EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 13, No. 5, 2020, 1162-1175 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Special Issue Dedicated to Professor Hari M. Srivastava On the Occasion of his 80th Birthday Modular stabilities of a reciprocal second power functional equation B.V. Senthil Kumar1,∗, Hemen Dutta2, S. Sabarinathan3 1 Department of Information Technology, University of Technology and Applied Sciences- Nizwa, Nizwa - 611, Oman 2 Department of Mathematics, Gauhati University, Guwahati - 781 014, Assam, India 3 Department of Mathematics, SRM Institute of Science & Technology, Kattankulthur-603 203, Tamil Nadu, India Abstract. In the present work, we propose a different reciprocal second power Functional Equa- tion (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 Key Words and Phrases: Reciprocal functional equation, quadratic functional equation, HUGR stability, non-Archimedean field 1. 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]. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v13i5.3709 Email addresses: senthilkumar@nct.edu.om; bvskumarmaths@gmail.com (B. V. Senthil Kumar), hemen dutta08@rediffmail.com (Hemen Dutta), ssabarimaths@gmail.com (S. Sabarinathan) https://www.ejpam.com 1162 c© 2020 EJPAM All rights reserved. B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1163 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, 11–15] 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 mq ( uv 2u+ v ) +mq ( uv 2u− v ) = 2mq(u) + 8mq(v). (1) 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. 2. 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 mq : R? −→ R is called a reciprocal second power function if it satisfies (1). Hence, (1) is said to be a reciprocal second power FE. Theorem 1. Let mq : R? −→ R be a function. Then, mq satisfies (1) if and only if there exists an identity function I : R? −→ R such that mq(u) = [I(1/u)]2, for all u ∈ R?. Proof. Let mq satisfies (1). Then mq is a reciprocal second power function and hence we can assume mq(u) = 1 u2 for all u ∈ R?. If I is an identity mapping, then [I(1/u)]2 = 1 u2 = mq(u) for all u ∈ R?. On the other hand, let there exists an identity function I : R? −→ R such that mq(u) = [I(1/u)]2 for all u ∈ R?. Thus, we have mq ( uv 2u+ v ) +mq ( uv 2u− v ) = [ I ( 2u+ v uv )]2 + [ I ( 2u− v uv )]2 = (2u+ v)2 u2v2 + (2u− v)2 u2v2 = 8 v2 + 2 u2 B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1164 = 2mq(u) + 8mq(v) for all u, v ∈ R?, which indicates mq satisfies (1). In the following results, for the purpose of easy computation, let us consider the dif- ference operator Γmq defined as follows: Γmq(u, v) = mq ( uv 2u+ v ) +mq ( uv 2u− v ) − 8mq(u)− 2mq(v). 3. Modular stability of equation (1) with ∆ 1 3 -condition In this present section, we explore the investigate stability results of equation (1) con- nected with modular theory with modular space Uµ 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 ∈ Uµ, 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 -condition with ∆ 1 3 -constant ` > 0. If ` < 1 3 , then µ(u) ≤ 1 `µ ( u 3 ) ≤ 1 3`µ(u), which implies µ = 0. When µ is convex modular, then we have ∆ 1 3 -constant ` ≥ 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 Uµ satisfies the ∆ 1 3 -condition. Let there exists a mapping φ : P × P −→ [0,∞) such that the mapping mq : P −→ Uµ satisfies µ ( Γmq(u, v) ) ≤ φ(u, v), (2) lim n→∞ `2nφ ( u 3n , v 3n ) = 0 and ∞∑ i=0 ( 3`3 )i φ ( u 3i , u 3i ) <∞ for all u, v ∈ P, then a unique reciprocal second power function D : P −→ Uµ exists and satisfies µ (mq(u)−D(u)) ≤ 3 ` ∞∑ i=0 ( 3`3 )i φ ( u 3i , u 3i ) (3) for all u ∈ P. Proof. By taking v = u in (2), we obtain µ ( mq (u 3 ) − 9mq(u) ) ≤ φ(u, u) for all u ∈ P . Employing ∆ 1 3 -condition of µ, one can find µ ( mq(u)− 1 9n mq ( u 3n )) = µ ( n∑ i=0 3i ( 1 33i−2 mq ( u 3i−1 ) − 1 33i mq ( u 3i ))) ≤ 1 `2 n∑ i=0 (3`3)iφ ( u 3i , u 3i ) (4) B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1165 for all u ∈ P . Now, shifting u to 3−mu in (4), we obtain µ ( 1 9m mq ( u 3m ) − 1 9n+m mq ( u 3n+m )) ≤ `−2mµ ( mq ( u 3m ) − 1 9n mq ( u 3n+m )) ≤ `−(2m+2) n∑ i=0 (3`3)iφ ( u 3i+m , u 3i+m ) ≤ 3−m `m+2 n+m∑ i=m+1 (3`3)iφ ( u 3i , u 3i ) for all u ∈ P . 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 Uµ, this sequence { 1 9n mq ( u 3n )} turns out to be Cauchy for all u ∈ P and hence it is µ−convergent in Uµ. Hence, we have a mapping D : P −→ Uµ given by D(u) = µ− lim n→∞ 1 9n mq ( u 3n ) , that is, limn→∞ µ ( 1 9nmq ( u 3n ) −D(u) ) = 0 for all u ∈ P . So, without using Fatou property, we observe from ∆ 1 3 -condition that the inequality µ (mq(u)−D(u)) ≤ 3µ ( 1 3 mq(u)− 1 3 · 1 9n mq ( u 3n )) + 3µ ( 1 3 · 1 9n mq ( u 3n ) − 1 3 D(u) ) ≤ 3 k µ ( mq(u)− 1 9n mq ( u 3n )) + 3kµ ( 1 9n mq ( u 3n ) −D(u) ) ≤ 3 ` n∑ i=0 ( 3`3 )i φ ( u 3i , u 3i ) + 3`µ ( 1 9n mq ( u 3n ) −D(u) ) 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−nu, 3−nv) in (2), we find that µ ( 3−nmq ( 3−2nuv 3−n(2u+ v) ) + 3−nmq ( 3−2nuv 3−n(2u− v) ) − 8 · 3−nmq(3 −nu)− 2 · 3−nmq(3 −nv) ) ≤ `2nφ ( u 3n , v 3n ) which approaches zero as n→∞ for all u, v ∈ P . Thus, in liue of the convexity of µ, we have µ ( 1 13 D ( uv 2u+ v ) + 1 13 D ( uv 2u− v ) − 8 13 D(u)− 2 13 D(v) ) B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1166 ≤ 1 13 µ ( 1 13 D ( uv 2u+ v ) − 3−nmq ( 3−nuv 2u+ v ) + 1 13 D ( uv 2u− v ) − 3−nmq ( 3−nuv 2u− v ) + 8 13 µ ( D(u)− 3−nmq ( 3−nu )) + 2 13 µ ( D(v)− 3−nmq ( 3−nv )) + 1 13 µ ( 3−nmq ( 3−nuv 2u+ v ) + 3−nmq ( 3−nuv 2u− v )) − 8 · 3−nmq ( 3−nu ) − 2 · 3−nmq ( 3−nv )) 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′ : P −→ Uµ satisfying µ ( mq(u)−D′(u) ) ≤ 3 k ∞∑ i=0 (3`3)iφ ( u 3i , u 3i ) . Then we see from the equalities: D(3−nu) = 9nD(u) and D ′ (3−nu) = 9nD ′ (u) that µ ( D(u)−D′(u) ) ≤ 3µ ( 1 3 · 1 9n D ( u 3n ) − 1 3 · 1 9n mq ( u 3n )) + 3µ ( 1 3 · 1 9n mq ( u 3n ) − 1 3 · 1 9n D′ ( u 3n )) ≤ 3`−(2n+1)µ ( D ( u 3n ) −mq ( u 3n )) + 3`−(2n+1)µ ( mq ( u 3n ) −D′ ( u 3n )) ≤ 3`−3n ∞∑ i=1 (3`3)iφ ( u 3(n+i) , u 3(n+i) ) ≤ 31−n `n ∞∑ i=0 (3`3)iφ ( u 3i , u 3i ) for all u ∈ P . It indicates from the above inequality that D is distinctive by allowing n→∞. Hence the proof is complete. 4. Modular stability of equation (1) without ∆ 1 3 -condition In this present section, we provide a different result related to modular stability of equation (1) without ∆ 1 3 -condition. B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1167 Theorem 3. Assume that Up is a p-complex modular space where p is convex. Also, let φ : U × U −→ [0,∞) be a function with the condition φ̂(u, v) = ∞∑ i=0 1 9i+1 φ(3−iu, 3−iu) <∞ (5) for all u, v ∈ U . Assume that mq : U −→ Up is a mapping such that p ( Γmq(u, v) ) ≤ φ(u, v) (6) for all u, v ∈ U . Then a unique reciprocal second power function T : U −→ Up exists and satisfies p (mq(u)− T (u)) ≤ φ̂(u, v) (7) for all u, v ∈ U . Proof. Putting (u, v) as (u, u) in (6) and then dividing by 9 on both sides, we obtain p ( 1 9 mq(3 −1u)−mq(u) ) ≤ 1 9 mq(u, u) (8) for all u ∈ U . Then by induction arguments, we arrive at p ( mq(3 −nu) 9n −mq(u) ) ≤ 1 9 n−1∑ i=0 1 9i φ(3−iu, 3−iu) (9) 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: p ( mq(3 −(n+1)u) 9n+1 −mq(u) ) = p ( 1 9 ( mq(3 −nu) 9n −mq(3 −1u) ) + 1 9 ( mq(3 −1u)− 9mq(u) )) ≤ 1 9 p ( mq(3 −nu)−mq(3 −1u) ) + 1 9 p ( mq(3 −1u)− 9mq(u) ) ≤ 1 9 . 1 9 n−1∑ i=0 φ(3−(i+1)u, 3−(i+1)u) 9i + 1 9 φ(u, u) ≤ 1 9 ( n−1∑ i=0 φ(3−(i+1)u, 3−(i+1)u) 9i+1 ) + 1 9 φ(u, u) = 1 9 n∑ i=0 φ ( 3−iu, 3−iu ) 9n B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1168 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 p ( mq(3 −nu) 9n − mq(3 −mu) 9m ) = p ( 1 9m ( mq(3 −nu) 9n−m −mq(3 −mu) )) ≤ 1 9m · 1 9 n−m−1∑ i=0 1 9i φ ( 3−(m+i)u, 3−(m+i)u ) ≤ 1 9 n−m−1∑ i=0 1 9m+i φ ( 3−(m+i)u, 3−(m+i)u ) ≤ 1 9 n−1∑ k=m 1 9k φ ( 3−ku, 3−ku ) (10) for all u ∈ U . By the application of (5) and (10), we observe that the the sequence{ mq(3 −nu) 9n } turns out to be Cauchy in Up. By virtue of completeness of Up, the sequence is convergent. This formulates that there exists a function T : U −→ Up defined by T (u) = p− lim mq(3 −nu) 9n . (11) To confirm that T satisfies (1), plugging (u, v) into (3−nu, 3−nv) in (6) and then multiplying by 9−n on both sides, we obtain 9−np ( mq ( 3−n ( uv 2u+ v )) +mq ( 3−n ( uv 2u− v )) − 8mq(3 −nu)− 2mq(3 −nv) ) ≤ 9−nφ(3−nu, 3−nu) (12) 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−np ( T ′ (3−nu)− T (3−nu) ) ≤ 9−n ( p ( T ′ (3−nu)− f(3−nu) ) + p ( f(3−nu)− T (3−nu) )) ≤ ∞∑ i=n+1 1 9i φ(3−iu, 3−iu) (13) 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. Corollary 1. Let mq : U −→ Up be a mapping with a constant c ≥ 0, not depending on the values of u, v such that the inequality p ( Γmq(u, v) ) ≤ c B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1169 holds for all u, v ∈ U . Then, T : U −→ Up is a unique reciprocal second power function satisfying (1) and p (mq(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 λ1 ≥ 0 be fixed and s 6= −2 if a function mq : U −→ Up fulfills the inequality p ( Γmq(u, v) ) ≤ λ1(|u|s + |v|s) holds for all u, v ∈ U . Then, there exists a unique reciprocal second power function T : U −→ Up satisfying (1) and p (mq(u)− T (u)) ≤ 2λ1 (9− 3−s) |u|s for all u ∈ U . Proof. The proof is obtained by taking φ(u, v) = λ1(|u|s + |v|s) in Theorem 3. Corollary 3. Let mq : U −→ Up be a mapping. If there exist x, y : s = x + y 6= −2 and λ2 ≥ 0 such that p ( Γmq(u, v) ) ≤ λ2(|u|x|v|y) holds for all u, v ∈ U . Then, there exists a unique reciprocal second power function T : U −→ Up satisfying (1) and p (mq(u)− T (u)) ≤ λ2 (9− 3−s) |u|s for all u ∈ U . Proof. The proof directly follows by taking φ(u, v) = c2(|u|a|v|b) in Theorem 3. Corollary 4. Let λ3 ≥ 0 be fixed and s 6= −1. If a function mq : U −→ Up satisfies the inequality p ( Γmq(u, v) ) ≤ λ3(|u|s|v|s + (|u|2s + |v|2s)) for all u, v ∈ U . Then, a unique reciprocal second power function T : U −→ Up exists and satisfies (1) and p (mq(u)−mq(v)) ≤ 3λ3 (9− 3−2s) |u|2s for all u ∈ U . Proof. The proof is achieved by considering φ(u, v) = λ3(|u|s|v|s + (|u|2s + |v|2s)) in Theorem 3. B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1170 5. Stability of equation (1) in β-homogeneous spaces In this section, we obtain the stability results of equation (1) in β-homogenous spaces. Theorem 4. Let V be a β-homogeneous complex Banach space (0 < β ≤ 1), and φ : U × U −→ (0,∞] be a function with φ̂(u, v) = 1 9β ∞∑ i=0 1 9βi φ ( 3−iu, 3−iu ) <∞ (14) for all u, v ∈ U . Assume that mq : U −→ V is a mapping such that∥∥Γmq(u, v) ∥∥ ≤ φ(u, v) (15) holds for all u, v ∈ U . Then there exists a unique reciprocal second power function T : U −→ V such that ‖mq(u)− T (u)‖ ≤ φ̂(u, u) (16) for all u ∈ U . Proof. Firstly, let us substitute v = u in (15). Then, we get∥∥mq(3 −1u)− 9mq(u) ∥∥ ≤ φ(u, u) (17) for all u ∈ U . By employing induction technique on k ∈ N and using (17), we acquire∥∥∥∥mq(3 −nu) 9n −mq(u) ∥∥∥∥ ≤ 1 9β n−1∑ i=0 φ(3−iu, 3−iu) 9iβ (18) for all u ∈ U . Let m and n be non-negative integers with n > m, Then using (18), we have ∥∥∥∥mq(3 −nu) 9n − mq(3 −nu) 9m ∥∥∥∥ = ∥∥∥∥ 1 9m ( mq(3 −n) 9n−m −mq(3 −m) )∥∥∥∥ ≤ 1 9mβ 1 9β n−m−1∑ i=0 φ(3−(i+m)u, 3−(i+m)u) 9iβ = 1 9β n−1∑ i=m 1 9iβ φ ( 3−iu, 3−iu ) (19) for all u ∈ U . Letting n → ∞ in the above inequality, we find that the sequence{ mq(3 −nu) 9n } becomes Cauchy in U . Due to the completeness of U , the sequence is convergent. Hence there exists a mapping T : U −→ V defined by T (u) = lim n→∞ mq(3 −nu) 9n (20) B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1171 for all u ∈ U . Putting m = 0 and taking the limit n → ∞ in the above inequality, we obtain (16) using (20). Next, consider an additional function S : U −→ V satisfying (16) and (20). Then, we get ‖T (u)− S(u)‖ ≤ ∥∥∥∥T (3−n)−mq(3 −n) 9n ∥∥∥∥+ ∥∥∥∥mq(3 −n)− S(3−n) 9n ∥∥∥∥ ≤ 1 9β ∞∑ i=0 1 9(i+n)β φ ( 3−(n+i)u, 3−(n+i)u ) = 1 9β ∞∑ i=n 1 9iβ φ ( 3−iu, 3−iu ) . From the above inequality, weobserve T is unique by letting n→∞, which completes the proof. Corollary 5. Let mq : U −→ V be a function with a constant λ4 ≥ 0, not depending on the values of u, v such that the inequality∥∥Γmq(u, v) ∥∥ ≤ λ4 holds for all u, v ∈ U . Then, T : U −→ V is a unique reciprocal second power function satisfying (1) and ‖mq(u)− T (u)‖ ≤ λ4 9β − 1 for all u ∈ U . Proof. Taking φ(u, v) = λ4 in Theorem 4, we arrive at the required result. Corollary 6. Let λ5 ≥ 0 be fixed and s 6= −2β. Suppose a function mq : U −→ V satisfies the inequality ∥∥Γmq(u, v) ∥∥ ≤ λ5(||u||s + ||v||s) for all u, v ∈ U . Then, a unique reciprocal second power function T : U −→ V exists and satisfies (1) and ‖mq(u)− T (u)‖ ≤ 2λ5 (9β − 3−s) ||u||s for all u ∈ U . Proof. Replacing φ(u, v) = λ5(||u||s + ||v||s) in Theorem 4 and proceeding further, we obtain the desired result. Corollary 7. Let mq : U −→ V be a function. If there exist x, y : s = x + y 6= −2β and λ6 ≥ 0 such that ∥∥Γmq(u, v) ∥∥ ≤ λ6(||u||x||v||y) B. V. Senthil Kumar, Hemen Dutta, S. Sabarinathan / Eur. J. Pure Appl. Math, 13 (5) (2020), 1162-1175 1172 holds for all u, v ∈ U . Then, there exists a unique reciprocal second power function T : U −→ V satisfying (1) and ‖mq(u)− T (u)‖ ≤ λ6 (9β − 3−s) ||u||s for all u ∈ U . Proof. Choosing φ(u, v) = λ6(|u|x|v|y) in Theorem 4, we achieve the result. Corollary 8. Let λ7 ≥ 0 be fixed and s 6= −β. Let a function mq : U −→ V satisfies the inequality ∥∥Γmq(u, v) ∥∥ ≤ λ7(||u||s||v||s + (||u||2s + ||v||2s)) for all u, v ∈ U . Then, a unique reciprocal second power function T : U −→ V exists and satisfies (1) and ‖mq(u)− T (u)‖ ≤ 3λ7 (9β − 3−2s) ||u||2s for all u ∈ U . Proof. Selecting φ(u, v) = λ7(||u||s||v||s + (||u||2s + ||v||2s)) in Theorem 4, we get the required result. 6. 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. Figure 1: Electrostatic force of attraction F between two point charges q1 and q2 That is, F = 1 4πε0 q1q2 r2 where F and r, respectively, are the force of attraction and distance between the point charges q1 and q2. Suppose the constant 1 4πε0 is taken as a constant c and unit point charges are assumed, then the electrocstatic force of attraction is given by F = c r2 REFERENCES 1173 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 mq ( uv 2u+ v ) = c(2u+ v)2 u2v2 . Also, if the distance is uv 2u−v , then the electrocstatic force of attraction is given by mq ( uv 2u− v ) = c(2u− v)2 u2v2 . Then using equation (1), we can relate that the sum of the above electrocstatic forces of attraction mq ( uv 2u+v ) and mq ( uv 2u−v ) is given by the sum of electrocstatic forces of attraction 2mq(u) = 2c u2 and 8mq(v) = 8c v2 . Hence equation (1) dealt in this study can be associated with the electrocstatic forces of attraction between the charges in different situations. 7. Conclusion In this investigation, we introduced a new reciprocal second power FE (1) and inves- tigated 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 asso- ciated equation (1) with Coloumb’s law to employ it in various situations to connect the electrocstatic forces of attraction in different assumptions. Acknowledgements The authors are thankful to the referees for their fruitful comments and suggestions. References [1] I Amemiya. On the representation of complemented modular lattices. J. Math. Soc., 9:263–279, 1957. [2] A Bodaghi and B V Senthil Kumar. Estimation of inexact reciprocal-quintic and reciprocal-sextic functional equations. Mathematica, 49(82)1-2:3–14, 2017. [3] K Cieplinski. Applications of fixed point theorems to the hyers-ulam stability of functional equations-a survey. Ann. Funct. Anal., 3:151–164, 2012. [4] H Dutta and B V Senthil Kumar. 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