1. Introduction
The stability problem for functional equations, a classical topic in functional analysis, concerns conditions under which a function that approximately satisfies an equation is close to an exact solution. This notion was first posed by Ulam [
1] in 1940 for group homomorphisms. Hyers [
2] provided the first partial solution in 1941 for additive functions in Banach spaces, establishing what is now known as Hyers–Ulam stability. Subsequent extensions include Aoki’s treatment of
p-power norms [
3], Rassias’ allowance of unbounded Cauchy differences [
4], and Gâvruta’s use of general control functions [
5], leading to what is now referred to as Hyers–Ulam–Rassias stability.
Definition 1 (Hyers–Ulam–Rassias Stability)
. Let X and Y be normed vector spaces and let be a mapping. We say that the functional equation is Hyers–Ulam–Rassias stable if the following holds: If there exists a control function such thatthen there exists an exact solution of the functional equation such thatwhere is a function that depends on Over time, stability theory has developed through distinct methodological approaches. One classical approach is the direct method, first investigated by Hyers [
2] and later extended by Rassias [
4]. This approach is constructive in nature: starting from a function that approximately satisfies a given functional equation, it derives exact solutions by explicitly generating sequences that converge to the desired solution.
In contrast, the fixed point method, systematically advanced by Radu [
6] and Cădariu and Radu [
7], reformulates stability problems as fixed point problems in suitable function spaces. By applying general fixed point theorems—such as the Banach fixed point theorem or the fixed point alternative—one can guarantee the existence and uniqueness of exact solutions. Brzdȩk and Ciepliński [
8] further refined this method in non-Archimedean spaces, thereby establishing a powerful framework for generalized settings. More recently, Koh [
9] applied Brzdȩk’s fixed point method to obtain stability estimates for an arithmetic functional equation, demonstrating the versatility of this approach within the broader stability theory.
Beyond these methodological advances, stability theory has also expanded to generalized structures. Fuzzy sets, introduced by Zadeh [
10], and non-Archimedean fuzzy norms, developed by Mirmostafaee and Moslehian [
11] and George and Veeramani [
12], provide natural settings for studying stability in contexts involving uncertainty or non-Archimedean valuations. In this paper, we adopt a slightly modified framework by defining fuzzy metrics and norms on the set of positive real numbers
, thereby aligning the analytic setting with the structure of the functional equations under consideration.
Definition 2 (Non-Archimedean Fuzzy Metric). Let X be a non-empty set. A non-Archimedean fuzzy metric on X is a 3-tuple where is a fuzzy metric and ∗ is a continuous t-norm, which is the minimum operator: for all The function M satisfies the following conditions for all and :
- (1)
if and only if ;
- (2)
;
- (3)
;
- (4)
is non-decreasing on and
Definition 3 (Non-Archimedean Fuzzy Norm). Let X be a real linear space. A function is said to be a non-Archimedean fuzzy norm on if for all and the following conditions hold:
- (1)
if and only if for all ;
- (2)
;
- (3)
;
- (4)
is non-decreasing on and
- (5)
for is continuous on
In this paper, we focus on the reciprocal functional equation
where
is defined on the set of nonzero real numbers. Reciprocal-type functional equations inherently exhibit symmetric structures, as the variables
x and
y play interchangeable roles in the functional relation. For each real number
the reciprocal function
is a solution of the functional Equation (
1) on the space of non-zero real numbers. Such equations arise naturally in various applications. For instance, in physics, the concept of the reduced mass
of a two-body system with masses
and
is given by
A similar structure appears in electrical engineering when computing the equivalent resistance of resistors connected in parallel. The stability of this equation has been investigated using several approaches. For instance, Jung [
13] successfully applied the fixed point alternative method to establish its stability. A significant body of work by Ravi and his collaborators has explored its stability using the direct method, extending the results to reciprocal difference and adjoint functional equations [
14] and to a multi-variable setting [
15,
16].
The aim of this paper is to study the Hyers–Ulam–Rassias stability of Equation (
1) in non-Archimedean fuzzy normed spaces using both the direct method and the fixed point alternative method. Furthermore, we will study the stability of a slightly modified reciprocal-type equation via Brzdȩk’s fixed point method. Our primary goal is to compare and analyze the distinctive features and effectiveness of these stability methods. To do so, we first need to clarify the definition of Hyers–Ulam–Rassias stability in our context.
Definition 4 (Hyers–Ulam–Rassias Stability)
. Let X be a linear vector space, and and be non-Archimedean fuzzy norm spaces. Let be a mapping. We say that the functional equation is Hyers–Ulam–Rassias stable if the following holds: If there exists a control function such thatthen there exists an exact solution of the functional equation such thatfor some function that depends on Throughout this paper, let be a non-Archimedean fuzzy norm on the real vector space . We restrict the domain of this norm to and denote the resulting structure by . This allows us to perform all proofs using the same norm N for both spaces.
Moreover, we denote
. For a mapping
, we define
for all
4. Brzdȩk’s Fixed Point Method
In this section, we investigate the reciprocal functional Equation (
1) using Brzdȩk’s fixed point method. To apply this method, we first reformulate the equation in the following form:
We begin by recalling Brzdȩk’s fixed point method. In their work, Brzdȩk and Ciepliński [
8] established an existence theorem for fixed points of nonlinear operators in metric spaces and subsequently applied this result to prove stability results for functional equations in non-Archimedean metric spaces. Their framework also yields fixed point results in arbitrary metric spaces. Specifically, Brzdȩk’s fixed point method can be viewed as a direct consequence of the theorems presented in [
8,
20].
Throughout this section, we denote by the set of all functions from a set X to a set Y.
Theorem 4 ([
20])
. Let X be a non-empty set, be a complete metric space and be given mappings. Suppose that and are two operators satisfying the following conditionsandfor all and If there exist and such thatfor all then the limit exists for each Moreover, the function is a fixed point of withfor all We now investigate the stability of the reciprocal functional Equation (
17) using Brzdȩk’s fixed point method in the setting of non-Archimedean fuzzy metric spaces and non-Archimedean fuzzy normed spaces.
Theorem 5. Let be a non-Archimedean fuzzy metric space. Suppose that the metric M is invariant, i.e., for all and ,and suppose there exists a function such that the setis non-empty, where Assume that for all , , and , the following inequality holds: Additionally, suppose there exists a function such thatfor all and . Then there exists a unique solution for the Equation (17) such that, for all and ,where Proof. Replacing
y with
where
and
in the inequality (
23), we obtain
for all
For each
we define the operators
and
by
for all
For each
,
has the form described in (
19) in Theorem 4 with
and
. Since
M is invariant, we have
and
where
Hence, for each
,
satisfies the inequalities (
18) in Theorem 4. It can be shown by mathematical induction on
k that
for all
and
.
For each
and
, we have
where
. Now, we can use Brzdȩk’s fixed point method as in Theorem 4. Hence, we obtain the limit
which exists for each
and
, and satisfies
Next, we show that
for all non-negative integers
n,
,
, and
. We note that
. The case
is covered by inequality (
23). Assume that the inequality holds for
. Then, for
, we have
for all
,
, and
.
As
, we have
for
. Hence, we obtain the following equality
for all
and
. That is, for each
, the mapping
is defined in this way and is a solution of the Equation (
17).
Let
be a constant. We will prove that each mapping
satisfying the Equation (
17) and
coincides with
for each
. Fix
. Since
is non-decreasing, we have
where
Hence, we obtain
for all
and
. For
, we will show that
for all
and
. The case
corresponds to inequality (
29). Assume that inequality (
30) holds for
. Then, we have
for all
and
. As
, we have
Hence, we conclude that
for all
. Thus, for each
, it follows that
as desired. □
The following corollary is a direct consequence of Theorem 5.
Corollary 2. Let M be as in Theorem 5, and let be a mapping satisfyingSuppose that satisfies the inequality (23). Then there exists a unique reciprocal mapping such thatfor all and . Proof. For each
, let
The assumption of Theorem 5 implies
Hence, we obtain
By the assumption that the sequence
has a subsequence
such that
, we get
From inequalities (
33) and (
34), we deduce
which implies
Thus, we conclude
Finally, by letting
as in Theorem 5, the inequality (
32) follows from inequality (
24). □
Remark 1. It should be emphasized that reciprocal-type functional equations inherently possess a symmetric property. This symmetry can be illustrated, for instance, by the substitutions
Case (1) or case (2) in the direct method and the fixed point alternative method, or
Case (1) or case (2) in the Brzdȩk’s fixed point method.
These substitutions highlight the underlying symmetric structure of the original functional equation. However, each case considered separately does not, in fact, preserve symmetry on its own. As a result, when stability methods such as the direct method, the fixed point alternative, or Brzdȩk’s fixed point approach are applied, the symmetric property intrinsic to the original equation is no longer maintained within the stability framework.
Remark 2. In this work, we have investigated the direct method alongside two fixed point frameworks—the classical fixed point alternative and Brzdȩk’s fixed point method—in studying stability problems. These approaches were selected to provide a more comprehensive analysis by leveraging their complementary strengths.
The direct method constructs explicit approximating functions for a given functional equation and estimates the deviation from exact solutions without relying on abstract fixed point arguments. While it offers an intuitive and constructive framework, Brzdȩk’s fixed point method guarantees existence and uniqueness from a more general theoretical framework.
The classical fixed point alternative typically requires a complete generalized metric space and restrictive Lipschitz-type conditions. In contrast, Brzdȩk’s theorem introduces a general control function, allowing generalized contractive conditions that drive the operator toward a fixed point even when conventional Lipschitz continuity fails. This generalization significantly extends the applicability of fixed point techniques to nonlinear functional equations.
For the reciprocal functional equation, the relation plays a central role in both the direct and classical fixed point approaches. In Brzdȩk’s method, the linear relation not only aids in establishing stability results but also enhances the effectiveness of analyzing a slightly modified reciprocal equation. Stability estimates further indicate that the underlying non-Archimedean fuzzy metric must satisfy an invariance property.
By integrating these methods, we obtain specific stability results under restrictive assumptions and more general results under relaxed conditions, thereby strengthening the completeness of the analysis.
An open question remains for future research: Can Brzdȩk’s fixed point method be systematically extended to more general classes of functional equations and more general normed or metric spaces?