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A New Approach to Involution in Fuzzy C★-Algebra via Functional Inequality and Python Implementation

A New Approach to Involution in Fuzzy C★-Algebra via Functional Inequality and Python... axioms Article A New Approach to Involution in Fuzzy C -Algebra via Functional Inequality and Python Implementation 1, 2, Ehsan Movahednia * and Manuel De la Sen * Department of Mathematics, Behbahan Khatam Alanbia University of Technology-Iran, Behbahan 6361663973, Iran Department of Electricity and Electronics, Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa (Bizkaia), 48080 Bilbao, Spain * Correspondences: movahednia@bkatu.ac.ir (E.M.); manuel.delasen@ehu.eus (M.D.l.S) Abstract: This article explores the stability of involution in fuzzy C -algebras through the use of a functional inequality. We present an approach to obtaining an approximate involution in fuzzy C -algebras by utilizing a fixed-point method. Moreover, for greater clarity, we implemented Python code for the main theorem. Keywords: functional inequality; Hyers–Ulam stability; fixed-point theorem; Python programming MSC: 39B05; 39B82; 46S40; 47H106 1. Introduction Hyers–Ulam stability is a fundamental concept in the field of functional equations. The issue of the stability of functional equations arises from the question: “Is a function that approximately satisfies a functional equation also close to a function that exactly satisfies the same functional equation”. In the field of mathematics, the Hyers–Ulam theorem has been a topic of great interest and discussion among scholars. This concept was first Citation: Movahednia, E.; De la Sen, introduced in 1940 by Stanislaw Ulam in his famous speech at the University of Wisconsin, M. A New Approach to Involution in ? where he presented a problem on the stability of functional equations [1]. This problem was Fuzzy C -Algebra via Functional solved by Donald Hyers in 1941 for additive mappings [2]. Hyers’s theorem was as follows: Inequality and Python Implementation. Axioms 2023, 12, 435. Theorem 1 ([2]). Assume that U and V are Banach spaces, and let f be a mapping from U to V https://doi.org/10.3390/ axioms12050435 that satisfies the inequality k f (u + v) f (u) f (v)k  d Academic Editors: Zhenyu Zhang, Vladimir Simic and Jing Li for some d > 0 and all u, v 2 U. Then, there exists a unique additive mapping T : U ! V such that Received: 3 April 2023 kT(u) f (u)k  d Revised: 20 April 2023 Accepted: 23 April 2023 holds for every u 2 U. Published: 27 April 2023 Ten years later, Takashi Aoki had a major impact on the development of the Hyers– Ulam stability theorem by introducing a modification to the control function, replacing d p p with K(kxk +kyk ) for K > 0 and 0  p < 1 [3]. In a later development, Rassias extended Copyright: © 2023 by the authors. the Hyers–Ulam stability theorem and Aoki’s results to linear mappings with different Licensee MDPI, Basel, Switzerland. control functions [4]. This article is an open access article Today, Hyers–Ulam stability has become an important topic of research in mathe- distributed under the terms and matics, as it has applications in several fields, including physics, engineering, economics, conditions of the Creative Commons and computer science. The results of Hyers–Ulam stability have been extended to various Attribution (CC BY) license (https:// types of functional equations, including quadratic, cubic, Jensen, differential, and integral creativecommons.org/licenses/by/ 4.0/). equations, among others. Axioms 2023, 12, 435. https://doi.org/10.3390/axioms12050435 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 435 2 of 14 Many mathematicians have contributed to the development of the theory of Hyers– Ulam stability by proposing and proving new theorems. They have changed the type of functional equation, control function, and space in the Hyers–Ulam stability theorem to investigate and prove new conditions [5–9]. The Hyers–Ulam stability theorem has been used to prove many other results in different branches of mathematics. In 1984, Katrasas introduced the concept of fuzzy norm spaces [10], where a fuzzy norm is defined in a linear space and the topological structure of the fuzzy vector is established. Several mathematicians have investigated fuzzy norms on linear spaces from a variety of perspectives over time. One important study [11] in 2003 added a fuzzy norm and created a fuzzy measure in the concept of Kramosil and Michalek [12]. They also formulated a theorem on decomposing a fuzzy norm into a set of crisp norms and examined certain features of fuzzy-normed spaces. In recent times, there has been a surge in the study of functional equations stability in fuzzy-normed spaces. Researchers have explored several fuzzy stability outcomes relating to Cauchy, Jensen, simple quadratic, and cubic functional equations. Additionally, the concept of intuitionistic fuzzy-normed spaces has been introduced, and stability results in this area have also been examined. An investigation into the relationship between Ulam’s stability and self-testing cor- recting programs has recently been carried out in [13]. Based on this research, we aim to implement Hyers–Ulam stability in fuzzy C algebra by using the Python program- ming language. Python is a high-level programming language that can be used in various ways. It has access to powerful libraries, such as Numpy, which enable efficient calculations in mathematics and science. Travis Oliphant founded the open-source Numpy project in 2005 [14] and a huge group of collaborators actively maintain and develop it today. We will attempt to write Python code to implement the Hyers–Ulam stability concept. This explains Hyers–Ulam stability for specialists in the computer science and mathematics fields. Definition 1. A function d : U U ! [0,¥] is said to be a generalized metric on the set U if it satisfies the following properties: ( M ) d(u, v) = 0 if and only if u = v ( M ) d(u, v) = d(v, u) for all u, v 2 U ( M ) d(u, w)  d(u, v) + d(v, w) for all u, v, w 2 U. We now introduce one of the fundamental results of the fixed-point theory. Theorem 2 ([15,16]). Suppose (U, d) is a generalized complete metric space and let G : U ! U be a strictly contractive operator with the Lipschitz constant L < 1. Assume there exists a non-negative n +1 n 0 0 integer n such that d G u, G u < ¥ for some u 2 U. Then, the following conclusions hold: (i) The sequence G u converges to a fixed point u of G, (ii) The fixed point u is unique in V = fv 2 U j d(G u, v) < ¥g, (iii) For any v 2 V, the inequality d(v, u )  d(Gv, v), 1 L holds. Definition 2 ([11]). A fuzzy norm on a real vector space U is a function N : U R ! [0, 1] that satisfies the following conditions for any u, v 2 U and s, t 2 R: (N ) N(u, t) = 0 for all t  0; (N ) u = 0 if and only if N(u, t) = 1 for all t > 0; (N ) N(cu, t) = N(u, ) if c 6= 0; jcj (N ) N(u + v, s + t)  minfN(u, s), N(v, t)g; 4 Axioms 2023, 12, 435 3 of 14 N N(u, .) is a non-decreasing function of R and lim N(u, t) = 1; ( ) 5 t!¥ (N ) For u 6= 0, N(u, .) is continuous on R. The pair (U, N) is then referred to as a fuzzy-normed vector space. Definition 3. Consider a fuzzy-normed vector space (U, N). (1) A sequence fu g in U is said to be convergent if there exists u 2 U such that for all t > 0, lim N(u u, t) = 1. The limit of fu g is denoted as N-limit, i.e., N lim u = u. n!¥ n n n!¥ n (2) A sequence fu g in U is called Cauchy if for every e > 0 and t > 0, there exists n 2 N such n 0 that for all n  n and p > 0, N u u , t > 1 e. 0 n+ p n It is a well-known fact that in a fuzzy-normed space, every convergent sequence is also Cauchy. If every Cauchy sequence in a fuzzy-normed space converges, then the fuzzy norm is said to be complete, and the fuzzy-normed space is referred to as a fuzzy Banach space. A mapping of f : U ! V between two fuzzy-normed vector spaces U and V is said to be continuous at a point u 2 U if, for every sequence fu g converging to u in 0 0 U, the sequence f f (u )g converges to f (u ). If f : U ! V is continuous at every point n 0 u 2 U, then f : U ! V is said to be continuous on U. Definition 4 ([17]). Let U be an algebra and (U, N) a fuzzy-normed space. The fuzzy-normed space (U, N) is called a fuzzy-normed algebra if 0 0 0 + N uu , st  N(u, s)N u , t 8u, u 2 U, s, t 2 R . Complete fuzzy-normed algebra is called a fuzzy Banach algebra. Example 1. Every normed algebra (U,k.k) defines a fuzzy-normed algebra (U, N), where N is defined by N(u, t) = 8u 2 U,8t > 0. t +kuk This space is called the induced fuzzy-normed algebra. Definition 5. Let U be a complex algebra. An involution on U is a function ? : U ! U defined by a 7! a , satisfying the following properties: ? ? ? (i) (aa + bb) = aa + bb for all a, b 2 U and a, b 2 C; ? ? ? (ii) (ab) = b a for all a, b 2 U; ?? (iii) a = a for all a 2 U. If U is a complex algebra with an involution, then it is called a ?-algebra. A C -algebra is a ? 2 Banach algebra with an involution ? such that ka ak = jjajj . Definition 6 ([18]). Let U be an ?-algebra and (U, N) a fuzzy-normed algebra. The fuzzy-normed algebra (U, N) is called a fuzzy-normed ?-algebra if ? + N(a , t) = N(a, t) 8a 2 U,8t 2 R . A complete fuzzy-normed ?-algebra is called a fuzzy Banach ?-algebra. Definition 7 ([18]). Let (U, N) be a fuzzy Banach ?-algebra. Then (U, N) is called a fuzzy C -algebra if ? ? + N(a a, st) = N(a , s)N(a, t) 8a 2 U,8s, t 2 R . 2. Results In this section, we will use the fixed-point theorem and functional inequalities to prove the existence of a unique involution for the fuzzy Banach algebra (U, N). We will also demonstrate under what conditions this fuzzy Banach algebra can be transformed into a Axioms 2023, 12, 435 4 of 14 C -algebra. For this purpose, we will first prove two simple lemmas and then proceed to prove the main theorems. Lemma 1. Let F : U ! U be a mapping satisfying the following N(F(2u) + F(2v) + 2F(w), t)  N F(u + v + w), , (1) for all u, v, w 2 U. Then, F is additive, i. e., F(u + v) = F(u) + F(v), for all u, v 2 U. Proof. Putting u = v = w = 0 in (1), we obtain t t t N(4F(0), t)  N F(0), =) N F(0),  N F(0), , 3 4 3 for all t > 0. After applying N(5) and N(6), it can be deduced that the value of N(F(0), t) is 1. It means that F(0) = 0. Letting v := u and w = 0 in (1) so we have N(F(2u) + F(2u), t)  N F(0), = 1, thus F(u) = F(u). Next, putting w := u and v = 0 in (1), we get N(F(2u) + 2F(u), t)  N F(0), = 1, therefore F(2u) = 2F(u). Finally, replacing w by u v in (1), we have N(F(2u) + F(2v) + 2F((u + v), t)  N 2F(u + v u v), = 1, so, F(2u) + F(2v) = 2F((u + v)), i. e., F is additive. Lemma 2. Let F : U ! U be a mapping satisfying N(F(2u) + m ¯ F(2v) + 2F(w), t)  N F(u + mv + w), , (2) for all m 2 T := fz 2 C : jzj = 1g, and for every u, v, w 2 U, then F is Clinear. Proof. We prove that F(au) = a ¯ F(u) for all a 2 C. If m = 1 in (2), then F is additive by Lemma 1. Substituting u, v, w by mu,v, 0 in (2), respectively. We have N(F(2mu) + m ¯ F(2u), t)  N F(mu mu + 0), = 1, therefore, F(2mu) + m ¯ F(2u) = 0 ! F(2mu) = m ¯ F(2u), that is, F(mu) = m ¯ F(u) for all m 2 T and u 2 U. We know that m + m ¯ jmj = jm ¯j = 1, = t (real part of m), m+m ¯ so jtj =  1. Thus, for all real number t with jtj  1 and u 2 U, we get F(2tu) = F(mu + m ¯ u) = F(mu) + F(m ¯ u) = mF(u) + m ¯ F(u) = 2tF(u) Axioms 2023, 12, 435 5 of 14 for all m 2 T and u 2 U. So, F(tu) = tF(u) for all u 2 U and jtj  1. Moreover, It is a n n known fact that if F(2u) = 2F(u), then it follows that F(2 u) = 2 F(u) for any positive integer n. Additionally, by utilizing the Archimedean property, we can determine that, for any real number t, there exists a positive integer n such that jtj  2 . Hence, t t t n n n F(tu) = F 2 . u = 2 F u = 2 . F(u) = tF(u), n n n 2 2 2 for all t 2 R. Suppose that a 2 C be an arbitrary complex number. Therefore, a = t + is for some real numbers t, s. Since jij = 1, so F(iu) = iF(u) for all u 2 U. Finally, we get F(au) = F((t + is)u) = F(tu + isu) = F(tu) + F(isu) (F is additive) = tF(u) + sF(iu) = tF(u) + isF(u) = tF(u) isF(u) = a ¯ F(u), Then, F is Clinear. 3 2 Theorem 3. Let j : U ! (0,¥) and F : U ! [0,¥) be functions such that there exists an L < 1 with j(u, v, w)  j(2u, 2v, 2w), (3) and u v lim 4 F( , ) = 0, (4) n n n!¥ 2 2 for all u, v, w 2 U. In addition, assume F : U ! U to be an odd function satisfying t t N(F(2u) + m ¯ F(2v) + 2F(w), t)  min N 2F(u + mv + w), , , (5) 3 t + j(u, v, w) for all u, v, w 2 U and t > 0. Moreover, suppose that N(F(uv) F(v)F(u), t)  , (6) t + F(u, v) and k k m m N lim 2 F 2 (N lim 2 F(2 u)) = u (7) m!¥ k!¥ for all u, v 2 U and t > 0 and all m 2 T . Then there exists a unique involution A : U ! U such that A(u) := N lim 2 F (8) k!¥ 2 and (2 2L)t N(F(u) A(u), t)  (9) (2 2L)t + j(0, u,u) for all u 2 U and t > 0. Theorem 4. Under the assumptions of Theorem (3), if N([N(F(u), t) N(u, t)]u, t)  (10) t + j(u, u, 0) Then, (U, N) is fuzzy Banach ?algebra. Moreover, if N([N(F(u).u, st) N(F(u), s).N(u, t)]u, t)  (11) t + j(u, u, 0) Axioms 2023, 12, 435 6 of 14 ? ? Then (U, N) is fuzzy C -algebra with involution u = A(u) for all u 2 U. Furthermore, if N(F(u) u, t)  , (12) t + F(u, u) then u is self-adjoint element of U. Proof of Theorem 3. Replacing m, u, v, w by 1, 0, 2u,2u in (5), we have t t N(F(0)) + F(4u) + 2F(2u), t)  min N(2F(0), ), 3 t + j(0, 2u,2u) = , t + j(0, 2u,2u) therefore, N(F(4u) 2F(2u), t)  , (13) t + j(0, 2u,2u) for all u 2 U and t > 0. So N(F(2u) 2F(u), t)  , t + j(0, u,u) for all u 2 U and t > 0. We define the set S and introduce the function d : S S ! [0,¥) as follows. S : fG : U ! U j G(0) = 0g d(G, H) = inf l 2 R : N(G(u) H(u), lt)  , (14) t + j(0, u,u) where, as usual, infÆ = +¥. The proof that (S , d) is a generalized complete metric space has been investigated in ([19]). Now we consider the linear mapping P : S ! S such that PG(u) := 2G for all u 2 U. We prove that P : G ! G is a strictly contractive mapping with Lipschitz constant L. For this purpose, Let G, H 2 S be given such that d(G, H) = e. So, according to the definition of metric d in (14), we have N(G(u) H(u), et) t + j(0, u,u) for all u 2 U and t > 0. Therefore, u u N(PG(u) P H(u), Let) = N 2G 2H , Let 2 2 u u L = N G H , et 2 2 2 Lt Lt u u + j(0, , ) 2 2 2 Lt (by definition of j) Lt L + j(0, u,u) 2 2 for all u 2 U and t > 0. Then, assuming d(G, H) = e, we have proved that d(PG,P H) = Le. This means that d(PG,P H)  Ld(G, H), Axioms 2023, 12, 435 7 of 14 for all G, H 2 S . Therefore P is a strictly contractive mapping with Lipschitz constant L. Replacing u by in (13) and applying (3), we obtain U t t N F(u) 2F , t u u t + j(0, , ) t + j(0, u,u) 2 2 Lt Substituting by t in the above inequality, we obtain U Lt t N F(u) 2F , 2 2 t + j(0, u,u) for all u 2 U and t > 0. Therefore, d(F,PF) < . (15) The conditions of Theorem (2) are satisfied. Hence, (I) There exists a mapping A : U ! U such that it is a fixed point P. This means that u 1 A = A(u), (16) 2 2 for all u 2 U. The mapping A is a unique fixed point of P in the set G : fG 2 S : d(F, G) < ¥g. This suggests that A : U ! U is a unique mapping satisfying (16), furthermore, there exists a l 2 (0,¥) satisfying N(F(u) A(u), lt) t + j(0, u,u) for all u 2 U. (II) d(P F, A) ! 0 as n ! ¥. This suggests that n n A(u) = lim 2 F(2 u) (17) n!¥ for every u 2 U. d(F,PF) (III) d(F, A)  . By (15), we have 1L d(F, A)  . 2 2L u v w This suggests that (9) is held. Replacing u, v, w by , , , respectively, in (5); therefore, n n n 2 2 2 2u 2v w n n n n N 2 F + m ¯ 2 F + 2 2F , 2 t n n n 2 2 2 u v w 2 t t n+1 min N(2 F + m + , ), , u v w n n n 2 2 2 3 t + j( , , ) n n n 2 2 2 for all u, v, w 2 U and t > 0. Replacing t by , we thus obtain 2u 2v w n n n N 2 F( ) + m2 F( ) + 2 2F( ), t (18) n n n 2 2 2 ( ) u v w t n+1 2 min N(2 F( + m + ), ), , n n n t u v w 2 2 2 3 + j( , , ) n n n n 2 2 2 2 Axioms 2023, 12, 435 8 of 14 for all u, v, w 2 U and t > 0. Note that, by using (3), we obtain u v w L j , ,  j(u, v, w). 2 2 2 2 u v w Putting u, v, w by , , in the above inequality, such that 2 2 2 u v w L u v w L j , ,  j , ,  j(u, v, w). 2 2 2 2 2 2 2 2 2 2 2 2 By continuing this process, we have u v w L j , ,  j(u, v, w). n n n n 2 2 2 2 Therefore t t n n 2 2 = . t u v w t L n t + L j(u, v, w) + j( , , ) + j(u, v, w) n n n n n n 2 2 2 2 2 2 Thus, lim > 1. t u v w n!¥ + j( , , ) n n n n 2 2 2 2 By passing n ! ¥ in (18) and using (17), we obtain N( A(2u) + m ¯ A(2v) + 2 A(w), t)  N(2 A(u + mv + w), ), for all u, v, w 2 U, t > 0 and m 2 T . By Lemma (2), the mapping A : U ! U is C-linear. u v We replace , with u, v, respectively, in (6). Hence, n n 2 2 u v v u t N F( ) F( )F( ), t  , u v n n n n 2 2 2 2 t + F( , ) n n 2 2 therefore, u v v u t n n n n N 4 F( . ) 2 F( ).2 F( ), 4 t  , u v n n n n 2 2 2 2 t + F( , ) n n 2 2 so, u v v u N F( ) F( )F( ), t  . t u v n n n n 2 2 2 2 + F( , ) n n n 4 2 2 Passing n ! ¥ and applying (4) and (17), we get N( A(u.v) A(v) A(u), t)  1 ! A(u.v) = A(v) A(u). Furthermore, by (7) we have k k m m A( A(u)) = N lim 2 f (2 (N lim 2 f (2 u))) = u. m!¥ k!¥ We proved that A : U ! U is Clinear, A(u.v) = A(v). A(u) and A( A(u)) = u. These mean that A is involution. Axioms 2023, 12, 435 9 of 14 Proof of Theorem 4. Putting u = in (10), we obtain h i u u u t n n N N(2 F( ), 2 t) N( , t) , t  , for all t > 0, u 2 U u u n n n 2 2 2 t + j( , , 0) n n 2 2 therefore, h i u t n n n n N N(2 F( ), 2 t) N(u, 2 t) u, 2 t  , u u 2 t + j( , , 0) n n 2 2 so, h i t n 2 N N(2 F( ), t) N(u, t) u, t n t u u + j( , , 0) n n n 2 2 2 t + L j(u, u, 0) by passing n ! ¥, we have N([N( A(u), t) N(u, t)]u, t)  1 ! N( A(u), t) N(u, t)]u = 0, for all u 2 U and t > 0. This suggests that N( A(u), t) = N(u, t). Then, (U, N) is fuzzy Banach ?algebra. Next, we suppose that A satisfies in (11), so we obtain h i u u t n 2n n n n n N [N(2 F( ).u, 2 st) N(2 F( ), 2 s).N(u, 2 t) u, 2 t  , u u n n 2 2 t + j( , , 0) n n 2 2 for all u 2 U and t, s > 0. Thus, h i t u u n n n N N(2 F( ).u, st) N(2 F( ), s).N(u, t) u, t  , n n t u u 2 2 + j( , , 0) n n n 2 2 2 t + L j(u, u, 0) for every u 2 U and t, s > 0. By passing n ! ¥, we have N([N( A(u).u, st) N( A(u), s).N(u, t)]u, t)  1. This means that N( A(u).u, st) = N( A(u), s).N(u, t) ? ? for all u 2 U and t > 0. Therefore, U is a C -algebra with involution u = A(u) for all u 2 U. Moreover, replacing u with in (12), we obtain u t n n N 2 F( ) u, 2 t  , u u 2 t + F( , ) n n 2 2 therefore n 2 N 2 F( ) u, t  , n t u u + F( , ) n n n 2 2 2 by passing n ! ¥ and using (4), we have N( A(u) u, t) = 1 ! u = A(u) = u it means that u is self-adjoint element of U. Axioms 2023, 12, 435 10 of 14 Corollary 1. Let p > 1, q > 2. Suppose that h, h : U ! [0,¥) are functions such that u 1 h = h(u) (19) 2 2 u 1 0 0 h = h (u) (20) 2 2 for all u 2 U. Suppose that F : U ! U is a odd function that satisfies N(F(2u) + m ¯ F(2v) + 2F(w), t)  (21) t + h(u) + h(v) + h(w) for all u, v, w 2 U. Moreover, assume that N(F(uv) F(v)F(u), t)  , (22) 0 0 t + h (u) + h (u) and k k m m N lim 2 F(2 (N lim 2 F(2 u))) = u (23) m!¥ k!¥ for all u, v 2 U and t > 0 and all m 2 T . Then, there exists a unique involution A : U ! U such that A(x) := N lim 2 F (24) k!¥ 2 and (2 4)t N(F(u) A(u), t)  (25) (2 4)t + h(2u) + h(2u) for all u 2 U and t > 0. 3 2 Proof. We define j : U ! (0,¥) and F : U ! [0,¥) by j(u, v, w) = h(u) + h(v) + h(w) for all u, v, w 2 U, 0 0 F(u, v) = h (u) + h (v) for all u, v 2 U. From (19), we have j(2u, 2v, 2w) = h(2u) + h(2v) + h(2w) = 2 h(u) + h(v) + h(w) = 2 j(u, v, w), for all u, v, w 2 U. Therefore, j(u, v, w) = j(2u, 2v, 2w) 1 p = j(2u, 2v, 2w), 2 Axioms 2023, 12, 435 11 of 14 1 p we put L := 2 , since p > 1, so L < 1. On the other u v u v n n 0 0 lim 4 F( , ) = lim 4 h ( ) + h ( ) n n n n n!¥ 2 2 n!¥ 2 2 n nq 0 0 = lim 4 ( ) h (u) + h (v) n!¥ 2 n(2q) 0 0 = lim 2 h (u) + h (v) = 0, since q > 2. n!¥ By Theorem (3), there exists a unique involution A : U ! U such that 1 p (2 2.2 )t N( f (u) A(u), t) 1 p p (2 2.2 )t + 2 h(0) + h(u) + h(u) 2 p (2 2 )t 2 p p (2 2 )t + 2 h(u) + h(u) (2 4)t = , for all u 2 U, t > 0 (2 4)t + h(2u) + h(2u) Note that h(2u) = 2 h(u). 3. Implement Python Code for Theorem 3 In this part, the Python code of Theorem (3) is provided. The purpose of writing the code is to first increase the clarity of the Theorem (3) by providing a practical representation of the assumptions and structure of the theorem. Secondly, it is to make a connection between the two topics of Hyers–Ulam stability and the field of computer science. In this code, the user is asked to enter N, F, j, and F functions. import numpy as np def N(u, t): return # The user defines the fuzzy-normed algebra N. def j(u, v, w): return # The user defines the j function, for example: return (u + v + w)**2 - 4*(u*v + v*w + w*u). def F(u, v): return # The user defines the F function, for example: return np.exp(u) * np.exp(2*v). def F(u): return # The user defines the F function, for example: return np.sin(u). def check_limit(F, u, v): n = 1 while True: term = 4**n * F(u/(2**n), v/(2**n)) if np.abs(term) < 1e-10: return True if n > 1e5: return False n =n+ 1 Axioms 2023, 12, 435 12 of 14 def inequality5(u, v, w, t, m = None): # the execution of inequality (5) return # True if inequality (5) holds, False otherwise def inequality6(u, v, t): # the execution of inequality (6) return # True if inequality (6) holds, False otherwise def Equation (7)(u): # the execution of Equation (7) return # True if Equation (7) holds, False otherwise def inequality9(u): # the execution of inequality (9) return # True if inequality (9) holds, False otherwise def check_assumptions(L, U,T ): # Check if the assumptions of the theorem are satisfied, for u, v, w 2 U: if j(u, v, w) > (L/2) j(2 u, 2 v, 2 w): return False for u, v 2 U: if not check_limit(F, u, v): return False for u, v, w 2 U: t = # The user defines the number t for m 2 T : if not inequality5 (u, v, w, t, m): return False for u, v 2 U: t = # The user defines the number t if not inequality(6) (u, v, t): return False for u 2 U: if not Equation (7)(u): return False return True def A(u,t): #Define the function A k = 0 tolerance= # The user enter tolerance while True: k k A_k = 2  F(u/(2 )) (k+1) (k+1) A_k_next = 2  F(u/(2 )) if N(A_k - A_k_next, t) > tolerance: break k = k+1 return A_k L = # The user defines the number L U = # define the set of values to check the assumptions 1 1 T = # define the T t= # The user enter any t>0 if check_assumptions (L, U,T ): print(f"{A(u,t)} is a unique involution") print(f" A satisfies in {inequality(9)(u)}") else: Axioms 2023, 12, 435 13 of 14 print("The conditions of the Theorem are not upheld") In this code, we first import the Numpy library so that we can use mathematical functions throughout the code. Now, we define the function of two variables N, the func- tion of three variables j, the function of two variables F, and the function F. These functions are given to the system by the user. To check assumption (4), we define the function “check_limit(F, u, v)”. This function has three variables and checks the limit of the function F at infinity. In the following, inequality(5) is defined. It takes five arguments u, v, w, t, and an optional m argument. Then, the inequalities (6), (9), and Equation (7) have been defined by the functions “def inequality(6)(u, v, t)”, “def Equation (7)(u)” and “def inequality(9)(u)”, respectively. Now, we will check whether the assumptions of the Theorem (3) are valid. For this purpose, we define the function “def check_assumptions(L, U,T )”. The task of this function is to check the validity of all five assumptions of the Theorem (3). In “de f A(u, t)”, we define the function A and check whether A is a Cauchy sequence. Note that since U is Banach space the Cauchyness of the sequence guarantees its conver- gence. We have defined the necessary functions. Now, to run the code, we first define the values of L, U, T, and t and call the function “check_assumptions” (L, U,T ). If this function is true, A is declared as an involution that applies to the inequality (9). 4. Conclusions In this article, we first investigated the stability of Hyers–Ulam involution in fuzzy C -algebras using functional inequality and the fixed-point method. We then implemented the code of the main theorem in Python with the aim of making the theorem clearer and making the connection between Hyers–Ulam stability and computer science. We hope to make Hyers–Ulam stability more accessible to researchers in mathematics and computer science and to encourage further research on the connection between these two fields. Author Contributions: Methodology, E.M. and M.D.l.S.; validation, M.D.l.S.; investigation, E.M. and M.D.l.S.; writing—original draft, E.M.; project administration, M.D.l.S.; funding acquisition, M.D.l.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Basque Government, Grant IT1555-22; Basque Govern- ment,Grant KK-2022/00090; MCIN/AEI 269.10.13039/501100011033, Grant PID2021-1235430B-C21. Data Availability Statement: All data required for this paper are included within this paper Conflicts of Interest: The authors declare no conflicts of interest. References 1. Ulam, S.M. Problems in Modern Mathematics; Courier Corporation: Chelmsford, MA, USA, 2004. 2. Hyers, D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27, 222–224. [CrossRef] [PubMed] 3. Aoki, T. On the Stability of the Linear Transformation in Banach Spaces. J. Math. Soc. Jpn. 1950, 2, 64–66. [CrossRef] 4. Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [CrossRef] 5. Forti, G. Hyers-Ulam stability of functional equations in several variables. Aequationes Math. 1995, 50, 143–190. [CrossRef] 6. Czerwik, S. Functional Equations and Inequalities in Several Variables; World Scientific Publishing Co.: Singapore, 1992. 7. Gavr ˘ u¸ ta, P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184, 431–436. [CrossRef] 8. Park, C.G. Hyers-Ulam stability of a Pexiderized trigonometric functional equation. J. Math. Anal. Appl. 2003, 281, 209–218. 9. Jung, S.M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis; Springer Science & Business Media: Berlin, Germany, 2004. 10. Katsaras, A.K. Fuzzy topological vector spaces II. Fuzzy Sets Syst. 1984, 12, 143–154. [CrossRef] 11. Bag, T.; Samanta, S.K. Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11, 687–706. 12. Kramosil, I.; Michálek, J. Fuzzy metrics and statistical metric spaces. Kybernetika 1975, 11, 336–344. 13. Movahednia, E.; Lolo, P. Self Testing Correcting Programs and Ulam Stability. In Proceedings of the The 51th Annual Iranian Mathematics Conference, Kashan, Iran, 15–20 February 2021; p. 4. 14. Oliphant, T.E. A Guide to NumPy; Trelgol Publishing USA, 2006; Volume 1. Axioms 2023, 12, 435 14 of 14 15. Diaz, J.; Margolis, B. A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74, 305–309. [CrossRef] 16. Cadariu, L.; Radu, V. Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math 2003, 4, 4. 17. Park, C.; Eshaghi Gordji, M.; Saadati, R. Random homomorphisms and random derivations in random normed algebras via fixed point method. J. Inequalities Appl. 2012, 2012, 194. [CrossRef] 18. Movahednia, E.; De la Sen, M. Hyers Stability and Multi-Fuzzy Banach Algebra. Mathematics 2022, 10, 106. 19. Cadariu, L.; Radu, V. On the stability of the Cauchy functional equation: A fixed point approach. Grazer Math. Ber. 2004, 346, 43–52. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Axioms Multidisciplinary Digital Publishing Institute

A New Approach to Involution in Fuzzy C&#9733;-Algebra via Functional Inequality and Python Implementation

Movahednia, Ehsan; De la Sen, Manuel
Axioms , Volume 12 (5) – Apr 27, 2023
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axioms Article A New Approach to Involution in Fuzzy C -Algebra via Functional Inequality and Python Implementation 1, 2, Ehsan Movahednia * and Manuel De la Sen * Department of Mathematics, Behbahan Khatam Alanbia University of Technology-Iran, Behbahan 6361663973, Iran Department of Electricity and Electronics, Institute of Research and Development of Processes, University of Basque Country, Campus of Leioa (Bizkaia), 48080 Bilbao, Spain * Correspondences: movahednia@bkatu.ac.ir (E.M.); manuel.delasen@ehu.eus (M.D.l.S) Abstract: This article explores the stability of involution in fuzzy C -algebras through the use of a functional inequality. We present an approach to obtaining an approximate involution in fuzzy C -algebras by utilizing a fixed-point method. Moreover, for greater clarity, we implemented Python code for the main theorem. Keywords: functional inequality; Hyers–Ulam stability; fixed-point theorem; Python programming MSC: 39B05; 39B82; 46S40; 47H106 1. Introduction Hyers–Ulam stability is a fundamental concept in the field of functional equations. The issue of the stability of functional equations arises from the question: “Is a function that approximately satisfies a functional equation also close to a function that exactly satisfies the same functional equation”. In the field of mathematics, the Hyers–Ulam theorem has been a topic of great interest and discussion among scholars. This concept was first Citation: Movahednia, E.; De la Sen, introduced in 1940 by Stanislaw Ulam in his famous speech at the University of Wisconsin, M. A New Approach to Involution in ? where he presented a problem on the stability of functional equations [1]. This problem was Fuzzy C -Algebra via Functional solved by Donald Hyers in 1941 for additive mappings [2]. Hyers’s theorem was as follows: Inequality and Python Implementation. Axioms 2023, 12, 435. Theorem 1 ([2]). Assume that U and V are Banach spaces, and let f be a mapping from U to V https://doi.org/10.3390/ axioms12050435 that satisfies the inequality k f (u + v) f (u) f (v)k  d Academic Editors: Zhenyu Zhang, Vladimir Simic and Jing Li for some d > 0 and all u, v 2 U. Then, there exists a unique additive mapping T : U ! V such that Received: 3 April 2023 kT(u) f (u)k  d Revised: 20 April 2023 Accepted: 23 April 2023 holds for every u 2 U. Published: 27 April 2023 Ten years later, Takashi Aoki had a major impact on the development of the Hyers– Ulam stability theorem by introducing a modification to the control function, replacing d p p with K(kxk +kyk ) for K > 0 and 0  p < 1 [3]. In a later development, Rassias extended Copyright: © 2023 by the authors. the Hyers–Ulam stability theorem and Aoki’s results to linear mappings with different Licensee MDPI, Basel, Switzerland. control functions [4]. This article is an open access article Today, Hyers–Ulam stability has become an important topic of research in mathe- distributed under the terms and matics, as it has applications in several fields, including physics, engineering, economics, conditions of the Creative Commons and computer science. The results of Hyers–Ulam stability have been extended to various Attribution (CC BY) license (https:// types of functional equations, including quadratic, cubic, Jensen, differential, and integral creativecommons.org/licenses/by/ 4.0/). equations, among others. Axioms 2023, 12, 435. https://doi.org/10.3390/axioms12050435 https://www.mdpi.com/journal/axioms Axioms 2023, 12, 435 2 of 14 Many mathematicians have contributed to the development of the theory of Hyers– Ulam stability by proposing and proving new theorems. They have changed the type of functional equation, control function, and space in the Hyers–Ulam stability theorem to investigate and prove new conditions [5–9]. The Hyers–Ulam stability theorem has been used to prove many other results in different branches of mathematics. In 1984, Katrasas introduced the concept of fuzzy norm spaces [10], where a fuzzy norm is defined in a linear space and the topological structure of the fuzzy vector is established. Several mathematicians have investigated fuzzy norms on linear spaces from a variety of perspectives over time. One important study [11] in 2003 added a fuzzy norm and created a fuzzy measure in the concept of Kramosil and Michalek [12]. They also formulated a theorem on decomposing a fuzzy norm into a set of crisp norms and examined certain features of fuzzy-normed spaces. In recent times, there has been a surge in the study of functional equations stability in fuzzy-normed spaces. Researchers have explored several fuzzy stability outcomes relating to Cauchy, Jensen, simple quadratic, and cubic functional equations. Additionally, the concept of intuitionistic fuzzy-normed spaces has been introduced, and stability results in this area have also been examined. An investigation into the relationship between Ulam’s stability and self-testing cor- recting programs has recently been carried out in [13]. Based on this research, we aim to implement Hyers–Ulam stability in fuzzy C algebra by using the Python program- ming language. Python is a high-level programming language that can be used in various ways. It has access to powerful libraries, such as Numpy, which enable efficient calculations in mathematics and science. Travis Oliphant founded the open-source Numpy project in 2005 [14] and a huge group of collaborators actively maintain and develop it today. We will attempt to write Python code to implement the Hyers–Ulam stability concept. This explains Hyers–Ulam stability for specialists in the computer science and mathematics fields. Definition 1. A function d : U U ! [0,¥] is said to be a generalized metric on the set U if it satisfies the following properties: ( M ) d(u, v) = 0 if and only if u = v ( M ) d(u, v) = d(v, u) for all u, v 2 U ( M ) d(u, w)  d(u, v) + d(v, w) for all u, v, w 2 U. We now introduce one of the fundamental results of the fixed-point theory. Theorem 2 ([15,16]). Suppose (U, d) is a generalized complete metric space and let G : U ! U be a strictly contractive operator with the Lipschitz constant L < 1. Assume there exists a non-negative n +1 n 0 0 integer n such that d G u, G u < ¥ for some u 2 U. Then, the following conclusions hold: (i) The sequence G u converges to a fixed point u of G, (ii) The fixed point u is unique in V = fv 2 U j d(G u, v) < ¥g, (iii) For any v 2 V, the inequality d(v, u )  d(Gv, v), 1 L holds. Definition 2 ([11]). A fuzzy norm on a real vector space U is a function N : U R ! [0, 1] that satisfies the following conditions for any u, v 2 U and s, t 2 R: (N ) N(u, t) = 0 for all t  0; (N ) u = 0 if and only if N(u, t) = 1 for all t > 0; (N ) N(cu, t) = N(u, ) if c 6= 0; jcj (N ) N(u + v, s + t)  minfN(u, s), N(v, t)g; 4 Axioms 2023, 12, 435 3 of 14 N N(u, .) is a non-decreasing function of R and lim N(u, t) = 1; ( ) 5 t!¥ (N ) For u 6= 0, N(u, .) is continuous on R. The pair (U, N) is then referred to as a fuzzy-normed vector space. Definition 3. Consider a fuzzy-normed vector space (U, N). (1) A sequence fu g in U is said to be convergent if there exists u 2 U such that for all t > 0, lim N(u u, t) = 1. The limit of fu g is denoted as N-limit, i.e., N lim u = u. n!¥ n n n!¥ n (2) A sequence fu g in U is called Cauchy if for every e > 0 and t > 0, there exists n 2 N such n 0 that for all n  n and p > 0, N u u , t > 1 e. 0 n+ p n It is a well-known fact that in a fuzzy-normed space, every convergent sequence is also Cauchy. If every Cauchy sequence in a fuzzy-normed space converges, then the fuzzy norm is said to be complete, and the fuzzy-normed space is referred to as a fuzzy Banach space. A mapping of f : U ! V between two fuzzy-normed vector spaces U and V is said to be continuous at a point u 2 U if, for every sequence fu g converging to u in 0 0 U, the sequence f f (u )g converges to f (u ). If f : U ! V is continuous at every point n 0 u 2 U, then f : U ! V is said to be continuous on U. Definition 4 ([17]). Let U be an algebra and (U, N) a fuzzy-normed space. The fuzzy-normed space (U, N) is called a fuzzy-normed algebra if 0 0 0 + N uu , st  N(u, s)N u , t 8u, u 2 U, s, t 2 R . Complete fuzzy-normed algebra is called a fuzzy Banach algebra. Example 1. Every normed algebra (U,k.k) defines a fuzzy-normed algebra (U, N), where N is defined by N(u, t) = 8u 2 U,8t > 0. t +kuk This space is called the induced fuzzy-normed algebra. Definition 5. Let U be a complex algebra. An involution on U is a function ? : U ! U defined by a 7! a , satisfying the following properties: ? ? ? (i) (aa + bb) = aa + bb for all a, b 2 U and a, b 2 C; ? ? ? (ii) (ab) = b a for all a, b 2 U; ?? (iii) a = a for all a 2 U. If U is a complex algebra with an involution, then it is called a ?-algebra. A C -algebra is a ? 2 Banach algebra with an involution ? such that ka ak = jjajj . Definition 6 ([18]). Let U be an ?-algebra and (U, N) a fuzzy-normed algebra. The fuzzy-normed algebra (U, N) is called a fuzzy-normed ?-algebra if ? + N(a , t) = N(a, t) 8a 2 U,8t 2 R . A complete fuzzy-normed ?-algebra is called a fuzzy Banach ?-algebra. Definition 7 ([18]). Let (U, N) be a fuzzy Banach ?-algebra. Then (U, N) is called a fuzzy C -algebra if ? ? + N(a a, st) = N(a , s)N(a, t) 8a 2 U,8s, t 2 R . 2. Results In this section, we will use the fixed-point theorem and functional inequalities to prove the existence of a unique involution for the fuzzy Banach algebra (U, N). We will also demonstrate under what conditions this fuzzy Banach algebra can be transformed into a Axioms 2023, 12, 435 4 of 14 C -algebra. For this purpose, we will first prove two simple lemmas and then proceed to prove the main theorems. Lemma 1. Let F : U ! U be a mapping satisfying the following N(F(2u) + F(2v) + 2F(w), t)  N F(u + v + w), , (1) for all u, v, w 2 U. Then, F is additive, i. e., F(u + v) = F(u) + F(v), for all u, v 2 U. Proof. Putting u = v = w = 0 in (1), we obtain t t t N(4F(0), t)  N F(0), =) N F(0),  N F(0), , 3 4 3 for all t > 0. After applying N(5) and N(6), it can be deduced that the value of N(F(0), t) is 1. It means that F(0) = 0. Letting v := u and w = 0 in (1) so we have N(F(2u) + F(2u), t)  N F(0), = 1, thus F(u) = F(u). Next, putting w := u and v = 0 in (1), we get N(F(2u) + 2F(u), t)  N F(0), = 1, therefore F(2u) = 2F(u). Finally, replacing w by u v in (1), we have N(F(2u) + F(2v) + 2F((u + v), t)  N 2F(u + v u v), = 1, so, F(2u) + F(2v) = 2F((u + v)), i. e., F is additive. Lemma 2. Let F : U ! U be a mapping satisfying N(F(2u) + m ¯ F(2v) + 2F(w), t)  N F(u + mv + w), , (2) for all m 2 T := fz 2 C : jzj = 1g, and for every u, v, w 2 U, then F is Clinear. Proof. We prove that F(au) = a ¯ F(u) for all a 2 C. If m = 1 in (2), then F is additive by Lemma 1. Substituting u, v, w by mu,v, 0 in (2), respectively. We have N(F(2mu) + m ¯ F(2u), t)  N F(mu mu + 0), = 1, therefore, F(2mu) + m ¯ F(2u) = 0 ! F(2mu) = m ¯ F(2u), that is, F(mu) = m ¯ F(u) for all m 2 T and u 2 U. We know that m + m ¯ jmj = jm ¯j = 1, = t (real part of m), m+m ¯ so jtj =  1. Thus, for all real number t with jtj  1 and u 2 U, we get F(2tu) = F(mu + m ¯ u) = F(mu) + F(m ¯ u) = mF(u) + m ¯ F(u) = 2tF(u) Axioms 2023, 12, 435 5 of 14 for all m 2 T and u 2 U. So, F(tu) = tF(u) for all u 2 U and jtj  1. Moreover, It is a n n known fact that if F(2u) = 2F(u), then it follows that F(2 u) = 2 F(u) for any positive integer n. Additionally, by utilizing the Archimedean property, we can determine that, for any real number t, there exists a positive integer n such that jtj  2 . Hence, t t t n n n F(tu) = F 2 . u = 2 F u = 2 . F(u) = tF(u), n n n 2 2 2 for all t 2 R. Suppose that a 2 C be an arbitrary complex number. Therefore, a = t + is for some real numbers t, s. Since jij = 1, so F(iu) = iF(u) for all u 2 U. Finally, we get F(au) = F((t + is)u) = F(tu + isu) = F(tu) + F(isu) (F is additive) = tF(u) + sF(iu) = tF(u) + isF(u) = tF(u) isF(u) = a ¯ F(u), Then, F is Clinear. 3 2 Theorem 3. Let j : U ! (0,¥) and F : U ! [0,¥) be functions such that there exists an L < 1 with j(u, v, w)  j(2u, 2v, 2w), (3) and u v lim 4 F( , ) = 0, (4) n n n!¥ 2 2 for all u, v, w 2 U. In addition, assume F : U ! U to be an odd function satisfying t t N(F(2u) + m ¯ F(2v) + 2F(w), t)  min N 2F(u + mv + w), , , (5) 3 t + j(u, v, w) for all u, v, w 2 U and t > 0. Moreover, suppose that N(F(uv) F(v)F(u), t)  , (6) t + F(u, v) and k k m m N lim 2 F 2 (N lim 2 F(2 u)) = u (7) m!¥ k!¥ for all u, v 2 U and t > 0 and all m 2 T . Then there exists a unique involution A : U ! U such that A(u) := N lim 2 F (8) k!¥ 2 and (2 2L)t N(F(u) A(u), t)  (9) (2 2L)t + j(0, u,u) for all u 2 U and t > 0. Theorem 4. Under the assumptions of Theorem (3), if N([N(F(u), t) N(u, t)]u, t)  (10) t + j(u, u, 0) Then, (U, N) is fuzzy Banach ?algebra. Moreover, if N([N(F(u).u, st) N(F(u), s).N(u, t)]u, t)  (11) t + j(u, u, 0) Axioms 2023, 12, 435 6 of 14 ? ? Then (U, N) is fuzzy C -algebra with involution u = A(u) for all u 2 U. Furthermore, if N(F(u) u, t)  , (12) t + F(u, u) then u is self-adjoint element of U. Proof of Theorem 3. Replacing m, u, v, w by 1, 0, 2u,2u in (5), we have t t N(F(0)) + F(4u) + 2F(2u), t)  min N(2F(0), ), 3 t + j(0, 2u,2u) = , t + j(0, 2u,2u) therefore, N(F(4u) 2F(2u), t)  , (13) t + j(0, 2u,2u) for all u 2 U and t > 0. So N(F(2u) 2F(u), t)  , t + j(0, u,u) for all u 2 U and t > 0. We define the set S and introduce the function d : S S ! [0,¥) as follows. S : fG : U ! U j G(0) = 0g d(G, H) = inf l 2 R : N(G(u) H(u), lt)  , (14) t + j(0, u,u) where, as usual, infÆ = +¥. The proof that (S , d) is a generalized complete metric space has been investigated in ([19]). Now we consider the linear mapping P : S ! S such that PG(u) := 2G for all u 2 U. We prove that P : G ! G is a strictly contractive mapping with Lipschitz constant L. For this purpose, Let G, H 2 S be given such that d(G, H) = e. So, according to the definition of metric d in (14), we have N(G(u) H(u), et) t + j(0, u,u) for all u 2 U and t > 0. Therefore, u u N(PG(u) P H(u), Let) = N 2G 2H , Let 2 2 u u L = N G H , et 2 2 2 Lt Lt u u + j(0, , ) 2 2 2 Lt (by definition of j) Lt L + j(0, u,u) 2 2 for all u 2 U and t > 0. Then, assuming d(G, H) = e, we have proved that d(PG,P H) = Le. This means that d(PG,P H)  Ld(G, H), Axioms 2023, 12, 435 7 of 14 for all G, H 2 S . Therefore P is a strictly contractive mapping with Lipschitz constant L. Replacing u by in (13) and applying (3), we obtain U t t N F(u) 2F , t u u t + j(0, , ) t + j(0, u,u) 2 2 Lt Substituting by t in the above inequality, we obtain U Lt t N F(u) 2F , 2 2 t + j(0, u,u) for all u 2 U and t > 0. Therefore, d(F,PF) < . (15) The conditions of Theorem (2) are satisfied. Hence, (I) There exists a mapping A : U ! U such that it is a fixed point P. This means that u 1 A = A(u), (16) 2 2 for all u 2 U. The mapping A is a unique fixed point of P in the set G : fG 2 S : d(F, G) < ¥g. This suggests that A : U ! U is a unique mapping satisfying (16), furthermore, there exists a l 2 (0,¥) satisfying N(F(u) A(u), lt) t + j(0, u,u) for all u 2 U. (II) d(P F, A) ! 0 as n ! ¥. This suggests that n n A(u) = lim 2 F(2 u) (17) n!¥ for every u 2 U. d(F,PF) (III) d(F, A)  . By (15), we have 1L d(F, A)  . 2 2L u v w This suggests that (9) is held. Replacing u, v, w by , , , respectively, in (5); therefore, n n n 2 2 2 2u 2v w n n n n N 2 F + m ¯ 2 F + 2 2F , 2 t n n n 2 2 2 u v w 2 t t n+1 min N(2 F + m + , ), , u v w n n n 2 2 2 3 t + j( , , ) n n n 2 2 2 for all u, v, w 2 U and t > 0. Replacing t by , we thus obtain 2u 2v w n n n N 2 F( ) + m2 F( ) + 2 2F( ), t (18) n n n 2 2 2 ( ) u v w t n+1 2 min N(2 F( + m + ), ), , n n n t u v w 2 2 2 3 + j( , , ) n n n n 2 2 2 2 Axioms 2023, 12, 435 8 of 14 for all u, v, w 2 U and t > 0. Note that, by using (3), we obtain u v w L j , ,  j(u, v, w). 2 2 2 2 u v w Putting u, v, w by , , in the above inequality, such that 2 2 2 u v w L u v w L j , ,  j , ,  j(u, v, w). 2 2 2 2 2 2 2 2 2 2 2 2 By continuing this process, we have u v w L j , ,  j(u, v, w). n n n n 2 2 2 2 Therefore t t n n 2 2 = . t u v w t L n t + L j(u, v, w) + j( , , ) + j(u, v, w) n n n n n n 2 2 2 2 2 2 Thus, lim > 1. t u v w n!¥ + j( , , ) n n n n 2 2 2 2 By passing n ! ¥ in (18) and using (17), we obtain N( A(2u) + m ¯ A(2v) + 2 A(w), t)  N(2 A(u + mv + w), ), for all u, v, w 2 U, t > 0 and m 2 T . By Lemma (2), the mapping A : U ! U is C-linear. u v We replace , with u, v, respectively, in (6). Hence, n n 2 2 u v v u t N F( ) F( )F( ), t  , u v n n n n 2 2 2 2 t + F( , ) n n 2 2 therefore, u v v u t n n n n N 4 F( . ) 2 F( ).2 F( ), 4 t  , u v n n n n 2 2 2 2 t + F( , ) n n 2 2 so, u v v u N F( ) F( )F( ), t  . t u v n n n n 2 2 2 2 + F( , ) n n n 4 2 2 Passing n ! ¥ and applying (4) and (17), we get N( A(u.v) A(v) A(u), t)  1 ! A(u.v) = A(v) A(u). Furthermore, by (7) we have k k m m A( A(u)) = N lim 2 f (2 (N lim 2 f (2 u))) = u. m!¥ k!¥ We proved that A : U ! U is Clinear, A(u.v) = A(v). A(u) and A( A(u)) = u. These mean that A is involution. Axioms 2023, 12, 435 9 of 14 Proof of Theorem 4. Putting u = in (10), we obtain h i u u u t n n N N(2 F( ), 2 t) N( , t) , t  , for all t > 0, u 2 U u u n n n 2 2 2 t + j( , , 0) n n 2 2 therefore, h i u t n n n n N N(2 F( ), 2 t) N(u, 2 t) u, 2 t  , u u 2 t + j( , , 0) n n 2 2 so, h i t n 2 N N(2 F( ), t) N(u, t) u, t n t u u + j( , , 0) n n n 2 2 2 t + L j(u, u, 0) by passing n ! ¥, we have N([N( A(u), t) N(u, t)]u, t)  1 ! N( A(u), t) N(u, t)]u = 0, for all u 2 U and t > 0. This suggests that N( A(u), t) = N(u, t). Then, (U, N) is fuzzy Banach ?algebra. Next, we suppose that A satisfies in (11), so we obtain h i u u t n 2n n n n n N [N(2 F( ).u, 2 st) N(2 F( ), 2 s).N(u, 2 t) u, 2 t  , u u n n 2 2 t + j( , , 0) n n 2 2 for all u 2 U and t, s > 0. Thus, h i t u u n n n N N(2 F( ).u, st) N(2 F( ), s).N(u, t) u, t  , n n t u u 2 2 + j( , , 0) n n n 2 2 2 t + L j(u, u, 0) for every u 2 U and t, s > 0. By passing n ! ¥, we have N([N( A(u).u, st) N( A(u), s).N(u, t)]u, t)  1. This means that N( A(u).u, st) = N( A(u), s).N(u, t) ? ? for all u 2 U and t > 0. Therefore, U is a C -algebra with involution u = A(u) for all u 2 U. Moreover, replacing u with in (12), we obtain u t n n N 2 F( ) u, 2 t  , u u 2 t + F( , ) n n 2 2 therefore n 2 N 2 F( ) u, t  , n t u u + F( , ) n n n 2 2 2 by passing n ! ¥ and using (4), we have N( A(u) u, t) = 1 ! u = A(u) = u it means that u is self-adjoint element of U. Axioms 2023, 12, 435 10 of 14 Corollary 1. Let p > 1, q > 2. Suppose that h, h : U ! [0,¥) are functions such that u 1 h = h(u) (19) 2 2 u 1 0 0 h = h (u) (20) 2 2 for all u 2 U. Suppose that F : U ! U is a odd function that satisfies N(F(2u) + m ¯ F(2v) + 2F(w), t)  (21) t + h(u) + h(v) + h(w) for all u, v, w 2 U. Moreover, assume that N(F(uv) F(v)F(u), t)  , (22) 0 0 t + h (u) + h (u) and k k m m N lim 2 F(2 (N lim 2 F(2 u))) = u (23) m!¥ k!¥ for all u, v 2 U and t > 0 and all m 2 T . Then, there exists a unique involution A : U ! U such that A(x) := N lim 2 F (24) k!¥ 2 and (2 4)t N(F(u) A(u), t)  (25) (2 4)t + h(2u) + h(2u) for all u 2 U and t > 0. 3 2 Proof. We define j : U ! (0,¥) and F : U ! [0,¥) by j(u, v, w) = h(u) + h(v) + h(w) for all u, v, w 2 U, 0 0 F(u, v) = h (u) + h (v) for all u, v 2 U. From (19), we have j(2u, 2v, 2w) = h(2u) + h(2v) + h(2w) = 2 h(u) + h(v) + h(w) = 2 j(u, v, w), for all u, v, w 2 U. Therefore, j(u, v, w) = j(2u, 2v, 2w) 1 p = j(2u, 2v, 2w), 2 Axioms 2023, 12, 435 11 of 14 1 p we put L := 2 , since p > 1, so L < 1. On the other u v u v n n 0 0 lim 4 F( , ) = lim 4 h ( ) + h ( ) n n n n n!¥ 2 2 n!¥ 2 2 n nq 0 0 = lim 4 ( ) h (u) + h (v) n!¥ 2 n(2q) 0 0 = lim 2 h (u) + h (v) = 0, since q > 2. n!¥ By Theorem (3), there exists a unique involution A : U ! U such that 1 p (2 2.2 )t N( f (u) A(u), t) 1 p p (2 2.2 )t + 2 h(0) + h(u) + h(u) 2 p (2 2 )t 2 p p (2 2 )t + 2 h(u) + h(u) (2 4)t = , for all u 2 U, t > 0 (2 4)t + h(2u) + h(2u) Note that h(2u) = 2 h(u). 3. Implement Python Code for Theorem 3 In this part, the Python code of Theorem (3) is provided. The purpose of writing the code is to first increase the clarity of the Theorem (3) by providing a practical representation of the assumptions and structure of the theorem. Secondly, it is to make a connection between the two topics of Hyers–Ulam stability and the field of computer science. In this code, the user is asked to enter N, F, j, and F functions. import numpy as np def N(u, t): return # The user defines the fuzzy-normed algebra N. def j(u, v, w): return # The user defines the j function, for example: return (u + v + w)**2 - 4*(u*v + v*w + w*u). def F(u, v): return # The user defines the F function, for example: return np.exp(u) * np.exp(2*v). def F(u): return # The user defines the F function, for example: return np.sin(u). def check_limit(F, u, v): n = 1 while True: term = 4**n * F(u/(2**n), v/(2**n)) if np.abs(term) < 1e-10: return True if n > 1e5: return False n =n+ 1 Axioms 2023, 12, 435 12 of 14 def inequality5(u, v, w, t, m = None): # the execution of inequality (5) return # True if inequality (5) holds, False otherwise def inequality6(u, v, t): # the execution of inequality (6) return # True if inequality (6) holds, False otherwise def Equation (7)(u): # the execution of Equation (7) return # True if Equation (7) holds, False otherwise def inequality9(u): # the execution of inequality (9) return # True if inequality (9) holds, False otherwise def check_assumptions(L, U,T ): # Check if the assumptions of the theorem are satisfied, for u, v, w 2 U: if j(u, v, w) > (L/2) j(2 u, 2 v, 2 w): return False for u, v 2 U: if not check_limit(F, u, v): return False for u, v, w 2 U: t = # The user defines the number t for m 2 T : if not inequality5 (u, v, w, t, m): return False for u, v 2 U: t = # The user defines the number t if not inequality(6) (u, v, t): return False for u 2 U: if not Equation (7)(u): return False return True def A(u,t): #Define the function A k = 0 tolerance= # The user enter tolerance while True: k k A_k = 2  F(u/(2 )) (k+1) (k+1) A_k_next = 2  F(u/(2 )) if N(A_k - A_k_next, t) > tolerance: break k = k+1 return A_k L = # The user defines the number L U = # define the set of values to check the assumptions 1 1 T = # define the T t= # The user enter any t>0 if check_assumptions (L, U,T ): print(f"{A(u,t)} is a unique involution") print(f" A satisfies in {inequality(9)(u)}") else: Axioms 2023, 12, 435 13 of 14 print("The conditions of the Theorem are not upheld") In this code, we first import the Numpy library so that we can use mathematical functions throughout the code. Now, we define the function of two variables N, the func- tion of three variables j, the function of two variables F, and the function F. These functions are given to the system by the user. To check assumption (4), we define the function “check_limit(F, u, v)”. This function has three variables and checks the limit of the function F at infinity. In the following, inequality(5) is defined. It takes five arguments u, v, w, t, and an optional m argument. Then, the inequalities (6), (9), and Equation (7) have been defined by the functions “def inequality(6)(u, v, t)”, “def Equation (7)(u)” and “def inequality(9)(u)”, respectively. Now, we will check whether the assumptions of the Theorem (3) are valid. For this purpose, we define the function “def check_assumptions(L, U,T )”. The task of this function is to check the validity of all five assumptions of the Theorem (3). In “de f A(u, t)”, we define the function A and check whether A is a Cauchy sequence. Note that since U is Banach space the Cauchyness of the sequence guarantees its conver- gence. We have defined the necessary functions. Now, to run the code, we first define the values of L, U, T, and t and call the function “check_assumptions” (L, U,T ). If this function is true, A is declared as an involution that applies to the inequality (9). 4. Conclusions In this article, we first investigated the stability of Hyers–Ulam involution in fuzzy C -algebras using functional inequality and the fixed-point method. We then implemented the code of the main theorem in Python with the aim of making the theorem clearer and making the connection between Hyers–Ulam stability and computer science. We hope to make Hyers–Ulam stability more accessible to researchers in mathematics and computer science and to encourage further research on the connection between these two fields. Author Contributions: Methodology, E.M. and M.D.l.S.; validation, M.D.l.S.; investigation, E.M. and M.D.l.S.; writing—original draft, E.M.; project administration, M.D.l.S.; funding acquisition, M.D.l.S. All authors have read and agreed to the published version of the manuscript. Funding: This research was funded by Basque Government, Grant IT1555-22; Basque Govern- ment,Grant KK-2022/00090; MCIN/AEI 269.10.13039/501100011033, Grant PID2021-1235430B-C21. Data Availability Statement: All data required for this paper are included within this paper Conflicts of Interest: The authors declare no conflicts of interest. References 1. Ulam, S.M. Problems in Modern Mathematics; Courier Corporation: Chelmsford, MA, USA, 2004. 2. Hyers, D.H. On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 1941, 27, 222–224. [CrossRef] [PubMed] 3. Aoki, T. On the Stability of the Linear Transformation in Banach Spaces. J. Math. Soc. Jpn. 1950, 2, 64–66. [CrossRef] 4. Rassias, T.M. On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 1978, 72, 297–300. [CrossRef] 5. Forti, G. Hyers-Ulam stability of functional equations in several variables. Aequationes Math. 1995, 50, 143–190. [CrossRef] 6. Czerwik, S. Functional Equations and Inequalities in Several Variables; World Scientific Publishing Co.: Singapore, 1992. 7. Gavr ˘ u¸ ta, P. A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 1994, 184, 431–436. [CrossRef] 8. Park, C.G. Hyers-Ulam stability of a Pexiderized trigonometric functional equation. J. Math. Anal. Appl. 2003, 281, 209–218. 9. Jung, S.M. Hyers-Ulam-Rassias Stability of Functional Equations in Nonlinear Analysis; Springer Science & Business Media: Berlin, Germany, 2004. 10. Katsaras, A.K. Fuzzy topological vector spaces II. Fuzzy Sets Syst. 1984, 12, 143–154. [CrossRef] 11. Bag, T.; Samanta, S.K. Finite dimensional fuzzy normed linear spaces. J. Fuzzy Math. 2003, 11, 687–706. 12. Kramosil, I.; Michálek, J. Fuzzy metrics and statistical metric spaces. Kybernetika 1975, 11, 336–344. 13. Movahednia, E.; Lolo, P. Self Testing Correcting Programs and Ulam Stability. In Proceedings of the The 51th Annual Iranian Mathematics Conference, Kashan, Iran, 15–20 February 2021; p. 4. 14. Oliphant, T.E. A Guide to NumPy; Trelgol Publishing USA, 2006; Volume 1. Axioms 2023, 12, 435 14 of 14 15. Diaz, J.; Margolis, B. A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 1968, 74, 305–309. [CrossRef] 16. Cadariu, L.; Radu, V. Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure Appl. Math 2003, 4, 4. 17. Park, C.; Eshaghi Gordji, M.; Saadati, R. Random homomorphisms and random derivations in random normed algebras via fixed point method. J. Inequalities Appl. 2012, 2012, 194. [CrossRef] 18. Movahednia, E.; De la Sen, M. Hyers Stability and Multi-Fuzzy Banach Algebra. Mathematics 2022, 10, 106. 19. Cadariu, L.; Radu, V. On the stability of the Cauchy functional equation: A fixed point approach. Grazer Math. Ber. 2004, 346, 43–52. Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Journal

Axioms – Multidisciplinary Digital Publishing Institute

Published: Apr 27, 2023

Keywords: functional inequality; Hyers–Ulam stability; fixed-point theorem; Python programming

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