Stability and nonstability of octadecic functional equation in multi-normed spaces

Arabian Journal of Mathematics, Sep 2017

In this paper, we introduce octadecic functional equation. Moreover, we prove the stability of the octadecic functional equation in multi-normed spaces by using the fixed point method.

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Stability and nonstability of octadecic functional equation in multi-normed spaces

Stability and nonstability of octadecic functional equation in multi-normed spaces M. Nazarianpoor 0 1 J. M. Rassias 0 1 Gh. Sadeghi 0 1 Mathematics Subject Classification 0 1 0 J. M. Rassias Pedagogical Department E. E, Section of Mathematics and Informatics, National and Capodistrian University of Athens , Athens , Greece 1 M. Nazarianpoor In this paper, we introduce octadecic functional equation. Moreover, we prove the stability of the octadecic functional equation in multi-normed spaces by using the fixed point method. In 1940, Ulam [17] proposed the following question concerning the stability of group homomorphisms: Let G1 be a group and (G2, d) be a metric group. Given ε > 0, does there exist a δ > 0, such that if a mapping h : G1 → G2 satisfies the inequality d(h(x y), h(x )h(y)) < δ for all x , y ∈ G1, then there exists a homomorphism H : G1 → G2, such that d(h(x ), H (x )) < ε for all x ∈ G1? In the next year, 1941, Hyers [8] solved the famous stability problem of Ulam in Banach spaces: Let X be a normed space and Y be a Banach space. Suppose that for some ε > 0, the mapping f : X → Y satisfies f (x + y) − f (x ) − f (y) ≤ ε for all x , y ∈ X . Then there exists a unique additive mapping T : X → Y , such that f (x ) − T (x ) ≤ ε for all x ∈ X . In 1978, Rassias [15] proved the following theorem: - 1 Introduction Let X and Y be real normed spaces with Y complete. Let f : X → Y be a mapping such that, for each fixed x ∈ X , the mapping h(t) = f (t x) is continuous on R, and let ε ≥ 0 and p ∈ [0, 1) be such that f (x + y) − f (x) − f (y) ≤ ε( x p + y p) holds for all x, y ∈ X . Then there exists a unique linear mapping T : X → Y , such that f (x) − T (x) ≤ ε x p 1 − 2 p−1 for all x ∈ X . Since the past few decades several stability problems of functional equations have been investigated [ 1,2,5–7,11,12,14,19 ]. Xu et al. [ 20 ] proved the general solution and the stability of the quintic functional equation f (x + 3y) − 5 f (x + 2y) + 10 f (x + y) − 10 f (x) + 5 f (x − y) − f (x − 2y) = 120 f (y) and the sextic functional equation f (x + 3y) − 6 f (x + 2y) + 15 f (x + y) − 20 f (x) + 15 f (x − y) − 6 f (x − 2y) + f (x − 3y) = 720 f (y) in quasi-β-normed spaces. The general solution and the stability of the septic functional equation f (x + 4y) − 7 f (x + 3y) + 21 f (x + 2y) − 35 f (x + y) − 35 f (x) − 21 f (x − y) + 7 f (x − 2y) − f (x − 3y) = 5040 f (y) and the octic functional equation f (x + 4y) − 8 f (x + 3y) + 28 f (x + 2y) − 56 f (x + y) − 70 f (x) − 56 f (x − y) + 28 f (x − 2y) − 8 f (x − 3y) + f (x − 4y) = 40320 f (y) in quasi-β-normed spaces were investigated by Xu and Rassias [ 18 ]. Rassias and Eslamian [ 13 ] investigated the general solution of a nonic functional equation f (x + 5y) − 9 f (x + 4y) + 36 f (x + 3y) − 84 f (x + 2y) − 126 f (x + y) − 126 f (x) + 84 f (x − y) − 36 f (x − 2y) + 9 f (x − 3y) − f (x − 4y) = 9! f (y) and proved the stability of nonic functional equation in quasi-β-normed spaces by using the fixed point method. A fixed point approach for the stability of decic functional equation f (x + 5y) − 10 f (x + 4y) + 45 f (x + 3y) − 120 f (x + 2y) − 210 f (x + y) − 252 f (x) + 210 f (x − y) − 120 f (x − 2y) + 45 f (x − 3y) − 10 f (x − 4y) + f (x − 5y) = 10! f (y) in quasi-β-normed spaces was investigated by Ravi et al. [ 16 ]. Let (X, . ) be a complex normed space, and k ∈ N. We denote the group of permutations on k symbols by Gk . Definition 1.1 [ 3,4,9 ] A multi-norm on {X k : k ∈ N} is a sequence ( . k ) = ( . k : k ∈ N) such that . k is a norm on X k for each k ∈ N, x 1 = satisfied for each k ∈ N with k ≥ 2: x for each x ∈ X , and the following axioms are (MN1) xσ (1), . . . , xσ (k) k = x1, . . . , xk k (σ ∈ Gk , x1, . . . , xk ∈ X ); In this case, we say that ((X k , . k ) : k ∈ N) is a multi-normed space. If (X, . ) is a Banach space, then (X k , . k ) is a Banach space for each k ∈ N, in this case ((X k , . k ) : k ∈ N) is a multi-Banach space. Example 1.2 Let (X, . ) be a Banach lattice, and let us define x1, . . . , xk k := |x1| ∨ · · · ∨ |xk | (x1, . . . , xk ∈ X ). Then ((X k , . k ) : k ∈ N) is a multi-Banach space. Let X and Y be real vector spaces and f : X → Y be a mapping. We define a mapping Df : X 2 → Y by Df(x, y) := f (x + 9y) − 18 f (x + 8y) + 153 f (x + 7y) − 816 f (x + 6y) + 3060 f (x + 5y) − 8568 f (x + 4y) + 18564 f (x + 3y) − 31824 f (x + 2y) + 43758 f (x + y) − 48620 f (x) + 43758 f (x − y) − 31824 f (x − 2y) + 18564 f (x − 3y) − 8568 f (x − 4y) + 3060 f (x − 5y) − 816 f (x − 6y) + 153 f (x − 7y) − 18 f (x − 8y) + f (x − 9y) − 18! f (y), for all x, y ∈ X , where 18!=6402373705728000. In this paper, we introduce the following octadecic functional equation: Df(x, y) = 0, (1.1) for all x, y ∈ X . Moreover, we prove the stability of the octadecic functional equation (1.1) in multi-normed spaces by using the standard fixed point method: Theorem 1.3 [ 10 ] If X and Y are real vector spaces and f : X → Y is a mapping satisfying octadecic functional equation (1.1) for all x, y ∈ X , then f is an octadecic mapping, i.e., f (x) = x18. 2 Stability of the functional equation (1.1) in multi-normed spaces In this section, we prove the generalized Hyers–Ulam stability of the octadecic functional equation (1.1) in multi-normed spaces. Throughout this section, we assume that X is a normed space and that Y is a Banach space. Let ((Y k , . k ) : k ∈ N) be a multi-Banach space. Theorem 2.1 [ 10 ] Let φ : X 2 → [0, ∞) be a mapping, such that there exists L < 1 with φ(2x, 2y) ≤ 218 Lφ(x, y) for all x, y ∈ X . Let f : X → Y be a mapping satisfying for all x, y ∈ X . Then there exists a unique octadecic mapping Q : X → Y , such that Df(x, y) ≤ φ(x, y), for all x ∈ X , where ψ(x) := (3201186852864000)−1(24310φ(0,x) + 43758φ(x,x) + 31824φ(2x,x) + 18564φ(3x,x) + 8568φ(4x,x) + 3060φ(5x,x) + 816φ(6x,x) + 153φ(7x,x) φ(0,2x) + 18φ(8x,x) + φ(9x,x) + 2 φT(h2exo1r,e.m. .2,.22xkL,e2t yk1, .∈. . N,2yakn)d ≤φ 2:18XL2φk(x1, . . . , xk , y1, . . . , yk ) for all x1, . . . , xk , y1, . . . , yk ∈ X . Let → [0, ∞) be a mapping, such that there exists L < 1 with f : X → Y be a mapping satisfying Df(x1, y1), . . . , Df(xk , yk ) k ≤ φ(x1, . . . , xk , y1, . . . , yk ), for all x1, . . . , xk , y1, . . . , yk ∈ X . Then there exists a unique octadecic mapping Q : X → Y , such that 1 f (x1) − Q(x1), . . . , f (xk ) − Q(xk ) k ≤ 218(1 − L) ψ(x1, . . . , xk ), for all x1, . . . , xk ∈ X , where ψ(x1, . . . , xk) := (3201186852864000)−1(24310φ(0, . . . ,0,x1, . . . ,xk) + 43758φ(x1, . . . ,xk,x1, . . . ,xk) + 31824φ(2x1, . . . ,2xk,x1, . . . ,xk) + 18564φ(3x1, . . . ,3xk,x1, . . . ,xk) + 8568φ(4x1, . . . ,4xk,x1, . . . ,xk) + 3060φ(5x1, . . . ,5xk,x1, . . . ,xk) + 816φ(6x1, . . . ,6xk,x1, . . . ,xk) + 153φ(7x1, . . . ,7xk,x1, . . . ,xk) + 18φ(8x1, . . . ,8xk,x1, . . . ,xk) φ(0, . . . ,0,2x1, . . . ,2xk) + φ(9x1, . . . ,9xk,x1, . . . ,xk) + 2 4537567325 48886173 + 6402373705728000 + 12705510619017216000 3234988548 1 1 + 6402373705728000 + 2324754432000 + 355687428096000 φ(0, . . . ,0,0, . . . ,0) × (φ(x1, . . . ,xk,x1, . . . ,xk) + φ(x1, . . . ,xk, − x1, . . . , − xk)) 1922913180 1 1 + 6402373705728000 + 41845579776000 + 292626432000 × (φ(2x1, . . . ,2xk,2x1, . . . ,2xk) + φ(2x1, . . . ,2xk, − 2x1, . . . , − 2xk)) 940105848 + 6402373705728000 (φ(3x1, . . . ,3xk,3x1, . . . ,3xk) 371153520 1 + φ(3x1, . . . ,3xk, − 3x1, . . . , − 3xk)) + 6402373705728000 + 402361344000 × (φ(4x1, . . . ,4xk,4x1, . . . ,4xk) + φ(4x1, . . . ,4xk, − 4x1, . . . , − 4xk)) 115306920 + 6402373705728000 (φ(5x1, . . . ,5xk,5x1, . . . ,5xk) Corollary 2.3 Let k ∈ N and α, L be positive real numbers, such that 2118 ≤ L < 1. Let f : X → Y be a mapping satisfying for all x1, . . . , xk , y1, . . . , yk ∈ X . Then there exists a unique octadecic mapping Q : X → Y , such that Df(x1, y1), . . . , Df(xk , yk ) k ≤ α, 1 f (x1) − Q(x1), . . . , f (xk ) − Q(xk ) k ≤ 218(1 − L) β, for all x1, . . . , xk ∈ X , where β := (3201186852864000)−1α(24310 + 43758 + 31824 + 18564 + 8568 + 3060 + 816 1 4537567325 48886173 + 153 + 18 + 1 + 2 + 6402373705728000 + 12705510619017216000 3234988548 1 1 + 2 6402373705728000 + 2324754432000 + 355687428096000 + 2 64012932723971035178208000 + 418455719776000 + 2926261432000 18802116960 371153520 1 + 6402373705728000 + 2 6402373705728000 + 402361344000 230613840 27123330 1 + 6402373705728000 + 2 6402373705728000 + 689762304000 9077796 481338 1 + 6402373705728000 + 2 6402373705728000 + 1494484992000 48620 2 2 + 6402373705728000 + 4184557977600 + 15692092416000 + 2(83691159552000)−1 + 2(711374856192000)−1 + 2(12804747411456000)−1). Corollary 2.4 Let k ∈ N and α, p, L be positive real numbers, such that L < 1 and 2p ≤ 218 L. Let f : X → Y be a mapping satisfying Df(x1, y1), . . . , Df(xk , yk ) k ≤ α( x1 p + · · · + xk p + y1 p + · · · + yk p), for all x1, . . . , xk, y1, . . . , yk ∈ X . Then there exists a unique octadecic mapping Q : X → Y , such that for all x1, . . . , xk ∈ X , where The following example shows that the assumption 2p ≤ 218 L cannot be omitted in Corollary 2.4. We know from Example 1.2 that if x1, . . . , xk k = sup{|x1|, . . . , |xk|}, then ((Rk , . k) : k ∈ N) is a multi-normed space. Example 2.5 Let k ∈ N. We define φ : R → R, by We consider the function f : R → R defined by Then f satisfies the following functional inequality: for all x1,..., xk, y1,..., yk ∈ R. Proof We have ⎧ 1 x ∈ [1,∞) φ(x) := ⎨⎪ x18 x ∈ (−1,1) ⎪⎩ − 1 x ∈ (−∞,−1]. f (x) = n∞=0 φ4(418nnx), (x ∈ R). 218 + 18!454(|x1|18 + ··· + |xk|18 + |y1|18 + ··· + |yk|18), (2.1) Df(x1, y1),...,Df(xk, yk) k ≤ 418 − 1 for all x ∈ R. Therefore, we see that f is bounded. Let x, y ∈ R. If |x|18 + |y|18 = 0 or |x|18 + |y|18 ≥ 4118, then |Df(x, y)| ≤ (2184+18 1−8!1)418 ≤ (2184+18 1−8!1)418418(|x|18 + |y|18). Now, suppose that 0 < |x|18 + |y|18 < 4118. Then there exists a nonnegative integer k such that for all n = 0,1,...,k − 1. Thus we get |x|D|18f(+x,|yy)|1|8 ≤ ∞ n=k Hence f satisfies (2.1) for all x1, . . . , xk , y1, . . . , yk ∈ R. Now, we claim that the octadecic functional equation (1.1) is not stable for p = 18 in Corollary 2.4. Suppose on the contrary that there exists an octadecic mapping C : R → R, such that f (x1) − C(x1), . . . , f (xk ) − C(xk ) k ≤ β(|x1|18 + · · · + |xk |18) for some β ∈ R and all x1, . . . , xk ∈ R. So | f (x) − C(x)| ≤ δ|x|18 for some constant δ > 0 and all x ∈ R. Then there exists γ ∈ R for which C(x) = γ x18 for all x ∈ Q. Therefore, Let M ∈ N be such that M > δ + |γ |. If x is a rational number in (0, 4M1−1 ), then we have 4n x ∈ (0, 1) for each n = 0, 1, 2, . . . , M − 1. Consequently, for such an x we have Corollary 2.6 Let k ∈ N and α, p, L be positive real numbers, such that L < 1 and 22kp ≤ 218 L. Let f : X → Y be a mapping satisfying Df(x1, y1), . . . , Df(xk , yk ) k ≤ α x1 p · · · xk p y1 p · · · yk p, for all x1, . . . , xk , y1, . . . , yk ∈ X . Then there exists a unique octadecic mapping Q : X → Y , such that 1 f (x1) − Q(x1), . . . , f (xk ) − Q(xk ) k ≤ 218(1 − L) ψ(x1, . . . , xk ), for all x1, . . . , xk ∈ X , where ψ(x1, . . . , xk) := (3201186852864000)−1α x1 2p . . . xk 2p(43758 + 31824(2)kp + 18564(3)kp + 8568(4)kp + 3060(5)kp + 816(6)kp + 153(7)kp + 18(8)kp + 9kp + x1 2kp + · · · + xk 2kp + y1 2kp + · · · + yk 2kp), for all x1, . . . , xk, y1, . . . , yk ∈ X . 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M. Nazarianpoor, J. M. Rassias, Gh. Sadeghi. Stability and nonstability of octadecic functional equation in multi-normed spaces, Arabian Journal of Mathematics, 2017, 1-10, DOI: 10.1007/s40065-017-0186-0