Stability and nonstability of octadecic functional equation in multinormed spaces
Stability and nonstability of octadecic functional equation in multinormed 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 multinormed 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 quasiβnormed 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 quasiβnormed 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 quasiβnormed 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 quasiβnormed 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 multinorm 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 multinormed 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 multiBanach 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 multiBanach 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 multinormed
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 multinormed spaces
In this section, we prove the generalized Hyers–Ulam stability of the octadecic functional equation (1.1) in
multinormed 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 multiBanach 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 multinormed 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(x118 + ··· + xk18 + y118 + ··· + yk18), (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 x18 + y18 = 0 or x18 + y18 ≥ 4118,
then
Df(x, y) ≤ (2184+18 1−8!1)418 ≤ (2184+18 1−8!1)418418(x18 + y18).
Now, suppose that 0 < x18 + y18 < 4118. Then there exists a nonnegative integer k such that
for all n = 0,1,...,k − 1. Thus we get
xD18f(+x,yy)18 ≤
∞
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(x118 + · · · + xk 18)
for some β ∈ R and all x1, . . . , xk ∈ R. So  f (x) − C(x) ≤ δx18 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 . Then there exists a unique octadecic mapping Q : X → Y , such that
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1. Baker , J.: A general functional equation and its stability . Proc. Am. Math. Soc . 133 , 1657  1664 ( 2005 )
2. Czerwik , S. : Stability of Functional Equations of UlamHyersRassias Type . Hadronic Press lnc, Florida ( 2003 )
3. Dales , H.G. ; Moslehian, M.S.: Stability of mapping on multinormed spaces . Glasg. Math. J . 49 , 321  332 ( 2007 )
4. Dales , H.G. ; Polyakov, M.E. : MultiNormed Spaces and Multibanach Algebras . University of Leeds, Leeds ( 2012 )
5. Eshaghi Gordji , M. ; Ghaemi , M.B. ; Rassias , J.M. ; Alizadeh , B. : Nearly ternary quadratic higher derivations on nonArchimedean ternary Banach algebras. A fixed point approach . Abstr. Appl. Anal . 2011 , 1  18 ( 2011 )
6. Eshaghi Gordji , M. ; Cho , Y.J. ; Ghaemi, M.B. ; Majani , H. : Approximately quintic and sextic mappings form r divisible groups into Sertnevic probabilistic Banach spaces. Fixed point method . Discrete Dyn. Nat. Soc. 5 , 1  16 ( 2011 )
7. Ghaemi , M.B.; Majani , H.; Eshaghi Gordji , M. : Approximately quintic and sextic mappings on the probabilistic normed spaces . Bull. Korean Math. Soc . 49 , 339  352 ( 2012 )
8. Hyers , D.H. : On the stability of the linear functional equation . Proc. Natl. Acad. Sci. USA 27 , 222  224 ( 1941 )
9. Moslehian , M.S. ; Nikodem , K. ; Popa , D. : Asymptotic aspect of the quadratic functional equation in multinormed spaces . J. Math. Anal. Appl . 355 , 717  724 ( 2009 )
10. Nazarianpoor , M. ; Rassias , J.M. ; Sadeghi , Gh.: Stability and non stability of octadecic functional equation in Banach spaces (submitted)
11. Park , C. ; Eshaghi Gordji, M. ; Ghaemi , M.B. ; Majani , H. : Fixed points and approximately octic mappings in nonArchimedean 2normed spaces . J. Inequal. Appl . 2012 , 11  12 ( 2012 )
12. Rassias , J.M. : On approximation of approximately linear mappings by linear mappings . J. Funct. Anal . 46 , 126  130 ( 1982 )
13. Rassias , J.M. ; Eslamian , M. : Fixed point and stability of nonic functional equation in quasiβnormed spaces . Contemp. Anal. Appl. Math. 3 , 293  309 ( 2015 )
14. Rassias , J.M. ; Kim , H.M. : Generalized HyersUlam stability for general additive functional equation in quasiβnormed spaces . J. Math. Anal. Appl . 356 , 302  309 ( 2009 )
15. Rassias , ThM: On the stability of the linear mapping in Banach spaces . Proc. Am. Math. Soc . 72 , 297  300 ( 1978 )
16. Ravi , K. ; Rassias , J.M. ; Pinelas , S. ; Sabarinathan , S.: A fixed point approach to the stability of decic functional equation in quasiβnormed spaces . Panam. Math. J . 25 , 42  52 ( 2015 )
17. Ulam , S.M.: A Collection of Mathematical Problems . Interscience Publishing, New York ( 1960 )
18. Xu , T. ; Rassias , J.M.: Approximate septic and octic mappings in quasiβnormed spaces . J. Comput. Anal. Appl . 15 , 1110  1119 ( 2013 )
19. Xu , T. ; Rassias , J.M. ; Xu , W.X.: A fixed point approach to the stability of a general mixed type additivecubic functional equation in quasi fuzzy normed spaces . Int. J. Phys. Sci. 6 , 313  324 ( 2011 )
20. Xu , T. ; Rassias , J.M. ; Xu , W.X.: A fixed point approach to the stability of quintic and sextic functional equations in quasiβnormed spaces . J. Inequal. Appl . 2010 , 1  23 ( 2010 )