CUBO, A Mathematical Journal Vol.22, N◦02, (233–255). August 2020 http://dx.doi.org/10.4067/S0719-06462020000200233 Received: 23 May, 2020 | Accepted: 20 July, 2020 Hyers-Ulam stability of an additive-quadratic functional equation Vediyappan Govindan1, Choonkil Park2, Sandra Pinelas3 and Themistocles M. Rassias4 1 Department of Mathematics, Sri Vidya Mandir Arts & Science College, Katteri, India. 2 Research Institute of Natural Sciences, Hanyang University, Seoul-04763, Korea. 3 Departamento de Ciências Exatas e Engenharia, Academia Militar, Portugal. 4 Department of Mathematics, National Technical University of Athens, Greece. govindoviya@gmail.com, baak@hanyang.ac.kr, sandra.pinelas@gmail.com, trassias@math.ntua.gr ABSTRACT In this paper, we introduce the following (a, b, c)-mixed type functional equation of the form g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2) where a, b, c are positive integers with a > 1, and investigate the solution and the Hyers-Ulam stability of the above functional equation in Banach spaces by using two different methods. RESUMEN En este art́ıculo introducimos la siguiente ecuación funcional de tipo (a, b, c)-mixta de la forma g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2) donde a, b, c son enteros positivos con a > 1, e investigamos la solución y la estabilidad de Hyers-Ulam de la ecuación funcional anterior en espacios de Banach usando dos métodos diferentes. Keywords and Phrases: Hyers-Ulam stability, mixed type functional equation, Banach space, fixed point. 2020 AMS Mathematics Subject Classification: 39B52, 32B72, 32B82. c©2020 by the author. This open access article is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License. http://dx.doi.org/10.4067/S0719-06462020000200233 234 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) 1 Introduction The stability problem of functional equations originated form a question of Ulam [28] concerning the stability of group homomorphisms. Hyers [12] gave a first affirmative partial answer to the question of Ulam [28] for Banach spaces. Hyers theorem was generalized by Aoki [3] for additive mappings and Rassias [12] for quadratic mappings. During the last three decades the stability theorem of Rassias [26] provided a lot of influence for the development of stability theory of a large variety of functional equations (see [1, 2, 4, 7, 9, 11, 14, 17, 18, 21, 22, 23, 27]). One of the most famous functional equations is the following additive functional equation g(x + y) = g(x) + g(y) (1.1) In 1821, it was first solved by Cauchy in the class of continuous real-valued functions. It is often called Cauchy additive functional equation in honour of Cauchy. The theory of additive func- tional equations is frequently applied to the development of theories of other functional equations. Moreover, the properties of additive functional equations are powerful tools in almost every field of natural and social science ([6, 24, 26]). Every solution of the additive functional equation (1.1) is called an additive mapping. The function g(x) = x2 satisfies the functional equation g(x + y) + g(x − y) = 2g(x) + 2g(y) (1.2) and therefore, the functional equation (1.2) is called quadratic functional equation. The Hyers- Ulam stability theorem for the quadratic functional equation (1.2) was proved by Skof [25] for the mapping g : E1 → E2, where E1 is a normed space and E2 is a Banach space. Moslehian and Rassias [20] studied the Hyers-Ulam stability problem in non-Archimedean normed spaces. Mirzavaziri and Moslehian [19] studied the Hyers-Ulam stability of a quadratic functional equation in Banach spaces by using the fixed point method and Ciepliński [5] sur- veyed the Hyers-Ulam stability of functional equations by using the fixed point method. Ebadian, Ghobadipour and Eshaghi Gordji [8] proved the Hyers-Ulam stability of bimultipliers and Jordan bimultipliers in C∗-ternary algebras by using the fixed point method for a three variable additive functional equation. Motivated by Ebadian et al. [8], we introduce the following three variable generalized additive- quadratic functional equation of the form Dg(x1, x2, x3) := g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 235 − [4g(ax1 + cx3) + 2g(−bx2) + 2g(bx2)] = 0 (1.3) where a, b, c are positive integers with a > 1, and investigate the solution and the Hyers-Ulam stability of the three variable generalized additive-quadratic functional equation (1.3) in Banach spaces by using the direct method and the fixed point method. 2 Solution of the functional equation (1.3): when g is odd In this section, we investigate the solution of the functional equation (1.3) for an odd mapping case. Throughout this section, let X and Y be real vector spaces. Theorem 1. If an odd mapping g : X → Y satisfies the functional equation (1.1) if and only if g : X → Y satisfies the functional equation (1.3). Proof. Assume that g : X → Y satisfies the functional equation (1.1). Since g is odd, g(0) = 0. Replacing (x, y) by (x, x) and by (x, 2x) respectively in (1.1), we obtain g(2x) = 2g(x) and g(3x) = 3g(x) (2.1) for all x ∈ X. In general for any positive integer d, we have g(dx) = dg(x) (2.2) for all x ∈ X. It is easy to verify from (1.1) that g(d2x) = d2g(x) and g(d3x) = d3g(x) (2.3) for all x ∈ X. Replacing (x, y) by (ax1 + bx2, cx3) in (1.1), we get g(ax1 + bx2 + cx3) = g(ax1 + bx2) + g(cx3) (2.4) for x1, x2, x3 ∈ X. Replacing x1 by −x1 in (2.4), we get g(−ax1 + bx2 + cx3) = g(−ax1 + bx2) + g(cx3) (2.5) for x1, x2, x3 ∈ X. Replacing x2 by −x2 in (2.4), we have g(ax1 − bx2 + cx3) = g(ax1 − bx2) + g(cx3) (2.6) for x1, x2, x3 ∈ X. Replacing x3 by −x3 in (2.4), we obtain g(ax1 + bx2 − cx3) = g(ax1 + bx2) + g(−cx3) (2.7) 236 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) for x1, x2, x3 ∈ X. By (2.4), (2.5), (2.6), (2.7), (1.1) and (2.3), we get g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) = 2ag(x1) − 2bg(x2) + 2cg(x3) (2.8) for x1, x2, x3 ∈ X. Adding 2ag(x1) − 2bg(x2) + 2cg(x3) + 2a 2g(x1) + 2b 2g(x2) + 2c 2g(x3) to both sides and using the oddness of g, we get (1.3). Conversely, assume that g satisfies (1.3). Letting x3 = 0 in (1.3), we have g(ax1 + bx2 + cx3) − g(−ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) + 2a2[g(x1) + g(−x1)] + 2b 2[g(x2) + g(−x2)] + 2c 2[g(x3) + g(−x3)] + a[g(x1) − g(−x1)] + b[g(x2) − g(−x2)] + c[g(x3) − g(−x3)] = 2g(ax1 − bx2) + 2ag(x1) + 2bg(x2) for all x1, x2 ∈ X, since g is odd. So 2g(ax1 − bx2) + 2ag(x1) + 2bg(x2) = 4g(ax1) (2.9) for all x1, x2 ∈ X. Letting x2 = 0 in (2.9), we have 2g(ax1) + 2ag(x1) = 4g(ax1) and so g(ax1) = ag(x1) for all x1 ∈ X. Letting x1 = 0 in (2.9), we have −2g(bx2) + 2bg(x2) = 0 and so g(bx2) = bg(x2) for all x2 ∈ X. It follows from (2.9) that 2g(ax1 − bx2) + 2g(ax1) + 2g(bx2) = 4g(ax1) for all x1, x2 ∈ X and so g(x − y) + g(y) = g(x) for all x, y ∈ X. Letting z = x − y in the above equation, we get g(z) + g(y) = g(z + y) for all z, y ∈ X. 3 Solution of the functional equation (1.3): when g is even In this section, we investigate the solution of the functional equation (1.3) for an even mapping case. Throughout this section, let X and Y to be real vector spaces. Theorem 2. If an even mapping g : X → Y satisfies the functional equation (1.2) if and only if g : X → Y satisfies the functional equation (1.3). Proof. Assume that g : X → Y satisfies the functional equation (1.2). Setting x = y = 0 in (1.2), we get g(0) = 0. CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 237 Replacing (x, y) by (x, x) and by (x, 2x), respectively, in (1.2), we obtain g(2x) = 4g(x) and g(3x) = 9g(x) (3.1) for all x ∈ X. In general for any positive integer d, we have g(dx) = d2g(x) (3.2) for all x ∈ X. It is easy to verify from (1.2) that g(d2x) = d4g(x) and g(d3x) = d6g(x) (3.3) for all x ∈ X. Replacing (x, y) by (ax1, cx3) in (1.2), we get g(ax1 + cx3) + g(ax1 − cx3) = 2g(ax1) + 2g(cx3) (3.4) for x1, x2, x3 ∈ X. Multiplying 2 on both sides and using (3.3), we get 2g(ax1 + cx3) + 2g(ax1 − cx3) = 4a 2g(x1) + 4c 2g(x3) (3.5) for x1, x2, x3 ∈ X. Adding 2g(ax1 + cx3) to (3.5) on both sides and using (3.3), we obtain 2g(ax1 + cx3) + 2g(ax1 − cx3) + 2g(ax1 + cx3) = 4a 2g(x1) + 4c 2g(x3) + 2g(ax1 + cx3) (3.6) for x1, x2, x3 ∈ X. So 4g(ax1 + cx3) = 4a 2 g(x1) + 4c 2 g(x3) + 2g(ax1 + cx3) − 2g(ax1 − cx3). (3.7) Adding and subtracting 2g(bx2) to (3.7), we get 4g(ax1 + cx3) = 4a 2g(x1) + 4c 2g(x3) + g(ax1 + cx3 + bx2) + g(ax1 + cx3 − bx2) − g(ax1 − cx3 + bx2) − g(ax1 − cx3 − bx2) (3.8) for x1, x2, x3 ∈ X. Adding 4g(bx2) to (3.8) on both sides, we obtain 4g(ax1 + cx3) + 4g(bx2) = 4a 2g(x1) + 4c 2g(x3) + g(ax1 + cx3 + bx2) + g(ax1 + cx3 − bx2) − g(ax1 − cx3 + bx2) − g(ax1 − cx3 − bx2) + 4g(bx2) (3.9) for x1, x2, x3 ∈ X. By (3.9) and (3.3), we get 4g(ax1 + cx3) + 4g(bx2) = 4a 2 g(x1) + 4c 2 g(x3) + 4b 2 g(x2) + g(ax1 + cx3 + bx2) + g(ax1 + cx3 − bx2) − g(ax1 − cx3 + bx2) − g(−ax1 + cx3 + bx2) (3.10) 238 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) for x1, x2, x3 ∈ X. Using (3.10), (3.3) and the evenness of g, we get g(ax1 + bx2 + cx3) + g(ax1 − bx2 + cx3) − g(ax1 + bx2 − cx3) − g(−ax1 + bx2 + cx3) + 4a2g(x1) + 4b 2g(x2) + 4c 2g(x3) = 4g(ax1 + cx3) + 4g(bx2) (3.11) for all x1, x2, x3 ∈ X. Conversely, assume that g : X → Y satisfies the functional equation (1.3). Replacing (x1, x2, x3) by ( x a , 0, y c ) in (1.3), we get g(x − y) − g(−x + y) + g(x + y) − g(x − y) + 4g(x) + 4g(y) = 4g(x + y) (3.12) for all x, y ∈ X. Using (1.3) and the evenness of g, we get g(x + y) + g(x − y) = 2g(x) + 2g(y), which is quadratic. 4 Stability results for (1.3): Odd case and direct method In this section, we present the Hyers-Ulam stability of the functional equation (1.3) for an odd mapping case. Theorem 3. Let j ∈ {−1, 1} and α : X3 → [0, ∞) be a function such that ∞ ∑ k=0 α(akjx1, a kjx2, a kjx3) akj < ∞ for all x1, x2, x3 ∈ X. Let g : X → Y be an odd mapping satisfying the inequality ‖Dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (4.1) for all x1, x2, x3 ∈ X. There exists a unique additive mapping A : X → Y which satisfies the functional equation (1.3) and ‖g(x) − A(x)‖ ≤ 1 2 ∞ ∑ k= 1−j 2 α(akjx1, 0, 0) akj (4.2) for all x1 ∈ X. The mapping A(x) is defined by, A(x) = lim k→∞ g(akjx1) akj for all x ∈ X Proof. Assume that j = 1. Replacing (x1, x2, x3) by (x, 0, 0) in (4.2) and using the oddness of g, we get ‖2g(ax) − 2ag(x)‖ ≤ α(x, 0, 0) (4.3) CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 239 for all x ∈ X. It follows from (4.3) that ∥ ∥ ∥ ∥ g(ax) a − g(x) ∥ ∥ ∥ ∥ ≤ 1 2a α(x, 0, 0) (4.4) for all x ∈ X. Replacing x by ax in (4.4) and dividing by a, we obtain ∥ ∥ ∥ ∥ g(a2x) a2 − g(ax) a ∥ ∥ ∥ ∥ ≤ 1 2a2 α(ax, 0, 0) (4.5) for all x ∈ X. It follows from (4.4) and (4.5) that ∥ ∥ ∥ ∥ g(a2x) a2 − g(x) ∥ ∥ ∥ ∥ ≤ 1 2a [ α(x, 0, 0) + α(ax, 0, 0) a ] (4.6) for all x ∈ X. Similarly, for any positive integer n, we have ∥ ∥ ∥ ∥ g(x) − g(anx) an ∥ ∥ ∥ ∥ ≤ 1 2a n−1 ∑ k=0 α(akx, 0, 0) ak ≤ 1 2a ∞ ∑ k=0 α(akx, 0, 0) ak (4.7) for all x ∈ X. In order to prove convergence of the sequence { g(akx) ak } , replacing x by amx and dividing am in (4.7) for any m, n > 0, we get ∥ ∥ ∥ ∥ g(amx) am − g(am+nx) am+n ∥ ∥ ∥ ∥ = 1 2am ∥ ∥ ∥ ∥ g(amx) − g(amanx) an ∥ ∥ ∥ ∥ ≤ 1 2a n−1 ∑ m=0 α(am+nx, 0, 0) am+n ≤ 1 2a n−1 ∑ m=0 α(am+nx, 0, 0) am+n → 0 as m → ∞. Hence the sequence { g(anx) an } is a Cauchy sequence. Since Y is complete, there exists a mapping A : X → Y such that A(x) = lim n→∞ g(anx) an , ∀x ∈ X. (4.8) Letting n → ∞ in (4.8), we see that (4.8) holds for x ∈ X. To prove that A satisfies (1.3), replacing (x1, x2, x3) by (a nx, anx, anx) and dividing an in (4.1), we obtain 1 an ‖Dg(anx, anx, anx)‖ ≤ 1 an α(anx, anx, anx) for all x1, x2, x3 ∈ X. Letting m → ∞ in the above inequality and using the definition of A(x), we see that DA(x1, x2, x3) = 0. Hence A satisfies (1.3) for all x1, x2, x3 ∈ X. To show that A is unique, let B(x) be another additive mapping satisfying (4.2). Then ‖A(x) − B(x)‖ = 1 an ‖A(anx) − B(anx)‖ ≤ 1 an {‖A(anx) − g(anx)‖ + ‖g(anx) − B(anx)‖} → 0 as n → ∞. 240 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) Hence A is unique. Assume that j = −1. Replacing x by x a in (4.3), we get ∥ ∥ ∥ ag(x) − a2g ( x a ) ∥ ∥ ∥ ≤ α ( x a , 0, 0 ) (4.9) for all x ∈ X. The rest of the proof is similar to the proof of the case j = 1. This completes the proof of the theorem. The following corollary is an immediate consequence of Theorem 3 concerning the stability of (1.3). Corollary 1. Let ǫ and p be nonnegative real numbers. Let g : X → Y be an odd mapping satisfiying the inequality ‖Dg(x1, x2, x3)‖ (4.10) ≤        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; p > 1 or p < 1 ǫ ( ‖x1‖ p + ‖x2‖ p + ‖x3‖ p + ‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p ) ; p > 1 3 or p < 1 3 for all x1, x2, x3 ∈ X. Then there exists a unique additive mapping A : X → Y such that ‖g(x) − A(x)‖ ≤        ǫ 2|a−1| ; ǫ‖x‖p 2|a−ap| ; p > 1 or p < 1 ǫ‖x‖3p 2|a−a3p| ; p > 1 3 or p < 1 3 (4.11) for all x ∈ X. Proof. Letting α(x1, x2, x3) =        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; ǫ ( ‖x1‖ p + ‖x2‖ p + ‖x3‖ p + ‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p ) for all x1, x2, x3 ∈ X, we can get the result. 5 Stability results for (1.3): Even case and direct method In this section, we discuss the Hyers-Ulam stability of the functional equation (1.3) for an even mapping case by using the direct method. CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 241 Theorem 4. Let j ∈ {−1, 1} and α : X3 → [0, ∞) be a function such that ∞ ∑ k=0 α(akjx1, a kjx2, a kjx3) akj < ∞ (5.1) for all x1, x2, x3 ∈ X. Let g : X → Y be an even mapping satisfying g(0) = 0 and the inequality ‖Dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (5.2) for all x1, x2, x3 ∈ X. There exists a unique additive mapping Q : X → Y which satisfies the functional equation (1.3) and ‖g(x) − Q(x)‖ ≤ 1 4a2 ∞ ∑ k= 1−j 2 α(akjx, 0, 0) a2kj (5.3) for all x ∈ X. The mapping Q(x) is defined by Q(x) = lim n→∞ g(akjx) a2kj (5.4) for all x ∈ X. Proof. Assume that j = 1. Replacing (x1, x2, x3) by (x, 0, 0) in (5.2), we get ‖4g(ax) − 4a2g(x)‖ ≤ α(x, 0, 0) (5.5) for all x ∈ X. It follows from (5.5) that ∥ ∥ ∥ ∥ g(ax) a2 − g(x) ∥ ∥ ∥ ∥ ≤ 1 4a2 α(x, 0, 0) (5.6) for all x ∈ X. Replacing x by ax in (5.6) and dividing by a2, we obtain ∥ ∥ ∥ ∥ g(a2x) a4 − g(ax) a2 ∥ ∥ ∥ ∥ ≤ 1 4a4 α(ax, 0, 0) (5.7) for all x ∈ X. It follows from (5.6) and (5.7) that ∥ ∥ ∥ ∥ g(a2x) a4 − g(x) ∥ ∥ ∥ ∥ ≤ 1 4a2 [ α(x, 0, 0) + α(ax, 0, 0) a2 ] (5.8) for all x ∈ X. Inductively, we have ∥ ∥ ∥ ∥ g(x) − g(anx) a2n ∥ ∥ ∥ ∥ ≤ 1 4a2 n−1 ∑ k=0 α(akx, 0, 0) a2k ≤ 1 a3 ∞ ∑ k=0 α(akx, 0, 0) a2k (5.9) 242 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) for all x ∈ X. In order to prove convergence of the sequence { g(akx) a2k } , replacing x by amx and dividing am in (5.9) for any m, n > 0, we get ∥ ∥ ∥ ∥ g(amx) a2m − g(am+nx) a2(m+n) ∥ ∥ ∥ ∥ = 1 a2m ∥ ∥ ∥ ∥ g(amx) − g(amanx) a2n ∥ ∥ ∥ ∥ ≤ 1 a3 n−1 ∑ m=0 α(am+nx, 0, 0) a2(m+n) ≤ 1 a3 n−1 ∑ m=0 α(am+nx, 0, 0) a2(m+n) → 0 as m → ∞. Hence the sequence { g(anx) a2n } is a Cauchy sequence. Since Y is complete, there exists a mapping Q : X → Y such that Q(x) = lim n→∞ g(anx) a2n , ∀x ∈ X. (5.10) Letting n → ∞ in (5.10) we see that (5.10) holds for x ∈ X. To prove that Q satisfies (1.3), replacing (x1, x2, x3) by (a nx, anx, anx) and dividing a2n in (5.2), we obtain 1 a2n ‖Dg(anx, anx, anx)‖ ≤ 1 a2n α(anx, anx, anx) for all x1, x2, x3 ∈ X. Letting n → ∞ in the above inequality and using the definition of Q(x), we see that DQ(x1, x2, x3) = 0. Hence Q satisfies (1.3) for all x1, x2, x3 ∈ X. To show that Q is unique, let B(x) be another quadratic mapping satisfying (5.4). Then ‖Q(x) − B(x)‖ = 1 a2n ‖Q(anx) − B(anx)‖ ≤ 1 a2n {‖Q(anx) − g(anx)‖ + ‖g(anx) − B(anx)‖} → 0 as n → ∞. Hence Q is unique. Assume that j = −1. Replacing x by x a in (5.5), we get ∥ ∥ ∥ ag(x) − a2g ( x a ) ∥ ∥ ∥ ≤ 1 4 α ( x a , 0, 0 ) (5.11) for all x ∈ X. The rest of the proof is similar to the proof of the case j = 1. This completes the proof of the theorem. The following corollary is an immediate consequence of Theorem 4 concerning the stability of (1.3). CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 243 Corollary 2. Let ǫ and p be nonnegative real numbers. Let gq : X → Y be an even mapping satisfiying g(0) = 0 and the inequality ‖Dg(x1, x2, x3)‖ (5.12) ≤        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; p > 2 or p < 2 ǫ ( ‖x1‖ p‖x2‖ p‖x3‖ p + {‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p} ) ; p > 2 3 or p < 2 3 for all x1, x2, x3 ∈ X. Then there exists a unique quadratic mapping Q : X → Y such that ‖g(x) − Q(x)‖ ≤        ǫ 4|a2−1| ǫ‖x‖p 4|a2−ap| ǫ‖x‖3p 4|a2−a3p| (5.13) for all x ∈ X. Proof. Letting α(x1, x2, x3) =        ǫ; ǫ (‖x1‖ p + ‖x2‖ p + ‖x3‖ p) ; ǫ ( ‖x1‖ p‖x2‖ p‖x3‖ p + {‖x1‖ 3p‖x2‖ 3p‖x3‖ 3p} ) ; for all x1, x2, x3 ∈ X, we get the result. 6 Stability results of (1.3): Mixed case In this section, we establish the Hyers-Ulam stability of the functional equation(1.3) for a mixed mapping case. Theorem 5. Let j ∈ {−1, 1} and α : X3 → [0, ∞) be a function satisfying (1.3) for all x1, x2, x3 ∈ X. Let g : X → Y be a mapping satisfying the inequality ‖Dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (6.1) for all x1, x2, x3 ∈ X. There exist a unique additive mapping A : X → Y and a unique quadratic mapping Q : X → Y which satisfies the functional equation (1.3) and ‖f(x) − A(x) − Q(x)‖ ≤ 1 2 {[ 1 2a ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) akj + α(−akjx, 0, 0) akj ] ] + 1 4n2 [ ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) a2kj + α(−akjx, 0, 0) a2kj ] ]} for all x ∈ X. The mapping A(x) and Q(x) are defined in (4.2) and (5.10), respectively. 244 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) Proof. Let go(x) = ga(x)−ga(−x) 2 for all x ∈ X. Then go(0) = 0 and go(−x) = −go(x) for all x ∈ X. Hence ‖Dgo(x1, x2, x3)‖ ≤ 1 2 { ‖Dga(x1, x2, x3)‖ + ‖Dga(−x1, −x2, −x3)‖ } ≤ α(x1, x2, x3) 2 + α(−x1, −x2, −x3) 2 for all x1, x2, x3 ∈ X. By Theorem 3, we have ‖go(x) − A(x)‖ ≤ 1 4a ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) akj + α(−akjx, 0, 0) akj ] (6.2) for all x ∈ X. Let ge(x) = gq(x)+gq(−x) 2 for all x ∈ X. Then ge(0) = 0 and ge(−x) = ge(x) for all x ∈ X. Hence, ‖Dge(x1, x2, x3)‖ ≤ 1 2 { ‖Dgq(x1, x2, x3)‖ + ‖Dgq(−x1, −x2, −x3)‖ } ≤ α(x1, x2, x3) 2 + α(−x1, −x2, −x3) 2 for all x1, x2, x3 ∈ X. By Theorem 4, we have ‖ge(x) − Q(x)‖ ≤ 1 8a2 ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) a2kj + α(−akjx, 0, 0) a2kj ] (6.3) for all x ∈ X. Then g(x) = ge(x) + go(−x) (6.4) for all x ∈ X. It follows from (6.2), (6.3) and (6.4) that ‖g(x) − A(x) − Q(x)‖ = ‖ge(x) + go(−x) − A(x) − Q(x)‖ ≤ ‖go(−x) − A(x)‖ + ‖ge(x) − Q(x)‖ ≤ 1 4a ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) akj + α(−akjx, 0, 0) akj ] + 1 8a2 ∞ ∑ k= 1−j 2 [α(akjx, 0, 0) a2kj + α(−akjx, 0, 0) a2kj ] for all x ∈ X. Hence the theorem is proved. Using Corollaries 1 and 2, we have the following corollary concerning the stability of (1.3). CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 245 Corollary 3. Let λ and s be a nonnegative real numbers. Let gq : X → Y be a mapping satisfiying the inequality ‖Dg(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); s 6= 1, 2 λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s}; s 6= 1 3 , 2 3 (6.5) for all x1, x2, x3 ∈ X. Then there exist a unique additive function A : X → Y and a unique quadratic mapping Q : X → Y such that ‖g(x) − A(x) − Q(x)‖ ≤          λ 2 [ 1 |a−1| + 1 2|a2−1| ] λ‖x‖S 2 [ 1 |a−aS| + 1 2|a2−aS| ] λ‖x‖3S 2 [ 1 |a−a3S| + 1 2|a2−a3S| ] (6.6) for all x ∈ X. 7 Fixed point stability of (1.3): Odd mapping case The following theorems are useful to prove our fixed point stability results. Theorem 6. [12] (Banach Contraction Principle) Let (X, d) be a complete metric space and consider a mapping T : X → X which is strictly contractive mapping. (A1) d(T x, T y) ≤ Ld(x, y) for some (Lipschitz constant) L < 1. (i) The mapping T has one and only fixed point x∗ = T (x∗); (ii) The fixed point for each given element x∗ is globally contractive, that is, (A2) limn→∞ T nx = x∗ for any starting point x ∈ X; (iii) One has the following estimation inequalities (A3) d(T nx, x∗) ≤ 1 1−L d(T nx, T n+1x), ∀n ≥ 0, ∀x ∈ X; (A4) d(x, x∗) = 1 1−L d(x, x∗), ∀x ∈ X. Theorem 7. [12] (Alternative Fixed Point Theorem) Suppose that for a complete generalized metric space (X, d) and a strictly contractive mapping T : X → X with Lipschitz constant L. Then for each given element x ∈ X, (B1) d(T nx, T n+1x) = ∞, ∀n ≥ 0; (B2) there exists a natural number n0 such that 246 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) (i) d(T nx, T n+1x) < ∞, ∀n ≥ 0; (ii) The sequence {T nx} is convergent to a fixed point y∗ of T ; (iii) y∗ is the unique fixed point of T in the set Y = {y ∈ Y : d(T n0, y) < ∞}; (iv) d(y∗, y) ≤ 1 1−L d(y, T y) for all y ∈ Y . In this method, we investigate the Hyers-Ulam stability of the functional equation (1.3) for an odd mapping case by using fixed point method. Theorem 8. Let g : W → B be an odd mapping for which there exists a function α : W 3 → [0, ∞) with the condition lim n→∞ α(aki x1, a k i x2, a k i x3) aki = 0, (7.1) for ai =      a i = 0 1 a i = 1, such that the functional inequality ‖Dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (7.2) for all x1, x2, x3 ∈ W. If there exists L = L(i) such that the function x → β(x) = 1 2 α ( x a , 0, 0 ) has the property 1 ai β(aix) = L (β(x)) (7.3) for all x ∈ W. Then there exists a unique additive function A : W → B satisfying the functional equation (1.3) and ‖g(x) − A(x)‖ ≤ L1−i 1 − L β(x) (7.4) for all x ∈ W. Proof. Consider the set X = {P |P : W → B, P(0) = 0} and introduce the generalized metric on X. d(p, q) = inf{k ∈ (0, ∞) : ‖p(x) − q(x)‖ ≤ β(x), x ∈ W} It is easy to see that (X, d) is complete. Define T : X → X by Tp(x) = 1 ai p(aix) for all x ∈ W . Now p, q ∈ X, d(p, q) ≤ k ⇒ ‖p(x) − q(x)‖ ≤ kβ(x), x ∈ W. ⇒ ∥ ∥ ∥ ∥ 1 ai p(aix) − 1 ai q(aix) ∥ ∥ ∥ ∥ ≤ 1 ai kβ(aix), ∀x ∈ W ⇒ ‖Tp(x) − Tq(x)‖ ≤ Lkβ(x), ∀x ∈ W ⇒ d(Tp, Tq) ≤ Lk. CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 247 This implies d(Tp, Tq) ≤ Ld(p, q) for all p, q ∈ X. That is, T is a strictly contractive mapping on X with Lipschitz constant L. It follows from (4.3) that ‖2g(ax) − 2ag(x)‖ ≤ α(x, 0, 0) (7.5) for all x ∈ W . It follows from (7.5) that, ∥ ∥ ∥ ∥ g(x) − g(ax) a ∥ ∥ ∥ ∥ ≤ 1 2a α(x, 0, 0) (7.6) for all x ∈ W . Using (6.2), for this case i = 0, it reduces to ∥ ∥ ∥ ∥ g(x) − g(ax) a ∥ ∥ ∥ ∥ ≤ 1 a β(x) (7.7) for all x ∈ W . Thus d(ga, T ga) ≤ 1 a = L = L1 < ∞. Again replacing x by x a in (7.5), we get ∥ ∥ ∥ g(x) − ag ( x a ) ∥ ∥ ∥ ≤ 1 2 α ( x a , 0, 0 ) (7.8) for all x ∈ W . By using (7.3) for the case i = 1, it reduces to ∥ ∥ ∥ g(x) − ag ( x a ) ∥ ∥ ∥ ≤ β(x). (7.9) That is, d(g, T g) ≤ 1 ⇒ d(g, T g) ≤ 1 = L0 < ∞. In the above case, we have d(g, T g) ≤ L1−i. Therefore (B2(i)) holds. From (B2(ii)), it follows that there exists a fixed point A of T in X such that A(x) = lim i→∞ ga(a k i x) aki , ∀x ∈ W. (7.10) In order to prove A : W → B is additive, replacing (x1, x2, x3) by (a k i x1, a k i x2, a k i x3) in (7.2) and dividing aki , it follows from (7.3) and (7.10) that A satisfies (1.3) for all x1, x2, x3 ∈ W . By (B2(iii)), A is the unique fixed point of T in the set, Y = {g ∈ X : d(T g, A) < ∞}. Using the fixed point alternative result, A is the unique function such that ‖g(x) − A(x)‖ ≤ kβ(x) for all x ∈ W and k > 0. Finally, by (B2(iv)), we obtain d(g, A) ≤ 1 1 − L d(g, T g). That is, d(g, A) ≤ L 1−i 1−L . Hence we conclude that ‖g(x) − A(x)‖ ≤ L1−i 1 − L β(x) for all x ∈ W . This completes the proof of the theorem. 248 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) Corollary 4. Let g : W → B be an odd mapping and assume that there exist real numbers λ and s such that ‖Dga(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s} (7.11) for all x1, x2, x3 ∈ X. Then there exists a unique additive mapping A : W → B such that ‖g(x) − A(x)‖ ≤        λ 2|a−1| ; λ‖x‖s 2|a−as| ; s 6= 1 λ‖x‖3s 2|a−a3s| ; s 6= 1 3 (7.12) for all x ∈ X. Proof. Let α(x1, x2, x3) =        λ; λ(‖x1‖ S + ‖x2‖ S + ‖x3‖ S); λ(‖x1‖ S + ‖x2‖ S + ‖x3‖ S) + {λ(‖x1‖ 3S + ‖x2‖ 3S + ‖x3‖ 3S)}; for all x1, x2, x3 ∈ W . Now, α(aki x1, a k i x2, a k i x3) aki =                λ aki ; λ aki (‖aki x1‖ S + ‖aki x2‖ S + ‖aki x3‖ S); λ ak i (‖aki x1‖ S + ‖aki x2‖ S + ‖aki x3‖ S) + {‖aki x1‖ 3S + ‖aki x2‖ 3S + ‖aki x3‖ 3S} =        → 0 as k → ∞ → 0 as k → ∞ → 0 as k → ∞. (7.13) That is, (7.1) holds. But we have β(x) = 1 2 α ( x a , 0, 0 ) . Hence β(x) = 1 2 α ( x a , 0, 0 ) =        λ 2 λ 2aS (‖x‖S) λ 2aS (‖x‖S). CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 249 Also 1 ai β(ai, x) =        λ 2ai λ 2ai (‖aix‖ S) λ 2ai (‖aix‖ S) =        a −1 i β(x) a S−1 i β(x) a 3S−1 i β(x). Hence the inequality (7.7) holds. Either L = a−1 for s = 0 if i = 0 and L = 1 a−1 for s = 0 if i = 1. Either L = as−1 for s < 1 if i = 0 and L = 1 as−1 for s > 1 if i = 1. Either L = a3s−1 for s < 1 if i = 0 and L = 1 a3s−1 for s > 1 if i = 1. Now from (7.2), we prove the following cases: Case: 1 L = a−1, i = 0 ‖ga(x) − A(x)‖ ≤ L1−i 1 − L β(x) = (a−1)1−0 1 − a−1 λ 2 = λ 2(a − 1) . (7.14) Case: 2 L = ( 1 a )−1 , i = 1 ‖ga(x) − A(x)‖ ≤ L1−i 1 − L β(x) = (a)1−1 1 − a λ 2 = λ 2(1 − a) . (7.15) Case: 3 L = as−1, s < 1, i = 0 ‖ga(x) − A(x)‖ ≤ L1−i 1 − L β(x) = (as−1)1−0 1 − aS−1 λ 2aS ‖x‖S = λ‖x‖S 2|a − aS| . (7.16) Case: 4 L = ( 1 a )S−1 , S > 1, i = 1 ‖ga(x) − A(x)‖ ≤ L1−i 1 − L β(x) = (a1−s)1−1 1 − a1−S λ 2aS ‖x‖S = λ‖x‖S 2(aS − a) . (7.17) Case: 5 L = a3s−1, S < 1 3 , i = 0 ‖ga(x) − A(x)‖ ≤ L1−i 1 − L β(x) = (a3S−1)1−0 1 − a3S−1 λ 2a3S ‖x‖S = λ‖x‖S 2(a − a3S) . (7.18) Case: 6 L = ( 1 a )−1 , i = 1 ‖ga(x) − A(x)‖ ≤ L1−i 1 − L β(x) = (a1−3S)1−1 1 − a1−3S λ 2a3S ‖x‖S = λ‖x‖S 2(a3S − a) . (7.19) Hence the proof of the corollary is completed. 8 Fixed point stability of (1.3): Even mapping case In this method, we investigate the Hyers-Ulam stability of the functional equation (1.3) for an even case mapping by using fixed point method. 250 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) Theorem 9. Let g : W → B be an even mapping for which there exists a function α : W 3 → [0, ∞) with the condition lim n→∞ α(aki x1, a k i x2, a k i x3) a2ki = 0 (8.1) for ai =      a i = 0 1 a i = 1, such that the functional inequality ‖Dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (8.2) for all x1, x2, x3 ∈ W. If there exists L = L(i) such that the function x → β(x) = 1 2 α ( x a , 0, 0 ) (8.3) has the property 1 a2i β(aix) = L (β(x)) (8.4) for all x ∈ W, then there exists a unique quadratic mapping Q : W → B satisfying the functional equation (1.3) and ‖g(x) − Q(x)‖ ≤ L1−i 1 − L β(x) (8.5) for all x ∈ W. Proof. Consider the set X = {P |P : W → B, P(0) = 0} and introduce the generalized metric on X. d(p, q) = inf{k ∈ (0, ∞) : ‖p(x) − q(x)‖ ≤ β(x), x ∈ W} It is easy to see that (X, d) is complete. Define T : X → X by Tp(x) = 1 a2 i p(aix) for all x ∈ W . Now p, q ∈ X, d(p, q) ≤ k ⇒ ‖p(x) − q(x)‖ ≤ kβ(x), x ∈ W. ⇒ ∥ ∥ ∥ ∥ 1 a2i p(aix) − 1 a2i q(aix) ∥ ∥ ∥ ∥ ≤ 1 a2i kβ(aix), ∀x ∈ W ⇒ ‖Tp(x) − Tq(x)‖ ≤ Lkβ(x), ∀x ∈ W ⇒ d(Tp, Tq) ≤ Lk. This implies d(Tp, Tq) ≤ Ld(p, q) for all p, q ∈ X. That is, T is a strictly contractive mapping on X with Lipschitz constant L. Replacing (x1, x2, x3) by (x, 0, 0) in (9.1) and using the evenness of g, we get ‖4g(ax) − 4a2g(x)‖ ≤ α(x, 0, 0), (8.6) ∥ ∥ ∥ ∥ g(x) − g(ax) n2 ∥ ∥ ∥ ∥ ≤ 1 4a2 α(x, 0, 0) (8.7) CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 251 for all x ∈ W . By using (8.4), for this case i = 0, it reduces to ∥ ∥ ∥ ∥ g(x) − g(ax) a2 ∥ ∥ ∥ ∥ ≤ 1 2a2 β(x) (8.8) for all x ∈ W . That is, d(g, T g) ≤ 1 a2 ⇒ d(g, T g) ≤ 1 a2 = L = L1 < ∞. Again replacing x by x a in (8.6), we get ∥ ∥ ∥ g(x) − a2g ( x a ) ∥ ∥ ∥ ≤ 1 4 α ( x a , 0, 0 ) (8.9) for all x ∈ W . That is, d(g, T g) ≤ 1 2 < 1 ⇒ d(g, T g) ≤ 1 = L0 < ∞. In above case, we get d(g, T g) ≤ L1−i. The rest of the proof is similar to that of the previous theorem. This completes the proof of the theorem. Corollary 5. Let g : W → B be an even mapping and assume that there exist real numbers λ and s such that ‖Dg(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); s 6= 2 λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s}; s 6= 1 3 (8.10) for all x1, x2, x3 ∈ X. Then there exists a unique quadratic mapping Q : W → B such that ‖gq(x) − Q(x)‖ ≤        λ 4|a2−1| ; λ‖x‖s 4|a2−as| λ‖x‖3s 4|a2−a3s| (8.11) for all x ∈ X. 9 Fixed point stability of (1.3): Mixed mapping case In this method, we present the Hyers-Ulam stability of the functional equation (1.3) for a mixed mapping case by using fixed point method. Theorem 10. Let g : W → B be a mapping for which there exists a function α : W 3 → [0, ∞) with the condition (7.1) and (8.1) for ai =      a i = 0 1 a i = 1, such that the functional inequality ‖Dg(x1, x2, x3)‖ ≤ α(x1, x2, x3) (9.1) 252 V. Govindan, C. Park, S. Pinelas & T. M. Rassias CUBO 22, 2 (2020) for all x1, x2, x3 ∈ W. If there exists L = L(i) such that the function x → β(x) = 1 2 α ( x a , 0, 0 ) satisfies (7.3) and (8.3) for all x ∈ W, then there exist a unique additive mapping A : W → B and a quadratic mapping Q : W → B satisfying the functional equation (1.3) and ‖g(x) − A(x) − Q(x)‖ ≤ L1−i 1 − L [β(x) + β(−x)] holds for all x ∈ W. Proof. It follows from (6.2) and Theorem 8 that ‖go(x) − A(x)‖ ≤ 1 2 L1−i 1 − L [β(x) + β(−x)]. (9.2) Similarly, it follows from (7.5) and Theorem 9 that ‖ge(x) − Q(x)‖ ≤ 1 2 L1−i 1 − L [β(x) + β(−x)] (9.3) for all x ∈ W . Then g(x) = go(x) + ge(x) for all x ∈ W . From (8.11), (9.2) and (9.3), we have ‖g(x) − A(x) − Q(x)‖ = ‖ge(x) + go(x) − A(x) − Q(x)‖ ≤ ‖go(x) − A(x)‖ + ‖ge(x) − Q(x)‖ = L1−i 1 − L [β(x) + β(−x)] for all x ∈ W . Hence the theorem is proved. Corollary 6. Let g : W → B be a mapping and assume that there exist real numbers λ and s such that ‖Dg(x1, x2, x3)‖ ≤        λ; λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s); s 6= 1, 2 λ(‖x1‖ s + ‖x2‖ s + ‖x3‖ s) + {‖x1‖ 3s + ‖x2‖ 3s + ‖x3‖ 3s}; s 6= 1 3 , 2 3 for all x1, x2, x3 ∈ X. Then there exist a unique additive mapping A : W → B and a unique quadratic mapping Q : W → B such that ‖g(x) − A(x) − Q(x)‖ ≤        λ 2|a−1| + λ 4|a2−1| λ‖x‖S 2|a−aS| + λ‖x‖S 4|a2−aS| λ‖x‖3S 2|a−a3S| + λ‖x‖3S 4|a2−a3S| for all x ∈ X. CUBO 22, 2 (2020) Hyers-Ulam stability of an additive-quadratic functional equation 253 Declarations Availablity of data and materials Not applicable. Competing interests The authors declare that they have no competing interests. Fundings This work was supported by Basic Science Research Program through the National Research Foun- dation of Korea funded by the Ministry of Education, Science and Technology (NRF-2017R1D1A1B04032937). 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Introduction Solution of the functional equation (1.3): when g is odd Solution of the functional equation (1.3): when g is even Stability results for (1.3): Odd case and direct method Stability results for (1.3): Even case and direct method Stability results of (1.3): Mixed case Fixed point stability of (1.3): Odd mapping case Fixed point stability of (1.3): Even mapping case Fixed point stability of (1.3): Mixed mapping case