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

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1 Arab. J. Math : 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 ghadir54@gmail.com; g.sadeghi@hsu.ac.ir M. Nazarianpoor 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 jrassias@primedu.uoa.gr; Ioannis.Rassias@primedu.uoa.gr

2 220 Arab. J. Math : 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 = 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;

3 Arab. J. Math : 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 f x y f x 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!= 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 := φ0,x 43758φx,x 3824φ2x,x

4 222 Arab. J. Math : φ3x,x 8568φ4x,x 3060φ5x,x 86φ6x,x 53φ7x,x 8φ8x,x φ9x,x φ0,2x φ0, φx,x φx, x φ2x,2x φ2x, 2x φ3x,3x φ3x, 3x φ4x,4x φ4x, 4x φ5x,5x φ5x, 5x φ6x,6x φ6x, 6x φ7x,7x φ7x, 7x φ8x,8x φ8x, 8x φ9x,9x φ9x, 9x φ0x,0x φ0x, 0x φ2x,2x φ2x, 2x φ4x,4x φ4x, 4x φ6x,6x φ6x, 6x 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 := φ0,...,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 φ0,...,0,0,...,

5 Arab. J. Math : φx,...,x k,x,...,x k φx,...,x k, x,..., x k φ2x,...,2x k,2x,...,2x k φ2x,...,2x k, 2x,..., 2x k φ3x,...,3x k,3x,...,3x k φ3x,...,3x k, 3x,..., 3x k φ4x,...,4x k,4x,...,4x k φ4x,...,4x k, 4x,..., 4x k φ5x,...,5x k,5x,...,5x k φ5x,...,5x k, 5x,..., 5x k φ6x,...,6x k,6x,...,6x k φ6x,...,6x k, 6x,..., 6x k φ7x,...,7x k,7x,...,7x k φ7x,...,7x k, 7x,..., 7x k φ8x,...,8x k,8x,...,8x k φ8x,...,8x k, 8x,..., 8x k 2430φ9x,...,9x k,9x,...,9x k φ9x,...,9x k, 9x,..., 9x k φ0x,...,0x k,0x,...,0x k φ0x,...,0x k, 0x,..., 0x k φ2x,...,2x k,2x,...,2x k φ2x,...,2x k, 2x,..., 2x k φ4x,...,4x k,4x,...,4x k φ4x,...,4x k, 4x,..., 4x k φ6x,...,6x k,6x,...,6x k φ6x,...,6x k, 6x,..., 6x k 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 := α

6 224 Arab. J. Math : 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 := α x p x k p p p p p 866 p 537 p 88 p 9 p 2p p p p p p p p p p p p p p 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 },

7 Arab. J. Math : 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! 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 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! ! 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 ! 4 8n 4 36 = 28 8! , n=0 Dfx, y 28 8! x 8 y ! 4 8n 4 8k2 x 8 y 8 436

8 226 Arab. J. Math : 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 := α x 2p... x k 2p kp kp kp kp 866 kp 537 kp 88 kp 9 kp kp kp kp kp kp kp kp kp kp kp kp kp kp 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

9 Arab. J. Math : 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 := α x 2p... x k 2p kp kp kp kp 866 kp 537 kp 88 kp 9 kp kp kp kp kp kp kp kp kp kp kp kp kp kp α x 2kp... x k 2kp kp kp kp kp 8636 kp 5349 kp 864 kp 8 kp 22kp kp kp kp kp kp kp kp kp kp kp kp kp kp Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License 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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