International Journal of Analysis and Applications Volume 17, Number 3 (2019), 369-387 URL: https://doi.org/10.28924/2291-8639 DOI: 10.28924/2291-8639-17-2019-369 ON APPROXIMATION SOLUTIONS OF THE CAUCHY-JENSEN AND THE ADDITIVE-QUADRATIC FUNCTIONAL EQUATION IN PARANORMED SPACES PRONDANAI KASKASEM1 AND CHAKKRID KLIN-EAM1,2,∗ 1Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok, 65000, Thailand 2Research center for Academic Excellence in Mathematics, Naresuan University, Thailand ∗Corresponding author: chakkridk@nu.ac.th Abstract. In this paper, we prove the generalized Hyers-Ulam-Rassias stability of the bi-Cauchy-Jensen functional equation and the bi-additive-quadratic functional equation in paranormed spaces. Moreover, we investigate the Hyers-Ulam-Rassias stability of the generalized Cauchy-Jensen equation in such spaces. 1. Introduction and Preliminaries The stability problem of functional equations was initiated by Ulam in 1940 [17] arising from concerning the stability of group homomorphisms. These question form is the object of the stability theory. In 1941, Hyers [7] provided a first affirmative partial answer to Ulam’s problem for the case of approximately additive mapping in Banach spaces. In 1978, Rassias [16] gave a generalization of Hyers’s theorem for linear mapping by considering an unbounded Cauchy difference. A generalization of Rassias’s result was developed by Găvruţa [6] in 1994 by replacing the unbounded Cauchy difference by a general control function. For more information on that subject and further references we refer to a survey paper [3] and to a recent monograph on Ulam stability [4]. Received 2018-12-21; accepted 2019-01-30; published 2019-05-01. 2010 Mathematics Subject Classification. 39B82; 39B52. Key words and phrases. Hyers-Ulam-Rassias stability; bi-Cauchy-Jensen functional equation; bi-additive-quadratic func- tional equation; generalized Cauchy-Jensen functional equation; paranormed space. c©2019 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 369 https://doi.org/10.28924/2291-8639 https://doi.org/10.28924/2291-8639-17-2019-369 Int. J. Anal. Appl. 17 (3) (2019) 370 Let R and N be the set of real numbers and the set of natural numbers, respectively. Next, let X and Y be vector spaces and k be a positive integer, a function f : Xk → Y is called k-additive functional equation if and only if f satisfies the equation f(x1,x2, . . . ,xi−1,xi + y,xi+1, . . . ,xk) =f(x1,x1, . . . ,xi) + f(x1,x2, . . . ,xi−1,y,xi+1, . . . ,xk) for all i ∈ N, 1 ≤ i ≤ k and for every x1,x2, . . . ,xk,y ∈ X, that is, f is additive in each of its variables xi ∈ X for all i = 1, 2, . . . ,k. Some fundamental properties on such mappings be mentioned in [10]. In particular, a 2-additive functional equation is called bi-additive functional equation. A mapping f : X × X → Y is called a bi-additive-quadratic functional equation (bi-AQE, shortly) if f satisfies the system equations f(x + y,z) = f(x,z) + f(y,z), f(x,y + z) + f(x,y −z) = 2f(x,y) + 2f(x,z) (1.1) for all x,y,z ∈ X. When X = Y = R, the solution of (1.1) is given by the function f(x,y) = cxy2 where x,y,c ∈ R. For mapping f : X ×X → Y satisfies f(x + y,z + w) + f(x + y,z −w) = 2f(x,z) + 2f(x,w) + 2f(y,z) + 2f(y,w) (1.2) for all x,y,z,w ∈ X. In 2005, Park, Bae and Chung [13] proved that the mapping f : X × X → Y satisfies (1.1) if and only if it satisfies (1.2) and provided the general solution of (1.1) which is given by f(x,y) = M(x,y,y) and M(x,y,z) = M(x,z,y) for all x,y,z ∈ X where M : X × X × X → Y is a multi-additive mapping. A mapping f : X×X → Y is called a bi-Cauchy-Jensen functional equation (bi-CJE, shortly) if f satisfies the system equations f(x + y,z) = f(x,z) + f(y,z) 2f ( x, y + z 2 ) = f(x,y) + f(x,z) (1.3) for all x,y,z ∈ X. In particular, For X = Y = R, The solution of (1.3) is given by the function f(x,y) = axy + bx where x,y,a,b ∈ R. For mapping f : X ×X → Y satisfies 2f ( x + y, z + w 2 ) = f(x,z) + f(x,w) + f(y,z) + f(y,w) (1.4) for all x,y,z,w ∈ X. In 2006, Park and Bae [12] showed that the mapping f : X ×X → Y satisfies (1.3) if and only if it satisfies (1.4) and gave the general solution of (1.4) which is given by f(x,y) = B(x,y) + A(x) for all x,y ∈ X where B : X ×X → Y is a bi-additive mapping and A : X → Y is an additive mapping. Int. J. Anal. Appl. 17 (3) (2019) 371 Next, we recall the concepts of paranormed space and some basic facts on the Fréchet spaces. Definition 1.1 ([18]). Let X be a vector space. A paranorm on X is a function P : X → R such that for all x,y ∈ X, the following conditions hold : (i) P(0) = 0; (ii) P(−x) = P(x); (iii) P(x + y) ≤ P(x) + P(y) (triangle inequality); (iv) If {tn} is a sequence of scalars with tn → t and {xn}⊆ X with P(xn−x) → 0, then P(tnxn−tx) → 0. (continuity of scalar multication) The pair (X,P) is called a paranormed space if P is a paranorm on X. Note that P(nx) ≤ nP(x) for all n ∈ N and all x ∈ X. The paranorm P on X is called total if, in addition, P satisfies (v) P(x) = 0 implies x = 0. A Fréchet space is a total and complete paranormed space. In 2015, Bae and Park [2] proved the Hyers-Ulam stability of the functional equation (1.2) and (1.4) in paranormed spaces in the sense of Rassias [16]. We refer to some works of stability of the functional equation (1.2) and (1.4) and various functional equations in paranormed spaces with [1, 8, 9, 11, 13–15]. In the first section of main results, we investigate stability of the functional equation (1.2) and (1.4) in paranormed spaces in the sense of Găvruţa [6]. In 2009, Gao et al. [5] introduced the generalized Cauchy-Jensen functional equation and gave some useful properties. Let G be an n-divisible abelian group where n ∈ N and X be a normed space with norm ‖·‖X. For a mapping f : G → X is called a generalized Cauchy-Jensen functional equation (GCJE, shortly) if it satisfies the equation αf ( x + y α + z ) = f(x) + f(y) + αf(z) (1.5) for all x,y,z ∈ X and for any fixed positive integer α ≥ 2. In particular, when α = 2, it is called a Cauchy-Jensen functional equation (CJE, shortly). Proposition 1.1 ([5]). Let G be an n-divisible abelian group for some positive integer n and X be a normed space with norm ‖·‖X. Then a mapping f : G → X is additive if and only if it satisfies ‖f(x) + f(y) + nf(z)‖X ≤ ∥∥∥∥nf(x + yn + z) ∥∥∥∥ X for all x,y,z ∈ G. The following corollary is an immediate consequence of Proposition 1.1. Int. J. Anal. Appl. 17 (3) (2019) 372 Corollary 1.1 ([5]). For a mapping f : G → X, the following statements are equivalent. (a) f is additive. (b) f(x) + f(y) + nf(z) = nf(x+y n + z), for all x,y,z ∈ G. (c) ‖f(x) + f(y) + nf(z)‖X ≤‖nf(x+yn + z)‖X, for all x,y,z ∈ G. Clearly, a vector space is n-divisible abelian group, so Corollary 1.1 is right when G is a vector space. In the second section of main results, we proved the stability of the functional equation (1.5) in paranormed spaces in the sense of Rassias [16]. Throughtout this paper, assume that (X,P) is a Fréchet space and that (E,‖·‖) is a Banach space. 2. The stability of the bi-Cauchy-Jensen functional equation and bi-additive-quadratic functional equation in paranormed spaces The following result is the generalized Hyers-Ulam-Rassias stability of the functional equation (1.4). Theorem 2.1. Let ϕ : E ×E ×E ×E → [0,∞) be a function and f : E ×E → X be a mapping satisfying f(x, 0) = 0 for all x ∈ E such that P ( 2f ( x + y, z + w 2 ) −f(x,z) −f(x,w) −f(y,z) −f(y,w) ) ≤ϕ(x,y,z,w) (2.1) for all x,y,z,w ∈ E. Then there exists a unique mapping F : E ×E → X satisfying (1.4) such that P (2f(x,y) −F(x,y)) ≤ ϕ̃(x,x,y,y) (2.2) for all x,y ∈ E where ϕ̃(x,y,z,w) (2.3) := ∞∑ j=0 6j [ 6ϕ ( x 2j+1 , y 2j+1 , z 3j+1 ,− w 3j+1 ) + 4ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j ) +ϕ ( x 2j+1 , y 2j+1 , z 3j , w 3j )] < ∞ for all x,y,z,w ∈ E and the mapping F : E ×E → X is given by F(x,y) = lim j→∞ 2·6jf ( x 2j , y 3j ) for all x,y ∈ E. Int. J. Anal. Appl. 17 (3) (2019) 373 Proof. Letting y = x in (2.1), we obtain that P ( 2f ( 2x, z + w 2 ) − 2f(x,z) − 2f(x,w) ) ≤ ϕ(x,x,z,w) (2.4) for all x,z,w ∈ E. Letting w = −z in (2.4), we get that P (2f(x,z) + 2f(x,−z)) ≤ ϕ(x,x,z,−z) (2.5) for all x,z ∈ E. Subtituting z by −z and w by −z in (2.4), we get P(2f(2x,−z) − 4f(x,−z)) ≤ ϕ(x,x,−z,−z) (2.6) for all x,z ∈ E. It follows from (2.5) and (2.6) that P(4f(x,z) + 2f(2x,−z)) (2.7) = P(4f(x,z) + 4f(x,−z) − 4f(x,−z) + 2f(2x,−z)) ≤ P(4f(x,z) + 4f(x,−z)) + P(2f(2x,−z) − 4f(x,−z)) ≤ 2P(2f(x,z) + 2f(x,−z)) + P(2f(2x,−z) − 4f(x,−z)) ≤ 2ϕ(x,x,z,−z) + ϕ(x,x,−z,−z) for all x,z ∈ E. Letting w = −3z in (2.4), we have P(2f(2x,−z) − 2f(x,−3z) − 2f(x,z)) ≤ ϕ(x,x,z,−3z) for all x,z ∈ E. By (ii) of definition 1.1, we have P(2f(x,−3z) + 2f(x,z) − 2f(2x,−z)) ≤ ϕ(x,x,z,−3z) (2.8) for all x,z ∈ E. By (2.7) and (2.8), we have P(6f(x,z) + 2f(x,−3z)) (2.9) =P(4f(x,z) + 2f(2x,−z) + 2f(x,−3z) + 2f(x,z) − 2f(2x,−z)) ≤P(4f(x,z) + 2f(2x,−z)) + P(2f(x,−3z) + 2f(x,z) − 2f(2x,−z)) ≤2ϕ(x,x,z,−z) + ϕ(x,x,−z,−z) + ϕ(x,x,z,−3z) for all x,z ∈ E. Putting z = 3z in (2.6), we obtain that P(2f(2x,−3z) − 4f(x,−3z)) ≤ ϕ(x,x,−3z,−3z) (2.10) Int. J. Anal. Appl. 17 (3) (2019) 374 for all x,z ∈ E. It follows from (2.9) and (2.10) P(12f(x,z) + 2f(2x,−3z)) (2.11) =P(12f(x,z) + 4f(x,−3z) − 4f(x,−3z) + 2f(2x,−3z)) ≤P(12f(x,z) + 4f(x,−3z)) + P(2f(2x,−3z) − 4f(x,−3z)) ≤2P(6f(x,z) + 2f(x,−3z)) + P(2f(2x,−3z) − 4f(x,−3z)) ≤4ϕ(x,x,z,−z) + 2ϕ(x,x,−z,−z) + 2ϕ(x,x,z,−3z) + ϕ(x,x,−3z,−3z) for all x,z ∈ E. Replacing z by −z in the above inequality, we get that P(12f(x,−z) + 2f(2x, 3z)) (2.12) ≤4ϕ(x,x,−z,z) + 2ϕ(x,x,z,z) + 2ϕ(x,x,−z, 3z) + ϕ(x,x, 3z, 3z) for all x,z ∈ E. By (2.5) and the above inequality, we have P(12f(x,z) − 2f(2x, 3z)) (2.13) =P(12f(x,z) + 12f(x,−z) − 12f(x,−z) − 2f(2x, 3z)) ≤P(12f(x,z) + 12f(x,−z)) + P(−12f(x,−z) − 2f(2x, 3z)) =P(12f(x,z) + 12f(x,−z)) + P(12f(x,−z) + 2f(2x, 3z)) ≤6P(2f(x,z) + 2f(x,−z)) + P(12f(x,−z) + 2f(2x, 3z)) ≤6ϕ(x,x,z,−z) + 4ϕ(x,x,−z,z) + 2ϕ(x,x,z,z) + 2ϕ(x,x,−z, 3z) + ϕ(x,x, 3z, 3z) for all x,z ∈ E. Replacing x by x 2j+1 and z by z 3j+1 in (2.13), we obtain that P ( 12f ( x 2j+1 , z 3j+1 ) − 2f ( x 2j , z 3j )) (2.14) ≤6ϕ ( x 2j+1 , x 2j+1 , z 3j+1 ,− z 3j+1 ) + 4ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 , z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j ) + ϕ ( x 2j+1 , x 2j+1 , z 3j , z 3j ) Int. J. Anal. Appl. 17 (3) (2019) 375 for all x,z ∈ E. By (2.14), for any integers l,m such that 0 ≤ l < m, we get that P ( 2·6mf ( x 2m , z 3m ) − 2·6lf ( x 2l , z 3l )) (2.15) =P ( 2·6mf ( x 2m , z 3m ) − 2·6m−1f ( x 2m−1 , z 3m−1 ) + 2·6m−1f ( x 2m−1 , z 3m−1 ) − 2·6m−2f ( x 2m−2 , z 3m−2 ) + 2·6m−2f ( x 2m−2 , z 3m−2 ) + · · · + 2·6l+1f ( x 2l+1 , z 3l+1 ) − 2·6lf ( x 2l , z 3l )) ≤P ( 2·6mf ( x 2m , z 3m ) − 2·6m−1f ( x 2m−1 , z 3m−1 )) + P ( 2·6m−1f ( x 2m−1 , z 3m−1 ) − 2·6m−2f ( x 2m−2 , z 3m−2 )) + · · · + P ( 2·6l+1f ( x 2l+1 , z 3l+1 ) − 2·6lf ( x 2l , z 3l )) ≤6m−1P ( 12f ( x 2m , z 3m ) − 2f ( x 2m−1 , z 3m−1 )) + 6m−2P ( 12f ( x 2m−1 , z 3m−1 ) − 2f ( x 2m−2 , z 3m−2 )) + · · · + 6lP ( 12f ( x 2l+1 , z 3l+1 ) − 2f ( x 2l , z 3l )) = m−1∑ j=l 6jP ( 12f ( x 2j+1 , z 3j+1 ) − 2f ( x 2j , z 3j )) ≤ ∞∑ j=l 6j [ 6ϕ ( x 2j+1 , x 2j+1 , z 3j+1 ,− z 3j+1 ) + 4ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 , z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j ) +ϕ ( x 2j+1 , x 2j+1 , z 3j , z 3j )] for all x,z ∈ E. It follows from (2.3) that lim l→∞ ∞∑ j=l 6j [ 6ϕ ( x 2j+1 , x 2j+1 , z 3j+1 ,− z 3j+1 ) + 4ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 , z 3j+1 , z 3j+1 ) + 2ϕ ( x 2j+1 , x 2j+1 ,− z 3j+1 , z 3j ) +ϕ ( x 2j+1 , x 2j+1 , z 3j , z 3j )] = 0 for all x,z ∈ E. This implies that the sequence {2·6jf ( x 2j , z 3j ) }∞j=0 is a Cauchy sequence in X for all x,z ∈ E. Since X is complete paranormed space, the sequence {2·6jf ( x 2j , z 3j ) }∞j=0 converges for all x,z ∈ E. Define F : E ×E → X by F(x,z) = lim j→∞ 2·6jf ( x 2j , z 3j ) (2.16) Int. J. Anal. Appl. 17 (3) (2019) 376 for all x,z ∈ E. By (2.3), we get that ∞∑ j=1 1 6 ·2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = ∞∑ j=0 1 6 ·2·6j+1ϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) = ∞∑ j=0 2·6jϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) ≤ ∞∑ j=0 6·6jϕ ( x 2j+1 , y 2j+1 , z 3j+1 ,− w 3j+1 ) + 4· ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j+1 ) + 2· ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) + 2· ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j ) + ∞∑ j=0 6jϕ ( x 2j+1 , y 2j+1 , z 3j , w 3j ) ≤ ∞∑ j=0 6j [ 6ϕ ( x 2j+1 , y 2j+1 , z 3j+1 ,− w 3j+1 ) + 4ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 , z 3j+1 , w 3j+1 ) + 2ϕ ( x 2j+1 , y 2j+1 ,− z 3j+1 , w 3j ) +ϕ ( x 2j+1 , y 2j+1 , z 3j , w 3j )] =ϕ̃(x,y,z,w) < ∞ for all x,y,z,w ∈ E. This implies that lim j→∞ 1 6 ·2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = 0 for all x,y,z,w ∈ E, which implies lim j→∞ 2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = 0 (2.17) for all x,y,z,w ∈ E. It follows from (2.1), (2.16) and (2.17) that we have P ( 2F ( x + y, z + w 2 ) −F(x,z) −F(x,w) −F(y,z) −F(y,w) ) ≤P ( 2 lim j→∞ 2·6jf ( x + y 2j , z+w 2 3j ) − lim j→∞ 2·6jf ( x 2j , z 3j ) − lim j→∞ 2·6jf ( x 2j , w 3j ) − lim j→∞ 2·6jf ( y 2j , z 3j ) − lim j→∞ 2·6jf ( y 2j , w 3j )) ≤ lim j→∞ 2·6jP ( 2f ( x 2j + y 2j , z 3j + w 3j 2 ) −f ( x 2j , z 3j ) −f ( x 2j , w 3j ) −f ( y 2j , z 3j ) −f ( y 2j , w 3j )) ≤ lim j→∞ 2·6jϕ ( x 2j , y 2j , z 3j , w 3j ) = 0 Int. J. Anal. Appl. 17 (3) (2019) 377 for all x,y,z,w ∈ E. Since X is total, we have 2F ( x + y, z + w 2 ) = F(x,z) + F(x,w) + F(y,z) + F(y,w) for all x,y,z,w ∈ E. Setting l = 0 and taking m → ∞ in (2.15), this implies that the inequality (2.2). Next, we will show that F is unique. Let G : E × E → X be another mapping satisfying (1.4) and (2.2). By [12], there exists bi-additive mapping B,B′ : E×E → X and additive mapping A,A′ : E → X such that F(x,y) = B(x,y) + A(x) and G(x,y) = B′(x,y) + A′(x) for all x,y ∈ E. Since B is bi-additive mapping, A is additive mapping and f(x, 0) = 0 for all x ∈ E, we have F(x,y) − 6F (x 2 , x 3 ) = [B(x,y) + A(x)] − 6 [ B (x 2 , y 3 ) + A (x 2 )] = B(x,y) + A(x) − 6B (x 2 , y 3 ) − 6A (x 2 ) = B(x,y) + A(x) −B(x,y) − 3A (x) = −2A(x) = −2B(x, 0) − 2A(x) = −2F(x, 0) = −2 lim j→∞ 2 · 6jf ( x 2j , 0 ) = 0 for all x,y ∈ E, that is, F(x,y) = 6F (x 2 , y 3 ) (2.18) for all x,y ∈ E. Replacing x by x 2 and y by y 3 in (2.18), we have F (x 2 , y 3 ) = 6F ( x 22 , y 32 ) for all x,y ∈ E. Continuing this process, we have F(x,y) = 6nF ( x 2n , y 3n ) for all x,y ∈ E and for all n ∈ N. Similarly, we get that G(x,y) = 6nG ( x 2n , y 3n ) for all x,y ∈ E and for all n ∈ N. For any n ∈ N, we obtain Int. J. Anal. Appl. 17 (3) (2019) 378 that P(F(x,y) −G(x,y)) (2.19) =P ( 6nF ( x 2n , y 3n ) − 6nG ( x 2n , y 3n )) =P ( 6nF ( x 2n , y 3n ) − 2·6nf ( x 2n , y 3n ) + 2·6nf ( x 2n , y 3n ) − 6nG ( x 2n , y 3n )) ≤P ( 6nF ( x 2n , y 3n ) − 2·6nf ( x 2n , y 3n )) + P ( 2·6nf ( x 2n , y 3n ) − 6nG ( x 2n , y 3n )) ≤6nP ( F ( x 2n , y 3n ) − 2f ( x 2n , y 3n )) + 6nP ( 2f ( x 2n , y 3n ) −G ( x 2n , y 3n )) ≤2·6nϕ̃ ( x 2n , x 2n , y 3n , y 3n ) =2·6n ∞∑ j=0 6j [ 6ϕ ( x 2n 2j+1 , x 2n 2j+1 , y 3n 3j+1 ,− y 3n 3j+1 ) + 4ϕ ( x 2n 2j+1 , x 2n 2j+1 ,− y 3n 3j+1 , y 3n 3j+1 ) + 2ϕ ( x 2n 2j+1 , x 2n 2j+1 , y 3n 3j+1 , y 3n 3j+1 ) + 2ϕ ( x 2n 2j+1 , x 2n 2j+1 ,− y 3n 3j+1 , y 3n 3j ) +ϕ ( x 2n 2j+1 , x 2n 2j+1 , y 3n 3j , y 3n 3j )] =2 ∞∑ j=0 6n+j [ 6ϕ ( x 2n+j+1 , x 2n+j+1 , y 3n+j+1 ,− y 3n+j+1 ) + 4ϕ ( x 2n+j+1 , x 2n+j+1 ,− y 3n+j+1 , y 3n+j+1 ) + 2ϕ ( x 2n+j+1 , x 2n+j+1 , y 3n+j+1 , y 3n+j+1 ) +2ϕ ( x 2n+j+1 , x 2n+j+1 ,− y 3n+j+1 , y 3n+j ) + ϕ ( x 2n+j+1 , x 2n+j+1 , y 3n+j , y 3n+j )] =2 ∞∑ i=n 6i [ 6ϕ ( x 2i+1 , x 2i+1 , y 3i+1 ,− y 3i+1 ) + 4ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i+1 ) + 2ϕ ( x 2i+1 , x 2i+1 , y 3i+1 , y 3i+1 ) + 2ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i ) +ϕ ( x 2i+1 , x 2i+1 , y 3i , y 3i )] for all x,y ∈ E. By (2.3), we obtain that lim n→∞ ∞∑ i=n 6i [ 6ϕ ( x 2i+1 , x 2i+1 , y 3i+1 ,− y 3i+1 ) + 4ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i+1 ) (2.20) + 2ϕ ( x 2i+1 , x 2i+1 , y 3i+1 , y 3i+1 ) + 2ϕ ( x 2i+1 , x 2i+1 ,− y 3i+1 , y 3i ) +ϕ ( x 2i+1 , x 2i+1 , y 3i , y 3i )] = 0 for all x,y ∈ E. From (2.20), taking limit n →∞ in (2.19), we obtain that lim n→∞ P(F(x,y) −G(x,y)) = 0 for all x,y ∈ E. Since paranorm P on X is total, we have F(x,y) −G(x,y) = 0 for all x,y ∈ E. Hence F is a unique mapping satisfying (1.4) and (2.2). � Int. J. Anal. Appl. 17 (3) (2019) 379 Remark 2.1. Let r,θ be positive real numbers with r > log2 6. If we set ϕ(x,y,z,w) = θ(‖x‖r + ‖y‖r + ‖z‖r + ‖w‖r) for all x,y,z,w ∈ E, then Theorem 2.1 recovers Theorem 2.1 in [2]. The following result is the generalized Hyers-Ulam-Rassias stability of the functional equation (1.2). Theorem 2.2. Let ϕ : E ×E ×E ×E → [0,∞) be a function and f : E ×E → X be a mapping satisfying f(x, 0) = 0 for all x ∈ E such that P (f(x + y,z + w) + f(x + y,z −w) − 2f(x,z) − 2f(x,w) − 2f(y,z) − 2f(y,w)) (2.21) ≤ϕ(x,y,z,w) for all x,y,z,w ∈ E. Then there exists a unique mapping F : E ×E → X satisfying (1.2) such that P (f(x,y) −F(x,y)) ≤ ϕ̃(x,x,y,y) (2.22) for all x,y ∈ E where ϕ̃(x,y,z,w) := ∞∑ j=0 8jϕ ( x 2j+1 , y 2j+1 , z 2j+1 , w 2j+1 ) < ∞ (2.23) for all x,y,z,w ∈ E where the mapping F : E ×E → X is given by F(x,y) = lim j→∞ 8jf ( x 2j , y 2j ) for all x,y ∈ E. Proof. Letting y = x and w = z in (2.21), we obtain that P(f(2x, 2z) − 8f(x,z)) ≤ ϕ(x,x,z,z) for all x,z ∈ E. Replacing x by x 2j+1 and z by z 2j+1 in the above inequality, we get that P ( f ( x 2j , z 2j ) − 8f ( x 2j+1 , z 2j+1 )) ≤ ϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) (2.24) for all nonnegative integer j and for all x,z ∈ E. It follows from (2.24) that we have P ( 8jf ( x 2j , z 2j ) − 8j+1f ( x 2j+1 , z 2j+1 )) (2.25) ≤8jP ( f ( x 2j , z 2j ) − 8f ( x 2j+1 , z 2j+1 )) ≤8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) Int. J. Anal. Appl. 17 (3) (2019) 380 for all nonnegative integer j and for all x,z ∈ E. By (2.25), for any integers l and m such that 0 ≤ l < m, we have P ( 8lf ( x 2l , y 2l ) − 8mf ( x 2m , y 2m )) (2.26) =P ( 8lf ( x 2l , y 2l ) − 8l+1f ( x 2l+1 , y 2l+1 ) + 8l+1f ( x 2l+1 , y 2l+1 ) + · · · + 8m−1f ( x 2m−1 , y 2m−1 ) − 8mf ( x 2m , y 2m )) ≤P ( 8lf ( x 2l , y 2l ) − 8l+1f ( x 2l+1 , y 2l+1 )) + P ( 8l+1f ( x 2l+1 , y 2l+1 ) − 8l+2f ( x 2l+2 , y 2l+2 )) + · · · + P ( 8m−1f ( x 2m−1 , y 2m−1 ) − 8mf ( x 2m , y 2m )) = m−1∑ j=l 8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) ≤ ∞∑ j=l 8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) for all x,z ∈ E. It follows from (2.23) that we obtain that lim l→∞ ∞∑ j=l 8jϕ ( x 2j+1 , x 2j+1 , z 2j+1 , z 2j+1 ) = 0 for all x,z ∈ E. This implies that {8jf ( x 2j , z 2j ) } is Cauchy seqeunce in X for all x,z ∈ E. By completeness of X, the sequence {8jf ( x 2j , z 2j ) } is convergent sequence for all x,y ∈ E. Define F : E ×E → X by F(x,z) = lim j→∞ 8jf ( x 2j , z 2j ) for all x,z ∈ E. By (2.21), we obtain that P(F(x + y,z + w) + F(x + y,z −w) − 2F(x,z) − 2F(x,w) − 2F(y,z) − 2F(y,w)) =P ( lim j→∞ 8jf ( x + y 2j , z + w 2j ) + lim j→∞ 8jf ( x + y 2j , z −w 2j ) − 2· lim j→∞ 8jf ( x 2j , z 2j ) −2· lim j→∞ 8jf ( x 2j , w 2j ) − 2· lim j→∞ 8jf ( y 2j , z 2j ) − 2· lim j→∞ 8jf ( y 2j , w 2j )) = lim j→∞ P ( 8jf ( x + y 2j , z + w 2j ) + 8jf ( x + y 2j , z −w 2j ) − 2·8jf ( x 2j , z 2j ) −2·8jf ( x 2j , w 2j ) − 2·8jf ( y 2j , z 2j ) − 2·8jf ( y 2j , w 2j )) ≤ lim j→∞ 8jP ( f ( x + y 2j , z + w 2j ) + f ( x + y 2j , z −w 2j ) − 2f ( x 2j , z 2j ) − 2f ( x 2j , w 2j ) −2f ( y 2j , z 2j ) − 2f ( y 2j , w 2j )) ≤ lim j→∞ 8jϕ ( x 2j , y 2j , z 2j , w 2j ) = 0 Int. J. Anal. Appl. 17 (3) (2019) 381 for all x,y,z,w ∈ E. Since X is total, we have F(x + y,z + w) + F(x + y,z −w) = 2F(x,z) + 2F(x,w) + 2F(y,z) + 2F(y,w) for all x,y,z,w ∈ E. Setting l = 0 and letting m → ∞ in (2.26), the inequality (2.26) holds. Next, we will show that F is unique. Let G : E × E → X be a another mapping satisfying (1.2) and (2.22). It follows from Theorem 3 in [13] that there exists multi-additive mapping M,M′ : E × E × E → X such that F(x,y) = M(x,y,y), G(x,y) = M′(x,y,y), M(x,y,z) = M(x,z,y) and M′(x,y,z) = M′(x,z,y) for all x,y,z ∈ E. For any n ∈ N, we get that P(F(x,y) −G(x,y)) =P (M(x,y,y) −M′(x,y,y)) =P ( M ( 2nx 2n , 2ny 2n , 2ny 2n ) −M′ ( 2nx 2n , 2ny 2n , 2ny 2n )) =P ( 8n [ M ( x 2n , y 2n , y 2n )] − 8n [ M′ ( x 2n , y 2n , y 2n )]) =P ( 8n [ M ( x 2n , y 2n , y 2n ) −M′ ( x 2n , y 2n , y 2n )]) ≤8nP ( M ( x 2n , y 2n , y 2n ) −M′ ( x 2n , y 2n , y 2n )) ≤8nP ( F ( x 2n , y 2n ) −f ( x 2n , y 2n ) + f ( x 2n , y 2n ) −G ( x 2n , y 2n )) ≤8n [ P ( F ( x 2n , y 2n ) −f ( x 2n , y 2n )) + P ( f ( x 2n , y 2n ) −G ( x 2n , y 2n ))] ≤2·8nϕ̃ ( x 2n , x 2n , y 2n , y 2n ) =2·8n ∞∑ j=0 8jϕ ( x 2n 2j+1 , x 2n 2j+1 , y 2n 2j+1 , y 2n 2j+1 ) =2· ∞∑ j=0 8n+jϕ ( x 2n+j+1 , x 2n+j+1 , y 2n+j+1 , y 2n+j+1 ) =2· ∞∑ i=n 8iϕ ( x 2i+1 , x 2i+1 , y 2i+1 , y 2i+1 ) for all x,y ∈ E. By (2.23), we get that lim n→∞ ∞∑ i=n 8iϕ ( x 2i+1 , x 2i+1 , y 2i+1 , y 2i+1 ) = 0 for all x,y ∈ E. Hence lim n→∞ P(F(x,y) −G(x,y)) = 0 for all x,y ∈ E. Since paranorm P on X is total, we have F(x,y) −G(x,y) = 0 for all x,y ∈ E. Hence F is a unique mapping satisfying (1.2) and (2.22). � Int. J. Anal. Appl. 17 (3) (2019) 382 Remark 2.2. Let r,θ be positive real numbers with r > 3. If we set ϕ(x,y,z,w) = θ(‖x‖r + ‖y‖r + ‖z‖r + ‖w‖r) for all x,y,z,w ∈ E, then Theorem 2.2 recovers Theorem 3.1 in [2]. 3. Stability of the generalized Cauchy-Jensen functional equation in paranormed space The following result is the Hyers-Ulam-Rassias stability of the functional equation (1.5). Theorem 3.1. Let r be a positive real number with r > 1, and let f : E → X be a mapping satisfying P ( αf ( x + y α + z ) −f(x) −f(y) −αf(z) ) ≤ θ (‖x‖r + ‖y‖r + ‖z‖r) (3.1) for all x,y,z ∈ E. Then there exists a unique mapping F : E → X satisfying (1.5) such that P (f(x) −F(x)) ≤ ( 3αr + 1 αr −α ) θ‖x‖r (3.2) for all x ∈ E where the mapping F : E → X is given by F(x) = lim n→∞ αnf ( x αn ) for all x ∈ E. Proof. Putting x = y = z = 0 in (3.1), we have P(f(0)) ≤ 0. Since X is total, we obtain that f(0) = 0. Letting x = −x α , y = x α and z = 0 in (3.1), we obtain that P ( f ( − x α ) + f (x α )) (3.3) =P ( −f ( − x α ) −f (x α )) =P ( αf ( −x α + x α α + 0 ) −f ( − x α ) −f (x α ) −αf(0) ) ≤θ (∥∥∥−x α ∥∥∥r + ∥∥∥x α ∥∥∥r + ‖0‖r) = 2θ αr ‖x‖r for all x ∈ E. Replacing x by αx in the inequality (3.3), we get that P (f(−x) + f(x)) ≤ 2θ‖x‖r (3.4) Int. J. Anal. Appl. 17 (3) (2019) 383 for all x ∈ E. Replacing x = −x, y = 0, and z = x α in (3.1), we have P ( f(−x) + αf (x α )) (3.5) =P ( −f(−x) −αf (x α )) =P ( αf ( −x + 0 α + ( − x α )) −f(−x) −f(0) −αf (x α )) ≤θ ( ‖−x‖r + ‖0‖r + ∥∥∥x α ∥∥∥r) = ( 1 + 1 αr ) θ‖x‖r for all x ∈ E. It follows from (3.4) and (3.5) that we have P ( αf (x α ) −f(x) ) = P ( αf (x α ) + f(−x) −f(−x) −f(x) ) (3.6) ≤ P ( αf (x α ) + f(−x) ) + P (f(−x) + f(x)) ≤ ( 1 + 1 αr ) θ‖x‖r + 2θ‖x‖r ≤ ( 3 + 1 αr ) θ‖x‖r for all x ∈ E. For i ∈ N, replacing x = x αi in (3.6), we get that P ( αi+1f ( x αi+1 ) −αif ( x αi )) ≤ αiP ( αf ( x αi+1 ) −f ( x αi )) (3.7) ≤ αi ( 3 + 1 αr ) θ ∥∥∥ x αi ∥∥∥r = ( 1 αr−1 )i ( 3 + 1 αr ) θ‖x‖r for all x ∈ E. For given nonnegative integer l,m such that l < m, we have P ( αmf ( x αm ) −αlf ( x αl )) (3.8) =P ( αmf ( x αm ) −αm−1f ( x αm−1 ) + αm−1f ( x αm−1 ) + · · · + αl+1f ( x αl+1 ) −αlf ( x αl )) ≤ m−1∑ j=l P ( αj+1f ( x αj+1 ) −αjf ( x αj )) ≤ m−1∑ j=l ( 1 αr−1 )i ( 3 + 1 αr ) θ‖x‖r ≤ ( 3 + 1 αr ) θ‖x‖r ∞∑ j=0 ( 1 αr−1 )i Int. J. Anal. Appl. 17 (3) (2019) 384 for all x ∈ E. Since r > 1, we have 1 αr−1 < 1. Since 1 αr−1 < 1, the sequence {αnf ( x αn ) } is Cauchy sequence for all x ∈ E. By completeness of X, the sequence {αnf ( x αn ) } converges. Define F : E → X by F(x) = lim n→∞ αnf ( x αn ) (3.9) for all x ∈ E. Moreover, letting l = 0 and taking limit m →∞ in (3.8), we can obtain that inequality (3.2). It follows from (3.1) and (3.9) that P ( αF ( x + y α + z ) −F(x) −F(y) −αF(z) ) =P ( α· lim n→∞ αnf ( x+y α + z αn ) − lim n→∞ αnf ( x αn ) − lim n→∞ αnf ( y αn ) −α lim n→∞ αnf ( z αn )) = lim n→∞ αnP ( αf ( x αn + y αn α + z αn ) −f ( x αn ) −f ( y αn ) −αf ( z αn )) ≤ lim n→∞ αnθ (∥∥∥ x αn ∥∥∥r + ∥∥∥ x αn ∥∥∥r + ∥∥∥ x αn ∥∥∥r) =θ‖x‖r lim n→∞ ( 1 αr−1 )n = 0 for all x,y,z ∈ E. Since X is total, we have αF ( x + y α + z ) = F(x) + F(y) + αF(z) for all x,y,z ∈ E. By Corollary 1.1, F is additive. Next, we will show that F is unique. Let G be another mapping satisfying (1.5) and (3.2). Then, we consider P (F(x) −G(x)) = P ( nF (x n ) −nf (x n ) + nf (x n ) −nG (x n )) (3.10) ≤ n ( P ( F (x n ) −f (x n )) + P ( f (x n ) −G (x n ))) ≤ 2nθ ( 3αr + 1 αr −α )∥∥∥x n ∥∥∥r = ( 1 nr−1 ) 2θ ( 3αr + 1 αr −α ) ‖x‖r for all x ∈ E. Since r − 1 > 0, taking limit n → ∞ in (3.10), we have P(F(x) − G(x)) = 0 for all x ∈ E. Since X is total, we have F(x) = G(x) for all x ∈ E, that is F is unique. � Theorem 3.2. Let r be a positive real number with r < 1 and let f : X → E be a mapping satisfying∥∥∥∥αf ( x + y α + z ) −f(x) −f(y) −αf(z) ∥∥∥∥ ≤ P (x)r + P (y)r + P (z)r (3.11) for all x,y,z ∈ X. Then there exists a unique mapping F : X → E satisfying (1.5) such that ‖f(x) −F(x)‖≤ 2 + 3αr α−αr P (x) r (3.12) Int. J. Anal. Appl. 17 (3) (2019) 385 for all x ∈ X where the mapping F : X → E is given by F(x) = lim n→∞ 1 αn f (αnx) for all x ∈ X. Proof. Letting x = y = z = 0 in (3.11), we get that ‖2f(0)‖ = ∥∥∥∥αf ( 0 + 0 α + 0 ) −f(0) −f(0) −αf(0) ∥∥∥∥ ≤P (0)r + P (0)r + P (0)r =0 So f(0) = 0. Subtituting x = −αx, y = 0 and z = x in (3.11), we obtain that ‖f(−αx) + αf(x)‖ = ∥∥∥∥αf ( −αx + 0 α + x ) −f(−αx) −f(0) −αf(x) ∥∥∥∥ ≤ P (−αx)r + P (0)r + P (x)r ≤ (1 + αr)P(x)r for all x ∈ X. Letting x = −αx, y = αx and z = x, we get that ‖f(−αx) + f(αx)‖ = ∥∥∥∥αf ( −αx + αx α + x ) −f(−αx) −f(αx) −αf(x) ∥∥∥∥ ≤ P (−αx)r + P (αx)r + P (x)r ≤ (1 + 2αr)P (x)r for all x ∈ X. Then we have ‖f(αx) −αf(x)‖ = ‖f(αx) + f(−αx) −f(−αx) −αf(x)‖ = ‖f(αx) + f(−αx)‖ + ‖f(−αx) + αf(x)‖ ≤ (1 + αr)P(x)r + (1 + 2αr)P (x)r = (2 + 3αr)P(x)r and so ∥∥∥∥ 1αf(αx) −f(x) ∥∥∥∥ ≤ 2 + 3αrα P(x)r (3.13) for all x ∈ X. Replacing x = αix and multiplying by 1 αi in (3.13), we have∥∥∥∥ 1αi+1 f(αi+1x) − 1αif(αix) ∥∥∥∥ ≤ 1αi · 2 + 3α r α P(αix)r (3.14) ≤ 2 + 3αr α P(x)r· ( 1 α1−r )i Int. J. Anal. Appl. 17 (3) (2019) 386 for all x ∈ X. By (3.14), for any integers l,m such that 0 ≤ l < m, we obtain that∥∥∥∥ 1αmf(αmx) − 1αlf(αlx) ∥∥∥∥ (3.15) = ∥∥∥∥ 1αmf(αmx) + 1αm−1 f(αm−1x) − 1αm−1 f(αm−1x) + · · · + 1αl+1 f(αl+1x) − 1 αl f(αlx) ∥∥∥∥ ≤ m−1∑ i=l 2 + 3αr α P(x)r· ( 1 α1−r )i ≤ 2 + 3αr α P(x)r· ∞∑ i=0 ( 1 α1−r )i for all x ∈ X. Since 1 α1−r < 1, we have ∑∞ i=0 ( 1 α1−r )i < ∞. It follows from (3.15) that the sequence { 1 αn f(αnx)} is Cauchy sequence for all x ∈ X. Since E is complete, the sequence { 1 αn f(αnx)} is convergent sequence. We define a mapping F : X → E by F(x) = lim n→∞ 1 αn f (αnx) (3.16) for all x ∈ X. Moreover, letting l = 0 and taking limit m → ∞ in (3.15), we can obtain that inequality (3.12). It follows from (3.11) and (3.16) that we have∥∥∥∥αF ( x + y α + z ) −F(x) −F(y) −αF(z) ∥∥∥∥ = ∥∥∥∥α limn→∞ 1αnf ( αn ( x + y α + z )) − lim n→∞ 1 αn f (αny) − lim n→∞ 1 αn f (αnz) −α lim n→∞ 1 αn f (αnz) ∥∥∥∥ ≤ lim n→∞ 1 αn ∥∥∥∥αf ( αnx + αny α + αnz ) −f (αny) −f (αnz) −αf (αnz) ∥∥∥∥ ≤ lim n→∞ 1 αn ∥∥∥∥αf ( αnx + αny α + αnz ) −f (αny) −f (αnz) −αf (αnz) ∥∥∥∥ ≤ lim n→∞ 1 αn (P(αnx)r + P(αny)r + P(αnz)r) ≤ lim n→∞ αnr αn (P(x)r + P(y)r + P(z)r) ≤(P(x)r + P(y)r + P(z)r) lim n→∞ ( 1 α1−r )n = 0 for all x,y,z ∈ X. Hence αF ( x + y α + z ) = F(x) + F(y) + αF(z) for all x,y,z ∈ X, that is F is the generalized Cauchy-Jensen functional equation. 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