On the Occasion of his 80th Birthday Modular stabilities of a reciprocal second power functional equation

In the present work, we propose a different reciprocal second power Functional Equation (FE) which involves the arguments of functions in rational form and determine its stabilities in the setting of modular spaces with and without using Fatou property. We also prove the stabilities in β-homogenous spaces. As an application, we associate this equation with the electrostatic forces of attraction between unit charges in various cases using Coloumb’s law. 2020 Mathematics Subject Classifications: 39B52, 39B62, 39B82


Introduction & Preliminaries
The hypothesis connected with linear spaces and the concepts of modular spaces were dealt in [20]. Later, this theory has been employed by many authors [1, 9,16,29,32]. The significant application of modular theory is that it is useful in interpolation ( [10,17]) and in numerous Orlicz spaces [21]. The common notions and properties related to modular theory are available in [18,19,21].
The detailed information about the evolution of theory of stability of FEs are available in [3,5,6,24,25,30]. There are many techniques of solving stability problems of FEs, such as the technique through the attribute of shadowing [28], the technique via fixed averages [27], the technique by virtue of sandwich hypothesis [22]. The dominant tools to determine classical stability problems are the direct method and the fixed point method [6,23].
Also, without the application of ∆ 2 -condition, proposed in [7], there are many stability problems via fixed point theorem of quasicontracion functions in the setting of modular spaces. By employing Khamis's invariant point theorem, the modular stabilities of additive FE alongwith with the Fatou property and ∆ 2 -condition are dealt in [26]. Moreover, the modular stability problems of quadratic FEs were discussed satisfying Fatou property without utilizing ∆ 2 -condition in [31]. One can refer [2,4,8,[11][12][13][14][15] for more details about stabilities of real and complex valued multiplicative inverse FEs.
In this present work, we propose a different reciprocal second power FE of the form We solve equation (1) for its solution and investigate its various stability results in modular spaces with and without using Fatou property and in β-homogenous spaces.

Solution of equation (1) in the domain of non-zero real numbers
In this section, we impose the definition of reciprocal second power function and then we solve equation (1) for its solution in the setting of non-zero real numbers.
Definition 1. A mapping m q : R −→ R is called a reciprocal second power function if it satisfies (1). Hence, (1) is said to be a reciprocal second power FE. Proof. Let m q satisfies (1). Then m q is a reciprocal second power function and hence we can assume m q (u) = 1 u 2 for all u ∈ R . If I is an identity mapping, then [I(1/u)] 2 = 1 u 2 = m q (u) for all u ∈ R .
On the other hand, let there exists an identity function I : R −→ R such that m q (u) = [I(1/u)] 2 for all u ∈ R . Thus, we have for all u, v ∈ R , which indicates m q satisfies (1).
In the following results, for the purpose of easy computation, let us consider the difference operator Γ mq defined as follows: 3. Modular stability of equation (1) with ∆ 1

-condition
In this present section, we explore the investigate stability results of equation (1) connected with modular theory with modular space U µ without applying the Fatou property. In this section, let P denote a linear space. In the following results, suppose there exists > 0 so that µ(3u) ≤ 1 µ(u), for all u ∈ U µ , then the modular µ is said to satisfy the ∆ 1 3 -condition. Also, we say this constant is a ∆ 1 3 -constant related to ∆ 1 3 -condition. One can notice that if µ is convex and satisfies ∆ 1 3 . In the following main results, let us consider U to be a normed linear space over the set of real numbers.
Theorem 2. Suppose U µ satisfies the ∆ 1 3 -condition. Let there exists a mapping φ : lim n→∞ 2n φ u 3 n , The right-hand side of the above inequality tends to 0 when m → ∞ since ≥ 1 3 , which indicates that the series is convergent. In lieu of completeness of U µ , this sequence 1 9 n m q u 3 n turns out to be Cauchy for all u ∈ P and hence it is µ−convergent in U µ . Hence, we have a mapping D : P −→ U µ given by that is, lim n→∞ µ 1 9 n m q u 3 n − D(u) = 0 for all u ∈ P . So, without using Fatou property, we observe from ∆ 1 3 -condition that the inequality is true for u ∈ P and all integers n > 1. Allowing n → ∞ in the above inequality indicates that (4) holds. Plugging (u, v) by (3 −n u, 3 −n v) in (2), we find that which approaches zero as n → ∞ for all u, v ∈ P . Thus, in liue of the convexity of µ, we have for all u, v ∈ P and all integer n > 1. Letting the limit n → ∞, one obtains that D is reciprocal inverse second power function. To show the uniqueness of D, let us assume that there is another reciprocal second power function D : Then we see from the equalities: D(3 −n u) = 9 n D(u) and D (3 −n u) = 9 n D (u) that for all u ∈ P . It indicates from the above inequality that D is distinctive by allowing n → ∞. Hence the proof is complete.

-condition
In this present section, we provide a different result related to modular stability of equation (1) without ∆ 1 3 -condition. Theorem 3. Assume that U p is a p-complex modular space where p is convex. Also, let φ : U × U −→ [0, ∞) be a function with the condition for all u, v ∈ U . Assume that m q : U −→ U p is a mapping such that for all u, v ∈ U . Then a unique reciprocal second power function T : U −→ U p exists and satisfies for all u, v ∈ U .
Proof. Putting (u, v) as (u, u) in (6) and then dividing by 9 on both sides, we obtain for all u ∈ U . Then by induction arguments, we arrive at for all u ∈ U . It is clear that the case n = 1 follows directly from (8). Assume that (9) is true for n ∈ N. Then, we obtain the ensuing inequality: for all u ∈ U . Hence (9) holds for every k ∈ N. Let m and n be non-negative integers with n > m. Then (9), we have for all u ∈ U . By the application of (5) and (10), we observe that the the sequence m q (3 −n u) 9 n turns out to be Cauchy in U p . By virtue of completeness of U p , the sequence is convergent. This formulates that there exists a function T : U −→ U p defined by To confirm that T satisfies (1), plugging (u, v) into (3 −n u, 3 −n v) in (6) and then multiplying by 9 −n on both sides, we obtain for all u, v ∈ U . We can find that T satisfies (1) by letting n → ∞ in the above inequality.
To prove that T is unique reciprocal second power function which satisfies (1) and also (7). It is clear that both T and T satisfy (7). Hence, we obtain p T (u) − T (u) = 9 −n p T (3 −n u) − T (3 −n u) for all u, v ∈ U . It is easy to find that T is distinctive by allowing n → ∞ in (13) and employing (5), which completes the proof. holds for all u, v ∈ U . Then, T : U −→ U p is a unique reciprocal second power function satisfying (1) and p (m q (u) − T (u)) ≤ c 8 , for all u ∈ U .
Proof. It is easy to prove this corollary by taking φ(u, v) = c, for all u, v ∈ U in Theorem 3.
Corollary 2. Let λ 1 ≥ 0 be fixed and s = −2 if a function m q : U −→ U p fulfills the inequality holds for all u, v ∈ U . Then, there exists a unique reciprocal second power function T : U −→ U p satisfying (1) and Proof. The proof is obtained by taking φ(u, v) = λ 1 (|u| s + |v| s ) in Theorem 3.
Corollary 3. Let m q : U −→ U p be a mapping. If there exist x, y : s = x + y = −2 and holds for all u, v ∈ U . Then, there exists a unique reciprocal second power function T : U −→ U p satisfying (1) and Proof. The proof directly follows by taking φ(u, v) = c 2 (|u| a |v| b ) in Theorem 3.
holds for all u, v ∈ U . Then, there exists a unique reciprocal second power function T : U −→ V satisfying (1) and Proof. Choosing φ(u, v) = λ 6 (|u| x |v| y ) in Theorem 4, we achieve the result. for all u, v ∈ U . Then, a unique reciprocal second power function T : U −→ V exists and satisfies (1) and for all u ∈ U .

Application of equation (1)
We close our investigation with an application of equation (1) using Coloumb's law. According to Coloumb, the electrostatic force of attraction between two point charges is directly proportional to the product of the charges and inversely proportional to the square of the distance between them. That is, where F and r, respectively, are the force of attraction and distance between the point charges q 1 and q 2 . Suppose the constant 1 4π 0 is taken as a constant c and unit point charges are assumed, then the electrocstatic force of attraction is given by which is a reciprocal second power function. Suppose the distance between two unit point charges is uv 2u+v , then the electrocstatic force of attraction is given by Also, if the distance is uv 2u−v , then the electrocstatic force of attraction is given by Then using equation (1), we can relate that the sum of the above electrocstatic forces of attraction m q uv 2u+v and m q uv 2u−v is given by the sum of electrocstatic forces of attraction 2m q (u) = 2c u 2 and 8m q (v) = 8c v 2 . Hence equation (1) dealt in this study can be associated with the electrocstatic forces of attraction between the charges in different situations.

Conclusion
In this investigation, we introduced a new reciprocal second power FE (1) and investigated its various classical stability results in modular spaces and β-homogenous spaces. We solved equation (1) for its solution in the setting of non-zero real numbers. We associated equation (1) with Coloumb's law to employ it in various situations to connect the electrocstatic forces of attraction in different assumptions.