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Choonkill Park (2019)
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APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 2023, VOL. 31, NO. 1, 2176851 https://doi.org/10.1080/27690911.2023.2176851 A system of biadditive functional equations in Banach algebras a a b Yamin Sayyari , Mehdi Dehghanian and Choonkil Park a b Department of Mathematics, Sirjan University of Technology, Sirjan, Iran; Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul, Korea ABSTRACT ARTICLE HISTORY Received 31 August 2022 In this paper, we obtain the general solution and the Hyers-Ulam Accepted 26 January 2023 stability of the system of biadditive functional equations KEYWORDS 2f (x + y, z + w) − g(x, z) − g(x, w) = g(y, z) + g(y, w) Hyers–Ulam stability; biadditive mapping; g(x + y, z + w) − 2f (x − y, z − w) = 4f (x, w) + 4f (y, z) f-biderivation; fixed point method; system of biadditive in complex Banach spaces. Furthermore, we prove the Hyers–Ulam functional equations stability of f-biderivations in complex Banach algebras. MATHEMATICS SUBJECT CLASSIFICATIONS Primary 47B47; 17B40; 39B72; 47H10 1. Introduction In the fall of 1940, Ulam [1] raised the rfi st stability problem. He proposed a question whether there exists an exact homomorphism near an approximate homomorphism. An answer to the problem was given by Hyers [2] in the setting of Banach spaces. Since then the stability problems have been extensively investigated for a variety of functional equa- tions and spaces. In most cases, a functional equation is algebraic in nature whereas the stabilityisrathermetrical. Hence, anormedlinearspace is asuitablechoicetoworkwith the stability of functional equations. We refer to [3–7] for results, references and examples. Hyers was the rfi st mathematician to present the concequence concerning the stability of functional equations. He answered the question of Ulam for the case of approximate additive mappings under the assumption that G and G are Banach spaces (see [2]). 1 2 The method provided by Hyers [2] which produces the additive function will be called a direct method. This method is the most important and powerful tool to concerning the sta- bility of dieff rent functional equations. That is, the exact solution of the functional equation is explicitly constructed as a limit of a sequence, starting from the given approximate solu- tion [6, 8]. The other significant method is fixed point theorem, that is, the exact solution of the functional equation is explicitly created as a xfi ed point of some certain map [ 9–12]. CONTACT Choonkil Park baak@hanyang.ac.kr Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 04763, Korea © 2023 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons. org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 2 Y. SAYYARI ET AL. Let B be a complex Banach algebra. A mapping f : B × B → B is biadditive if f is additive in each variable. Furthermore, f is called a C-bilinear mapping if f is C-linear in each variable. Recently, the Hyers–Ulam stability and the hyperstability of biadditive functional equations were proved in [13]. Moreover, many mathematicians have studied the Hyers–Ulam stability of some derivations in algebras and rings (see [14]). Lemma 1.1 ([15]): Let A and B be complex Banach algebras and f : A × A → B be a bi- additive mappingsuchthatf (λx, μy) = λμf (x, y) for all λ, μ ∈ T and all x, y ∈ A,thenf is C-bilinear. Remark 1.1: In Lemma 1.1, T is toobig.Wecan take asmaller setsuchasapart of the unit circle T or we cantakeaconnectedpathtoobtainthe same result as in Lemma1.1 (see [16]). Definition 1.2 ([ 17, 18]): Let B be aring. Abiadditive mapping g : B × B → B is called a symmetric biderivation on B if g satisefi s g(xy, z) = g(x, z)y + xg(y, z) g(x, z) = g(z, x) for all x, y, z ∈ B. Definition 1.3 ([ 19]): Let B be a complex Banach algebra. A C-bilinear mapping g : B × B → B is called a biderivation on B if g satisefi s g(xy, z) = g(x, z)y + xg(y, z) g(x, zw) = g(x, z)w + zg(x, w) for all x, y, z, w ∈ B. See [11, 20–22] for more information on biderivations in several spaces. In this paper, we introduce f -biderivations in a Banach algebra. Definition 1.4: Let B be acomplex Banach algebraand f : B × B → B be a C-bilinear mapping. A C-bilinear mapping g : B × B → B is called an f -biderivation on B if g satisefi s g(xy, z ) = g(x, z)f (y, z) + f (x, z)g(y, z) g(x , zw) = g(x, z)f (x, w) + f (x, z)g(x, w) for all x, y, z, w ∈ B. Example 1.5: Let f : R × R → R be den fi ed by f (x, y) = xy and g : R × R → R be defined by g(x, y) = xy for all x, y ∈ R.Then g is an f -biderivation. In this paper, we consider the following system of functional equations 2f (x + y, z + w) − g(x, z) − g(x, w) = g(y, z) + g(y, w) (1) g(x + y, z + w) − 2f (x − y, z − w) = 4f (x, w) + 4f (y, z) for all x, y, z, w ∈ B. APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 3 The aim of the present paper is to solve the system of biadditive functional equations (1) and prove the Hyers–Ulam stability of f -biderivations in complex Banach algebras by using the xfi ed point method. Throughout this paper, assume that B isacomplex Banach algebra. 2. Stability of the system of biadditive functional equations (1) We solve and investigate the system of biadditive functional equations (1) in complex Banach algebras. Lemma 2.1: Let f , g : B × B → B be mappings satisfying (1) for all x, y, z, w ∈ B. Then the mappings f , g : B × B → B are biadditive. Proof: Setting x = y = z = w = 0 in (1), we have f (0, 0) = g(0, 0) = 0. Putting y = z = w = 0in(1),wehave 2f (x,0) = 2g(x,0) g(x,0) = 6f (x,0) for all x ∈ B.Hence f (x,0) = g(x,0) = 0 for all x ∈ B.Taking x = y = w = 0 in (1), we obtain 2f (0, z) = 2g(0, z) g(0, z) = 6f (0, z) and so f (0, z) = g(0, z) = 0 for all z ∈ B. Letting y = w = 0 in (1), we have g(x, z) = 2f (x, z) (2) for all x, z ∈ B.So g(x + y, z + w) = 2f (x + y, z + w) = g(x, z) + g(x, w) + g(y, z) + g(y, w) for all x, y, z, w ∈ B. Therefore the mapping g : B × B → B is biadditive and thus by (2) the mapping f : B × B → B is biadditive. Using the fixed point method, we prove the Hyers–Ulam stability of the system of biadditive functional equations (1) in complex Banach algebras. 4 Y. SAYYARI ET AL. Theorem 2.2: Suppose that : B → [0, ∞) is afunctionsuchthatthere exists an L < 1 with x y z w L , , , ≤ (x, y, z, w) (3) 2 2 2 2 4 for all x, y, z, w ∈ B.Let f , g : B × B → B be mappings satisfying 2f (x + y, z + w) − g(x, z) − g(x, w) − g(y, z) − g(y, w)≤ (x, y, z, w) (4) g(x + y, z + w) − 2f (x − y, z − w) − 4f (x, w) − 4f (y, z)≤ (x, y, z, w) for all x, y, z, w ∈ B. Then there exist unique biadditive mappings F, G : B × B → B such that L+L F(x, z) − f (x, z)≤ (x, x, z, z) 8(1−L) (5) L+L G(x, z) − g(x, z)≤ (x, x, z, z) 4(1−L) for all x, z ∈ B. Proof: Letting x = y = z = w = 0in(4),weobtain 2f (0, 0) − 4g(0, 0)≤ (0, 0, 0, 0) = 0 g(0, 0) − 10f (0, 0)≤ (0, 0, 0, 0) = 0 and hence f (0, 0) = g(0, 0) = 0. Letting y = x and w = z in (4), we get 2f (2x,2z) − 4g(x, z)≤ (x, x, z, z) g(2x,2z) − 8f (x, z)≤ (x, x, z, z) for all x, z ∈ B.So x z 1 x x z z x x z z L+L f (x, z) − 16f , ≤ , , , + 2( , , , ) ≤ (x, x, z, z) 4 4 2 2 2 2 2 4 4 4 4 8 (6) x z x x z z x x z z L+L g(x, z) − 16g , ≤ , , , + 4 , , , ≤ (x, x, z, z) 4 4 2 2 2 2 4 4 4 4 4 for all x, z ∈ B. We define H ={h : B × B → B : h(0, 0) = 0} and introduce the generalized metric on H as follows: d : H × H → [0, ∞]by d(δ, h) = inf {k ∈ R : δ(x, z) − h(x, z)≤ k(x, x, z, z), ∀x, z ∈ B} and we consider inf ∅=+∞.Then d is a complete generalized metric on H (see [23]). Now, we define the mapping J : (H, d) → (H, d) such that x z J δ(x, z) := 16δ , 4 4 for all x, z ∈ B. APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 5 Assume that δ, h ∈ H such that d(δ, h) = k.Then x z x z J δ(x, z) − J h(x, z)= 16δ , − 16h , 4 4 4 4 x x z z ≤ 16k , , , ≤ L k(x, x, z, z) 4 4 4 4 for all x, z ∈ B. It follows that d(J δ(x, z),J h(x, z)) ≤ L k.Thus d(J δ(x, z),J h(x, z)) ≤ L d(δ, h) for all x, z ∈ B and all δ, h ∈ H. 2 2 L+L L+L From (6), we have d(f ,J f ) ≤ and d(g,J g) ≤ . 8 4 Using the fixed point alternative (see [ 24]), we deduce the existence of unique fixed points of J ,thatis, theexistence of mappings F, G : B × B → B,respectively, such that x z x z F(x, z) = 16F , , G(x, z) = 16G , 4 4 4 4 with the following property: there exist k , k ∈ (0, ∞) satisfying 1 2 f (x, z) − F(x, z)≤ k (x, x, z, z), g(x, z) − G(x, z)≤ k (x, x, z, z) 1 2 for all x, z ∈ B. n n Since lim d(J f , F) = 0and lim d(J g, G) = 0, n→∞ n→∞ x z x z 2n 2n lim 4 f , = F(x, z),lim 4 g , = G(x, z) n n n n n→∞ n→∞ 4 4 4 4 for all x, z ∈ B. 1 1 Next, d(f , F) ≤ d(f ,J f ) and d(g, G) ≤ d(g,J g) which imply 1−L 1−L L + L f (x, z) − F(x, z)≤ (x, x, z, z), 8(1 − L) L + L g(x, z) − G(x, z)≤ (x, x, z, z) 4(1 − L) for all x, z ∈ B. Using (3) and (4), we conclude that 2F(x + y, z + w) − G(x, z) − G(x, w) − G(y, z) − G(y, w) x + y z + w x z x w 2n = lim 4 2f , − g , − g , n n n n n n n→∞ 4 4 4 4 4 4 y z y w − g , − g , n n n n 4 4 4 4 x y z w 2n 2n ≤ lim 4 , , , ≤ lim L (x, y, z, w) = 0 n n n n n→∞ n→∞ 4 4 4 4 6 Y. SAYYARI ET AL. and G(x + y, z + w) − 2F(x − y, z − w) − 4F(x, w) − 4F(y, z) x + y z + w x − y z − w x w y z 2n = lim 4 G , − 2f , − 4f , − 4f , n n n n n n n n n→∞ 4 4 4 4 4 4 4 4 x y z w 2n 2n ≤ lim 4 , , , ≤ lim L (x, y, z, w) = 0 n n n n n→∞ n→∞ 4 4 4 4 for all x, y, z, w ∈ B,since L < 1. Hence 2F(x + y, z + w) − G(x, z) − G(x, w) = G(y, z) + G(y, w) G(x + y, z + w) − 2F(x − y, z − w) = 4F(x, w) + 4F(y, z) for all x, y, z, w ∈ B. So by Lemma 2.1, the mappings F, G : B → B are biadditive. Corollary 2.3: Let θ,pbenonnegative real numberswithp > 2 and f , g : B × B → B be mappings satisfying 2f (x + y, z + w) − g(x, z) − g(x, w) − g(y, z) − g(y, w) p p p p ≤ θ(x +y +z +w ) g(x + y, z + w) − 2f (x − y, z − w) − 4f (x, w) − 4f (y, z) ⎩ p p p p ≤ θ(x +y +z +w ) for all x, y, z, w ∈ B. Then there exist unique biadditive mappings F, G : B × B → B such that (2 + 8)θ p p F(x, z) − f (x, z)≤ (x +z ) p+1 p 2 (2 − 4) (7) (2 + 8)θ ⎪ p p G(x, z) − g(x, z)≤ (x +z ) p p 2 (2 − 4) for all x, z ∈ B. 2−p Proof: The proof follows from Theorem 2.2 by taking L = 2 and (x, y, z, w) = p p p p θ(x +y +z +w ) for all x, y, z, w ∈ B. Corollary 2.4: Letp,q,r,sbe nonnegativerealnumbers with p + q + r + s >4and f , g : B × B → B be mappings satisfying 2f (x + y, z + w) − g(x, z) − g(x, w) − g(y, z) − g(y, w) p q r s ≤x y z w g(x + y, z + w) − 2f (x − y, z − w) − 4f (x, w) − 4f (y, z) ⎩ p q r s ≤x y z w APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 7 for all x, y, z, w ∈ B. Then there exist unique biadditive mappings F, G : B × B → B such that 2(2 + 16) p+q+r+s F(x, z) − f (x, z)≤ x p p 2 (2 − 16 (8) ⎪ 2 + 16 ⎪ p+q+r+s G(x, z) − g(x, z)≤ x p p 2 (2 − 16 for all x, z ∈ B. 4−(p+q+r+s) Proof: The proof follows from Theorem 2.2 by taking L = 2 and (x, y, z, w) = p q r s x y z w for all x, y, z, w ∈ B. 3. Stability of f-biderivations in Banach algebras In this section, by using the fixed point technique, we prove the Hyers–Ulam stability of f -biderivations in complex Banach algebras. Lemma 3.1: Let f , g : B × B → B be mappings satisfying 2f (λ(x + y), μ(z + w)) − λμg(x, z) − λμg(x, w) = λμg(y, z) + λμg(y, w) (9) g(λ(x + y), μ(z + w)) − 2f (λ(x − y), μ(z − w)) = 4λμf (x, w) + 4λμf (y, z) for all x, y, z, w ∈ B and all λ, μ ∈ T . Then the mappings f , g : B × B → B are C-bilinear. Proof: If we put λ = μ = 1 in (9), then f and g are biadditive and g(x, z) = 2f (x, z) (10) for all x, z ∈ B by Lemma 2.1. Now, taking y = x and w = z in (9), we have 8f (λx, μz) = 4λμg(x, z) (11) 4g(λx, μy) = 8λμf (x, z) for all x, z ∈ B and all λ, μ ∈ T . From (10) and (11), we obtain f (λx, μz) = λμf (x, z), g(λx, μz) = λμg(x, z) for all x, z ∈ B and all λ, μ ∈ T . Thus by Lemma 1.1 the mappings f and g are C-bilinear. 8 Y. SAYYARI ET AL. Theorem 3.2: Suppose that : B → [0, ∞) is afunctionsuchthatthere exists an L < 1 with x y z w L ( , , , ) ≤ (x, y, z, w) (12) 2 2 2 2 16 for all x, y, z, w ∈ B.Let f , g : B × B → B be mappings satisfying 2f (λ(x + y), μ(z + w)) − λμg(x, z) − λμg(x, w) − λμg(y, z) − λμg(y, w) ≤ (x, y, z, w) g(λ(x + y), μ(z + w)) − 2f (λ(x − y), μ(z − w)) − 4λμf (x, w) − 4λμf (y, z) ≤ (x, y, z, w) (13) and g(xy, z ) − g(x, z)f (y, z) − f (x, z)g(y, z) ≤ (x, y, z, z) (14) g(x , zw) − g(x, z)f (x, w) − f (x, z)g(x, w) ≤ (x, x, z, w) for all x, y, z, w ∈ B and all λ, μ ∈ T . Then there exist unique C-bilinear mappings F, G : B × B → B such that G is an F-biderivation and satisfying (5). Proof: Let λ = μ = 1 in (13). By the same reasoning as in the proof of Theorem 2.2, there exist unique biadditive mappings F, G : B × B → B satisfying (5), which are given by x z x z 2n 2n F(x, z) = lim 4 f , , G(x, z) = lim 4 g , n n n n n→∞ n→∞ 4 4 4 4 for all x, z ∈ B. From (12) and (13), we have 2F(λ(x + y), μ(z + w)) − λμG(x, z) − λμG(x, w) − λμG(y, z) − λμG(y, w) λ(x + y) μ(z + w) x z x w 2n = lim 4 2f , − λμg , − λμg , n n n n n n n→∞ 4 4 4 4 4 4 y z y w − λμg , − λμg , n n n n 4 4 4 4 2n x y z w L 2n ≤ lim 4 , , , ≤ lim (x, y, z, w) = 0 n n n n 2n n→∞ n→∞ 4 4 4 4 4 for all x, y, z, w ∈ B and all λ, μ ∈ T .Similarly,one canshowthat G(λ(x + y), μ(z + w)) − F(λ(x − y), μ(z − w)) − 4λμF(x, w) − 4λμF(y, z) 2n x y z w L 2n ≤ lim 4 , , , ≤ lim (x, y, z, w) = 0 n n n n 2n n→∞ n→∞ 4 4 4 4 4 for all x, y, z, w ∈ B and all λ, μ ∈ T .So 2F(λ(x + y), μ(z + w)) − λμG(x, z) − λμG(x, w) = λμG(y, z) + λμG(y, w) G(λ(x + y), μ(z + w)) − F(λ(x − y), μ(z − w)) = 4λμF(x, w) + 4λμF(y, z) for all x, y, z, w ∈ B and all λ, μ ∈ T . Therefore, by Lemma 3.1, the mappings F, G : B × B → B are C-bilinear. APPLIED MATHEMATICS IN SCIENCE AND ENGINEERING 9 Next, from (12) and (14) it follows that G(xy, z ) − G(x, z)F(y, z) − F(x, z)G(y, z) xy z x z y z x z y z 4n = lim 4 g , − g , f , − f , g , 2n 2n n n n n n n n n n→∞ 4 4 4 4 4 4 4 4 4 4 x y z z 4n 2n ≤ lim 4 , , , ≤ lim L (x, y, z, z) = 0 n n n n n→∞ n→∞ 4 4 4 4 and G(x , zw) − G(x, z)F(x, w) − F(x, z)G(x, w) x zw x z x w x z x w 4n = lim 4 g , − g , f , − f , g , 2n 2n n n n n n n n n n→∞ 4 4 4 4 4 4 4 4 4 4 x x z w 4n 2n ≤ lim 4 , , , ≤ lim L (x, x, z, w) = 0 n n n n n→∞ n→∞ 4 4 4 4 for all x, y, z, w ∈ B and all λ, μ ∈ T ,since L < 1. Thus G(xy, z ) = G(x, z)F(y, z) + F(x, z)G(y, z) G(x , zw) = G(x, z)F(x, w) + F(x, z)G(x, w) for all x, y, z, w ∈ B.Hence themapping G : B × B → B is an F-biderivation. Corollary 3.3: Let θ,pbenonnegative real numberswithp >4and f , g : B × B → B be mappings satisfying 2f (λ(x + y), μ(z + w)) − λμg(x, z) − λμg(x, w) − λμg(y, z) − λμg(y, w) p p p p ≤ θ(x +y +z +w ) g(λ(x + y), μ(z + w)) − 2f (λ(x − y), μ(z − w)) − 4λμf (x, w) − 4λμf (y, z) p p p p ≤ θ(x +y +z +w ) for all x, y, z, w ∈ B and all λ, μ ∈ T . Then there exist unique C-bilinear mappings F, G : B × B → B such that G is an F-biderivation and satisfying (7). Proof: The proof follows from Theorem 3.2 and Corollary 2.3. Corollary 3.4: Letp,q,r,sbenonnegative real numberswithp + q + r + s > 16 and f , g : B × B → B be mappings satisfying 2f (λ(x + y), μ(z + w)) − λμg(x, z) − λμg(x, w) − λμg(y, z) − λμg(y, w) p q r s ≤x y z w g(λ(x + y), μ(z + w)) − 2f (λ(x − y), μ(z − w)) − 4λμf (x, w) − 4λμf (y, z) ⎩ p q r s ≤x y z w for all x, y, z, w ∈ B. Then there exist unique C-bilinear mappings F, G : B × B → B such that G is an F-biderivation and satisfying (8). 10 Y. SAYYARI ET AL. Proof: The proof follows from Theorem 3.2 and Corollary 3.4. Remark 3.1: We can obtain some results on asymptotically generalized Lie bi-derivations corresponding to the results given in [25]. Acknowledgments We wouldliketoexpress oursincere gratitudetothe anonymousreferee forhis/her helpful comments that will help to improve the quality of the manuscript. Declarations Human and animal rights: We would like to mention that this article does not contain any studies with animals and does not involve any studies over human being. Authors’ contributions The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the n fi al manuscript. Disclosure statement No potential conflict of interest was reported by the author(s). ORCID Choonkil Park http://orcid.org/0000-0001-6329-8228 References [1] Ulam SM. A collection of the mathematical problems. New York: Interscience Publ; 1960. 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Applied Mathematics in Science and Engineering – Taylor & Francis
Published: Dec 31, 2023
Keywords: Hyers–Ulam stability; biadditive mapping; f -biderivation; fixed point method; system of biadditive functional equations; Primary 47B47; 17B40; 39B72; 47H10
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