Addressing an open problem of Choudhury et al. “Fuzzy Sets Syst. 8 (2013),84-97”

Abstract

The question of relaxing the compatible hypothesis of the pair of mappings in fixed point theory has always been remained an open problem. We address such an open problem raised by Choudhury et al. [4] and also explicitly settles the issue of monotone and continuity hypotheses of the involved mappings in coupled coincidence point results. Moreover, we state a gap in an example given in [3] and repair it. Application to the dynamic programming problem shows the usability of present work. Finally, we also propose an open problem for further investigation.

1 Introduction and preliminaries

The theory of coupled fixed points (CFP) was pioneered by Guo and Lakshmikantham [11] but got familiarized with the work of Bhaskar and Lakshmikantham [2], which was later extended in the work of Lakshmikantham and Ćirić [17] for commutative mappings. The mappings Γ : Y × Y → Y and h : Y → Y are called commutative if Γ (h (p) , h (z)) = h (Γ (p, z)) for all p, z ∈ Y. Afterwards, coupled fixed point (we call, CFP) results in various abstract spaces got popularized.

In 1975, Kramosil and Michalek [18] introduced fuzzy metric spaces. Subsequently, Grabiec [8] defined completeness in fuzzy metric spaces (now known as a G-complete fuzzy metric spaces) and provided an extension of Banach’s contraction theorem in these spaces. Later on, George and Veeramani [7] refined the notion of Cauchy sequences innovated by Grabiec [8]. George and Veeramani [7] thought M (a, b, t) as the degree of nearness between a and b with respect to t and identified a = b with M (a, b, t) =1 for t > 0 and M (a, b, t) =0 with t =∞, meanwhile, they slightly improved the concept of fuzzy metric space introduced by Kramosil and Michalek [18], thereby, explicated the first countable and Hausdorff topology in these spaces. Afterwards, the notion of complete fuzzy metric spaces of George and Veeramani [7] (now known as M-complete fuzzy metric spaces) have been emanated as another characterization of completeness. Interestingly, G-completeness of fuzzy metric space implies M-completeness. Further, the formulation of fixed point results in M-complete fuzzy metric spaces appeared as of much importance.

In present times, CFP problems are analysed under fuzzy study. Sedghi et al. [23] manifested a CFP result in GV-fuzzy metric spaces. Hu [13] obtained a coupled common fixed point result for compatible mappings under a φ-contraction in these spaces. In a fuzzy metric space (Y, M, *), mappings Γ : Y × Y → Y and h : Y → Y are called compatible if $\begin{matrix} lim_{n \to \infty} M (h (Γ (p_{n}, z_{n})), Γ (h (p_{n}), h (z_{n})), t) & = & 1, \\ lim_{n \to \infty} M (h (Γ (z_{n}, p_{n})), Γ (h (z_{n}), h (p_{n})), t) & = & 1 \end{matrix}$ for t > 0 whenever the sequences {p_n} and {z_n} in Y are such that $lim_{n \to \infty} Γ (p_{n}, z_{n}) = lim_{n \to \infty} h (p_{n}) = p$ and $lim_{n \to \infty} Γ (z_{n}, p_{n}) = lim_{n \to \infty} h (z_{n}) = z$ for any p, z ∈ Y.

Using the pair of compatible mappings, Choudhury et al. [3] proved the ensuing result:

Theorem 1.1. [3] Let (Y, M, *) be a complete fuzzy metric space with a Hadɘić type t-norm such that M (p, z, t) →1 as t→ ∞ for all p, z ∈ Y. Let ⪯ be a partial order defined on Y. Let Γ : Y × Y → Y and h : Y → Y be two mappings such that Γ has a mixed h-monotone property and satisfies the following conditions:

Γ (Y × Y) ⊂ h (Y);

h is continuous and monotone increasing;

(h, Γ) is a compatible pair;

$\begin{matrix} M (Γ (p, z), Γ (u, w), kt) \geq γ (M (h (p), \\ h (u), t) * M (h (z), h (w), t)) \end{matrix}$ (1) for all p, z, u, w ∈ Y and t > 0 with h (p) ⪯ h (u) and h (z) ≽ h (w), where k ∈ (0, 1) and γ ; [0, 1] → [0, 1] is a continuous function such that γ (c) * γ (c) ≥ c for all c ∈ [0, 1].

Suppose that Y has the following properties:

For a non-decreasing sequence {p_n} → p, p_n ⪯ p for all $n \in ℕ \cup {0}$ ;

For a non-increasing sequence {z_n} → z, z ⪯ z_n for all $n \in ℕ \cup {0}$ .

If there exist p₀, z₀ ∈ Y such that h (p₀) ⪯ Γ (p₀, z₀) , h (z₀) ≽ Γ (z₀, p₀), then h and Γ have a coupled coincidence point in Y.

Further, Choudhury et al. [3] proposed an open problem given below:

Problem 1.2. [3] Whether conditions weaker than compatibility can be defined which can replace compatibility in Theorem 1.1?

In this current work, we not only address Problem 1.2 but also provide a check to the continuity and monotone hypotheses of the self-mapping h. Further, we also provide relaxation to the continuity hypothesis of the control function γ in our results. An application to dynamic programming problem highlights the utility of our work. The proper inferring has been drawn in the conclusion part. Before answering the Problem 1.2, first, we detect a fallacy in an example provided in support of [3, Theorem 1.1] and provide a suitable revised example in Section 2.

In present work, denote by $ℝ^{+}$ , $ℝ_{0}^{+}$ , J and $ℕ$ , the set of non-negative real numbers, the set of positive real numbers, the interval [0, 1] and the set of positive integers, respectively.

Many authors have obtained different results for φ-contractions in probabilistic as well as in fuzzy metric spaces (see [4 , 26]). There are several interesting results presented in the literature that utilized the gauge function (or auxiliary function) $φ : ℝ^{+} \to ℝ^{+}$ , where φ assumes any one of the following conditions:

φ (t) = kt for t > 0, where k ∈ (0, 1);

$\sum_{n = 1}^{\infty} φ^{n} (t) < \infty$ for t > 0.

Ćirić [5] signified the typicality of condition (b) and consequently, suggested the following condition:

φ (0) =0,φ (t) < t and $lim inf_{ρ \to t^{+}} φ (ρ) < t$ for t > 0.

Subsequently, Jachymski [14] instigated the below condition:

0 < φ (t) < t and $lim_{n \to \infty} φ^{n} (t) = 0$ for t > 0.

Let Φ denote the set of all functions $φ : ℝ^{+} \to ℝ^{+}$ satisfying the condition (d).

To further weaken condition (d), Fang [6] developed the condition:

For each t > 0, there exists ρ ≥ t so that $lim_{n \to \infty} φ^{n} (ρ) = 0$ .

Fang [6] obtained some results in context of Menger probabilistic metric spaces and fuzzy metric spaces.

Denote by Φ_w, the set of all functions $φ : ℝ^{+} \to ℝ^{+}$ with the condition (e).

In their work, Wang et al. [25] simulated the class of the gauge functions satisfying the following weakest condition to be known till now:

For each t₁, t₂ > 0, there exist ρ ≥ max {t₁, t₂} and $N \in ℕ$ so that φⁿ (ρ) < min {t₁, t₂} for n > N.

Denote by Φ_w*, the set of all functions $φ : ℝ^{+} \to ℝ^{+}$ with the condition (f). Clearly, if φ ∈ Φ_w*, then for each t > 0, there exist ρ ≥ t and $N \in ℕ$ so that φⁿ (ρ) < t for n > N.

Wang et al. [25] proved that Φ ⊂ Φ_w ⊂ Φ_w* and these inculsions are proper. In [25], the following important lemma was also established:

Lemma 1.3. [25] Let φ ∈ Φ_w*. Then for t > 0, there exists ρ ≥ t such that φ (ρ) < t.

Below, we state some already existing concepts useful for us in proving our results:

Definition 1.4. [27] A fuzzy set U in Y is a function with domain Y and values in J.

Definition 1.5. [22] A binary operation * : J × J → J is a continuous t-norm if it satisfies the following conditions:

associativity and commutativity;

continuity;

e * 1 = e for e ∈ J;

e * f ≤ g * w whenever e ≤ g and f ≤ w for e, f, g, w ∈ J.

Note that e * f can be written as * (e, f). Sometimes, “Δ” is used in place of “*”. Common examples of the continuous t-norm are e * ₁f = ef and e * ₂f = min {e, f} for all e, f ∈ J.

Definition 1.6. [12] Let $sup_{0 < s < 1} Δ (s, s) = 1$ . A t-norm Δ is said to be Hadɘić type t-norm (in short, H-type t-norm) if the family of functions ${Δ^{k} (s)}_{k = 1}^{\infty}$ is equicontinuous at s = 1, where $Δ^{1} (s) = s, Δ^{k + 1} (s) = s Δ (Δ^{k} (s))$ for all s ∈ J and k = 1, 2, 3, ⋯.

A t-norm Δ is an H-type t-norm iff for any λ ∈ (0, 1), there exists δ (λ) ∈ (0, 1) such that Δ^k (s) >1 - λ for all $k \in ℕ$ , when s > 1 - δ.

Clearly, *₂ is an example of H-type t-norm.

Definition 1.7. [7] The triplet (Y, M, *) is called a GV-fuzzy metric space, if Y is an arbitrary non-empty set, * is a continuous t-norm and M is a fuzzy set on $Y^{2} \times ℝ_{0}^{+}$ that satisfy followings: for all a, b, c ∈ Y and t > 0, r > 0,

M (a, b, c) >0;

M (a, b, t) =1 if and only if a = b;

M (a, b, t) = M (b, a, t);

M (a, b, t) * M (b, c, r) ≤ M (a, c, t + r);

$M (a, b, \cdot) : ℝ_{0}^{+} \to J$ is continuous.

Definition 1.8. [7] Let (Y, M, *) be a fuzzy metric space. A sequence {p_n} in Y is said to be

convergent to a point p ∈ Y if $lim_{n \to \infty} M (p_{n}, p, t) = 1$ for t > 0, that is, for 0 < ζ < 1 and t > 0, there exists a positive integer n₀ such that M (p_n, p, t) >1 - ζ for all n > n₀;

a Cauchy sequence if for each 0 < ζ < 1 and t > 0, there exists a positive integer n₀ so that M (p_n, p_m, t) >1 - ζ for all n, m > n₀.

A fuzzy metric space in which every Cauchy sequence is convergent is said to be complete.

Lemma 1.9. [8] Let (Y, M, *) be a fuzzy metric space. Then M (p, z, ·) is non-decreasing for all p, z ∈ Y.

Lemma 1.10. [19] Let (Y, M, *) be a fuzzy metric space. Then M is a continuous function on $Y^{2} \times ℝ_{0}^{+}$ .

Lakshmikantham and Ćirić [17] instigated the following notion of coupled coincidence point:

Definition 1.11. [17] An element (p, z) ∈ Y × Y is called a coupled coincidence point (we call, CCP) of the mappings Γ : Y × Y → Y and h : Y → Y if Γ (p, z) = h (p) and Γ (z, p) = h (z).

In Definition 1.11, taking h to be the identity mapping on Y, we can obtain the definition of Coupled Fixed Point (CFP) for the mapping Γ.

Definition 1.12. [17] Let (Y, ⪯) be a partially ordered set and Γ : Y × Y → Y and h : Y → Y be two mappings. The mapping Γ is said to have mixed h-monotone property (we call, MhMP) if for p, z ∈ Y, $p_{1}, p_{2} \in Y, h (p_{1}) ⪯ h (p_{2}) \Rightarrow Γ (p_{1}, z) ⪯ Γ (p_{2}, z),$ $z_{1}, z_{2} \in Y, h (z_{1}) ⪯ h (z_{2}) \Rightarrow Γ (p, z_{1}) ≽ Γ (p, z_{2}) .$

In Definition 1.12, taking h to be the identity mapping on Y, we can obtain the definition of mixed monotone property (we call, MMP) for the mapping Γ.

Before we give our main results and address Problem 1.2, we shall state the gap in the form of a fallacy in an example formulated by Choudhury et al. [3] in support of Theorem 1.1 and repair it in the ensuing section.

2 Fallacy in supportive example

Choudhury et al. [3] supported Theorem 1.1 by using the following example:

Example 2.1. [3] Let (Y, ⪯) be a partially ordered set with Y = [0, 1] = J and the natural ordering ≤ of the real numbers as the partial ordering ⪯. Let $M (p, z, t) = e^{- \frac{| p - z |}{t}}$ for all t > 0, p, z ∈ Y and e * f = min {e, f} for e, f ∈ J. Then (Y, M, *) is a complete fuzzy metric space such that M (p, z, t) →1 as t→ ∞ for p, z ∈ Y. Let γ : J → J be defined as γ (c) = c for c ∈ J. Define the mapping Γ : Y × Y → Y by $Γ (p, z) = \frac{p^{2} - z^{2}}{3} + \frac{2}{3}$ for p, z ∈ Y and the mapping h : Y → Y by $h (p) = p^{2}$ for p ∈ Y. Then Γ (Y × Y) ⊂ h (Y) and Γ satisfies the MhMP. Also, authors in [3] claimed that the pair (Γ, h) is compatible. Further, in [3], the inequality (1.1) was shown to be satisfied for $k = \frac{2}{3}$ and p₀ = 0.6 and z₀ = 0.9, for which h (p₀) ⪯ Γ (p₀, z₀) and h (z₀) ≽ Γ (z₀, p₀) hold. Then, on applying Theorem 1.1, it was concluded in [3] that the mappings Γ and h have the CCP $(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}})$ .

Remark 2.2. In Example 2.1, Choudhury et al. [3] claimed that the pair (Γ, h) is compatible. But actually, the pair (Γ, h) is not compatible as shown below:

On choosing the sequences ${p_{n} : = \sqrt{\frac{2}{3}} + \frac{1}{100 n}, n \in ℕ}$ and ${z_{n} : = \sqrt{\frac{2}{3}} - \frac{1}{100 n}, n \in ℕ}$ in Y, we get $lim_{n \to \infty} h (p_{n}) = lim_{n \to \infty} Γ (p_{n}, z_{n}) = \frac{2}{3} \in Y,$ $lim_{n \to \infty} h (z_{n}) = lim_{n \to \infty} Γ (z_{n}, p_{n}) = \frac{2}{3} \in Y .$ Now, $\begin{matrix} h (Γ (p_{n}, z_{n})) = {[\frac{p_{n}^{2} - z_{n}^{2}}{3} + \frac{2}{3}]}^{2} = \\ {[\frac{{(\sqrt{\frac{2}{3}} + \frac{1}{100 n})}^{2} - {(\sqrt{\frac{2}{3}} - \frac{1}{100 n})}^{2}}{3} + \frac{2}{3}]}^{2}, \end{matrix}$ $\begin{matrix} Γ (h (p_{n}), h (z_{n})) = \frac{p_{n}^{4} - z_{n}^{4}}{3} + \frac{2}{3} = \\ \frac{{(\sqrt{\frac{2}{3}} + \frac{1}{100 n})}^{4} - {(\sqrt{\frac{2}{3}} - \frac{1}{100 n})}^{4}}{3} + \frac{2}{3} . \end{matrix}$ So, we have $\begin{matrix} M (h (Γ (p_{n}, z_{n})), Γ (h (p_{n}), h (z_{n})), t) = \\ e^{- \frac{| h (Γ (p_{n}, z_{n})) - Γ (h (p_{n}), h (z_{n}))}{t}} notto 1 \end{matrix}$ as n→ ∞. Therefore, (h, Γ) is not a compatible pair.

Consequently, Example 2.1 has a gap and requires correction. We now rectify Example 2.1 as follows:

Example 2.3. Let (Y, ⪯) be a partially ordered set with Y = [0, 1] = J and the natural ordering ≤ of the real numbers as the partial ordering ⪯. Let $M (p, z, t) = e^{- \frac{| p - z |}{t}}$ for all t > 0, p, z ∈ Y and e * f = min {e, f} for e, f ∈ J. Then (Y, M, *) is a complete fuzzy metric space with the H-type t-norm such that M (p, z, t) →1 as t→ ∞ for p, z ∈ Y. Define the mapping Γ : Y × Y → Y as $Γ (p, z) = {\begin{matrix} \frac{p^{2} - z^{2}}{12}, & p \geq z \\ 0, & p < z \end{matrix}$ for p, z ∈ Y and the mapping h : Y → Y as $h (p) = p^{2}$ for p ∈ Y. Then Γ (Y × Y) ⊂ h (Y) and Γ satisfies MhMP. The function h is continuous and monotone increasing. Define γ : J → J by $γ (c) = \sqrt{c}$ for all c ∈ J. Then γ (c) * γ (c) ≥ c for all c ∈ J. Let the sequences {p_n} and {z_n} in Y be such that $lim_{n \to \infty} h (p_{n}) = lim_{n \to \infty} Γ (p_{n}, z_{n}) = a \in Y$ and $lim_{n \to \infty} h (z_{n}) = lim_{n \to \infty} Γ (z_{n}, p_{n}) = b \in Y$ . Now, for all n ≥ 0, we have $h (p_{n}) = p_{n}^{2}, h (z_{n}) = z_{n}^{2},$ $\begin{matrix} Γ (p_{n}, z_{n}) = {\begin{matrix} \frac{p_{n}^{2} - z_{n}^{2}}{12}, & p_{n} \geq z_{n} \\ 0, & p_{n} < z_{n} \end{matrix}, \\ Γ (z_{n}, p_{n}) = {\begin{matrix} \frac{z_{n}^{2} - p_{n}^{2}}{12}, & z_{n} \geq p_{n} \\ 0, & z_{n} < p_{n} \end{matrix} . \end{matrix}$ Then, obviously, a = 0 and b = 0. It follows immediately that $M (h (Γ (p_{n}, z_{n})), Γ (h (p_{n}), h (z_{n})), t) \to 1,$ $M (h (Γ (z_{n}, p_{n})), Γ (h (z_{n}), h (p_{n})), t) \to 1$ as n→ ∞. Therefore, the pair (Γ, h) is compatible. Also, the following can be observed easily: $\sqrt{min {c, d}} = min {\sqrt{c}, \sqrt{d}}$ (2) for all c, d ∈ J. Take p₀ = 0 and z₀ = 0.1 in Y. Then h (p₀) ≤ Γ (p₀, z₀) and h (z₀) ≥ Γ (z₀, p₀). Let $k = \frac{2}{3}$ . Then, we verify the following inequality, which in fact, is the inequality (1.1) of Theorem 1.1: $\begin{matrix} M (Γ (p, z), Γ (u, w), kt) \geq γ (M (h (p), h (u), t) \\ * M (h (z), h (w), t)) \end{matrix}$ (3) for all p, z, u, w ∈ Y and t > 0 with h (p) ⪯ h (u) and h (z) ≽ h (w), that is, $p^{2} \leq u^{2}, z^{2} \geq w^{2} .$ (4) Consider the below cases:

Case 1: p ≥ z and u ≥ w.

We claim that (2.2) holds. If not, then there exists some t > 0 so that $\begin{matrix} M (Γ (p, z), Γ (u, w), kt) < γ (M (h (p), h (u), t) \\ * M (h (z), h (w), t)) . \end{matrix}$ So $\begin{matrix} M (Γ (p, z), Γ (u, w), kt) \\ < \sqrt{min {M (h (p), h (u), t), M (h (z), h (w), t)}} \\ = min {\sqrt{M (h (p), h (u), t)}, \sqrt{M (h (z), h (w), t)}}, \end{matrix}$ that is, $e^{- \frac{| \frac{p^{2} - z^{2}}{12} - \frac{u^{2} - w^{2}}{12} |}{kt}} < min {e^{- \frac{| p^{2} - u^{2} |}{2 t}}, e^{- \frac{| z^{2} - w^{2} |}{2 t}}},$ therefore $\begin{matrix} e^{- \frac{| \frac{p^{2} - z^{2}}{12} - \frac{u^{2} - w^{2}}{12} |}{kt}} < e^{- \frac{| p^{2} - u^{2} |}{2 t}}, \\ e^{- \frac{| \frac{p^{2} - z^{2}}{12} - \frac{u^{2} - w^{2}}{12} |}{kt}} < e^{- \frac{| z^{2} - w^{2} |}{2 t}} . \end{matrix}$ Thus, $\begin{matrix} \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{12 kt} > \frac{| p^{2} - u^{2} |}{2 t}, \\ \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{12 kt} > \frac{| z^{2} - w^{2} |}{2 t}, \end{matrix}$ therefore $\begin{matrix} \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{6 k} > | p^{2} - u^{2} |, \\ \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{6 k} > | z^{2} - w^{2} | . \end{matrix}$ Since, we have taken $k = \frac{2}{3}$ , the last two inequalities become $\begin{matrix} \frac{1}{4} | (p^{2} - z^{2}) - (u^{2} - w^{2}) | > | p^{2} - u^{2} |, \\ \frac{1}{4} | (p^{2} - z^{2}) - (u^{2} - w^{2}) | > | z^{2} - w^{2} | . \end{matrix}$ Adding the above two last inequalities, we get $\frac{1}{2} | (p^{2} - z^{2}) - (u^{2} - w^{2}) | > | p^{2} - u^{2} | + | z^{2} - w^{2} |,$ that is, $\frac{1}{2} | (p^{2} - u^{2}) - (z^{2} - w^{2}) | > | p^{2} - u^{2} | + | z^{2} - w^{2} |,$ which is impossible. Thus (2.2) holds in this case.

Case 2: p ≥ z and u < w.

Since p ≥ z and u < w, $p^{2} \geq z^{2}, u^{2} < w^{2},$ (5) but by (2.3), we have $p^{2} \leq u^{2}, z^{2} \geq w^{2} .$ (6) Combining (2.4) and (2.5), we have p² ≥ z² ≥ w² > u² ≥ p², which is impossible.

Case 3: p < z and u ≥ w.

Then we have p² < z² and u² ≥ w².

We claim that (2.2) holds. If not, there exists t > 0 such that $\begin{matrix} M (Γ (p, z), Γ (u, w), kt) < γ (M (h (p), h (u), t) \\ * M (h (z), h (w), t)) . \end{matrix}$ So $\begin{matrix} M (Γ (p, z), Γ (u, w), kt) \\ < \sqrt{min {M (h (p), h (u), t), M (h (z), h (w), t)}} \\ = min {\sqrt{M (h (p), h (u), t)}, \sqrt{M (h (z), h (w), t)}}, \end{matrix}$ that is, $e^{- \frac{| \frac{u^{2} - w^{2}}{12} |}{kt}} < min {e^{- \frac{| p^{2} - u^{2} |}{2 t}}, e^{- \frac{| z^{2} - w^{2} |}{2 t}}},$ and so $e^{- \frac{| \frac{u^{2} - w^{2}}{12} |}{kt}} < e^{- \frac{| p^{2} - u^{2} |}{2 t}}, e^{- \frac{| \frac{u^{2} - w^{2}}{12} |}{kt}} < e^{- \frac{| z^{2} - w^{2} |}{2 t}} .$ Thus, $\frac{| u^{2} - w^{2} |}{12 kt} > \frac{| p^{2} - u^{2} |}{2 t}, \frac{| u^{2} - w^{2} |}{12 kt} > \frac{| z^{2} - w^{2} |}{2 t},$ therefore $\frac{| u^{2} - w^{2} |}{6 k} > | p^{2} - u^{2} |, \frac{| u^{2} - w^{2} |}{6 k} > | z^{2} - w^{2} | .$ Since we have taken $k = \frac{2}{3}$ , the last two inequalities become $\frac{1}{4} | u^{2} - w^{2} | > | p^{2} - u^{2} |, \frac{1}{4} | u^{2} - w^{2} | > | z^{2} - w^{2} | .$ Adding the above two last inequalities, we get $\frac{1}{2} | u^{2} - w^{2} | > | p^{2} - u^{2} | + | z^{2} - w^{2} |,$ that is, $\begin{matrix} | p^{2} - u^{2} | + | z^{2} - w^{2} | \\ < \frac{1}{2} | u^{2} - w^{2} | = \frac{1}{2} (u^{2} - w^{2}) \\ = \frac{1}{2} (u^{2} - p^{2} + p^{2} - w^{2}) \\ < \frac{1}{2} (u^{2} - p^{2}) + \frac{1}{2} (z^{2} - w^{2}) \\ = \frac{1}{2} (| p^{2} - u^{2} | + | z^{2} - w^{2} |), \end{matrix}$ which is impossible. Hence (2.2) holds.

Case 4: p < z and u < w.

Then Γ (p, z) =0 and Γ (u, w) =0, that is, $M (Γ (p, z), Γ (u, w), kt) = e^{- \frac{| 0 - 0 |}{t}} = 1$ and so (2.2) holds.

Thus all the conditions of Theorem 1.1 hold. Applying Theorem 1.1, we obtain that (0, 0) is the CCP of the mappings Γ and h.

3 Main results and answer to the open problem

In this section, we give our main results and address Problem 1.2 proffered by Choudhury et al. [3].

Theorem 3.1. Let (Y, M, *) be a GV-fuzzy metric space having H-type t-norm with M (p, z, t) →1 as t→ ∞ for p, z ∈ Y. Let ⪯ be a partial order defined on Y. Let Γ : Y × Y → Y and h : Y → Y be the mappings such that Γ has MhMP with the following conditions:

Γ (Y × Y) ⊂ h (Y);

h (Y) is complete;

there exists φ ∈ Φ_w* such that $\begin{matrix} M (Γ (p, z), Γ (u, w), φ (t)) \geq γ (M (h (p), \\ h (u), t) * M (h (z), h (w), t) \end{matrix}$ (7) for p, z, u, w ∈ Y and t > 0 with h (p) ⪯ h (u) and h (z) ≽ h (w), where γ : J → J is a function with γ (c) * γ (c) ≥ c for all c ∈ J.

Let Y satisfies (H₁) and (H₂). If there exist p₀, z₀ ∈ Y so that h (p₀) ⪯ Γ (p₀, z₀) and h (z₀) ≽ Γ (z₀, p₀), then h and Γ have a CCP in Y.

Proof. Let p₀, z₀ ∈ Y so that h (p₀) ⪯ Γ (p₀, z₀) and h (z₀) ≽ Γ (z₀, p₀). Using the condition (i), we choose p₁, z₁ ∈ Y so that h (p₁) = Γ (p₀, z₀) and h (z₁) = Γ (z₀, p₀). Then, we have h (p₀) ⪯ h (p₁) and h (z₀) ≽ h (z₁). Again, by the condition (i), we choose p₂, z₂ ∈ Y so that h (p₂) = Γ (p₁, z₁) and h (z₂) = Γ (z₁, p₁). Continuing likewise, we construct sequences {p_n} and {z_n} in Y so that h (p_n+1) = Γ (p_n, z_n) and h (z_n+1) = Γ (z_n, p_n) for all n ≥ 0. By repeatedly using (i) and MhMP of Γ, the following holds inductively: $h (p_{n}) ⪯ h (p_{n + 1}), h (z_{n}) ≽ h (z_{n + 1})$ (8) for all n ≥ 0. If for some $n \in ℕ$ , we have h (p_n-1) = h (p_n) and h (z_n-1) = h (z_n), that is, h (p_n-1) = Γ (p_n-1, z_n-1) , h (z_n-1) = Γ (z_n-1, p_n-1), then (p_n-1, z_n-1) is a CCP of the mappings Γ and h.

Suppose h (p_n-1) ≠ h (p_n) or h (z_n-1) ≠ h (z_n) for all $n \in ℕ$ .

Since * is an H-type t-norm, for any η > 0, there exists ζ > 0 such that $*^{q} (1 - ζ) = \frac{(1 - ζ) * (1 - ζ) * \dots * (1 - ζ)}{q} > 1 - η$ (9) for all $q \in ℕ$ .

Using (3.1) - (3.2), for t > 0, we acquire $\begin{matrix} M (h (p_{n + 1}), h (p_{n}), φ (t)) \\ = M (Γ (p_{n}, z_{n}), Γ (p_{n - 1}, z_{n - 1}), φ (t)) \\ \geq γ (M (h (p_{n - 1}), h (p_{n}), t) * M (h (z_{n - 1}), h (z_{n}), t)) . \end{matrix}$ (10)

Similarly, for t > 0, we acquire $\begin{matrix} M (h (z_{n + 1}), h (z_{n}), φ (t)) \geq γ (M (h (p_{n - 1}), h (p_{n}), t) \\ * M (h (z_{n - 1}), h (z_{n}), t)) . \end{matrix}$ (11)

Let ⋀_n (t) = M (h (p_n-1) , h (p_n) , t) * M (h (z_n-1) , h (z_n) , t) .

Now, using (3.4) - (3.5) and the fact that γ (c) * γ (c) ≥ c for all c ∈ J, we obtain for t > 0, that $\begin{matrix} M (h (p_{n + 1}), h (p_{n}), φ (t)) * M (h (z_{n + 1}), h (z_{n}), φ (t)) \\ \geq γ (M (h (p_{n - 1}), h (p_{n}), t) * M (h (z_{n - 1}), h (z_{n}), t)) \\ * γ (M (h (p_{n - 1}), h (p_{n}), t) * M (h (z_{n - 1}), h (z_{n}), t)) \\ \geq M (h (p_{n - 1}), h (p_{n}), t) * M (h (z_{n - 1}), h (z_{n}), t), \end{matrix}$ that is, ⋀_n+1 (φ (t)) ≥ ⋀ _n (t) for t > 0. Now, again applying (3.1), for t > 0 and n ≥ 1, we obtain $\begin{matrix} M (h (p_{n + 1}), h (p_{n}), φ^{n} (t)) \\ \geq γ (M (h (p_{n - 1}), h (p_{n}), φ^{n - 1} (t)) \\ * M (h (z_{n - 1}), h (z_{n}), φ^{n - 1} (t))), \end{matrix}$ (12) $\begin{matrix} M (h (z_{n + 1}), h (z_{n}), φ^{n} (t)) \\ \geq γ (M (h (p_{n - 1}), h (p_{n}), φ^{n - 1} (t)) \\ * M (h (z_{n - 1}), h (z_{n}), φ^{n - 1} (t))) . \end{matrix}$ (13) Combining (3.6) and (3.7), for t > 0, we obtain that $\begin{matrix} M (h (p_{n + 1}), h (p_{n}), φ^{n} (t)) * M (h (z_{n + 1}), h (z_{n}), φ^{n} (t)) \\ \geq γ (M (h (p_{n - 1}), h (p_{n}), φ^{n - 1} (t)) \\ * M (h (z_{n - 1}), h (z_{n}), φ^{n - 1} (t))) \\ * γ (M (h (p_{n - 1}), h (p_{n}), φ^{n - 1} (t)) \\ * M (h (z_{n - 1}), h (z_{n}), φ^{n - 1} (t))) \\ \geq M (h (p_{n - 1}), h (p_{n}), φ^{n - 1} (t)) \\ * M (h (z_{n - 1}), h (z_{n}), φ^{n - 1} (t)), \end{matrix}$ that is, ⋀_n+1 (φⁿ (t)) ≥ ⋀ _n (φ^n-1 (t)) for t > 0 and n ≥ 1.

Hence, for all t > 0 and n ≥ 1, we obtain that $\begin{array}{l} Λ_{n + 1} (ϕ^{n} (t)) \geq Λ_{n} (ϕ^{n - 1} (t)) \geq \underset{n - 1}{Λ} (ϕ^{n - 2} (t)) \\ \geq \dots \geq M (h (p_{0}), h (p_{1}), t) * M (h (z_{0}), h (z_{1}), t) \\ = Λ 1^{(t)} \end{array}$ (14)

Next, we prove that $lim_{n \to \infty} M (h (p_{n}), h (p_{n + 1}), t) = 1$ and $lim_{n \to \infty} M (h (z_{n}), h (z_{n - 1}), t) = 1$ for t > 0.

Since $lim_{t \to \infty} M (p, z, t) = 1$ for p, z ∈ Y, for ζ > 0, there exists t₀ > 0 so that $\begin{matrix} M (h (p_{0}), h (p_{1}), t_{0}) > 1 - ζ, \\ M (h (z_{0}), h (z_{1}), t_{0}) > 1 - ζ . \end{matrix}$ Also, since φ ∈ Φ_w*, for t > 0, there exists t₁ ≥ max {t, t₀} and $n_{0} \in ℕ$ so that φⁿ (t₁) < min {t, t₀} for n > n₀. By the monotonicity of M (x, z, ·) and using (3.8), for t > 0, we acquire $\begin{matrix} M (h (p_{n}), h (p_{n + 1}), t) * M (h (z_{n}), h (z_{n + 1}), t) nonumber \\ nonumber \geq M (h (p_{n}), h (p_{n + 1}), φ^{n} (t_{1})) \\ * M (h (z_{n}), h (z_{n+1}), φ^{n} (t_{1})) = ⋀ n + 1 (φ^{n} (t_{1})) \\ nonumber \geq M (h (p_{0}), h (p_{1}), t_{1}) * M (h (z_{0}), h (z_{1}), t_{1}) \\ nonumber \geq M (h (p_{0}), h (p_{1}), t_{0}) * M (h (z_{0}), h (z_{1}), t_{0}) \\ > (1 - ζ) * (1 - ζ) > 1 - η . \end{matrix}$ (15) From this, we conlude that $lim_{n \to \infty} M (h (p_{n}), h (p_{n + 1}), t) * M (h (z_{n}), h (z_{n + 1}), t) = 1$ (16) for t > 0. Since φ ∈ Φ_w*, using Lemma 1.3, for t > 0, there exists ρ ≥ t so that φ (ρ) < t.

We claim that the sequences {h (p_n)} and {h (z_n)} are Cauchy sequences.

Let n ≥ 1 be given. Then for any k ≥ 1, first of all, we prove the following inequality inductively: $\begin{matrix} M (h (p_{n}), h (p_{n + k}), t) * M (h (z_{n}), h (z_{n + k}), t) nonumber \\ nonumber \geq *^{k - 1} (M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) \\ * M (h (z_{n}), h (z_{n + 1}), t - φ (ρ))) \\ = *^{k-1} ⋀ n + 1 (t - φ (ρ)) \end{matrix}$ (17) for all k ≥ 1.

For k = 1, $\begin{matrix} M (h (p_{n}), h (p_{n + 1}), t) * M (h (z_{n}), h (z_{n + 1}), t) \\ \geq M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) \\ * M (h (z_{n}), h (z_{n+1}), t - φ (ρ)) \\ = ⋀_{n + 1} (t - φ (ρ)) = *^{0} ⋀ n + 1 (t - φ (ρ)) . \end{matrix}$ So (3.11) holds for k = 1.

Let (3.11) holds for some k = q ≥ 1. We shall show the result for k = q + 1. Using (2.1) for h (p_n) ⪯ h (p_n+q) and h (z_n) ≽ h (z_n+q) and using γ (c) * γ (c) ≥ c for c ∈ J and ρ ≥ t, we get $\begin{array}{l} M (h (p_{n}), h (p_{n + q + 1}), t) * M (h (z_{n}), h (z_{n + q + 1}), t) \\ = M (h (p_{n}), h (p_{n + q + 1}), t - φ (ρ) + φ (ρ)) \\ * M (h (z_{n}), h (z_{n + q + 1}), t - φ (ρ) + φ (ρ)) \\ \geq (M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) \\ * M (h (p_{n + 1}), h (p_{n + q + 1}), φ (ρ))) \\ * (M (h (z_{n}), h (z_{n + 1}), t - φ (ρ)) \\ * M (h (z_{n + 1}), h (z_{n + q + 1}), φ (ρ))) \\ = M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) \\ * M (h (z_{n}), h (z_{n + 1}), t - φ (ρ)) \\ * M (h (p_{n + 1}), h (p_{n + q + 1}), φ (ρ)) \\ * M (h (z_{n + 1}), h (z_{n+q+1}), φ (ρ)) \\ = ⋀ n + 1 (t - φ (ρ)) * M (Γ (p_{n}, z_{n}), \\ Γ (p_{n + q}, z_{n + q}), φ (ρ)) \\ * M (Γ (z_{n}, p_{n}), Γ (z_{n + q}, p_{n+q}), φ (ρ)) \\ \geq ⋀ n + 1 (t - φ (ρ)) * γ (M (h (p_{n}), h (p_{n + q}), ρ) \\ * M (h (z_{n}), h (z_{n + q}), ρ)) * γ (M (h (p_{n}), h (p_{n + q}), ρ) \\ * M (h (z_{n}), h (z_{n+q}), ρ)) \\ \geq ⋀ n + 1 (t - φ (ρ)) * (M (h (p_{n}), h (p_{n + q}), ρ) \\ * M (h (z_{n}), h (z_{n+q}), ρ)) \\ \geq ⋀ n + 1 (t - φ (ρ)) * (M (h (p_{n}), h (p_{n + q}), t) \end{array} * M (h (z_{n}), h (z_{n+q}), t)) \geq ⋀ n + 1 (t - φ (ρ)) * (*^{q-1} ⋀_{n+1} (t - φ (ρ))) = *^{q-1} ⋀_{n+1} (t - φ (ρ)) .$

Thus (3.11) holds for k = q + 1 and so by induction (3.11) holds for k ≥ 1.

Since * is an H-type t-norm, for ζ ∈ (0, 1), there exists λ ∈ (0, 1) so that for s > 1 - λ, we have $*^{n} (s) > 1 - ζ$ (18) for all n ≥ 1. On the other hand, by (3.10), we have $\begin{matrix} lim_{n \to \infty} (M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) \\ * M (h (z_{n}), h (z_{n + 1}), t - φ (ρ))) = 1, \end{matrix}$ so there exists $N_{1} (ζ, t) \in ℕ$ such that $\begin{matrix} M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) * M (h (z_{n}), h (z_{n + 1}), \\ t - φ (ρ)) > 1 - λ, \end{matrix}$ for n > N₁ (ζ, t). Then, by (3.11)–(3.12), we get $\begin{matrix} M (h (p_{n}), h (p_{n + k}), t) * M (h (z_{n}), h (z_{n + k}), t) \\ \geq *^{k - 1} (M (h (p_{n}), h (p_{n + 1}), t - φ (ρ)) \\ * M (h (z_{n}), h (z_{n + 1}), t - φ (ρ))) > 1 - ζ, \end{matrix}$ that is, $\begin{matrix} M (h (p_{n}), h (p_{n + k}), t) * M (h (z_{n}), h (z_{n + k}), t) \\ > 1 - ζ > 1 - η \end{matrix}$ for all k ≥ 1 and n > N₁ (ζ, t). This implies that the sequences {h (p_n)} and {h (z_n)} are Cauchy in h (Y). Then, by the completeness of h (Y), there exist p, z ∈ Y so that $lim_{n \to \infty} h (p_{n}) = h (p), lim_{n \to \infty} h (z_{n}) = h (z) .$ (19) Since {h (p_n)} is a non-decreasing sequence converging to h (p) and {h (z_n)} is a non-increasing sequence converging to h (z), by (H₁) and (H₂), we get h (p_n) ⪯ h (p) and h (z_n) ≽ h (z) for n ≥ 0.

Since φ ∈ Φ_w*, using Lemma 1.3, for t > 0, there exists ρ ≥ t so that φ (ρ) < t. Since h (p_n) ⪯ h (p) and h (z_n) ≽ h (z) for all n ≥ 0, by (3.1), we have $\begin{matrix} M (h (p_{n + 1}), Γ (p, z), t) \geq M (h (p_{n + 1}), Γ (p, z), φ (ρ)) \\ = M (Γ (p_{n}, z_{n}), Γ (p, z), φ (ρ)) nonumber \\ \geq γ (M (h (p_{n}), h (p), ρ) * M (h (z_{n}), h (z), ρ)) . \end{matrix}$ (20) Similarly, we get $\begin{matrix} M (g (z_{n + 1}), Γ (z, p), t) \geq γ (M (h (p_{n}), h (p), ρ) \\ * M (h (z_{n}), h (z), ρ)) . \end{matrix}$ (21) Combining (3.14) and (3.15), we have $\begin{matrix} M (h (p_{n + 1}), Γ (p, z), t) * M (h (z_{n + 1}, Γ (z, p), t) \\ \geq nonumber γ (M (h (p_{n}), h (p), ρ) * M (h (z_{n}), h (z), ρ)) \\ * γ (M (h (p_{n}), h (p), ρ) * M (h (z_{n}), h (z), ρ)) \\ \geq nonumber M (h (p_{n}), h (p), ρ) * M (h (z_{n}), h (z), ρ) . \end{matrix}$ (22) Letting n→ ∞ in (3.16) and using (3.13), we have $M (h (p), Γ (p, z), t) * M (h (z), Γ (z, p), t) \geq 1 * 1 = 1,$ for t > 0. Hence, it follows that Γ (p, z) = h (p) and Γ (z, p) = h (z). This completes the proof.□

The following example supports Theorem 3.1:

Example 3.2. Let (Y, ⪯) be the partially ordered set with Y = (-1, 1] having natural ordering ≤ of the real numbers as partial ordering ⪯. Define a mapping M on $Y^{2} \times ℝ^{+}$ by $M (p, z, t) = 1 - \frac{| p - z |}{t + 2}$ , for all p, z ∈ Y, t > 0 and consider e * f = max {e + f - 1, 0} for e, f ∈ J, then (Y, M, *) is a GV-fuzzy metric space (which is not complete) but not a KM-fuzzy metric space such that M (p, z, t) →1 as t→ ∞, for all p, z ∈ Y and * being a H-type t-norm. Let us define the mappings h : Y → Y and Γ : Y × Y → Y respectively by h (p) = p² and $Γ (p, z) = \frac{p^{2} - z^{2} + 4}{6}$ for p, z ∈ Y. Interestingly, the pair (h, Γ) of mappings is not compatible. The mapping h is neither monotonically increasing nor monotonically decreasing on Y. Also, the mapping Γ has MhMP. Clearly h (Y) = [0, 1] is complete and Γ (Y × Y) ⊆ h (Y). Also, for p₀ = 0.80 and z₀ = 0.85, we have h (p₀) < Γ (p₀, z₀) and h (z₀) > Γ (z₀, p₀).

We define the functions $φ : ℝ^{+} \to ℝ^{+}$ by $φ (t) = {\begin{matrix} t, & 0 \leq t \leq 1 \\ t - 1, & t > 1 \end{matrix}$ so that, φ ∈ Φ_{w
^*} and φ ∉ Φ_w. Also, define γ : J → J by $γ (c) = \frac{c + 1}{2}$ for each c ∈ J, so that γ (c) * γ (c) ≥ c for c ∈ J.

We next verify the inequality (3.1) for all t > 0 and p, z, u, w ∈ Y with h (p) ⪯ h (u) and h (z) ≽ h (w). We discuss the following cases:

Case I. For 0 ≤ t ≤ 1, φ (t) = t, we have

M (Γ (p, z) , Γ (u, w) , φ (t)) ≥ γ (M (h (p) , h (u) , t) * M (h (z) , h (w) , t))

iff $M (\frac{p^{2} - z^{2} + 4}{6}, \frac{u^{2} - w^{2} + 4}{6}, t) \geq \frac{M (p^{2}, u^{2}, t) * M (z^{2}, w^{2}, t) + 1}{2}$

iff $1 - \frac{| (\frac{p^{2} - z^{2} + 4}{6}) - (\frac{u^{2} - w^{2} + 4}{6}) |}{t + 2} \geq \frac{1}{2} {max (M (p^{2}, u^{2}, t) + M (z^{2}, w^{2}, t) - 1, 0) + 1}$

iff $1 - \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{6 (t + 2)} \geq \frac{1}{2} {max (1 - \frac{| p^{2} - u^{2} |}{t + 2} + 1 - \frac{| z^{2} - w^{2} |}{t + 2} - 1, 0) + 1}$

iff $1 - \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{6 (t + 2)} \geq \frac{1}{2} {2 - \frac{| p^{2} - u^{2} |}{t + 2} - \frac{| z^{2} - w^{2} |}{t + 2}}$

iff $1 - \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{6 (t + 2)} \geq 1 - \frac{| p^{2} - u^{2} | + | z^{2} - w^{2} |}{2 (t + 2)}$

iff | (p² - z²) - (u² - w²) |≤3 (|p² - u²| + |z² - w²|), which is true.

Case II. For t > 1, φ (t) = t - 1, then, we have

M (Γ (p, z) , Γ (u, w) , φ (t)) ≥ γ (M (h (p) , h (u) , t) * M (h (z) , h (w) , t))

iff $1 - \frac{| (p^{2} - z^{2}) - (u^{2} - w^{2}) |}{6 (t + 1)} \geq 1 - \frac{| p^{2} - u^{2} | + | z^{2} - w^{2} |}{2 (t + 2)}$ (as in Case I)

iff $| (p^{2} - z^{2}) - (u^{2} - w^{2}) | \leq \frac{3 (t + 1)}{t + 2} (| p^{2} - u^{2} | + | z^{2} - w^{2} |)$ ,

which is true, since 1 < t, so t + 2 <2t + 2, consequently $1 < \frac{2 t + 2}{t + 2} < \frac{3 (t + 1)}{t + 2}$ and hence, the above inequality holds true.

Therefore, inequality (3.1) holds for all t > 0 and p, z, u, w ∈ Y with h (p) ⪯ h (u) and h (z) ≽ h (w).

Thus, all the conditions of Theorem 3.1 are satisfied, using Theorem 3.1 we can obtain that Γ and h have a CCP in Y × Y, which indeed is $(\sqrt{\frac{2}{3}}, \sqrt{\frac{2}{3}})$ .

Corollary 3.3. Let (Y, M, *) be a GV-fuzzy metric space having H-type t-norm with M (p, z, t) →1 as t→ ∞ for p, z ∈ Y. Let ⪯ be a partial order defined on Y. Let Γ : Y × Y → Y and h : Y → Y be the mappings such that Γ has MhMP with the following conditions:

Γ (Y × Y) ⊂ h (Y);

h (Y) is complete;

there exists k ∈ (0, 1) such that $\begin{matrix} M (Γ (p, z), Γ (u, w), kt) \geq γ (M (h (p), h (u), t) \\ * M (h (z), h (w), t)) \end{matrix}$ (23) for p, z, u, w ∈ Y and t > 0 with h (p) ⪯ h (u) and h (z) ≽ h (w), where γ : J → J is a function with γ (c) * γ (c) ≥ c for all c ∈ J.

Let Y satisfies (H₁) and (H₂). If there exist p₀, z₀ ∈ Y so that h (p₀) ⪯ Γ (p₀, z₀) and h (z₀) ≽ Γ (z₀, p₀), then h and Γ have a CCP in Y.

Proof. Considering φ (t) = kt where k ∈ (0, 1), in Theorem 3.1, we attain the result.□

Remark 3.4. (i) In our results, we observed the following:

The compatible hypothesis of the pair (Γ, h) has been relaxed;

The mapping h is neither continuous nor monotone;

The function γ is not assumed to be continuous;

Finally, the completeness hypothesis of entire space Y has been replaced by completeness assumption of the range subspace of mapping h.

(ii) Interestingly, the proof of Theorem 3.1 uses the fact that M (p, z, t) →1 as t→ ∞ for p, z ∈ Y only to imply that for ζ > 0 there exists t₀ > 0 such that $\begin{matrix} M (h (p_{0}), h (p_{1}), t_{0}) > 1 - ζ, \\ M (h (z_{0}), h (z_{1}), t_{0}) > 1 - ζ, \end{matrix}$ that is, $M (h (p_{0}), h (p_{1}), t) \to 1, M (h (z_{0}), h (z_{1}), t) \to 1$ as t→ ∞.

Therefore, in Theorem 3.1 (and hence, in Corollary 3.3), we can replace the condition: $M (p, z, t) \to 1 as t \to \infty for p, z \in Y$ by a weaker condition that $\begin{matrix} M (h (p_{0}), h (p_{1}), t) \to 1 and \\ M (h (z_{0}), h (z_{1}), t) \to 1 as t \to \infty . \end{matrix}$

Hence, the theoretic results (particularly, Corollary 3.3) presented by us improve and generalize Theorem 1.1.

Corollary 3.5. Let (Y, M, *) be a complete GV-fuzzy metric space having H-type t-norm with M (p, z, t) →1 as t→ ∞ for all p, z ∈ Y. Let ⪯ be a partial order defined on Y. Let Γ : Y × Y → Y be a mapping satisfying MMP. Also, suppose that there exists $φ \in Φ_{w}^{*}$ such that $M (Γ (p, z), Γ (u, w), φ (t)) \geq γ (M (p, u, t) * M (z, w, t)),$ (24) for all t > 0 and p, z, u, w ∈ Y with p ⪯ u and z ≽ w, and the function γ : J → J is a function with γ (c) * γ (c) ≥ c for all c ∈ J. Let Y satisfies (H₁) and (H₂). If there exist p₀, z₀ ∈ Y so that p₀ ⪯ Γ (p₀, z₀) and z₀ ≽ Γ (z₀, p₀), then Γ has a CFP in Y.

Proof. Considering h = I_Y (identity mapping on Y) in Theorem 3.1, we can obtain the result.□

4 Application in dynamic programming

As an application of our results, in this section, we provide the solution of the following type of system of functional equations that appear in dynamic programming problems: $\begin{matrix} p (ζ) = sup_{ϑ \in ᗡ} {Λ (ζ, ϑ) + L (ζ, ϑ, p (T (ζ, ϑ)), z (T (ζ, ϑ)))}, \\ z (ζ) = sup_{ϑ \in ᗡ} {Λ (ζ, ϑ) + L (ζ, ϑ, z (T (ζ, ϑ)), p (T (ζ, ϑ)))}, \end{matrix}}$ (25) where ζ ∈ ₰ and is a state space, ᗡ is a decision space, T : ₰ × ᗡ → ₰, $Λ : ₰ \times ᗡ \to ℝ$ and $L : ₰ \times ᗡ \times ℝ \times ℝ \to ℝ$ are given mappings.

Let ℘ (₰) denote the set of all bounded real-valued functions on a non-empty set ₰, and for any p ∈ ℘ (₰), define $∥ p ∥ = sup_{ζ \in ₰} | p (ζ) | .$ (26) Then, ℘ (₰) endowed with the following supremum metric is a complete metric space: $ϱ (p, z) = sup_{ζ \in ₰} | p (ζ) - z (ζ) |,$ (27) for all p, z ∈ ℘ (₰).

Further, ℘ (₰) has been equipped with the partial order ≼_℘ defined by:

p ≼ _℘z iff p (ζ) ≤ z (ζ) for all ζ ∈ ₰ (where, ≤ being the usual ordering of real number).

It can be easily seen that ℘ (₰) satisfy properties (H₁) and (H₂) in Corollary 3.5.

We denote by $H : R^{+} \to R^{+}$ the function satisfying followings:

$H$ is non-decreasing;

$lim_{n \to \infty} H^{n} (ℓ) = 0$ , for all ℓ≥0.

Clearly, if

H

satisfies (F₁) and (F₂), we have

H (ℓ) < ℓ

for

ℓ \in ℝ^{+}

Theorem 4.1. Consider the following hypotheses:

$Λ : ₰ \times ᗡ \to ℝ$ and $L : ₰ \times ᗡ \times ℝ \times ℝ \to ℝ$ are bounded functions;

for arbitrary points ζ ∈ ₰, ϑ ∈ ᗡ and $ℓ, s, ℓ_{1}, s_{1} \in ℝ$ such that $\begin{matrix} | L (ζ, ϑ, ℓ, s) - L (ζ, ϑ, ℓ_{1}, s_{1}) | \leq H \\ (\frac{1}{4} max {| ℓ - ℓ_{1} |, | s - s_{1} |}), \end{matrix}$

with L (ζ, ϑ, · , s) is non-decreasing and L (ζ, ϑ, ℓ , ·) is non-increasing;

there exist bounded functions $p_{0}, z_{0} : ₰ \to ℝ$ such that for each ζ ∈ ₰, we have $\begin{matrix} p_{0} (ζ) \leq sup_{ϑ \in ᗡ} {Λ (ζ, ϑ) + L (ζ, ϑ, p_{0} (T (ζ, ϑ)), \\ z_{0} (T (ζ, ϑ)))}, \\ z_{0} (ζ) \geq sup_{ϑ \in ᗡ} {Λ (ζ, ϑ) + L (ζ, ϑ, z_{0} (T (ζ, ϑ)), \\ p_{0} (T (ζ, ϑ)))} . \end{matrix}$

Then, problem (4.1) has a solution.

Proof. Define the operator Γ as $\begin{matrix} Γ (p, z) (ζ, ϑ) = sup_{ϑ \in ᗡ} {Λ (ζ, ϑ) + L (ζ, ϑ, p (T (ζ, ϑ)), \\ z (T (ζ, ϑ)))}, \end{matrix}$ (28) for all (p, z) ∈ ℘ (₰) × ℘ (₰) and ζ ∈ ₰. Since functions Λ and L are bounded, then Γ is well-defined. Further, since L (ζ, ϑ, · , s) is non-decreasing and L (ζ, ϑ, ℓ , ·) is non-increasing, so the function Γ satisfies MMP.

Define $M (p, z, t) = e^{- \frac{ϱ (p, z)}{t}} for p, z \in ℘ (₰) and t > 0 .$ (29) Then, (℘ (₰) , M, *) is a GV-fuzzy metric space which is complete.

Next, we verify the inequality (3.18) appearing in Corollary 3.5, for $φ (t) = \frac{t}{2}$ and $γ (a) = \sqrt{a}$ .

Consider (p, z) , (p₁, z₁) ∈ ℘ (₰) × ℘ (₰). Then, making use of (ii), it can be seen that $\begin{matrix} ϱ (Γ (p, z), Γ (p_{1}, z_{1})) \\ = sup_{ζ \in ₰} | Γ (p, z) (ζ, ϑ) - Γ (p_{1}, z_{1}) (ζ, ϑ) | \\ = sup_{ζ \in ₰} | sup_{ϑ \in ᗡ} {Λ (ζ, ϑ) + L (ζ, ϑ, p (T (ζ, ϑ)), \\ z (T (ζ, ϑ)))} \\ - sup_{ϑ \in ₰} {Λ (ζ, ϑ) + L (ζ, ϑ, p_{1} (T (ζ, ϑ)), \\ z_{1} (T (ζ, ϑ)))} | \\ = sup_{ζ \in ₰} | sup_{ϑ \in ᗡ} {L (ζ, ϑ, p (T (ζ, ϑ)), z (T (ζ, ϑ))) \\ - L (ζ, ϑ, p_{1} (T (ζ, ϑ)), z_{1} (T (ζ, ϑ)))} | \\ \leq sup_{ζ \in ₰} {sup_{ϑ \in ᗡ} | L (ζ, ϑ, p (T (ζ, ϑ)), z (T (ζ, ϑ))) \\ - L (ζ, ϑ, p_{1} (T (ζ, ϑ)), z_{1} (T (ζ, ϑ))) |} \\ \leq sup_{ζ \in ₰} {sup_{ϑ \in ᗡ} H (\frac{1}{4} max {| p (T (ζ, ϑ)) - p_{1} (T (ζ, ϑ)) |, \\ | z (T (ζ, ϑ)) - z_{1} (T (ζ, ϑ)) |})} \\ \leq sup_{ζ \in ₰} {sup_{ϑ \in ᗡ} (\frac{1}{4} max {| p (T (ζ, ϑ)) - p_{1} (T (ζ, ϑ)) |, \\ | z (T (ζ, ϑ)) - z_{1} (T (ζ, ϑ)) |})} \\ (since H (ℓ) < ℓ for ℓ \in ℝ^{+}) \\ \leq sup_{ζ \in ₰} (\frac{1}{4} max {∥ p (T (ζ, ϑ)) - p_{1} (T (ζ, ϑ)) ∥, \\ ∥ z (T (ζ, ϑ)) - z_{1} (T (ζ, ϑ)) ∥}) \\ \leq \frac{1}{4} max {ϱ (p, p_{1}), ϱ (z, z_{1})}, \end{matrix}$ so that, we have $\begin{matrix} \begin{matrix} - ϱ (Γ (p, z), Γ (p_{1}, z_{1})) \geq - \frac{1}{4} max {ϱ (p, p_{1}), ϱ (z, z_{1})} \\ = \frac{1}{4} min {- ϱ (p, p_{1}), - ϱ (z, z_{1})}, \end{matrix} \end{matrix}$ or $\begin{matrix} \begin{matrix} - 2 ϱ (Γ (p, z), Γ (p_{1}, z_{1})) \geq \\ \frac{1}{2} min {- ϱ (p, p_{1}), - ϱ (z, z_{1})}, \end{matrix} \end{matrix}$ that is, $\begin{matrix} \begin{matrix} \frac{- 2 ϱ (Γ (p, z), Γ (p_{1}, z_{1}))}{t} \geq \\ \frac{min {- ϱ (p, p_{1}), - ϱ (z, z_{1})}}{2 t} = \\ min {\frac{- ϱ (p, p_{1})}{2 t}, \frac{- ϱ (z, z_{1})}{2 t}}, \end{matrix} \end{matrix}$ therefore, we have $\begin{matrix} e^{\frac{- 2 ϱ (Γ (p, z), Γ (p_{1}, z_{1}))}{t}} & \geq e^{min {- \frac{ϱ (p, p_{1})}{2 t}, - \frac{ϱ (z, z_{1})}{2 t}}} \\ = {(min {e^{- \frac{ϱ (p, p_{1})}{t}}, e^{- \frac{ϱ (z, z_{1})}{t}}})}^{\frac{1}{2}}, \end{matrix}$ or $\begin{matrix} \begin{matrix} M (Γ (p, z), Γ (p_{1}, z_{1}), \frac{t}{2}) \\ \geq (min {M (p, p_{1}, t), M (z, z_{1}, t)})^{\frac{1}{2}}, \end{matrix} \end{matrix}$ or $\begin{matrix} \begin{matrix} M (Γ (p, z), Γ (p_{1}, z_{1}), \frac{t}{2}) \\ \geq (M (p, p_{1}, t) * M (z, z_{1}, t))^{\frac{1}{2}}, \end{matrix} \end{matrix}$ or $\begin{matrix} M (Γ (p, z), Γ (p_{1}, z_{1}), φ (t)) \geq γ (M (p, p_{1}, t) \\ * M (z, z_{1}, t)) . \end{matrix}$ (30) which is inequality (3.18) of Corollary 3.5. Using Corollary 3.5, we can obtain the existence of a CFP of the mapping Γ, so that the system (4.1) has a solution in ℘ (₰) × ℘ (₰).□

Remark 4.2. To illustrate Theorem 4.1, we consider the following system of functional equations which appear in dynamic programming with the state space ₰ = J = [0, 1] and the decision space $ᗡ = ℝ$ : $\begin{matrix} p (ζ) = sup_{ϑ \in D} {e^{- (ζ + | ϑ |)} + arctan (\frac{1}{8} (ζ + | ϑ | + \frac{1}{2} | p (| sin (ζ + 2 ϑ) |) | - \frac{1}{3} | z (| sin (ζ + 2 ϑ) |) |))}, \\ z (ζ) = sup_{ϑ \in D} {e^{- (ζ + | ϑ |)} + arctan (\frac{1}{8} (ζ + | ϑ | + \frac{1}{2} | z (| sin (ζ + 2 ϑ) |) | - \frac{1}{3} | c (| sin (ζ + 2 ϑ) |) |))} \end{matrix}} . (4.7)$ Notice that the system (4.7) is a particular case of (4.1), where ₰ = J = [0, 1], $ᗡ = ℝ$ , $Λ : J \times ℝ \to ℝ$ is defined as Λ (ζ, ϑ) = e^-(ζ+|ϑ|), $T : J \times ℝ \to J$ is T (ζ, ϑ) = | sin(ζ + 2ϑ) |, and $L : J \times ℝ \times ℝ \times ℝ \to ℝ$ is defined as $L (ζ, ϑ, ℓ, s) = arctan (\frac{1}{8} (ζ + | ϑ | + \frac{1}{2} | ℓ | - \frac{1}{3} | s |))$ and $H (ℓ) = arctan (ℓ)$ . It can be seen easily that all the hypotheses of Theorem 4.1 are satisfied. Consequently, on applying Theorem 4.1, the system (4.70 has a solution in ℘ (₰) × ℘ (₰).

5 Conclusion and open problem

Choudhury et al. [3] proposed Problem 1.2 openly for exploration.

On comparing Theorem 1.1 with Corollary 3.3, it can easily be asserted that Corollary 3.3 affirmatively addressed Problem 1.2. In fact, in present work, we have addressed this open problem not by replacing the compatibility assumption of the pair (Γ, h) with some weaker condition but by relaxing the compatible hypothesis entirely.

In view of Remark 3.4, a new avenue is opened for further enhancement of research by relaxing the continuity and monotone hypothesis of the self-mapping h and by replacing the completeness of entire space Y with completeness of range subspace of mapping h.

Further, the application part of our work to the system of functional equations in dynamic programming opened new gateways of research in fuzzy metric spaces.

Consequently, a multitude of the existing results in coupled fixed point theory are hereby sharpened and enriched.

At last, we also propose an open problem: “Whether the properties (H₁) and (H₂) be replaced by some weaker conditions or relaxed entirely".

Declarations

Availability of data and materials

Not applicable.\hrule \hrule

Competing interests

The authors declare that they have no competing interests.

Funding

The authors declare that there is no funding available for this paper.

Authors’ contributions

The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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