Proving new fixed point results in fuzzy metric spaces employing simulation function

Abstract

In this paper, the idea of MA-simulation function is introduced which has further been utilized to establish new fixed point results in fuzzy metric spaces. Moreover, it is shown that the proven result is quite a unified one to generalize several existing result. Furthermore, some new results are established to show the utility of our results. Some illustrative examples are also given which exhibit the usability of our results. Finally, we provide an application of our main results.

Keywords

fuzzy metric space MA-simulation function fixed point

1 Introduction

In [28], Zadeh initiated the idea of fuzzy set (associated to a set M) defined as a function D with domain M and co-domain [0, 1]. Thereafter, the existing available literature has witnessed a very intense and rapid fuzzification of almost entire mathematics. A host of researchers utilized this idea very extensively to develop several remarkable results in topology and analysis besides presenting fruitful applications. By now, such fuzzification remains a perceived framework to grasp uncertainties emerging in different physical circumstances. Like other notions, researchers developed the idea of fuzzy metric spaces in several ways such as, Erceg [6], Kalewa and Seikkala [15] and Kramosil and Michalek [18]. Later on, George and Veeramani [7] modified the idea of fuzzy metric spaces introduced by Kramosil and Michalek [18] and came forward with a view to have Hausdorff topology on such spaces. Here, it can be pointed out that such modified metric spaces were utilized very fruitfully by various researchers to prove fixed point results. In 1988, Grabiec [10] presented a fuzzy version of Banach contraction principle. Thereafter, there came a multitude of fixed point results in fuzzy metric spaces and to mention a few, one can be referred [9 , 21] and referencestherein.

Technically speaking, Gregori and Sapena [11] introduced the notion of fuzzy contractive mapping, which is as follows: Let $(M, M, *)$ be a fuzzy metric space and S : M → M a self-mapping satisfying the following condition (for all distinct z, y ∈ M and t > 0): $(\frac{1}{M (Sz, Sy, t)} - 1) \leq k (\frac{1}{M (z, y, t)} - 1),$ (1.1) where k ∈ (0, 1). Then S has a unique fixed point.

In recent past, many authors tried to generalize the fixed point theorems by modifying and varying the contraction conditions (see [20 , 27]). In 2015, Khojasteh [17] introduced a new approach to study the theory of fixed points in metric spaces via simulation functions defined as follows:

A mapping $ζ : [0, \infty) \times [0, \infty) \to ℝ$ is said to be a simulation function if it satisfies the following:

ζ (0, 0) =0;

ζ (t, s) < s - t, ∀t, s > 0;

if {t_n} and {s_n} are sequences in (0, ∞) such that $lim_{n \to \infty} t_{n} = lim_{n \to \infty} s_{n} > 0$ , then $\underset{n \to \infty}{lim sup} ζ (t_{n}, s_{n}) < 0 .$

After that, the concept of simulation function was used with some modification to establish fixed point results in some other settings, e.g., b-metric spaces, θ-metric spaces. In the same direction, Argoubi [2] modified this concept by withdrawing (ζ₁) and at the same time Roldan et al. [22] improved condition (ζ₃) as follows:

if {t_n} and {s_n} are sequences in (0, ∞) such that $lim_{n \to \infty} t_{n} = lim_{n \to \infty} s_{n} > 0$ and t_n < s_n, $\forall n \in ℕ$ , then $\underset{n \to \infty}{lim sup} ζ (t_{n}, s_{n}) < 0 .$

In 2015, Karapinar [16] presented a more generalized version of the result of Khojasteh [17] by involving α-admissible mappings.

In this paper, we introduce a new simulation function, namely MA-simulation function and employ the same to prove a new fixed point result in fuzzy metric spaces (in the sense of George and Veeramani [7]). We prove that our result is general enough to unify several existing results and also some new results are established as corollaries. We also provide an example to support our result. Finally, as an application of our result, we establish the existence and uniqueness of the solution of Fredholm non-linear integral equation.

2 Preliminaries

In this section, we recall some basic definitions in order to make our discussion self-contained.

Definition 2.1. [23] A binary relation * : [0, 1] × [0, 1] → [0, 1] is said to be a continuous t-norm if ([0, 1] , *) is an abelian topological monoid, i.e.,

* is associative and commutative;

* is continuous;

p * 1 = p, ∀p ∈ [0, 1];

p ≤ q and r ≤ s implies p * r ≤ q * s, ∀p, q, r, s ∈ [0, 1].

Example 2.1. [9] Some basic t-norms are as follows:

p * q = pq;

p * q = min {p, q};

p * q = max {p + q - 1, 0},

∀p, q ∈ [0, 1].

Gregori and Veeramani [11] defined the notion of fuzzy metric space as follows:

Definition 2.2. [7] Let M be a non-empty set, * a continuous t-norm and $M$ a fuzzy set on M × M × (0, ∞). If for all z, y, w ∈ M and s, t > 0, the following conditions are satisfied:

$M (z, y, t) > 0$ ;

$M (z, y, t) = 1 \Leftrightarrow z = y$ ;

$M (z, y, t) = M (y, z, t)$ ;

$M (z, w, t) * M (w, y, s) \leq M (z, y, t + s)$ ;

$M (z, y, .) : (0, \infty) \to [0, 1]$ is continuous,

then the triplet

(M, M, *)

is known as fuzzy metric space. Notice that a fuzzy metric

M (z, y, t)

can be visualized as the measure of adjacency between z and y with respect to t.

Remark 2.1. [7] For z, y ∈ M, $M (z, y, .) : (0, \infty) \to [0, 1]$ is a non-decreasing function. Each fuzzy metric $M$ on M generates a first countable Hausdorff topology $τ_{M}$ on M with base, the family of open $M$ -balls ${B_{M} (z, \in, t) : z \in M, \in \in (0, 1), t > 0}$ , where $B_{M} (z, \in, t) = {y \in M : M (z, y, t) > 1 - \in} .$

Example 2.2. [8] Let (M, d) be a metric space. Define p * q = pq (or p * q = min {p, q}) and for all z, y ∈ M and t > 0 $M (z, y, t) = \frac{t}{t + d (z, y)} .$ Then, the triplet $(M, M, *)$ is a fuzzy metric space and the fuzzy metric $M$ induced by metric d is a standard fuzzy metric.

Definition 2.3. [10 , 26] Let $(M, M, *)$ be a fuzzy metric space.

A sequence {z_n} ⊆ M is said to be convergent to z ∈ M if $lim_{n \to \infty} M (z_{n}, z, t) = 1, \forall t > 0 .$

A sequence {z_n} ⊆ M is said to be Cauchy (G-Cauchy) if $\begin{matrix} lim_{m, n \to \infty} M (z_{m}, z_{n}, t) \\ = 1 (lim_{n \to \infty} M (z_{n}, z_{n + m}, t) = 1, m \in ℕ), \forall t > 0 . \end{matrix}$

$(M, M, *)$ is said to be complete (G-complete) if every Cauchy (G-Cauchy) sequence in M converges to a point of M.

Lemma 2.1. [10] In a fuzzy metric space $(M, M, *)$ , the mapping $M$ is continuous on M × M × (0, ∞).

We need the following in the subsequent discussion.

Definition 2.4. [9] Let $(M, M, *)$ be a fuzzy metric space. A mapping S : M → M is said to be α-admissible if there exists a function α : M × M × (0, ∞) → [0, ∞) such that ∀t > 0 $z, y \in M, α (z, y, t) \geq 1 \Rightarrow α (Sz, Sy, t) \geq 1 .$

Definition 2.5. [5] Let $(M, M, *)$ be a fuzzy metric space. An α-admissible mapping S : M → M is said to be triangular α-admissible if ∀t > 0 $\begin{matrix} z, y, w & \in & M, α (z, y, t) \geq 1 and α (y, w, t) \\ \geq & 1 \Rightarrow α (z, w, t) \geq 1 . \end{matrix}$

Lemma 2.2. [5] Let $(M, M, *)$ be a fuzzy metric space and S : M → M a triangular α-admissible mapping. Assume that there exists a point z₀ ∈ M such that α (z₀, Sz₀, t) ≥1. Define a sequence {z₀} ⊆ M by z_n = Sz_n-1, $\forall n \in ℕ$ . Then we have $α (z_{n}, z_{m}, t) \geq 1, \forall m, n \in ℕ, n < m .$

3 Main Results

Inspired by Khojasteh et al. [17], we hereby introduce a new simulation function, namely: MA-simulation function. By utilizing this function, we define a new type of contraction, namely: α-admissible Ξ_MA-contraction, which will be utilized to establish a new result unifying several results of the existing literature besides deducing some new ones.

Definition 3.1. A mapping $ξ : (0, 1] \times (0, 1] \to ℝ$ is said to be MA-simulation function if it satisfies the following:

$ξ (t, s) < \frac{1}{t} - \frac{1}{s}$ , ∀s, t ∈ (0, 1);

if {t_n} and {s_n} are sequences in (0, 1] such that $lim_{n \to \infty} t_{n} = lim_{n \to \infty} s_{n} = l \in (0, 1)$ and t_n < s_n, $\forall n \in ℕ$ , then $\underset{n \to \infty}{lim sup} ξ (t_{n}, s_{n}) < 0 .$

We denote the set of all MA-simulation functions by Ξ_MA.

In the following lines, we furnish some examples of MA-simulation function.

Example 3.1. Let $ξ : (0, 1] \times (0, 1] \to ℝ$ be defined as: $\begin{matrix} ξ (t, s) & = & k (\frac{1}{t} - 1) - (\frac{1}{s} - 1), \forall s, t \in (0, 1] \\ and k \in (0, 1) . \end{matrix}$

Example 3.2. Let $ξ : (0, 1] \times (0, 1] \to ℝ$ be defined as: $ξ (t, s) = ψ (\frac{1}{t} - 1) - (\frac{1}{s} - 1), \forall s, t \in (0, 1],$ where ψ : [0, ∞) → [0, ∞) is right continuous functions such that ψ (r) < r, ∀r > 0.

Example 3.3. Let $ξ : (0, 1] \times (0, 1] \to ℝ$ be defined as: $\begin{matrix} ξ (t, s) = (\frac{1}{t} - 1) - ψ (\frac{1}{t} - 1) - (\frac{1}{s} - 1), \\ \forall s, t \in (0, 1], \end{matrix}$ where ψ : [0, ∞) → [0, ∞) is a given function such that ψ (r) >0 for r > 0 and ψ (0) =0.

Example 3.4. Let $ξ : (0, 1] \times (0, 1] \to ℝ$ be defined as: $ξ (t, s) = s - ψ (t), \forall s, t \in (0, 1],$ where ψ : (0, 1] → (0, 1] is non-decreasing and left-continuous such that ψ (r) > r, ∀r ∈ (0, 1).

Example 3.5. Let $ξ : (0, 1] \times (0, 1] \to ℝ$ be defined as: $ξ (t, s) = (\frac{1}{t} - 1) ψ (\frac{1}{t} - 1) - (\frac{1}{s} - 1), \forall s, t \in (0, 1],$ where ψ : [0, ∞) → [0, 1) is a given function such that $lim_{r \to s^{+}} ψ (r) < 1$ , ∀s > 0.

Example 3.6. Let $ξ : (0, 1] \times (0, 1] \to ℝ$ be defined as: $ξ (t, s) = (\frac{1}{t} - 1) - \int_{0}^{\frac{1}{s} - 1} ψ (s) ds, \forall s, t \in (0, 1],$ where ψ : [0, ∞) → [0, ∞) is a given function such that $\int_{0}^{r} ψ (s) ds$ exists and $\int_{0}^{r} ψ (s) ds > r$ , for each r > 0.

Now, we introduce the notion of α-admissible Ξ_MA-contraction.

Definition 3.2. Let $(M, M, *)$ be a fuzzy metric space and ξ ∈ Ξ_MA. A mapping S : M → M is said to be an α-admissible Ξ_MA-contraction if there exists a ξ ∈ Ξ_MA such that ∀t > 0, it satisfies the following: $z, y \in M, α (z, y, t) \geq 1 \Rightarrow ξ (M (z, y, t), M (Sz, Sy, t)) \geq 0 .$ (3.1)

Now, we are equipped to present our main result.

Theorem 3.1. Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M an α-admissible Ξ_MA-contraction with respect to ξ. Assume that the following conditions are satisfied:

there exists z₀ ∈ M such that α (z₀, Sz₀, t) ≥1;

S is triangular α-admissible;

S is continuous or if {z_n} is a sequence in M such that α (z_n, z_n+1, t) ≥1, $\forall n \in ℕ$ , t > 0 and {z_n} → z, for some z ∈ M, there exists a subsequence {z_{n
_k}} of {z_n} such that α (z_{n
_k}, z, t) ≥1, $\forall k \in ℕ$ and t > 0.

Then S has a fixed point.

Proof. Let z₀ ∈ M be an arbitrary point. Define a Picard sequence {z_n = Sⁿz₀}. Suppose there exists some $m_{0} \in ℕ$ such that S^{m
₀} (z₀) = S^m₀+1z₀, i.e., z_{m
₀} = z_m₀+1, then z_{m
₀} is a fixed point of S. Now, assume that S^n-1z₀ ≠ Sⁿz₀, $\forall n \in ℕ$ . Then, using Lemma 2.2, we have $α (z_{n}, z_{m}, t) \geq 1, \forall m, n \in ℕ, n < m .$ (3.2) Thus, using 3.2 and 3.1, for z = z_n-1 and y = z_n, we obtain $\begin{matrix} 0 & \leq ξ (M (z_{n - 1}, z_{n}, t), M ({Sz}_{n - 1}, {Sz}_{n}, t)) \\ = ξ (M (z_{n - 1}, z_{n}, t), M (z_{n}, z_{n + 1}, t)) \\ < \frac{1}{M (z_{n - 1}, z_{n}, t)} - \frac{1}{M (z_{n}, z_{n + 1}, t)}, \end{matrix}$ which implies $M (z_{n - 1}, z_{n}, t) < M (z_{n}, z_{n + 1}, t) .$ Therefore, ${M (z_{n}, z_{n + 1}, t)}$ is an increasing sequence of positive real numbers in (0, 1]. Let $r (t) = lim_{n \to \infty} M (z_{n}, z_{n + 1}, t)$ . We assert that r (t) =1, ∀t > 0. Suppose on contrary that r (t₀) <1, for some t₀ > 0. Then, as ${t_{n} = M (z_{n - 1}, z_{n}, t_{0})} \to r (t_{0})$ and ${s_{n} = M (z_{n}, z_{n + 1}, t_{0})} \to r (t_{0})$ so using (ξ₂), we obtain $\begin{matrix} 0 & \leq \underset{n \to \infty}{lim sup} ξ (M (z_{n - 1}, z_{n}, t_{0}), M (z_{n}, z_{n + 1}, t_{0})) \\ < 0, \end{matrix}$ a contradiction. Thus, r (t) =1, ∀t > 0, i.e., we get (∀ t > 0) $lim_{n \to \infty} M (z_{n}, z_{n + 1}, t) = 1 .$ (3.3) Next, we have to prove that {z_n} is a Cauchy sequence. Suppose it is not so, then there exists 0 < ∈₀ < 1, t₀ > 0 and two subsequences {z_{n
_k}} and {z_{m
_k}} of {z_n} such that k ≤ n (k) < m (k) and $M (z_{n (k)}, z_{m (k)}, t_{0}) \leq 1 - \in_{0} .$ By Remark 2, we get $M (z_{n (k)}, z_{m (k)}, \frac{t_{0}}{2}) \leq 1 - \in_{0} .$ (3.4) Now, suppose that m (k) is the smallest integer corresponding to n (k) satisfying (3.4). Then, we get $M (z_{n (k)}, z_{m (k) - 1}, \frac{t_{0}}{2}) > 1 - \in_{0} .$ (3.5) Now, using condition (F4), (3.4) and (3.5), we obtain $\begin{matrix} 1 - \in_{0} & \geq M (z_{n (k)}, z_{m (k)}, t_{0}) \\ \geq M (z_{n (k)}, z_{m (k) - 1}, \frac{t_{0}}{2}) * M (z_{m (k) - 1}, z_{m (k)}, \frac{t_{0}}{2}) \\ > (1 - \in_{0}) * M (z_{m (k) - 1}, z_{m (k)}, \frac{t_{0}}{2}) . \end{matrix}$ Letting k→ ∞ and applying (t₃), it yields $1 - \in_{0} \geq lim_{k \to \infty} M (z_{n (k)}, z_{m (k)}, t_{0}) \geq 1 - \in_{0}$ and hence $lim_{k \to \infty} M (z_{n (k)}, z_{m (k)}, t_{0}) = 1 - \in_{0} .$ (3.6) Also, again by 3.1 and (ξ₂), for z = z_{n_k-1}, y = z_{m_k-1} and t = t₀, we get $\begin{matrix} 0 & \leq ξ (M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0}), M (z_{n (k)}, z_{m (k)}, t_{0})) \\ < \frac{1}{M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0})} - \frac{1}{M (z_{n (k)}, z_{m (k)}, t_{0})}, \end{matrix}$ so that $\begin{matrix} M (z_{n (k)}, z_{m (k)}, t_{0}) > M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0}) \\ \geq M (z_{n (k) - 1}, z_{n (k)}, \frac{t_{0}}{2}) * M (z_{n (k)}, z_{m (k) - 1}, \frac{t_{0}}{2}) \\ > M (z_{n (k) - 1}, z_{n (k)}, \frac{t_{0}}{2}) * (1 - \in_{0}), \end{matrix}$ which on letting k→ ∞ and using (t₃) yields $1 - \in_{0} \geq lim_{k \to \infty} M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0}) \geq 1 - \in_{0} .$ Hence, we have $lim_{k \to \infty} M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0}) = 1 - \in_{0} .$ (3.7) Henceforth, by 3.2, we have α (z_n(k)-1, z_m(k)-1, t₀) ≥1, thus taking ${t_{k} = M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0})}$ and ${s_{k} = M (z_{n (k)}, z_{m (k)}, t_{0})}$ and applying (ξ₂), we obtain $\begin{matrix} 0 \leq \underset{k \to \infty}{lim sup} ξ (M (z_{n (k) - 1}, z_{m (k) - 1}, t_{0}), \\ M (z_{n (k)}, z_{m (k)}, t_{0})) < 0, \end{matrix}$ a contradiction. Thus, {z_n} is a Cauchy sequence in $(M, M, *)$ . Now, by the completeness of M, there exists z ∈ M such that {z_n} → z. If S is continuous, then we have {Sz_n} → Sz, which by uniqueness of limit implies that Sz = z. This completes the proof.

In the next theorem, we present the uniqueness of fixed point.

Theorem 3.2. In addition to the hypothesis of Theorem 3, if the following condition is fulfilled:

for every z, y ∈ Fix (S), there exists w ∈ M such that α (z, w, t) ≥1 and α (y, w, t) ≥1, ∀t > 0,

then the fixed point of S is unique.

Proof. The existence part is followed by Theorem 3.1. For the uniqueness of fixed point, assume that z and z^* are two distinct fixed points of S. Then by condition (d), there exists a point w ∈ M such that α (z, w, t) ≥1 and α (z^*, w, t) ≥1, ∀t > 0. Construct a sequence {w_n} ⊆ M by w₀ = w and w_n+1 = Sw_n, $\forall n \in ℕ \cup {0}$ . By triangular α-admissibility, we have $\begin{matrix} α (z, w_{n}, t) & \geq & 1 and α (z^{*}, w_{n}, t) \\ \geq & 1, \forall n \in ℕ \cup {0} and t > 0 . \end{matrix}$ (3.8) Now, using (3.8) and applying 3.1 (for z = z and y = w_n), we derive $M (z, w_{n + 1}, t) > M (z, w_{n}, t), \forall n \in ℕ \cup {0} and t > 0,$ (3.9) which shows that ${M (z, w_{n}, t)}$ is an increasing sequence of positive real numbers in (0, 1]. Let $L (t) = lim_{n \to \infty} M (z, w_{n}, t)$ . Our claim is that L (t) =1, ∀t > 0. Assume on contrary that there exists some t₀ > 0 such that L (t₀) <1. Thus, for ${t_{n} = M (z, w_{n}, t_{0})}$ and ${s_{n} = M (z, w_{n + 1}, t_{0})}$ , by (ξ₂) and applying 3.1, we obtain $0 \leq lim_{n \to \infty} ξ (M (z, w_{n}, t_{0}), M (z, w_{n + 1}, t_{0})) < 0,$ a contradiction. Hence, L (t) =1, ∀t > 0. Thus, we have $lim_{n \to \infty} M (z, w_{n}, t) = 1$ , ∀t > 0, i.e., $lim_{n \to \infty} w_{n} = z$ . In the similar way, we can prove that $lim_{n \to \infty} w_{n} = z^{*}$ . By uniqueness of limit, we obtain z = z^* and it completes the proof.

Now, we present the following example which exhibits the utility of Theorem 3.1.

Example 3.7. Let M = [0, 1]. Define t-norm * : [0, 1] × [0, 1] → [0, 1] by p * q = min {p, q} and define fuzzy metric $M$ by $M (z, y, t) = \frac{t}{t + | z - y |} .$ Then $(M, M, *)$ is a complete fuzzy metric space. Define mappings α : M × M × (0, ∞) → [0, ∞) and S : M → M by: $α (z, y, t) = {\begin{matrix} 1, & if z, y \in [0, \frac{1}{2}], \\ 0, & otherwise \end{matrix}$ and $Sz = {\begin{matrix} \frac{az}{1 + z}, & if z \in [0, \frac{1}{2}], \\ z, & otherwise, \end{matrix}$ where a ∈ (0, 1). Then, we have (∀z, y ∈ M and t > 0) $\begin{matrix} \frac{1}{M (z, y, t)} - 1 & = \frac{t + | z - y |}{t} - 1 \\ = \frac{| z - y |}{t} . \end{matrix}$ Also, for z, y ∈ M such that α (z, y, t) ≥1, we have $\begin{matrix} \frac{1}{M (Sz, Sy, t)} - 1 & = \frac{t + | Sz - Sy |}{t} - 1 \\ = \frac{| Sz - Sy |}{t} \\ = \frac{| \frac{az}{1 + z} - \frac{ay}{1 + y} |}{t} \\ = \frac{a}{(1 + z) (1 + y)} (\frac{| z - y |}{t}) . \end{matrix}$ Then, taking $ξ (t, s) = k (\frac{1}{t} - 1) - (\frac{1}{s} - 1)$ , for any k ∈ [a, 1), for each a ∈ (0, 1), we obtain (for z, y ∈ M) $\begin{matrix} α (z, y, t) & \geq & 1 \Rightarrow ξ (M (z, y, t), M (Sz, Sy, t)) \\ = k \frac{| z - y |}{t} - \frac{a}{(1 + z) (1 + y)} (\frac{| z - y |}{t}) \geq 0, \end{matrix}$ for all t > 0. Hence, all the conditions of Theorem 3.1 are satisfied and the conclusion of the theorem holds, i.e., S has a unique fixed point (namely z = 0). But the result of Gregori and Sapena [11] can not be applied. Indeed, for any $z, y \in (\frac{1}{2}, 1]$ , there does not exist any k ∈ (0, 1) such that (1.1) is satisfied.

Next, Theorem 3.1 can be improved as follows:

Theorem 3.3. The conclusions of Theorems 3.1 and 3.2 hold if we replace α-admissible Ξ_MA-contraction by the following (retaining the other conditions same): $\begin{matrix} z, y \in M, α (z, y, t) \geq 1 \\ \Rightarrow ξ (M (z, y, t), M (S^{n} z, S^{n} y, t)) \geq 0, \end{matrix}$ for some $n \in ℕ$ and ∀t > 0.

Proof. By Theorem 3.1, Sⁿ has a unique fixed point, z ∈ M (say), i.e., Sⁿz = z. Also Sⁿ (Sz) = S (Sⁿz) = Sz, i.e., Sz is the fixed point of S. But, by the uniqueness of fixed point of S, we have Sz = z. Since fixed point of S is also that of Sⁿ, so S has a unique fixed point.

The following example exhibits that Theorem 3.3 is a genuine extension of Theorem 3.1.

Example 3.8. Let M = [-1, 1] and define a mapping $M$ on M × M × (0, ∞) as: $M (z, y, t) = {\begin{matrix} 1, & if z = y, \\ min {1 - \frac{| z |}{2}, 1 - \frac{| y |}{2}}, & otherwise . \end{matrix}$ The space $(M, M, *)$ is a complete fuzzy metric space with * : [0, 1] × [0, 1] → [0, 1], defined as p * q = min {p, q}. Define mappings α : M × M × (0, ∞) → [0, ∞) and S : M → M as: $α (z, y, t) = 1, \forall z, y \in M and t > 0$ and $Sz = {\begin{matrix} 0, & if - 1 < z < 0, \\ 1, & if 0 \leq z \leq 1, \end{matrix}$ respectively. We have $S^{2} z = 1, \forall z \in M .$ Then, with $ξ (t, s) = k (\frac{1}{t} - 1) - (\frac{1}{s} - 1)$ , ∀t, s ∈ (0, 1] and any k ∈ (0, 1), all the conditions of Theorem 3.3 are satisfied and S has a unique fixed point (namely z = 1). But Theorem 3.1 can not be applied here, since S is not an α-admissible Ξ_MA-contraction mapping. Indeed, for $z = \frac{- 1}{2}$ and $y = \frac{1}{2}$ , $M (Sz, Sy, t) < M (z, y, t)$ , ∀t > 0, which is a contradiction.

4 Consequences

In this section, we deduce some corollaries as consequences of Theorem 3.1 starting with the following one.

Corollary 4.1. (Banach [3] type) Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $\begin{matrix} z, y \in M, α (z, y, t) & \geq & 1 \Rightarrow \frac{1}{M (Sz, Sy, t)} - 1 \\ \leq & k (\frac{1}{M (z, y, t)} - 1), \end{matrix}$ for all t > 0 and k ∈ (0, 1). Then S has a unique fixed point.

Proof. The proof follows from Theorem 3.1 and Example 3.1.

By taking α (z, y, t) =1, ∀z, y ∈ M and t > 0, Corollary 4.1 reduces to the following result by Gregori and Sapena [11].

Corollary 4.2. [11] Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $\frac{1}{M (Sz, Sy, t)} - 1 \leq k (\frac{1}{M (z, y, t)} - 1),$ for all z, y ∈ M and t > 0 and k ∈ (0, 1). Then S has a unique fixed point.

In the next corollary, we are going to present Boyd Wong [4] type result in the setting of fuzzy metric spaces.

Corollary 4.3. (Boyd Wong [4] type) Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $\begin{matrix} z, y \in M, α (z, y, t) & \geq & 1 \Rightarrow \frac{1}{M (Sz, Sy, t)} - 1 \\ \leq & ψ (\frac{1}{M (z, y, t)} - 1), \end{matrix}$ for all t > 0, where ψ : [0, ∞) → [0, ∞) is a given function such that ψ (r) < r, for all r > 0 and ψ (0) =0. Then S has a unique fixed point.

Proof. In view of Theorem 3.1 and Example 3.2, the result follows.

Following is [1]-type fixed point result.

Corollary 4.4. (Abbas et al. [1] type) Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $\begin{matrix} z, y \in M, α (z, y, t) & \geq & 1 \Rightarrow \frac{1}{M (Sz, Sy, t)} - 1 \\ \leq & (\frac{1}{M (z, y, t)} - 1) \\ - ψ (\frac{1}{M (z, y, t)} - 1), \end{matrix}$ for all t > 0, where ψ : [0, ∞) → [0, ∞) is a given function such that ψ (r) >0, for all r > 0 and ψ (0) =0. Then S has a unique fixed point.

Proof. Taking into account of Theorem 3.1 and Example 3.3, the result follows.

Next, we present the following results, which are known in some natural settings but seems new to the fuzzy setting.

Corollary 4.5. Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $z, y \in M, α (z, y, t) \geq 1 \Rightarrow M (Sz, Sy, t) \geq ψ (M (z, y, t)),$ for all t > 0, where ψ : (0, 1] → (0, 1] is non-decreasing and left-continuous function such that ψ (r) > r, for all r ∈ (0, 1). Then S has a unique fixed point.

Proof. The result follows from Theorem 3.1 and Example 3.4.

Corollary 4.6. Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $\begin{matrix} z, y \in M, α (z, y, t) & \geq & 1 \Rightarrow \frac{1}{M (Sz, Sy, t)} - 1 \\ \leq & (\frac{1}{M (z, y, t)} - 1) \\ ψ (\frac{1}{M (z, y, t)} - 1), \end{matrix}$ for all t > 0, where ψ : [0, ∞) → [0, ∞) is a given function such that $lim_{r \to s^{+}} ψ (r) > 0$ , for all s > 0. Then S has a unique fixed point.

Proof. The result follows from Theorem 3.1 and Example 3.5.

Corollary 4.7. Let $(M, M, *)$ be a complete fuzzy metric space and S : M → M a mapping satisfying $\begin{matrix} z, y \in M, α (z, y, t) & \geq & 1 \Rightarrow \int_{0}^{\frac{1}{M (Sz, Sy, t)} - 1} ψ (s) ds \\ \leq & \frac{1}{M (z, y, t)} - 1, \end{matrix}$ for all t > 0, where ψ : [0, ∞) → [0, ∞) is a given function such that $\int_{0}^{r} ψ (s) ds$ exists and $\int_{0}^{r} ψ (s) ds > r$ , for each r > 0. Then S has a unique fixed point.

Proof. In view of Theorem 3.1 and Example 3.6, this result follows.

5 Application

In recent past, many authors used various sufficient conditions to find the existence and uniqueness of solutions of integral equations in varied settings. In this section, we consider a Fredholm non-linear integral equation and utilize our proved result in the setting of fuzzy metric spaces to find its unique solution. We see that by applying Theorem 3.1, this Fredholm non-linear integral equation has a unique solution under certain specific conditions and without these conditions, we cannot apply our results to find the unique solution.

To accomplish this, we consider the following:

$z (t) = \int_{a}^{b} K (t, s) h (z (s)) ds + g (t),$ (5.1) for all t ∈ Ω = [a, b] ( $a, b \in ℝ$ ), $K \in C (Ω \times Ω, ℝ)$ and $g, h \in C (Ω, ℝ)$ . Let varPhi be the collection of all mappings φ : [0, ∞) → [0, ∞) satisfying the following conditions:

φ is non-decreasing;

φ (t) ≤ t, ∀t ∈ [0, ∞).

Now, we are equipped to present our theorem in this section as follows:

Theorem 5.1. Consider the integral equation (5.1) with $K \in C (Ω \times Ω, ℝ)$ and $g \in C (Ω, ℝ)$ . If the following conditions are satisfied:

there exists a positive number λ and φ ∈ varPhi such that $\forall z, y \in C (Ω, ℝ)$ , the following condition holds: $h (z) - h (y) \leq λ φ (z - y);$ (5.2)

$λ sup_{t \in Ω} \int_{a}^{b} | K (t, s) | ds \leq \frac{1}{2}$ .

Then, the equation (5.1) has a unique solution in

C (Ω, ℝ)

Proof. Observe that $M = C (Ω, ℝ)$ is a complete metric space with respect to sup-metric $d (z, y) = sup_{t \in Ω} | z (t) - y (t) | .$ Also, the space $(M, M, *)$ with $M (z, y, t) = \frac{t}{t + d (z, y)}, \forall z, y \in M and t > 0$ and p * q = pq, ∀p, q ∈ [0, 1] is a complete fuzzy metric space. Now, define a mapping S : M → M as: $Sz (t) = \int_{a}^{b} K (t, s) h (z (s)) ds + g (t),$ (5.3) ∀t ∈ Ω. Using (5.2) and (5.3), we have $\begin{matrix} Sz (t) - Sy (t) & = \int_{a}^{b} K (t, s) [h (z (s)) - h (y (s))] ds \\ \leq λ \int_{a}^{b} K (t, s) φ (z (s) - y (s)) ds . \end{matrix}$ (5.4) Using (φ₁), we have $φ (z (s) - y (s)) \leq φ (sup_{s \in Ω} | z (s) - y (s) |) = φ (d (z, y)) .$ (5.5) Applying 3.11 in 3.10, we obtain $Sz (t) - Sy (t) \leq λ \int_{a}^{b} K (t, s) φ (d (z, y)) ds .$ Taking supremum over t ∈ Ω, using conditions (A₂) and (φ₂), we get $\begin{matrix} d (Sz, Sy) & \leq λ φ (d (z, y)) \int_{a}^{b} | K (t, s) | ds \\ \leq \frac{1}{2} φ (d (z, y)) \\ \leq \frac{1}{2} d (z, y) . \end{matrix}$ (5.6) Now, we have $\begin{matrix} \frac{1}{M (Sz, Sy, t)} - 1 & = \frac{d (Sz, Sy)}{t} \\ \leq \frac{d (z, y)}{2 t} \\ = \frac{1}{2} (\frac{1}{M (z, y, t)} - 1) . \end{matrix}$ By taking $ξ (t, s) = \frac{1}{2} (\frac{1}{t} - 1) - (\frac{1}{s} - 1)$ , ∀t, s ∈ (0, 1], all the conditions of Theorem 3.1 are satisfied with α (z, y, t) =1, ∀z, y ∈ M and t > 0. Hence, by the conclusions of Theorems 3.1 and 3.2, (5.1) has a unique solution in $C (Ω, ℝ)$ .

6 Conclusion

In this paper, motivated by the work of Khojasteh et al. [17] and Karapınar [16], we introduce a new simulation function besides proposing the concept of a new contraction namely, α-admissible Ξ_MA-contraction and utilize the same to prove fixed point results ensuring the existence and uniqueness of fixed points. Furthermore, via some corollaries, we demonstrate that our main result is general enough to unify several results of the existing literature. Finally, by presenting an application, we exhibit the usability of our main result.

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