from inequalities 3.3, 3.4 and 3.5 we have $\begin{matrix} d (η_{2 a + 1}, η_{2 b}), d (η_{2 a}, η_{2 b}), d (η_{2 a + 1}, η_{2 b + 1}) \\ \to 0 as a, b \to \infty . \end{matrix}$ Hence, it follows that {η_q} is a Cauchy sequence. As $\underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)}$ is complete, there exists u ∈ [S $(ω)]^{{(α, β)}_{S} (ω)}$ for some ω ∈ W such that η_q → u, this holds also if $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)} \cup^{} {[G (ξ)]}^{{(α, β)}_{G} (ξ)})$ is complete with $\begin{matrix} \lim_{q \to \infty} η_{q} & \in & \underset{ξ \in W}{\cup} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)} \cup^{} {[G (ξ)]}^{{(α, β)}_{G} (ξ)}) \\ \subseteq & \underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)} . \end{matrix}$ Moreover, Lemma 2.7 and inequality 3.2 implies $\begin{matrix} d (u, {[F (ω)]}^{{(α, β)}_{F} (ω)}) \\ \leq & d (u, η_{2 n + 2}) + d (η_{2 n + 2}, {[F (ω)]}^{{(α, β)}_{F} (ω)}) \\ \leq & d (u, η_{2 n + 2}) \\ + d_{H} ({[F (ω)]}^{{(α, β)}_{F} (ω)}, {[G (ξ_{2 n + 1})]}^{{(α, β)}_{G} (ξ_{2 n + 1})}) \\ \leq & d (u, η_{2 n + 2}) \\ + k d ({[S (ω)]}^{{(α, β)}_{S} (ω)}, {[S (ξ_{2 n + 1})]}^{{(α, β)}_{S} (ξ_{2 n + 1})}) \\ \leq & d (u, η_{2 n + 2}) + k d (u, η_{2 n + 1}) . \end{matrix}$ In the limiting case, when n→ ∞, we have $u \in {[F (ω)]}^{{(α, β)}_{F} (ω)} \cap^{} {[S (ω)]}^{{(α, β)}_{S} (ω)} .$ Again by Lemma 2.7 and inequality 3.2, we obtain $\begin{matrix} d (u, {[G (ω)]}^{{(α, β)}_{G} (ω)}) \\ \leq & d (u, η_{2 n + 1}) + d (η_{2 n + 1}, {[G (ω)]}^{{(α, β)}_{G} (ω)}) \\ \leq & d (u, η_{2 n + 1}) \\ + d_{H} ({[F (ξ_{2 n})]}^{{(α, β)}_{F} (ξ_{2 n})}, {[G (ω)]}^{{(α, β)}_{G} (ω)}) \\ \leq & d (u, η_{2 n + 1}) \\ + k d ({[S (ξ_{2 n})]}^{{(α, β)}_{S} (ξ_{2 n})}, {[S (ω)]}^{{(α, β)}_{S} (ω)}) \\ \leq & d (u, η_{2 n + 1}) + k d (η_{2 n}, u) . \end{matrix}$ In the limiting case, when n→ ∞, we have $u \in {[G (ω)]}^{{(α, β)}_{G} (ω)} \cap^{} {[S (ω)]}^{{(α, β)}_{S} (ω)} .$ Thus, $\begin{matrix} {[F (ω)]}^{{(α, β)}_{F} (ω)} & \cap^{} & {[G (ω)]}^{{(α, β)}_{G} (ω)} \\ \cap^{} & {[S (ω)]}^{{(α, β)}_{S} (ω)} \neq φ . \end{matrix}$ Hence, this completes the proof.

From theorem 3.1, we deduce the following corollaries

Corollary 3.2. [9] Let W be a nonempty set, (Z, d) be a metric space and F, S : W → (IFS) ^Z be a pair of intuitionistic fuzzy mappings. Suppose that for each ξ ∈ W there exists (α, β) _F(ξ), (α, β) _S(ξ) ∈ (0, 1] × [0, 1) such that ${[F (ξ)]}^{{(α, β)}_{F} (ξ)},$ ${[S (ξ)]}^{{(α, β)}_{S} (ξ)} \in C B (2^{Z}) .$ $\underset{ξ \in W}{\cup} {[F (ξ)]}^{{(α, β)}_{F} (ξ)} \subseteq \underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)}$ and either $\underset{ξ \in W}{\cup} {[F (ξ)]}^{{(α, β)}_{F} (ξ)}$ or $\underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)}$ is compete. If k ∈ [0, 1) such that for all ξ, η ∈ W, $\begin{matrix} d_{H} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)}, {[F (η)]}_{{(α, β)}_{F} (η)}) \\ \leq & k d ({[S (ξ)]}^{{(α, β)}_{S} (ξ)}, {[S (η)]}^{{(α, β)}_{S} (η)}) . \end{matrix}$ Thus, there exists ω ∈ W such that ${[F (ω)]}^{{(α, β)}_{F} (ω)} \cap^{} {[S (ω)]}^{{(α, β)}_{S} (ω)} \neq φ .$

Corollary 3.3. [7] Let W be a nonempty set, (Z, d) be a metric space and F, S : W → I^Z be two fuzzy mappings. Assume that for each ξ ∈ W there exists, such that [Fξ] _{α_F(ξ)}, [Sξ] _{α_S(ξ)} ∈ CB (2^Z). $\underset{ξ \in W}{\cup} {[F (ξ)]}_{α_{F} (ξ)} \subseteq \underset{ξ \in W}{\cup} {[S (ξ)]}_{α_{S} (ξ)}$ and either $\underset{ξ \in W}{\cup} {[F (ξ)]}_{α_{F} (ξ)} or \underset{ξ \in W}{\cup} {[S (ξ)]}_{α_{S} (ξ)}$ is complete. If there exists k ∈ [0, 1) such that for all ξ, η ∈ W $\begin{matrix} d_{H} ({[F (ξ)]}_{α_{F} (ξ)}, {[F (η)]}_{α_{F} (η)}) \\ \leq & kd ({[S (ξ)]}_{α_{S} (ξ)}, {[S (η)]}_{α_{S} (η)}) . \end{matrix}$ Thus, there exists ω ∈ W such that [F (ω)] _{α_F(ω)} ∩ [S (ω)] _{α_S(ω)} ≠ φ.

Example 3.4. Let W = Z = [0, ∞), d (ξ, η) = |ξ - η|, whenever ξ, η ∈ W and γ₁, λ₁, γ₂, λ₂, γ₃, λ₃ ∈ (0, 1]. Define intuitionistic fuzzy mappings P, Q, T : W → (IFS) ^Z as follows: $\begin{array}{l} μ_{p (ξ)} (t) = {\begin{array}{l} γ_{1}, & if 0 \leq t \leq 4 ξ \\ \frac{γ_{1}}{4}, & if 4 ξ < t \leq 6 ξ \\ \frac{γ_{1}}{6}, & if 6 ξ < t \leq 10 ξ \\ 0, & if 10 ξ < t < \infty, \end{array} \\ υ_{p (ξ)} (t) = {\begin{array}{l} 0, & if 0 \leq t \leq 8 ξ \\ \frac{λ_{1}}{6}, & if 8 ξ < t \leq 10 ξ \\ \frac{λ_{1}}{3}, & if 10 ξ < t \leq 12 ξ \\ λ_{1}, & if 12 ξ < t < \infty \end{array} \end{array}$ and $\begin{array}{l} μ_{Q (ξ)} (t) = {\begin{array}{l} γ_{2}, & if 0 \leq t \leq 6 ξ \\ \frac{γ_{2}}{5}, & if 6 ξ < t \leq 8 ξ \\ \frac{γ_{2}}{7}, & if 8 ξ < t \leq 10 ξ \\ 0, & if 10 ξ < t < \infty, \end{array} \\ υ_{Q (ξ)} (t) = {\begin{array}{l} 0, & if 0 \leq t \leq 3 ξ \\ \frac{λ_{2}}{8}, & if 3 ξ < t \leq 6 ξ \\ \frac{λ_{2}}{4}, & if 6 ξ < t \leq 10 ξ \\ λ_{2}, & if 10 ξ < t < \infty \end{array} \end{array}$ and $\begin{array}{l} μ_{T (ξ)} (t) = {\begin{array}{l} γ_{3}, & if t = 5^{2} ξ \\ \frac{γ_{3}}{4}, & if t = 5^{3} ξ \\ \frac{γ_{3}}{8}, & if t = 5^{4} ξ \\ 0, & otherwise, \end{array} \\ υ_{T (ξ)} (t) = {\begin{array}{l} 0, & if t = 5^{2} ξ \\ \frac{λ_{2}}{8}, & if t = 5^{3} ξ \\ \frac{λ_{2}}{4}, & if t = 5^{4} ξ \\ λ_{2}, & otherwise . \end{array} \end{array}$ Now define F, G, S : W → (IFS) ^Z as follows: $\begin{array}{l} μ_{F (ξ)} (t) = {\begin{array}{l} Γ_{{0}} & if ξ = 0 \\ μ_{P (ξ)} & if ξ > 0, \end{array} \\ υ_{F (ξ)} (t) = {\begin{array}{l} Φ_{{0}} & if ξ = 0 \\ υ_{P (ξ)} & if ξ > 0, \end{array} \\ μ_{G (ξ)} (t) = {\begin{array}{l} Γ_{{0}} & if ξ = 0 \\ μ_{Q (ξ)} & if ξ > 0, \end{array} \\ υ_{G (ξ)} (t) = {\begin{array}{l} Φ_{{0}} & if ξ = 0 \\ υ_{Q (ξ)} & if ξ > 0 \end{array} \end{array}$ and $\begin{array}{l} μ_{S (ξ)} (t) = {\begin{array}{l} Γ_{{0}} & if ξ = 0 \\ μ_{T (ξ)} & if ξ > 0, \end{array} \\ υ_{S (ξ)} (t) = {\begin{array}{l} Φ_{{0}} & if ξ = 0 \\ υ_{T (ξ)} & if ξ > 0. \end{array} \end{array}$ If ${(α, β)}_{F (ξ)} = (\frac{γ_{1}}{6}, \frac{λ_{1}}{6}),$ ${(α, β)}_{G (ξ)} = (\frac{γ_{2}}{7}, \frac{λ_{2}}{4})$ and (α, β) _S(ξ) = (γ₃, 0) then we obtain $\begin{array}{l} {[F_{(ξ)}]}_{(α, β) F (ξ)} = {\begin{array}{l} {0} & if ξ = 0 \\ [0, 10 ξ] & if ξ > 0, \end{array} \\ {[G (ξ)]}_{(α, β) G (ξ)} = {\begin{array}{l} {0} & if ξ = 0 \\ {5^{2} ξ} & if ξ > 0. \end{array} \end{array}$ and ${[S (ξ)]}_{(α, β) S (ξ)} = {\begin{array}{l} {0} & if ξ = 0 \\ {5^{2} ξ} & if ξ > 0. \end{array}$ Therefore, $\begin{matrix} \underset{ξ \in W}{\cup} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)} \cup^{} {[G (ξ)]}^{{(α, β)}_{G} (ξ)}) \\ = & [0, \infty) = \underset{ξ \in W}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)} . \end{matrix}$ Moreover, if ξ = η = 0, then we have $\begin{matrix} d_{H} ({[F (ξ)]}^{{(α, β)}_{F} (ξ)}, {[G (η)]}_{{(α, β)}_{G} (η)}) \\ = & k d ({[S (ξ)]}^{{(α, β)}_{S} (ξ)}, {[S (η)]}^{{(α, β)}_{S} (η)}) . \end{matrix}$ if ξ > 0 and η > 0, then we obtain $\begin{matrix} d_{H} ({[F (ξ)]}_{{(α, β)}_{F} (ξ)}, {[G (η)]}_{{(α, β)}_{G} (η)}) \\ = & 10 | ξ - η | . \end{matrix}$ Thus, for $k = \frac{2}{5},$ all the assumptions of theorem 3.1 are satisfied to obtain $\begin{matrix} {[F (0)]}^{{(α, β)}_{F} (0)} & \cap^{} & {[G (0)]}^{{(α, β)}_{G} (0)} \\ \cap^{} & {[S (0)]}^{{(α, β)}_{S} (0)} \neq φ . \end{matrix}$

Theorem 3.5. Let W be a nonempty set, (Z, d) be a metric space and F, G, S : W → M (Z). Assume that $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{(M, m)} \cup^{} {[G (ξ)]}^{(M, m)}) \subseteq \underset{ξ \in W}{\cup} {[S (ξ)]}^{(M, m)}$ and either $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{(M, m)} \cup^{} {[G (ξ)]}^{(M, m)})$ or $\underset{ξ \in W}{\cup} {[S (ξ)]}^{(M, m)}$ is complete. If k ∈ [0, 1) such that for all ξ, η ∈ W, $\begin{matrix} d_{(\infty, \infty)} (F (ξ), G (η)) \\ \leq & k d ({[S (ξ)]}^{(M, m)}, {[S (η)]}^{(M, m)}) . \end{matrix}$ Thus, there exists ω ∈ W such that ${[F (ω)]}^{(M, m)} \cap^{} {[G (ω)]}^{(M, m)} \cap^{} {[S (ω)]}^{(M, m)} \neq φ .$

Proof. Suppose that ξ ∈ W, then by assumption ${[F (ξ)]}^{(M, m)}$ , ${[G (ξ)]}^{(M, m)}$ and ${[S (ξ)]}^{(M, m)}$ are nonempty compact subsets of Z. It follows that $\begin{matrix} d_{H} ({[F (ξ)]}^{(M, m)}, {[G (η)]}_{(M, m)}) \\ = & D_{(M, m)} (F (ξ), G (η)) \\ \leq & d_{(\infty, \infty)} (F (ξ), G (η)) \\ \leq & k d ({[S (ξ)]}^{(M, m)}, {[S (η)]}^{(M, m)}) . \end{matrix}$ Then, by theorem 3.1 there exists ω ∈ W such that ${[F (ω)]}^{(M, m)} \cap^{} {[G (ω)]}^{(M, m)} \cap^{} {[S (ω)]}^{(M, m)} \neq φ .$

In theorem 3.5, if we assigns values to (M, m) = (1, 0) then the following corollary is obtained.

Corollary 3.6. Let W be a nonempty set, (Z, d) be a metric space and F, G, S : W → M (Z). Assume that $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{(1, 0)} \cup^{} {[G (ξ)]}^{(1, 0)}) \subseteq \underset{ξ \in W}{\cup} {[S (ξ)]}^{(1, 0)}$ and either $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{(1, 0)} \cup^{} {[G (ξ)]}^{(1, 0)}) or \underset{ξ \in W}{\cup} {[S (ξ)]}^{(1, 0)}$ is compete. If k ∈ [0, 1) such that for all ξ, η ∈ W, $d_{(\infty, \infty)} (F (ξ), G (η)) \leq k d ({[S (ξ)]}^{(1, 0)}, {[S (η)]}^{(1, 0)}) .$ Thus, there exists ω ∈ W such that ${[F (ω)]}^{(1, 0)} \cap^{} {[G (ω)]}^{(1, 0)} \cap^{} {[S (ω)]}^{(1, 0)} \neq φ .$

We deduce the following corollary by replacing M (Z) by C (Z) in corollary 3.6.

Corollary 3.7. Let W be a nonempty set, (Z, d) be a metric space and F, G, S : W → C (Z). Assume that $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{1} \cup^{} {[G (ξ)]}^{1}) \subseteq \underset{ξ \in W}{\cup} {[S (ξ)]}^{1}$ and either $\underset{ξ \in W}{\cup} ({[F (ξ)]}^{1} \cup^{} {[G (ξ)]}^{1}) or \underset{ξ \in W}{\cup} {[S (ξ)]}^{1}$ is compete. If k ∈ [0, 1) such that for all ξ, η ∈ W, $d_{\infty} (F (ξ), G (η)) \leq k d ({[S (ξ)]}^{1}, {[S (η)]}^{1}) .$ Thus, there exists ω ∈ W such that ${[F (ω)]}^{1} \cap^{} {[G (ω)]}^{1} \cap^{} {[S (ω)]}^{1} \neq φ .$

4 Applications

Many mathematical physics models such as scattering in quantum mechanics, conformal mapping, water waves, Volterra’s population growth model, heat transfer, electrostatic and propagation of acoustical and elastic waves contributed to give rise to non-linear integral equations. Moreover, a large family of initial and boundary value problems can be easily solved by converting them into Volterra or Fredholm integral equations.

In this section, theorem 3.1 has been applied to establish some hypothesis which gives the existence of solution of a system of nonlinear integral equations.

Theorem 4.1. Let f : Rⁿ → Rⁿ, g : [a, b] → Rⁿ and K₁, K₂ : [a, b] × [a, b] × Rⁿ → Rⁿ be continuous mappings, where f is injective and K₁, K₂ are the kernels of integral equations. If there exists l ∈ R such that $| | K_{1} (t, s, ξ_{1}) - K_{2} (t, s, ξ_{2}) | |_{\infty} \leq l | | f ξ_{1} - f ξ_{2} | |_{\infty}$ for all t, s ∈ [a, b] ⊆ R and ξ₁, ξ₂ ∈ Rⁿ. Assume that for each ξ ∈ (C [a, b], Rⁿ) there exists η ∈ (C [a, b], Rⁿ) such that $f (η (t)) = g (t) + q \int_{a}^{b} K_{1} (t, s, ξ (s)) ds$ or $f (η (t)) = g (t) + q \int_{a}^{b} K_{2} (t, s, ξ (s)) ds$ , where q is a parameter and {f (ξ (t)) : ξ∈ (C [a, b], Rⁿ) } is closed. Then the system of integral equations $f (ξ (t)) = g (t) + q \int_{a}^{b} K_{1} (t, s, ξ (s)) ds 4.1$ (4.1) $f (ξ (t)) = g (t) + q \int_{a}^{b} K_{2} (t, s, ξ (s)) ds 4.2$ (4.2) have a common solution in (C [a, b], Rⁿ) for any $q \in (- \frac{1}{l (b - a)}, \frac{1}{l (b - a)}) .$

Proof. Let Z = W = (C [a, b], Rⁿ) and d (ξ, η) = ||ξ (t) - η (t) ||_∞ for each ξ, η ∈ Z. Let A, B, P, Q, T, D be six arbitrary mappings from Z into (0, 1].

Assume that K₁, K₂ are such that ω_ξ, Ψ_ξ ∈ Z for each ξ ∈ Z, where $ω_{ξ} (t) = g (t) + q \int_{a}^{b} K_{1} (t, s, ξ (s)) ds$ $Ψ_{ξ} (t) = g (t) + q \int_{a}^{b} K_{2} (t, s, ξ (s)) ds .$ for all t ∈ [a, b].

Now we define intuitionistic fuzzy mappings F, G, S : Z → (IFS) ^Z as: $\begin{array}{l} μ_{F (ξ)} (u) = {\begin{array}{l} A (ξ) & if u (t) = ω_{ξ} (t) & for all t \in [a, b] \\ 0 & otherwise, \end{array} \\ υ_{F (ξ)} (u) = {\begin{array}{l} 0 & if u (t) = ω_{ξ} (t) & for all t \in [a, b] \\ B ((ξ)) & otherwise, \end{array} \\ μ_{G (ξ)} (u) = {\begin{array}{l} P (ξ) & if u (t) = Ψ_{ξ} (t) & for all t \in [a, b] \\ 0 & otherwise, \end{array} \\ υ_{G (ξ)} (u) = {\begin{array}{l} 0 & if u (t) = Ψ_{ξ} (t) & for all t \in [a, b] \\ 0 (ξ) & otherwise \end{array} \end{array}$ and $\begin{array}{l} μ_{S (ξ)} (u) = {\begin{array}{l} T (ξ) & if u (t) = f (ξ (t)) & for all t \in [a, b] \\ 0 & otherwise, \end{array} \\ υ_{S (ξ)} (u) = {\begin{array}{l} 0 & if u (t) = f (ξ (t)) & for all t \in [a, b] \\ D (ξ) & otherwise . \end{array} \end{array}$

If we take $α_{F (ξ)} = A_{F (ξ)}, β_{F (ξ)} = 0, α_{G (ξ)} = P_{(ξ)}, β_{G (ξ)} = 0$ and $α_{S (ξ)} = T_{(ξ)}, β_{S (ξ)} = 0$ , then we obtain $\begin{matrix} \in Z \end{matrix} \cup^{} {[F (ξ)]}^{{(α, β)}_{F} (ξ)} = \underset{ξ \in Z}{\cup} {u \in Z : μ_{F (ξ)} (u) = A (ξ) and υ_{F (ξ)} (u) = 0} = \underset{ξ \in Z}{\cup} {ω_{ξ}},$ $\begin{matrix} \in Z \end{matrix} \cup^{} {[G (ξ)]}^{{(α, β)}_{G} (ξ)} = \underset{ξ \in Z}{\cup} {u \in Z : μ_{G (ξ)} (u) = P (ξ) and υ_{G (ξ)} (u) = 0} = \underset{ξ \in Z}{\cup} {Ψ_{ξ}}$ and $\begin{matrix} \in Z \end{matrix} \cup^{} {[S (ξ)]}^{{(α, β)}_{S} (ξ)} = \underset{ξ \in Z}{\cup} {u \in Z : μ_{S (ξ)} (u) = T (ξ) and υ_{S (ξ)} (u) = 0} = \underset{ξ \in Z}{\cup} {f o ξ : ξ \in Z} .$ This implies that ${[S (ξ)]}_{(α, β) S (ξ)}$ is complete. However, for $u \in \underset{ξ \in Z}{\cup} {[F (ξ)]}_{(α, β) F (ξ)}$ or $u \in \underset{ξ \in Z}{\cup} {[G (ξ)]}_{(α, β) G (ξ)}$ , we obtain $ξ \in Z$ such that $u (t) = ω_{ξ} (t) or u (t) = Ψ_{ξ} (t), for all t \in [a, b] .$

Thus, by assumption, there exists $η \in Z$ such that $u = f o η$ .

Therefore, $\begin{matrix} \underset{ξ \in Z}{\cup} [(\underset{ξ \in Z}{\cup^{}} {[F (ξ)]}^{{(α, β)}_{F} (ξ)}) \cup^{} (\underset{ξ \in Z}{\cup} {[G (ξ)]}^{{(α, β)}_{G} (ξ)})] \\ = \underset{ξ \in Z}{\cup} ({ω_{ξ}} \cup^{} {Ψ_{ξ}}) \\ \subseteq \underset{ξ \in Z}{\cup} {[S (ξ)]}^{{(α, β)}_{S} (ξ)} . \end{matrix}$ Moreover, $\begin{matrix} | | ω_{ξ} (t) - Ψ_{η} (t) | |_{\infty} \\ = | q | | | \int_{a}^{b} K_{1} (t, s, ξ (s)) d s - \int_{a}^{b} K_{2} (t, s, η (s)) d s | |_{\infty} \\ \leq | q | \int_{a}^{b} | | K_{1} (t, s, ξ (s)) - K_{2} (t, s, η (s)) | |_{\infty} d s \\ \leq | q | \int_{a}^{b} l | | f (ξ (s)) - f (η (s)) | |_{\infty} d s \\ \leq | q | l (\sup_{t \in [a, b]} | | (S ξ) (t) - (S η) (t) | |) (b - a) \\ \leq | q | l (b - a) d ({[S (ξ)]}^{{(α, β)}_{S} (ξ)}, {[S (η)]}^{{(α, β)}_{S} (η)}) . \end{matrix}$ Thus, for $k = | q | l (b - a)$ , all the conditions of Theorem 3.1 are satisfied. Hence, there exists a $ω \in Z$ such that $\begin{matrix} {[F (ω)]}^{{(α, β)}_{F} (ω)} & \cap^{} & {[G (ω)]}^{{(α, β)}_{G} (ω)} \\ \cap^{} & {[S (ω)]}^{{(α, β)}_{S} (ω)} \neq φ . \end{matrix}$ Therefore, ω is a solution of system of integral Equations 4.1 and 4.2.

5 Conclusion

The current day databases is full of uncertainties, imprecision and vagueness. In order to handle such problems, classical mathematical tools may not be applied effectively. Some innovative models such as probability theory, fuzzy set theory [30], IF-set theory [4], rough set theory [23, 32], soft set theory [17 , 31] and vague set theory [10] are beneficial in exhibiting and extracting the hidden facts in uncertain data collection. These models have been successfully applied to intelligent systems, machine learning, pattern recognition, game theory, metrology, image processing, signal analysis, decision analysis, expert systems and many other fields. On the basis of the above research, we proved common coincidence theorems by extending the notion of fuzzy maps to intuitionistic fuzzy maps in connection with (α,) -level and (M,m) -level sets. We deduced a few corollaries to generalize many important results of intuitionistic fuzzy mappings, fuzzy mappings and multivalued mappings in the current literature. As applications, an existence theorem for the solution of a system of nonlinear integral equations has been established. Moreover, our work is based on IF-set with the induction of level sets of an IF-set to define an associated sequence of crisp structures which is flexible approach to solve multi-functions domain problems.

Footnotes

Acknowledgments

The authors are highly thankful to Associate Editor and anonymous referees for their valuable comments and constructive suggestions which helped to improve the quality of this paper significantly.

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