Common fixed point theorems for hybrid pairs of L -fuzzy mappings in non-Archimedean modified intuitionistic fuzzy metric spaces

Abstract

In this paper, we adopt the notion of CLR property for hybrid pairs of L-fuzzy mappings in non-Archimedean modified intuitionistic fuzzy metric spaces and utilize the same to prove some common fixed point theorems for L-fuzzy mappings in non-Archimedean modified intuitionistic fuzzy metric spaces. Our results generalize and improve several earlier relevant known results. We also substantiate our results using illustrative examples.

Keywords

Modified IFMS hybrid pairs of mappings common fixed point

1 Introduction

In 2004, Park [10] introduced the notion of intuitionistic fuzzy metric spaces as a generalization of fuzzy metric spaces. Thereafter, Saadati et al. [13] introduced the notion of modified intuitionistic fuzzy metric spaces (in short: modified IFMS) using continuous t-representable and defined the notion of compatible mappings in modified IFMS and utilize the same to prove several results in such spaces in varied ways.

Goguen [9] introduced the notion of L-fuzzy sets as a generalization of fuzzy sets [18]. Fuzzy set, L-fuzzy set and rough set are some variants of a classical set. For instance, Zhan et al. [7] studied roughness in soft hemirings with respect to Pawlak approximation spaces. Rashid et al. [11] proved the existence results on common L-fuzzy fixed point in complete metric spaces. Several results of Heilpern [5] can be utilized to prove L-fuzzy analogues in non-Archimedean modified IFMS.

In this paper, we adopt the notion of CLR property for hybrid pairs of L-fuzzy mappings in non-Archimedean modified IFMS and use the same to prove common fixed point theorems via an implicit relation involving integral contractive condition.

2 Preliminaries

Lemma 2.1.[3] Consider the set L^∗ and operation ≤_{L
^∗} defined by $L^{*} = {(x_{1}, x_{2}) : (x_{1}, x_{2}) \in [0, 1]^{2}, x_{1} + x_{2} \leq 1},$ (x₁, x₂) ≤ _{L
^∗} (y₁, y₂) ⇔ x₁ ≤ y₁ and x₂ ≥ y₂, for every (x₁, x₂), (y₁, y₂) ∈ L^∗. Then (L^∗, ≤ _{L
^∗}) is a complete lattice.

Definition 2.1. [2] An intuitionistic fuzzy set A_ζ,η in a universe U is an object A_ζ,η = {(ζ_A (u), η_A (u)) : u ∈ U}, where, for all u ∈ U, ζ_A (u) ∈ [0, 1] and η_A (u) ∈ [0, 1] are called the membership degree and the non-membership degree, respectively, of u in A_ζ,η and furthermore they satisfy ζ_A (u) + η_A (u) ≤1. We denote its units by 0_{L
^∗} = (0, 1) and 1_{L
^∗} = (1, 0). 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].

Definition 2.2. [4] A triangular norm (t-norm) on L^∗ is a mapping $T : (L^{*})^{2} \to L^{*}$ satisfying the following conditions, ∀x, x′, y, y′ ∈ L^∗, x ≤ _{L
^∗}x′, y ≤ _{L
^∗}y′:

$T (x, 1_{L^{*}}) = x$ , (boundary condition)

$T (x, y) = T (y, x)$ , (commutativity)

$T (x, T (y, z)) = T (T (x, y), z)$ , (associativity)

$T (x, y) \leq_{L^{*}} T (x^{'}, y^{'})$ . (monotonicity)

Definition 2.3. [3, 4] A continuous t-norm $T$ 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 $T (x, y) = (x_{1} * y_{1}, x_{2} ⋄ y_{2}), for all x = (x_{1}, x_{2}), y = (y_{1}, y_{2}) \in L^{*}$ .

Definition 2.4. [3, 4] A negator on L^∗ is any decreasing mapping $N : L^{*} \to L^{*}$ satisfying $N (0_{L^{*}}) = 1_{L^{*}}$ and $N (1_{L^{*}}) = 0_{L^{*}}$ . If $N (N (x)) = x$ , for all x ∈ L^∗, then $N$ 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 whereas N_s denotes the standard negator on [0, 1] which is defined as N_s (x) =1 - x for all x ∈ [0, 1].

Definition 2.5. [13] Let M and N be 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 triplet $(X, M_{M, N}, T)$ is said to be a modified IFMS if X is an arbitrary (non-empty) set, $T$ is a continuous t-representable and $M_{M, N}$ is a mapping X² × (0, ∞) → L^∗ (an intuitionistic fuzzy set), satisfying the following conditions for every x, y ∈ X and t, s > 0:

$M_{M, N} (x, y, t) >_{L^{*}} 0_{L^{*}}$ ,

$M_{M, N} (x, y, t) = 1_{L^{*}}$ if and only if x = y,

$M_{M, N} (x, y, t) = M_{M, N} (y, x, t)$ ,

$M_{M, N} (x, y, t + s) \geq_{L^{*}} T (M_{M, N} (x, z, t), M_{M, N} (z, y, s))$ ,

$M_{M, N} (x, y, -) : (0, \infty) \to L^{*}$ is continuous.

If we replace (IV) with $M_{M, N} (x, y, max {t, s}) \geq_{L^{*}} T (M_{M, N} (x, z, t), M_{M, N} (z, y, s))$ , for all t, s > 0, then the triplet $(X, M_{M, N}, T)$ is said to be a non-Archimedean modified IFMS.

Remark 2.1. Every non-Archimedean modified IFMS is obviously a modified IFMS.

Remark 2.2. [17] In an intuitionistic fuzzy metric space $(X, M_{M, N}, T)$ , M (x, y, .) is non-decreasing and N (x, y, .) is non-increasing for all x, y ∈ X. Hence $(X, M_{M, N}, T)$ is non-decreasing function for all x, y ∈ X.

Example 2.1. [13] Let (X, d) be a metric space. Denote $T (a, b) = (a_{1} b_{1}, min {a_{2} + b_{2}, 1})$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L^∗. Let M and N be fuzzy sets on X² × (0, ∞) defined as follows: $\begin{matrix} M_{M, N} (x, y, t) = (M (x, y, t), N (x, y, t)) \\ = (\frac{{ht}^{n}}{{ht}^{n} + md (x, y)}, \frac{md (x, y)}{{ht}^{n} + md (x, y)}), \end{matrix}$ for all $t, h, m, n \in ℝ^{+}$ . Then $(X, M_{M, N}, T)$ is a modified IFMS.

Definition 2.6. [13] Let $(X, M_{M, N}, T)$ be a modified IFMS. For t > 0, define the open ball B (x, r, t) with center x ∈ X and radius 0 < r < 1, as $B (x, r, t) = {y \in X : M_{M, N} (x, y, t) >_{L^{*}} (N_{s} (r), r)}$ . A subset $A \subset X$ is called open if for each $x \in A$ , there exist t > 0 and 0 < r < 1 such that $B (x, r, t) \subseteq A$ . Let $τ_{M_{M, N}}$ be the family of all open subsets of X. Often, $τ_{M_{M, N}}$ is called the topology induced by intuitionistic fuzzy metric $M_{M, N}$ .

Definition 2.7. [13] A sequence {x_n} in a modified IFMS $(X, M_{M, N}, T)$ is called a Cauchy sequence if for each 0 < ɛ < 1 and t > 0, there exists n₀ ∈ N such that $M_{M, N} (x_{n}, x_{m}, t) >_{L^{*}} (N_{s} (ɛ), ɛ)$ , and for each n, m ≥ n₀, where N_s is the standard negator. The sequence {x_n} is said to be convergent to x ∈ X in a modified IFMS $(X, M_{M, N}, T)$ and denoted by x_n → x if $M_{M, N} (x_{n}, x, t) \to 1_{L^{*}}$ whenever n→ ∞ for every t > 0. A modified IFMS is said to be complete if and only if every Cauchy sequence is convergent.

Lemma 2.2. [12] Let $M_{M, N}$ be an intuitionistic fuzzy metric. Then for any t > 0, $M_{M, N} (x, y, t)$ is non-decreasing with respect to t in (L^∗, ≤ _{L
^∗}), for all x, y ∈ X.

Definition 2.8. [13] Let $(X, M_{M, N}, T)$ be a modified IFMS. Then $M_{M, N}$ is said to be continuous on X² × (0, ∞), if $lim_{n \to \infty} M_{M, N} (x_{n}, y_{n}, t_{n}) = M_{M, N} (x, y, t)$ , whenever a sequence {(x_n, y_n, t_n)} in X² × (0, ∞) converges to a point (x, y, t) ∈ X² × (0, ∞), $i . e ., lim_{n \to \infty} M_{M, N} (x_{n}, x, t) = lim_{n \to \infty} M_{M, N} (y_{n}, y, t) = 1_{L^{*}}$ , and $lim_{n \to \infty} M_{M, N} (x, y, t_{n}) = M_{M, N} (x, y, t)$ .

Lemma 2.3. [13] Let $(X, M_{M, N}, T)$ be a modified IFMS. Then $M_{M, N}$ is continuous function on X × X × (0, ∞).

Definition 2.9. [13] Let (f, g) be a pair of mappings from a modified IFMS $(X, M_{M, N}, T)$ into itself. Then the pair of mappings (f, g) is said to be compatible if $lim_{n \to \infty} M_{M, N} ({fgx}_{n}, {gfx}_{n}, t) = 1_{L^{*}}$ , for all t > 0, whenever {x_n} is a sequence in X such that $lim_{n \to \infty} {fx}_{n} = lim_{n \to \infty} {gx}_{n} = x \in X$ .

Definition 2.10. [16] Let (f, g) be a pair of mappings from a modified IFMS $(X, M_{M, N}, T)$ into itself. Then the pair of mappings (f, g) is said to be non compatible if there exists at least one sequence {x_n} in X such that $lim_{n \to \infty} {fx}_{n} = lim_{n \to \infty} {gx}_{n} = x \in X$ , but $lim_{n \to \infty} M_{M, N} ({fgx}_{n}, {gfx}_{n}, t) \neq 1_{L^{*}}$ , or non-existent for at least one t > 0.

Definition 2.11. [13] Let (f, g) be a pair of self mappings of a non-empty set X. Then the pair of mappings (f, g) is said to be weakly compatible if they commute at their coincidence points, that is, fx = gx implies that fgx = gfx.

Remark 2.3. [13] Every pair of compatible self mappings (f, g) of a modified IFMS $(X, M_{M, N}, T)$ is weakly compatible but the converse implication is not true in general.

Definition 2.12. [1] Let CL (X) be the set of all nonempty closed subsets of a modified IFMS $(X, M_{M, N}, T)$ and F : Y ⊆ X → CL (X). Then the mapping f : Y → X is said to be F-weakly commuting at x ∈ X if ffx ∈ Ffx provided that fx ∈ Y for all x ∈ Y.

Definition 2.13. [1] A mapping f : Y ⊆ X → X is said to be coincidentally idempotent w.r.t. a mapping F : Y → CL (X) if f is idempotent at the coincidence points C (f, F) of f and F, i.e., ffx = fx for all x ∈ Y with fx ∈ Fx provided that fx ∈ Y.

Definition 2.14. [14] A mapping f : Y ⊆ X → X is said to be occasionally coincidentally idempotent w.r.t. a mapping F : Y → CL (X) if ffx = fx for some x ∈ C (f, F).

Remark 2.4. [14] Every coincidentally idempotent pairs of mappings are occasionally coincidentally idempotent, but the converse implication is not necessarily true.

Definition 2.15. [8] Let (X, M, N, ∗, ⋄) be an intuitionistic fuzzy metric space. Then one can construct the corresponding Hausdorff intuitionistic fuzzy metric as follows: $\begin{matrix} M_{M, N} (A, B, t) \\ = min {inf_{a \in A} M_{M, N} (a, B, t), inf_{b \in B} M_{M, N} (A, b, t)} . \end{matrix}$

Definition 2.16. [9] A partially ordered set (L, ≤ _L) is called

a lattice, if a ∨ b ∈ L and a ∧ b ∈ L, for any a, b ∈ L,

a complete lattice, if ∨A ∈ L and ∧A ∈ L, for any A ⊆ L,

distributive if a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c), a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c), for any a, b, c ∈ L.

Definition 2.17. [9] An L-fuzzy set A on a nonempty set X is a function A : X → L, where L is complete distributive lattice with 1_L and 0_L. In L-fuzzy sets, if we set L = [0, 1], then we obtain fuzzy sets.

Definition 2.18. [11] The α_L-level set of L-fuzzy set A is denoted by A_{α_L} and is defined as follows:

$A_{α_{L}} = {x : α_{L} ⪯_{L} A (x)} if α_{L} \in L ∖ {0_{L}}, A_{0_{L}} = \bar{{x : 0_{L} ⪯_{L} A (x)}}$ . Here $\bar{B}$ denotes the closure of the set B. The characteristic function χ_{L
_A} of an L-fuzzy set A runs as follows: $χ_{L_{A}} (x) = {\begin{matrix} 0_{L}, & if x \notin A, \\ 1_{L}, & if x \in A . \end{matrix}$

From Definitions 2.17 and 2.18, if 0_L = 0 and 1_L = 1, then we have a fuzzy set.

Definition 2.19. [11] Let X and Y be two arbitrary nonempty sets. A mapping T is called L-fuzzy mapping if T is a mapping from X into ℑ_L (Y). An L-fuzzy mapping T is an L-fuzzy subset on X × Y with membership function T (x) (y). The function T (x) (y) is the grade of membership of y in T (x).

Definition 2.20. [11] Let (F, G) be two L-fuzzy mappings from an arbitrary nonempty set X into ℑ_L (X). A point z ∈ X is called an L-fuzzy fixed point of F if z ∈ {Fz} _{α_L}, where α_L ∈ L ∖ {0_L}. The point z ∈ X is called a common L-fuzzy fixed point of F and G if z ∈ {Fz} _{α_L} ∩ {Gz} _{α_L}.

Definition 2.21. [1] Let $(X, M_{M, N}, T)$ be a modified IFMS, Y ⊆ X, f, g : Y → X and F, G : Y → CL (X). The hybrid pairs (f, F) and (g, G) are said to satisfy the JCLR-property at u ∈ Y with respect to F and G if there exist two sequences {x_n}, {y_n} in Y and A, B ∈ CL (X) such that $lim_{n \to \infty} M_{M, N} ({fx}_{n}, u, t) = lim_{n \to \infty} M_{M, N} ({gy}_{n}, u, t) = lim_{n \to \infty} M_{M, N} ({Fx}_{n}, A, t) = lim_{n \to \infty} M_{M, N} ({Gy}_{n}, B, t) = 1_{L^{*}},$ with u = fv = gw for some u, v, w ∈ Y.

Definition 2.22. [6] Let (X, M, ∗) be a metric space whereas Y is an arbitrary non empty set with F, G : Y → CB (X) and f, g : Y → X. Then the hybrid pairs (f, F) and (g, G) are said to have the JCLR-property if there exist two sequences {x_n}, {y_n} in Y and A, B ∈ CB (X) such that $lim_{n \to \infty} {Fx}_{n} = A, lim_{n \to \infty} {Gy}_{n} = B, lim_{n \to \infty} {fx}_{n} = lim_{n \to \infty} {gy}_{n} = u \in A \cap B \cap f (Y) \cap g (Y)$ with u = fv = gw, u ∈ A ∩ B, for some v, w ∈ Y.

We rewrite some definitions in non-Archimedean modified IFMS.

Definition 2.23. Let $(X, M_{M, N}, T)$ be a non-Archimedean modified IFMS, Y ⊆ X, f, g : Y → X and F, G : Y → CL (X). The hybrid pairs (f, F) and (g, G) are said to satisfy the JCLR-property at u ∈ Y with respect to F and G if there exist two sequences {x_n}, {y_n} in Y and A, B ∈ CL (X) such that $\begin{matrix} lim_{n \to \infty} M_{M, N} ({fx}_{n}, u, t) \\ = lim_{n \to \infty} M_{M, N} ({gy}_{n}, u, t) \\ = lim_{n \to \infty} M_{M, N} ({Fx}_{n}, A, t) \\ = lim_{n \to \infty} M_{M, N} ({Gy}_{n}, B, t) = 1_{L^{*}}, \end{matrix}$ with u = fv = gw, u ∈ A ∩ B for some v, w ∈ Y.

Remark 2.5. If f = g and F = G in Definition 2.23, then JCLR-property reduces to CLR-property.

Definition 2.24. Let $(X, M_{M, N}, T)$ be a non-Archimedean modified IFMS, Y ⊆ X. A mapping f : Y → X is said to be occasionally coincidentally idempotent w.r.t. a mapping F : Y → CL (X) if ffx = fx for some x ∈ C (f, F).

Definition 2.25. Let $(X, M_{M, N}, T)$ be a non-Archimedean modified IFMS, Y ⊆ X. A mapping f : Y → X is said to be F-weakly commuting at x ∈ Y if ffx ∈ Ffx provided that fx ∈ Y for all x ∈ Y.

Let Φ be the family of all continuous mappings φ : L^∗⁶ → L^∗, which are non-increasing in the 3rd, 4th, 5th, 6th coordinate variables and also satisfy the following properties:

φ (u, 1, 1, u, u, 1) ≥ _L
^∗0_L
^∗ or φ (u, 1, u, 1, 1, u) ≥ _L
^∗0_L
^∗,

$\int_{0}^{φ (u, 1, 1, u, u, 1)} φ (s) ds \geq_{L^{*}} 0_{L^{*}}$ or $\int_{0}^{φ (u, 1, u, 1, 1, u)} φ (s) ds \geq_{L^{*}} 0_{L^{*}}$ ,

$\int_{0}^{φ (\int_{0}^{u} ψ (s) ds, 1, \int_{0}^{u} ψ (s) ds, 1, 1, \int_{0}^{u} ψ (s) ds)} φ (s) ds \geq_{L^{*}} 0_{L^{*}}$ or $\int_{0}^{φ (\int_{0}^{u} ψ (s) ds, 1, 1, \int_{0}^{u} ψ (s) ds, \int_{0}^{u} ψ (s) ds, 1)} φ (s) ds \geq_{L^{*}} 0_{L^{*}}$ .

∀u ∈ L^∗ implies u = 1_{L
^∗}, where φ, ψ : [0, ∞) → [0, ∞) are summable non negative lebesgue integrable functions such that for each ɛ ∈ [0, 1],

\int_{0}^{ɛ} φ (s) ds > 0

and

\int_{0}^{ɛ} ψ (s) ds > 0

. Observe that restrictions ψ (s) =1; (φ₃) ⇒ (φ₂); φ (s) =1, (φ₂) ⇒ (φ₁) and φ (s) = ψ (s) =1, (φ₃) ⇒ (φ₁).

3 Main results

Firstly, we rewrite some definitions for hybrid pairs of L-fuzzy mappings in non-Archimedean fuzzy metric spaces. In the sequel ℑ (X) (resp. ℑ_L (X)) stands for the collection of all fuzzy sets (resp. L-fuzzy sets) of X.

Definition 3.1. Let $(X, M_{M, N}, T)$ be a non-Archime-dean modified IFMS, Y ⊆ X, f, g : Y → X and F, G : Y → ℑ _L (X) (two L-fuzzy mappings) such that for each x ∈ Y,α_L ∈ L ∖ {0_L}, {Fx} _{α_L} and {Gx} _{α_L} are nonempty closed subsets of X. Then the hybrid pairs (f, F) and (g, G) are said to have JCLR-property if there exist two sequences {x_n}, {y_n} in Y such that $\begin{matrix} lim_{n \to \infty} M ({fx}_{n}, u, t) \\ = lim_{n \to \infty} M ({gy}_{n}, u, t) \\ = lim_{n \to \infty} M ({{Fx}_{n}}_{α_{L}}, A, t) \\ = lim_{n \to \infty} M ({{Gy}_{n}}_{α_{L}}, B, t) = 1_{L^{*}} \end{matrix}$ with u = fv = gw, u ∈ A ∩ B, for some u, v, w ∈ Y.

Definition 3.2. Let $(X, M_{M, N}, T)$ be a non-Archimedean modified IFMS, Y ⊆ X, f : Y → X and F : Y → ℑ _L (X) such that for each x ∈ Y, α_L ∈ L ∖ {0_L}, {Fx} _{α_L} is nonempty closed subset of X. The mapping f is said to be occasionally coincidentally idempotent w.r.t. an L-fuzzy mapping F if ffx = fx for some x ∈ C (f, F).

Definition 3.3. Let $(X, M_{M, N}, T)$ be a non-Archimedean modified IFMS, Y ⊆ X and F : Y → ℑ _L (X). A mapping f : Y → X is said to be F-weakly commuting at x ∈ Y if ffx ∈ {Ffx} _{α_L} provided that fx ∈ Y for all x ∈ Y.

Now, we are equipped to prove our main result as follows:

Theorem 3.1. Let $(X, M_{M, N}, T)$ be a non-Archime-dean modified IFMS, Y ⊆ X, f, g : Y → X and F, G : Y → ℑ _L (X) be two L-fuzzy mappings such that for each x ∈ Y, α_L ∈ L ∖ {0_L}, and {Fx} _{α_L} as well as {Gx} _{α_L} are nonempty closed subsets of X. Suppose that for all x, y ∈ Y, there exists φ ∈ Φ such that $\begin{matrix} φ (\begin{matrix} M_{M, N} ({Fx}_{α_{L}}, {Gy}_{α_{L}}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {Fx}_{α_{L}}, t), \\ M_{M, N} (gy, {Gy}_{α_{L}}, t), \\ M_{M, N} (fx, {Gy}_{α_{L}}, t), \\ M_{M, N} (gy, {Fx}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$ If the pairs (f, F) and (g, G) satisfy JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F and G have a common fixed point.

Proof 3.1. Since the hybrid pairs (f, F) and (g, G) satisfy the JCLR-property, there exist two sequences {x_n}, {y_n} in Y and A, B ∈ CL (X) such that $\begin{matrix} lim_{n \to \infty} M_{M, N} ({fx}_{n}, u, t) & = & lim_{n \to \infty} M_{M, N} ({gy}_{n}, u, t) \\ = & lim_{n \to \infty} M_{M, N} ({{Fx}_{n}}_{α_{L}}, A, t) \\ = & lim_{n \to \infty} M_{M, N} ({{Gy}_{n}}_{α_{L}}, B, t) \\ = & 1_{L^{*}} \end{matrix}$ with u = fv = gw, u ∈ A ∩ B, for some u, v, w ∈ Y. We show that A = B. As $\begin{matrix} φ (\begin{matrix} M_{M, N} ({{Fx}_{n}}_{α_{L}}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{n}, {gy}_{n}, t), \\ M_{M, N} ({fx}_{n}, {{Fx}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Fx}_{n}}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ letting n→ ∞, we get $\begin{matrix} φ (\begin{matrix} (M_{M, N} (A, B, t), 1_{L^{*}}, \\ M_{M, N} (fv, A, t), M_{M, N} (fv, B, t), \\ M_{M, N} (fv, B, t), M_{M, N} (fv, A, t)) \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ so that $\begin{matrix} φ (M_{M, N} (A, B, t), 1_{L^{*}}, 1_{L^{*}}, M_{M, N} (A, B, t), \\ M_{M, N} (A, B, t), 1_{L^{*}}) \\ \geq_{L^{*}} φ (M_{M, N} (A, B, t), 1_{L^{*}}, M_{M, N} (fv, A, t), \\ M_{M, N} (fv, B, t), M_{M, N} (fv, B, t), \\ M_{M, N} (fv, A, t)) \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

Using (φ₁), we have $M_{M, N} (A, B, t) = 1_{L^{*}}$ , so that A = B. Next, we wish to show that fv ∈ {Fv} _{α_L}. To prove this, as fv ∈ A, we need to show that A = {Fv} _{α_L}. As $\begin{matrix} φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} (fv, {gy}_{n}, t), \\ M_{M, N} (fv, {Fv}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} (fv, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {Fv}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ letting n→ ∞, we have $\begin{matrix} φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, A, t), 1_{L^{*}}, \\ M_{M, N} (fv, {Fv}_{α_{L}}, t), \\ M_{M, N} (fv, A, t), \\ M_{M, N} (fv, A, t), \\ M_{M, N} (fv, {Fv}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$ so that $\begin{matrix} φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, A, t), 1_{L^{*}}, \\ M_{M, N} (A, {Fv}_{α_{L}}, t), 1_{L^{*}}, \\ 1_{L^{*}}, M_{M, N} (A, {Fv}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} \\ φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, A, t), \\ 1_{L^{*}}, M_{M, N} (fv, {Fv}_{α_{L}}, t), \\ M_{M, N} (fv, A, t), \\ M_{M, N} (fv, A, t), \\ M_{M, N} (fv, {Fv}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

Using (φ₁), we have $M_{M, N} (A, {Fv}_{α_{L}}, t) = 1_{L^{*}}$ so that A = {Fv} _{α_L}, and fv ∈ {Fv} _{α_L}. Similarly, one can also show that gw ∈ {Gw} _{α_L} for w ∈ Y. Further, ffv = fv and ffv ∈ {Ffv} _{α_L} so that u = fu ∈ {Fu} _{α_L}. Also, ggw = gw and ggw ∈ {Ggw} _{α_L} implies u = gu ∈ {Gu} _{α_L}. Then f, g, F and G have a common fixed point. This concludes the proof.

The following example illustrates the proceeding theorem.

Example 3.1. Consider Y = X = [0, 1]. Define $T (a, b) = (max {0, a_{1} + b_{1} - 1}, a_{2} + b_{2} - a_{2} b_{2})$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L^∗. Let M, N be fuzzy sets on X² × (0, ∞) defined as follows: $\begin{matrix} M_{M, N} (x, y, t) & = & (M (x, y, t), N (x, y, t)) \\ = & {\begin{matrix} (\frac{x}{y}, \frac{y - x}{y}), & if x \leq y, \\ (\frac{y}{x}, \frac{x - y}{x}), & if x \geq y, \end{matrix} \end{matrix}$ for all x, y ∈ Y and t > 0. Then $(X, M_{M, N}, T)$ is an intuitionistic fuzzy metric space. Define the mappings f, g, F, G on Y as $fx = \frac{2 x}{3}$ , $gx = \frac{x}{4}$ , $(Fx) (y) = {\begin{matrix} 0, & if 0 \leq y \leq \frac{1}{5}, \\ \frac{1}{3}, & if \frac{1}{5} < y < \frac{2 x}{3}, \\ \frac{2}{3}, & if \frac{2 x}{3} \leq y \leq 1 . \end{matrix}$ and $(Gx) (y) = {\begin{matrix} 0, & if 0 \leq y \leq \frac{1}{4}, \\ \frac{1}{6}, & if \frac{1}{4} < y < \frac{x}{4}, \\ \frac{1}{4}, & if \frac{x}{4} \leq y \leq 1 . \end{matrix}$ for all x, y ∈ Y. Consider two sequences {x_n} and {y_n} in Y such that ${x_{n}} = {\frac{1}{n}}, {y_{n}} = {\frac{1}{2 n}}, n \in ℕ$ . Since ${Fx}_{\frac{2}{3}} = [\frac{2 x}{3}, 1]$ , then $lim_{n \to \infty} {{Fx}_{n}}_{\frac{2}{3}} = lim_{n \to \infty} [\frac{2}{3 n}, 1] = [0, 1]$ . Similarly, ${Gx}_{\frac{1}{4}} = [\frac{x}{4}, 1]$ , then $lim_{n \to \infty} {{Gy}_{n}}_{\frac{1}{4}} = lim_{n \to \infty} [\frac{1}{8 n}, 1] = [0, 1]$ . Now, $lim_{n \to \infty} {fx}_{n} = lim_{n \to \infty} {gy}_{n} = 0 \in [0, 1] = lim_{n \to \infty} {{Fx}_{n}}_{\frac{2}{3}} = lim_{n \to \infty} {{Gy}_{n}}_{\frac{1}{4}}, i . e .,$ the hybrid pairs (f, F) and (g, G) satisfy the JCLR-property. Further, define φ (t₁, ..., t₆) = t₃ so that $M ({fx}_{n}, {{Fx}_{n}}_{\frac{2}{3}}) = 1$ . Thus all the conditions of the preceding theorem are satisfied. Observe that squo0’ is a common fixed point of f, g, F and $G as f 0 = g 0 = 0 \in [0, 1] = {F 0}_{\frac{2}{3}} = {G 0}_{\frac{1}{4}}$ .

With L = I = [0, 1] in Theorem 3.1, we have the following corollary.

Corollary 3.1. Let $(X, M_{M, N}, T)$ be a non-Archimedean modified IFMS, Y ⊆ X, f, g : Y → X and F, G : Y → ℑ (X) (two fuzzy mapping) such that for each x ∈ Y,α ∈ (0, 1], {Fx} _α and {Gx} _α are nonempty closed subsets of X. Suppose that for all x, y ∈ Y there exists φ ∈ Φ such that $\begin{matrix} φ (\begin{matrix} M_{M, N} ({Fx}_{α}, {Gy}_{α}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {Fx}_{α}, t), \\ M_{M, N} (gy, {Gy}_{α}, t), \\ M_{M, N} (fx, {Gy}_{α}, t), \\ M_{M, N} (gy, {Fx}_{α}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

If (f, F) and (g, G) satisfy the JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F, G have a common fixed point.

Next, we prove the following:

Theorem 3.2. Let $(X, M_{M, N}, T)$ be a non-Archime-dean modified IFMS, Y ⊆ X with f, g : Y → X. Let {F_n} be a sequence of L-fuzzy mappings from Y into ℑ_L (X) such that for each x ∈ Y, α_L ∈ L ∖ {0_L}, {Fx} _{α_L} is nonempty closed subsets of X which satisfy the following condition, there exists φ ∈ Φ such that for all x, y ∈ Y, k = 2n - 1 and l = 2n, n ∈ N $\begin{matrix} φ (\begin{matrix} M_{M, N} ({F_{k} x}_{α_{L}}, {F_{l} y}_{α_{L}}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {F_{k} x}_{α_{L}}, t), \\ M_{M, N} (gy, {F_{l} y}_{α_{L}}, t), \\ M_{M, N} (fx, {F_{l} y}_{α_{L}}, t), \\ M_{M, N} (gy, {F_{k} x}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$

If the pairs (f, F_k) and (g, F_l) satisfy the JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F_k and F_l have a common fixed point.

Proof 3.2. Since the hybrid pairs (f, F_k) and (g, F_l) satisfy the JCLR-property, there exist two sequences {x_kn}, {y_kn} in Y and A_k, B_k ∈ CL (X) such that $\begin{matrix} lim_{n \to \infty} {fx}_{kn} & = & lim_{n \to \infty} {gy}_{kn} = u_{k}, \\ lim_{n \to \infty} {F_{k} x_{kn}}_{α_{L}} & = & A_{k}, lim_{n \to \infty} {F_{l} y_{kn}}_{α_{L}} = B_{k} . \end{matrix}$ with u_k = fv_k = gw_l, u_k ∈ A_k ∩ B_k, for some u_k, v_k, w_k ∈ X. Now, we show that A_k = B_k. As $\begin{matrix} φ (\begin{matrix} M_{M, N} ({F_{k} x_{kn}}_{α_{L}}, {F_{l} y_{kn}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{kn}, {gy}_{kn}, t), \\ M_{M, N} ({fx}_{kn}, {F_{k} x_{kn}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{kn}, {F_{l} y_{kn}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{kn}, {F_{l} y_{kn}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{kn}, {F_{k} x_{kn}}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ letting n→ ∞, we get $φ (\begin{matrix} M_{M, N} (A_{k}, B_{k}, t), 1_{L^{*}}, \\ 1_{L^{*}}, M_{M, N} ({fv}_{k}, B_{k}, t), \\ M_{M, N} ({fv}_{k}, B_{k}, t), 1_{L^{*}} \end{matrix}) \geq_{L^{*}} 0_{L^{*}}$ so that $\begin{matrix} φ (\begin{matrix} M_{M, N} (A_{k}, B_{k}, t), 1_{L^{*}}, \\ 1_{L^{*}}, M_{M, N} (A_{k}, B_{k}, t), \\ M_{M, N} (A_{k}, B_{k}, t), 1_{L^{*}} \end{matrix}) \geq_{L^{*}} \\ φ (\begin{matrix} M_{M, N} (A_{k}, B_{k}, t), 1_{L^{*}}, \\ M_{M, N} ({fv}_{k}, A_{k}, t), \\ M_{M, N} ({fv}_{k}, B_{k}, t), \\ M_{M, N} ({fv}_{k}, B_{k}, t), \\ M_{M, N} ({fv}_{k}, A_{k}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

Using (φ₁), we have $M_{M, N} (A_{k}, B_{k}, t) = 1_{L^{*}}$ , i.e., A_k = B_k. As u_k ∈ A_k. Next, we show that {F_kv_k} _{α_L} = A_k. As $\begin{matrix} φ (\begin{matrix} M_{M, N} ({F_{k} v_{k}}_{α_{L}}, {F_{l} y_{kn}}_{α_{L}}, t), \\ M_{M, N} ({fv}_{k}, {gy}_{kn}, t), \\ M_{M, N} ({fv}_{k}, {F_{k} v_{k}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{kn}, {F_{l} y_{kn}}_{α_{L}}, t), \\ M_{M, N} ({fv}_{k}, {F_{l} y_{kn}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{kn}, {F_{k} v_{k}}_{α_{L}}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ which on making n→ ∞ reduces to $\begin{matrix} φ (\begin{matrix} M_{M, N} ({F_{k} v_{k}}_{α_{L}}, A_{k}, t), 1_{L^{*}}, \\ 1_{L^{*}}, M_{M, N} ({F_{k} v_{k}}_{α_{L}}, A_{k}, t), \\ M_{M, N} ({F_{k} v_{k}}_{α_{L}}, A_{k}, t), 1_{L^{*}} \end{matrix}) \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ so that {F_kv_k} _{α_L} = A_k. The rest of the proof can be completed on the lines of the proof of Theorem 3.1. This concludes the proof.

Remark. Theorem 3.1 and Theorem 3.2 are generalizations of Theorem 2.1 and Theorem 2.2 respectively (contained in [1]) to non-Archimedean modified IFMS.

If L = I = [0, 1] in Theorem 3.2 we have the following corollary.

Corollary 3.2. Let $(X, M_{M, N}, T)$ be a non-Archime-dean modified IFMS, Y ⊆ X with f, g : Y → X. Let {F_n} be a sequence of fuzzy mappings from Y into ℑ (X) such that for each x ∈ Y, α ∈ (0, 1] and {Fx} _α is nonempty closed subsets of X. Suppose that there exists φ ∈ Φ such that for all x, y ∈ Y, k = 2n - 1 and l = 2n, n ∈ N $\begin{matrix} φ (\begin{matrix} M_{M, N} ({F_{k} x}_{α}, {F_{l} y}_{α}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {F_{k} x}_{α}, t), \\ M_{M, N} (gy, {F_{l} y}_{α}, t), \\ M_{M, N} (fx, {F_{l} y}_{α}, t), \\ M_{M, N} (gy, {F_{k} x}_{α}, t) \end{matrix}) \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

If the pairs (f, F_k) and (g, F_l) satisfy the JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F_k and F_l have a common fixed point.

4 Integral type results

In fact, we prove the following:

Theorem 4.1. Let f and g be two mappings from a subset Y of a non-Archimedean modified IFMS $(X, M_{M, N}, T)$ into X and F, G be two L-fuzzy mappings from Y into ℑ_L (X) such that for each x ∈ Y,α_L ∈ L ∖ {0_L}, {Fx} _{α_L} and {Gx} _{α_L} are nonempty closed subsets of X. Suppose that for all x, y ∈ Y there exists φ ∈ Φ such that $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fx}_{α_{L}}, {Gy}_{α_{L}}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {Fx}_{α_{L}}, t), \\ M_{M, N} (gy, {Gy}_{α_{L}}, t), \\ M_{M, N} (fx, {Gy}_{α_{L}}, t), \\ M_{M, N} (gy, {Fx}_{α_{L}}, t) \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

If (f, F) and (g, G) satisfy the JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F and G have a common fixed point.

Proof 4.1. Since the hybrid pairs (f, F) and (g, G) satisfy the JCLR-property, there exist two sequences {x_n}, {y_n} in Y and A, B ∈ CL (X) such that $\begin{matrix} lim_{n \to \infty} {fx}_{n} = lim_{n \to \infty} {gy}_{n} = u, \\ lim_{n \to \infty} {{Fx}_{n}}_{α_{L}} = A, lim_{n \to \infty} {{Gy}_{n}}_{α_{L}} = B \end{matrix}$ with u = fv = gw, u ∈ A ∩ B, for some u, v, w ∈ Y.

Next, we show that A = B. As $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({{Fx}_{n}}_{α_{L}}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{n}, {gy}_{n}, t), \\ M_{M, N} ({fx}_{n}, {{Fx}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Fx}_{n}}_{α_{L}}, t) \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$ letting n→ ∞, we have $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} (A, B, t), \\ M_{M, N} (u, u, t), \\ M_{M, N} (u, A, t), \\ M_{M, N} (u, B, t), \\ M_{M, N} (u, B, t), \\ M_{M, N} (u, A, t) \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}} \end{matrix}$ so that $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} (A, B, t), \\ 1_{L^{*}}, M_{M, N} (u, A, t), \\ M_{M, N} (u, B, t), \\ M_{M, N} (u, B, t), \\ M_{M, N} (u, A, t) \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$ or $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} (A, B, t), 1_{L^{*}}, \\ 1_{L^{*}}, M_{M, N} (A, B, t), \\ M_{M, N} (A, B, t), 1_{L^{*}} \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

Using (φ₂) we have $M_{M, N} (A, B, t) = 1_{L^{*}}$ so hat A = B. As fv ∈ A, we show that A = {Fv} _{α_L} since $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} (fv, {gy}_{n}, t), \\ M_{M, N} (fv, {Fv}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} (fv, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {Fv}_{α_{L}}, t) \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ letting n→ ∞, we have $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, A, t), \\ M_{M, N} (fv, u, t), \\ M_{M, N} (fv, {Fv}_{α_{L}}, t), \\ M_{M, N} (u, A, t), \\ M_{M, N} (fv, A, t), \\ M_{M, N} (u, {Fv}_{α_{L}}, t) \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ so that $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, A, t), 1_{L^{*}}, \\ M_{M, N} (u, {Fv}_{α_{L}}, t), \\ M_{M, N} (u, A, t), \\ M_{M, N} (u, A, t), \\ M_{M, N} (u, {Fv}_{α_{L}}, t) \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$ or $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fv}_{α_{L}}, A, t), 1_{L^{*}}, \\ M_{M, N} (A, {Fv}_{α_{L}}, t), 1_{L^{*}}, \\ 1_{L^{*}}, M_{M, N} (A, {Fv}_{α_{L}}, t) \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}}, \end{matrix}$ which shows that $M_{M, N} (A, {Fv}_{α_{L}}, t) = 1_{L^{*}}$ , so that A = {Fv} _{α_L} and fv ∈ {Fv} _{α_L}. This concludes the proof.

Corollary 4.1. Let f and g be two mappings from a subset Y of a non-Archimedean modified IFMS $(X, M_{M, N}, T)$ into X while F, G be two fuzzy mappings from Y into ℑ (X) such that for each x ∈ Y,α ∈ (0, 1], {Fx} _α and {Gx} _α are nonempty closed subsets of X. Suppose that for all x, y ∈ Y there exists φ ∈ Φ such that $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fx}_{α}, {Gy}_{α}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {Fx}_{α}, t), \\ M_{M, N} (gy, {Gy}_{α}, t), \\ M_{M, N} (fx, {Gy}_{α}, t), \\ M_{M, N} (gy, {Fx}_{α}, t) \end{matrix})} φ (s) ds \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

If (f, F) and (g, G) satisfy the JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F and G have a common fixed point.

Theorem 4.2. Let f, g : Y ⊆ X → X be two mappings from a subset Y of a non-Archimedean modified IFMS $(X, M_{M, N}, T)$ into X while F, G be two L-fuzzy mappings from Y into ℑ_L (X) such that for each x ∈ Y,α_L ∈ L ∖ {0_L}, {Fx} _{α_L} and {Gx} _{α_L} are nonempty closed subsets of X. Suppose that for all x, y ∈ Y there exists φ ∈ Φ such that $\begin{matrix} \int_{0}^{φ (\begin{matrix} \int_{0}^{M_{M, N} ({Fx}_{α_{L}}, {Gy}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (fx, gy, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (fx, {Fx}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (gy, {Gy}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (fx, {Gy}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (gy, {Fx}_{α_{L}}, t)} ψ (s) ds \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

If (f, F) and (g, G) are satisfy the JCLR-property, weakly commuting property and occasionally coincidentally idempotent property, then f, g, F and G have a common fixed point.

Proof 4.2. The proof is similar to that of Theorem 4.1 in view of (φ₃).

Remark 4.1. By setting ψ (s) =1 in Theorem 4.2, we get Theorem 4.1 while choosing ψ (s) = φ (s) =1 in Theorem 4.2, we deduce Theorem 3.1. Finally, notice that by putting φ (s) =1 in Theorem 4.1, we deduce Theorem 3.1.

Theorem 4.3. Let $(X, M_{M, N}, T)$ be a non-Archime-dean modified IFMS, Y ⊆ X, f, g : Y → X, {F_n} be a sequence of L-fuzzy mappings from Y into ℑ_L (X) such that for each x ∈ Y, α_L ∈ L ∖ {0_L} {Fx} _{α_L} is nonempty closed subsets of X which satisfy the following condition, there exists φ ∈ Φ such that for all x, y ∈ Y, k = 2n - 1 and l = 2n, n ∈ N $\begin{matrix} \int_{0}^{φ (\begin{matrix} \int_{0}^{M_{M, N} ({F_{k} x}_{α_{L}}, {F_{l} y}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (fx, gy, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (fx, {F_{k} x}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (gy, {F_{l} y}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (fx, {F_{l} y}_{α_{L}}, t)} ψ (s) ds, \\ \int_{0}^{M_{M, N} (gy, {F_{k} x}_{α_{L}}, t)} ψ (s) ds \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

If the pairs (f, F_k) and (g, F_l) are satisfy the JCLR-property, weakly commuting and occasionally coincidentally idempotent, then (f, F_k) and (g, F_l) have a common fixed point.

Remark 4.2. Put ψ (s) =1 and F_k = F, F_l = G in Theorem 4.3, we have Theorem 4.1. Put ψ (s) =1 and F_k = F, F_l = G and L = I = [0, 1] in Theorem 4.3, we have Corollary 4.1. Put F_k = F, F_l = G in Theorem 4.3, we have Theorem 4.2. Put ψ (s) = φ (s) =1 in Theorem 4.2, we have Theorem 3.1 and put φ (s) =1 in Theorem 4.1, we have Theorem 3.2.

The following example illustrates Theorem 4.1:

Example 4.1. Consider X = Y = [0, 1]. Denote $T (a, b) = (a_{1} b_{1}, min {a_{2} + b_{2}, 1})$ for all a = (a₁, a₂) and b = (b₁, b₂) ∈ L^∗. Let M and N be fuzzy sets on X² × (0, ∞) defined as follows: $\begin{matrix} M_{M, N} (x, y, t) & = & (M (x, y, t), N (x, y, t)) \\ = & (\frac{t}{t + d (x, y)}, \frac{d (x, y)}{t + d (x, y)}), \end{matrix}$

Then $(X, M_{M, N}, T)$ is a modified IFMS. Define the mappings f, g, F, G on X as $fx = \frac{x}{2}$ , $gx = \frac{x}{3}$ , $(Fx) (y) = {\begin{matrix} 0, & if 0 \leq y \leq \frac{1}{3}, \\ \frac{1}{3}, & if \frac{1}{3} < y < \frac{x}{2}, \\ \frac{2}{3}, & if \frac{x}{2} \leq y \leq 1, \end{matrix}$ and $(Gx) (y) = {\begin{matrix} 0, & if 0 \leq y \leq \frac{1}{4}, \\ \frac{1}{6}, & if \frac{1}{4} < y < \frac{x}{3}, \\ \frac{1}{4}, & if \frac{x}{3} \leq y \leq 1, \end{matrix}$ for all x, y ∈ X. Define two sequences {x_n} and {y_n} in Y such that ${x_{n}} = {\frac{1}{3} + \frac{1}{n}}, {y_{n}} = {\frac{1}{2} + \frac{1}{2 n}}, for all n \in ℕ$ . Now, we show that for x, y ∈ Y there exists φ ∈ Φ such that $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({Fx}_{α_{L}}, {Gy}_{α_{L}}, t), \\ M_{M, N} (fx, gy, t), \\ M_{M, N} (fx, {Fx}_{α_{L}}, t), \\ M_{M, N} (gy, {Gy}_{α_{L}}, t), \\ M_{M, N} (fx, {Gy}_{α_{L}}, t), \\ M_{M, N} (gy, {Fx}_{α_{L}}, t) \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

Since $lim_{n \to \infty} {fx}_{n} = \frac{1}{6}$ and $lim_{n \to \infty} {Fx}_{n} = [\frac{1}{6}, 1]$ , therefore $lim_{n \to \infty} {fx}_{n} = \frac{1}{6} \in [\frac{1}{6}, 1] = lim_{n \to \infty} {Fx}_{n}$ . Similarly, $lim_{n \to \infty} {gy}_{n} = \frac{1}{6}$ and $lim_{n \to \infty} {Gy}_{n} = [\frac{1}{6}, 1]$ . Now, $lim_{n \to \infty} {fx}_{n} = lim_{n \to \infty} {gy}_{n} = \frac{1}{6} \in [\frac{1}{6}, 1] = lim_{n \to \infty} {Fx}_{n} = lim_{n \to \infty} {Gy}_{n}$ so that the hybrid pairs (f, F) and (g, G) satisfy the JCLR-property. Further, $\begin{matrix} \int_{0}^{φ (\begin{matrix} M_{M, N} ({{Fx}_{n}}_{α_{L}}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{n}, {gy}_{n}, t), \\ M_{M, N} ({fx}_{n}, {{Fx}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({fx}_{n}, {{Gy}_{n}}_{α_{L}}, t), \\ M_{M, N} ({gy}_{n}, {{Fx}_{n}}_{α_{L}}, t) \end{matrix})} φ (s) ds \\ \geq_{L^{*}} 0_{L^{*}} . \end{matrix}$

Thus the mappings f, g, F and G satisfy all the conditions of Theorem 4.1. Observe that the mappings f, g, F and G have a common fixed point.

5 Conclusion

Modified intuitionistic fuzzy metric spaces offer a setting which can be utilized to describe phenomenons containing vague and imprecise informations. Often, fixed point theorems ensuring uniqueness offer a method to construct the fixed point via a suitable iteration and such methods can be practically implemented by developing suitable algorithms often employed to find solutions of certain functional equations, differential equations, random differential equations arising in physical and engineering sciences, and even very recently in economic models and similar other processes. As a sample, one may recall that on the lines of Theorem 3.1 of Rao et al. [15] under a suitable set of conditions, as an application of our Theorem 3.1, an existence and uniqueness result on the solution involving the fuzzy analogue of the following Fredholm nonlinear integral equation can be proved: $\begin{matrix} x (p) & = & \int_{a}^{b} (K_{1} (p, q) + K_{2} (p, q)) [f (q, x (q)) \\ + g (q, x (q))] dq + h (p), \end{matrix}$ for all $p \in I = [a, b], K_{1}, K_{2} \in C (I \times I, ℝ) and h \in C (I, ℝ) .$

Overall this approach can be utilized to define a dynamical framework to classify objects into various classes. By now, it is established that intuitionistic framework is versatile enough to cover complex processes with no concrete mathematical description chiefly systems involving unpredictable or inaccurate informations. Generally, one can try to ascertain optimal approach for the classification of elements or objects into classes and decisions regarding the “closeness degree” (often described by the fuzzy set M) and “remoteness degree” (often described by the fuzzy set N).

Footnotes

Acknowledgments

All the authors are grateful to both the reviewers for their useful comments and suggestions.

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