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

Arab. J. Math. 208 7:29 228 https://doi.org/0.007/s40065-07-086-0 Arabian Journal of Mathematics M. Nazarianpoor J. M. Rassias Gh. Sadeghi Stability and nonstability of octadecic functional equation in multi-normed spaces Received: March 207 / Accepted: 9 September 207 / Published online: 30 September 207 The Authors 207. This article is an open access publication Abstract 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. Mathematics Subject Classification 39A 39B52 Introduction In 940, Ulam [7] proposed the following question concerning the stability of group homomorphisms: Let G be a group and G 2, d be a metric group. Given ε>0, does there exist a δ>0, such that if a mapping h : G G 2 satisfies the inequality dhxy, hxhy < δ for all x, y G, then there exists a homomorphism H : G G 2, such that dhx, Hx < ε for all x G? In the next year, 94, 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 978, Rassias [5] proved the following theorem: M. Nazarianpoor Gh. Sadeghi B Department of Mathematics and Computer Sciences, Hakim Sabzevari University, P.O. Box 397, Sabzevar, Iran E-mail: ghadir54@gmail.com; g.sadeghi@hsu.ac.ir M. Nazarianpoor E-mail: mehdi.nazarianpoor@yahoo.com; m.nazarianpoor@hsu.ac.ir J. M. Rassias Pedagogical Department E. E, Section of Mathematics and Informatics, National and Capodistrian University of Athens, Athens, Greece E-mail: jrassias@primedu.uoa.gr; Ioannis.Rassias@primedu.uoa.gr

220 Arab. J. Math. 208 7:29 228 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 ht = f tx is continuous on R, andletε 0andp [0, 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 x p f x T x ε 2 p for all x X. Since the past few decades several stability problems of functional equations have been investigated [,2,5 7,,2,4,9]. Xu et al. [20] proved the general solution and the stability of the quintic functional equation f x 3y 5 f x 2y 0 f x y 0 f x 5 f x y f x 2y = 20 f y and the sextic functional equation f x 3y 6 f x 2y 5 f x y 20 f x 5 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 2 f x 2y 35 f x y 35 f x 2 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 [8]. Rassias and Eslamian [3] investigated the general solution of a nonic functional equation f x 5y 9 f x 4y 36 f x 3y 84 f x 2y 26 f x y 26 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 0 f x 4y 45 f x 3y 20 f x 2y 20 f x y 252 f x 20 f x y 20 f x 2y 45 f x 3y 0 f x 4y f x 5y = 0! f y in quasi-β-normed spaces was investigated by Ravi et al. [6]. Let X,. be a complex normed space, and k N. We denote the group of permutations on k symbols by G k. Definition. [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 = x for each x X, and the following axioms are satisfied for each k N with k 2: MN x σ,...,x σk k = x,...,x k k σ G k, x,...,x k X;

Arab. J. Math. 208 7:29 228 22 MN2 MN3 α x,...,α k x k k max i N k α i x,...,x k k α,...,α k C, x,...,x k X; x,...,x k, 0 k = x,...,x k k x,...,x k X; MN4 x,...,x k, x k k = x,...,x k k x,...,x k 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.2 Let X,. be a Banach lattice, and let us define x,...,x k k := x x k x,...,x k 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 Dfx, y := f x 9y 8 f x 8y 53 f x 7y 86 f x 6y 3060 f x 5y 8568 f x 4y 8564 f x 3y 3824 f x 2y 43758 f x y 48620 f x 43758 f x y 3824 f x 2y 8564 f x 3y 8568 f x 4y 3060 f x 5y 86 f x 6y 53 f x 7y 8 f x 8y f x 9y 8! f y, for all x, y X, where 8!=6402373705728000. In this paper, we introduce the following octadecic functional equation: Dfx, y = 0,. for all x, y X. Moreover, we prove the stability of the octadecic functional equation. in multi-normed spaces by using the standard fixed point method: Theorem.3 [0] If X and Y are real vector spaces and f : X Y is a mapping satisfying octadecic functional equation. for all x, y X, then f is an octadecic mapping, i.e., f x = x 8. 2 Stability of the functional equation. in multi-normed spaces In this section, we prove the generalized Hyers Ulam stability of the octadecic functional equation. 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. [0] Let φ : X 2 [0, be a mapping, such that there exists L < with φ2x, 2y 2 8 Lφx, y for all x, y X. Let f : X Y be a mapping satisfying Dfx, y φx, y, for all x, y X. Then there exists a unique octadecic mapping Q : X Y, such that for all x X, where f x Qx 2 8 L ψx, ψx := 32086852864000 2430φ0,x 43758φx,x 3824φ2x,x

222 Arab. J. Math. 208 7:29 228 8564φ3x,x 8568φ4x,x 3060φ5x,x 86φ6x,x 53φ7x,x 8φ8x,x φ9x,x φ0,2x 2 4537567325 6402373705728000 4888673 φ0,0 270550690726000 6402373705728000 2324754432000 φx,x φx, x 9229380 6402373705728000 4845579776000 φ2x,2x 94005848 φ2x, 2x φ3x,3x φ3x, 3x 6402373705728000 3753520 6402373705728000 φ4x,4x φ4x, 4x 40236344000 5306920 φ5x,5x φ5x, 5x 6402373705728000 27330 6402373705728000 φ6x,6x φ6x, 6x 4538898 φ7x,7x φ7x, 7x 6402373705728000 48338 6402373705728000 φ8x,8x φ8x, 8x 494484992000 2430φ9x,9x φ9x, 9x φ0x,0x φ0x, 0x 6402373705728000 484557977600 φ2x,2x φ2x, 2x φ4x,4x φ4x, 4x 569209246000 836959552000 φ6x,6x φ6x, 6x 737485692000 28047474456000 φ8x,8x φ8x, 8x. Theorem 2.2 Let k N and φ : X 2k [0, be a mapping, such that there exists L < with φ2x,...,2x k, 2y,...,2y k 2 8 Lφx,...,x k, y,...,y k for all x,...,x k, y,...,y k X. Let f : X Y be a mapping satisfying Dfx, y,...,dfx k, y k k φx,...,x k, y,...,y k, for all x,...,x k, y,...,y k X. Then there exists a unique octadecic mapping Q : X Y, such that for all x,...,x k X, where f x Qx,..., f x k Qx k k 2 8 L ψx,...,x k, ψx,...,x k := 32086852864000 2430φ0,...,0,x,...,x k 43758φx,...,x k,x,...,x k 3824φ2x,...,2x k,x,...,x k 8564φ3x,...,3x k,x,...,x k 8568φ4x,...,4x k,x,...,x k 3060φ5x,...,5x k,x,...,x k 86φ6x,...,6x k,x,...,x k 53φ7x,...,7x k,x,...,x k 8φ8x,...,8x k,x,...,x k φ9x,...,9x k,x,...,x k φ0,...,0,2x,...,2x k 2 4537567325 6402373705728000 4888673 φ0,...,0,0,...,0 270550690726000 6402373705728000 2324754432000

Arab. J. Math. 208 7:29 228 223 φx,...,x k,x,...,x k φx,...,x k, x,..., x k 9229380 6402373705728000 4845579776000 φ2x,...,2x k,2x,...,2x k φ2x,...,2x k, 2x,..., 2x k 94005848 6402373705728000 φ3x,...,3x k,3x,...,3x k 3753520 φ3x,...,3x k, 3x,..., 3x k 6402373705728000 40236344000 φ4x,...,4x k,4x,...,4x k φ4x,...,4x k, 4x,..., 4x k 5306920 6402373705728000 φ5x,...,5x k,5x,...,5x k 27330 φ5x,...,5x k, 5x,..., 5x k 6402373705728000 φ6x,...,6x k,6x,...,6x k φ6x,...,6x k, 6x,..., 6x k 4538898 6402373705728000 φ7x,...,7x k,7x,...,7x k 48338 φ7x,...,7x k, 7x,..., 7x k 6402373705728000 494484992000 φ8x,...,8x k,8x,...,8x k φ8x,...,8x k, 8x,..., 8x k 2430φ9x,...,9x k,9x,...,9x k φ9x,...,9x k, 9x,..., 9x k 6402373705728000 φ0x,...,0x k,0x,...,0x k φ0x,...,0x k, 0x,..., 0x k 484557977600 φ2x,...,2x k,2x,...,2x k φ2x,...,2x k, 2x,..., 2x k 569209246000 φ4x,...,4x k,4x,...,4x k φ4x,...,4x k, 4x,..., 4x k 836959552000 φ6x,...,6x k,6x,...,6x k φ6x,...,6x k, 6x,..., 6x k 737485692000 28047474456000 φ8x,...,8x k,8x,...,8x k φ8x,...,8x k, 8x,..., 8x k. Proof The proof is similar to the proof of Theorem 2.. From Theorem 2.2, we get the following corollaries: Corollary 2.3 Let k N and α, L be positive real numbers, such that mapping satisfying Dfx, y,...,dfx k, y k k α, 2 8 L <. Let f : X Ybea for all x,...,x k, y,...,y k X. Then there exists a unique octadecic mapping Q : X Y, such that for all x,...,x k X, where f x Qx,..., f x k Qx k k 2 8 L β, β := 32086852864000 α2430 43758 3824 8564 8568 3060 86 53 8 2 4537567325 6402373705728000 4888673 270550690726000 2 6402373705728000 2324754432000

224 Arab. J. Math. 208 7:29 228 9229380 2 6402373705728000 4845579776000 88026960 6402373705728000 2 3753520 6402373705728000 40236344000 23063840 6402373705728000 2 27330 6402373705728000 9077796 6402373705728000 2 48338 6402373705728000 494484992000 48620 6402373705728000 2 484557977600 2 569209246000 2836959552000 2737485692000 228047474456000. Corollary 2.4 Let k N and α, p, L be positive real numbers, such that L < and 2 p f : X Y be a mapping satisfying 2 8 L. Let Dfx, y,...,dfx k, y k k α x p x k p y p y k p, for all x,...,x k, y,...,y k X. Then there exists a unique octadecic mapping Q : X Y, such that for all x,...,x k X, where f x Qx,..., f x k Qx k k 2 8 L ψx,...,x k, ψx,...,x k := 32086852864000 α x p x k p 2430 8756 38242 p 85643 p 85684 p 30605 p 866 p 537 p 88 p 9 p 2p 2 4 6402373705728000 2324754432000 42 p 9229380 6402373705728000 4845579776000 43p 940058480 6402373705728000 3753520 44p 6402373705728000 40236344000 45p 5306920 6402373705728000 27330 46p 6402373705728000 47p 4538898 6402373705728000 48338 48p 6402373705728000 494484992000 49 p 2430 6402373705728000 40 p 484557977600 42 p 569209246000 44 p 836959552000 46 p 737485692000 48 p 28047474456000. The following example shows that the assumption 2 p 2 8 L cannot be omitted in Corollary 2.4. We know from Example.2 that if then R k,. k : k N is a multi-normed space. x,...,x k k = sup{ x,..., x k },

Arab. J. Math. 208 7:29 228 225 Example 2.5 Let k N.Wedefineφ : R R, by x [, φx := x 8 x, x, ]. We consider the function f : R R defined by f x = n=0 Then f satisfies the following functional inequality: φ4 n x 4 8n, x R. Dfx, y,...,dfx k, y k k 28 8! 4 8 454 x 8 x k 8 y 8 y k 8, 2. for all x,...,x k, y,...,y k R. Proof We have f x 48 4 8 for all x R. Therefore, we see that f is bounded. Let x, y R. If x 8 y 8 = 0or x 8 y 8 4 8, then Dfx, y 28 8!4 8 4 8 28 8!4 8 4 8 4 8 x 8 y 8. Now, suppose that 0 < x 8 y 8 < 4 8. Then there exists a nonnegative integer k such that Hence, and 4 8k2 x 8 y 8 < 4 k x < 4 and 4k y < 4, 4 8k. 4 n x 9y, 4 n x 8y, 4 n x 7y, 4 n x 6y, 4 n x 5y, 4 n x 4y, 4 n x 3y, 4 n x 2y, 4 n x y, 4 n x, 4 n x y, 4 n x 2y, 4 n x 3y, 4 n x 4y, 4 n x 5y, 4 n x 6y, 4 n x 7y, 4 n x 8y, 4 n x 9y, 4 n y, for all n = 0,,...,k. Thus we get or Dfx, y x 8 y 8 n=0 n=k 2 8 8! 4 8n x 8 y 8 2 8 8! 4 8n 4 36 = 28 8! 4 8 454, n=0 Dfx, y 28 8! 4 8 454 x 8 y 8. 2 8 8! 4 8n 4 8k2 x 8 y 8 436

226 Arab. J. Math. 208 7:29 228 Hence f satisfies 2.forallx,...,x k, y,...,y k R. Now, we claim that the octadecic functional equation. is not stable for p = 8 in Corollary 2.4. Suppose on the contrary that there exists an octadecic mapping C : R R, such that f x Cx,..., f x k Cx k k β x 8 x k 8 for some β R and all x,...,x k R. So f x Cx δ x 8 for some constant δ>0andallx R. Then there exists γ R for which Cx = γ x 8 for all x Q. Therefore, f x x 8 δ γ, x Q. Let M N be such that M >δ γ. Ifx is a rational number in 0,,thenwehave4 n x 0, for 4 each n = 0,, 2,...,M. Consequently, for such an x we have M f x x 8 = n=0 φ4 n M x 4 8n x 8 4 8n x 8 4 8n = M >δ γ, x8 n=0 which leads to a contradiction. Corollary 2.6 Let k N and α, p, L be positive real numbers, such that L < and 2 2kp 2 8 L. Let f : X Y be a mapping satisfying Dfx, y,...,dfx k, y k k α x p x k p y p y k p, for all x,...,x k, y,...,y k X. Then there exists a unique octadecic mapping Q : X Y, such that for all x,...,x k X, where f x Qx,..., f x k Qx k k 2 8 L ψx,...,x k, ψx,...,x k := 32086852864000 α x 2p... x k 2p 43758 38242 kp 85643 kp 85684 kp 30605 kp 866 kp 537 kp 88 kp 9 kp 2 6402373705728000 2324754432000 24 kp 9229380 6402373705728000 4845579776000 29kp 940058480 6402373705728000 3753520 26kp 6402373705728000 40236344000 225kp 5306920 6402373705728000 27330 236kp 6402373705728000 249kp 4538898 6402373705728000 264kp 48338 6402373705728000 494484992000 28kp 2430 6402373705728000 200 kp 484557977600 244 kp 569209246000 296 kp 836959552000 2256 kp 737485692000 2324 kp 28047474456000. Corollary 2.7 Let k N and α, p, L be positive real numbers, such that L < and 4 kp f : X Y be a mapping satisfying Dfx, y,...,dfx k, y k k α x p... x k p y p... y k p x 2kp x k 2kp y 2kp y k 2kp, 2 8 L. Let

Arab. J. Math. 208 7:29 228 227 for all x,...,x k, y,...,y k X. Then there exists a unique octadecic mapping Q : X Y, such that for all x,...,x k X, where f x Qx,..., f x k Qx k k 2 8 L ψx,...,x k, ψx,...,x k := 32086852864000 α x 2p... x k 2p 43758 38242 kp 85643 kp 85684 kp 30605 kp 866 kp 537 kp 88 kp 9 kp 2 6402373705728000 2324754432000 24 kp 9229380 6402373705728000 4845579776000 29kp 940058480 6402373705728000 3753520 26kp 6402373705728000 40236344000 225kp 5306920 6402373705728000 27330 236kp 6402373705728000 249kp 4538898 6402373705728000 48338 264kp 6402373705728000 494484992000 28kp 2430 6402373705728000 200 kp 484557977600 244 kp 569209246000 296 kp 836959552000 2256 kp 737485692000 2324 kp 28047474456000 32086852864000 α x 2kp... x k 2kp 2430 8756 38244 kp 85649 kp 85686 kp 306025 kp 8636 kp 5349 kp 864 kp 8 kp 22kp 2 4 6402373705728000 2324754432000 44 kp 9229380 6402373705728000 4845579776000 49kp 940058480 6402373705728000 3753520 46kp 6402373705728000 40236344000 425kp 5306920 6402373705728000 27330 436kp 6402373705728000 449kp 4538898 6402373705728000 48338 464kp 6402373705728000 494484992000 48kp 2430 6402373705728000 400 kp 484557977600 444 kp 569209246000 496 kp 836959552000 4256 kp 737485692000 4324 kp 28047474456000. Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License http:// creativecommons.org/licenses/by/4.0/, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors and the source, provide a link to the Creative Commons license, and indicate if changes were made.

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