(ProQuest: ... denotes non-US-ASCII text omitted.)
Tian Zhou Xu 1 and John Michael Rassias 2
Recommended by Krzysztof Cieplinski
1, School of Mathematics, Beijing Institute of Technology, Beijing 100081, China
2, Pedagogical Department E.E., Section of Mathematics and Informatics, National and Kapodistrian University of Athens, 4 Agamemnonos Street, Aghia Paraskevi, Athens 15342, Greece
Received 7 January 2012; Accepted 15 February 2012
1. Introduction and Preliminaries
A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?
If the problem accepts a unique solution, we say the equation is stable (see [1]). The study of stability problems for functional equations is related to a question of Ulam [2] concerning the stability of group homomorphisms and affirmatively answered for Banach spaces by Hyers [3]. The result of Hyers was generalized by Aoki [4] for approximate additive mappings and by Rassias [5] for approximate linear mappings by allowing the Cauchy difference operator CDf(x,y)=f(x+y)-[f(x)+f(y)] to be controlled by ...(||x||p +||y||p ) . In 1994, a generalization of Rassias' theorem was obtained by G a vruta [6], who replaced ...(||x||p +||y||p ) by a general control function [straight phi](x,y) . On the other hand, several further interesting discussions, modifications, extensions, and generalizations of the original problem of Ulam have been proposed (see, e.g. [7-12] and the references therein).
Recently, Park [9] investigated the approximate additive mappings, approximate Jensen mappings, and approximate quadratic mappings in 2-Banach spaces and proved the generalized Hyers-Ulam stability of the Cauchy functional equation, the Jensen functional equation, and the quadratic functional equation in 2-Banach spaces. This is the first result for the stability problem of functional equations in 2-Banach spaces.
In [11, 12], we introduced the following mixed additive-cubic functional equation for fixed integers k with k...0;0 , ±1 : [figure omitted; refer to PDF] with f(0)=0 , and investigated the generalized Hyers-Ulam stability of (1.1) in quasi-Banach spaces and non-Archimedean fuzzy normed spaces, respectively.
In this paper, we investigate, approximate mixed additive-cubic mappings in n -Banach spaces. That is, we prove the generalized Hyers-Ulam stability of a general mixed additive-cubic equation (1.1) in n -Banach spaces by the direct method.
The concept of 2-normed spaces was initially developed by Gähler [13, 14] in the middle of 1960s, while that of n -normed spaces can be found in [15, 16]. Since then, many others have studied this concept and obtained various results; see for instance [15, 17-19].
We recall some basic facts concerning n -normed spaces and some preliminary results.
Definition 1.1.
Let n∈... , and let X be a real linear space with dim X...5;n and ||·,...,·||:Xn [arrow right]... a function satisfying the following properties:
(N1) ||x1 ,x2 ,...,xn ||=0 if and only if x1 ,x2 ,...,xn are linearly dependent,
(N2) ||x1 ,x2 ,...,xn || is invariant under permutation,
(N3) ||αx1 ,x2 ,...,xn ||=|α|||x1 ,x2 ,...,xn || ,
(N4) ||x+y,x2 ,...,xn ||...4;||x,x2 ,...,xn ||+||y,x2 ,...,xn ||
for all α∈... and x,y,x1 ,x2 ,...,xn ∈X . Then the function ||·,...,·|| is called an n -norm on X and the pair (X,||·,...,·||) is called an n -normed space.
Example 1.2.
For x1 ,x2 ,...,xn ∈...n , the Euclidean n -norm ||x1 ,x2 ,...,xn ||E is defined by [figure omitted; refer to PDF] where xi =(xi1 ,...,xin )∈...n for each i=1,2,...,n .
Example 1.3.
The standard n -norm on X , a real inner product space of dimension dim X...5;n , is as follows: [figure omitted; refer to PDF] where ...·,·... denotes the inner product on X . If X=...n , then this n -norm is exactly the same as the Euclidean n -norm ||x1 ,x2 ,...,xn ||E mentioned earlier. For n=1 , this n -norm is the usual norm ||x1 ||=...x1 ,x1 ...1/2 .
Definition 1.4.
A sequence {xk } in an n -normed space X is said to converge to some x∈X in the n -norm if [figure omitted; refer to PDF] for every y2 ,...,yn ∈X .
Definition 1.5.
A sequence {xk } in an n -normed space X is said to be a Cauchy sequence with respect to the n -norm if [figure omitted; refer to PDF] for every y2 ,...,yn ∈X . If every Cauchy sequence in X converges to some x∈X , then X is said to be complete with respect to the n -norm. Any complete n -normed space is said to be an n -Banach space.
Now we state the following results as lemma (see [9] for the details).
Lemma 1.6.
Let X be an n -normed space. Then,
(1) For xi ∈X(i=1,...,n) and γ , a real number, [figure omitted; refer to PDF] for all 1...4;i...0;j...4;n ,
(2) |||x,y2 ,...,yn ||-||y,y2 ,...,yn |||...4;||x-y,y2 ,...,yn || for all x,y,y2 ,...,yn ∈X ,
(3) if ||x,y2 ,...,yn ||=0 for all y2 ,...,yn ∈X , then x=0 ,
(4) for a convergent sequence {xj } in X , [figure omitted; refer to PDF] for all y2 ,...,yn ∈X .
2. Approximate Mixed Additive-Cubic Mappings
In this section, we investigate the generalized Hyers-Ulam stability of the generalized mixed additive-cubic functional equation in n -Banach spaces. Let X be a linear space and Y an n -Banach space. For convenience, we use the following abbreviation for a given mapping f:X[arrow right]Y : [figure omitted; refer to PDF] for all x,y∈X .
Theorem 2.1.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there is a unique additive mapping A:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [figure omitted; refer to PDF]
Proof.
Letting x=0 in (2.3), we get [figure omitted; refer to PDF] for all y,u2 ,...,un ∈X . Putting y=x in (2.3), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Thus [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Letting y=kx in (2.3), we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Letting y=(k+1)x in (2.3), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Letting y=(k-1)x in (2.3), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Replacing x and y by 2x and x in (2.3), respectively, we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Replacing x and y by 3x and x in (2.3), respectively, we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Replacing x and y by 2x and kx in (2.3), respectively, we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Setting y=(2k+1)x in (2.3), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Letting y=(2k-1)x in (2.3), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Letting y=3kx in (2.3), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By (2.6), (2.7), (2.13), (2.15), and (2.16), we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By (2.6), (2.10), and (2.11), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . It follows from (2.12) and (2.19) that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By (2.14) and (2.20), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By (2.6), (2.15), (2.16), and (2.17), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . It follows from (2.6), (2.8), (2.9), and (2.22) that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Hence, [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By (2.9), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . From (2.23) and (2.25), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Also, from (2.18) and (2.26), we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X .
On the other hand, it follows from (2.21) and (2.27) that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Therefore by (2.24) and (2.28), we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X .
Now, let g:X[arrow right]Y be the mapping defined by g(x):=f(2x)-8f(x) for all x,u2 ,...,un ∈X . Then, (2.29) means that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Also, we get [figure omitted; refer to PDF] for all x∈X . Replacing x by 2j x in (2.31) and dividing both sides of (2.31) by 2j+1 , we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X and all integers j...5;0 . For all integers l,m with 0...4;l<m , we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . So, we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . This shows that the sequence {(1/2j )g(2j x)} is a Cauchy sequence in Y . Since Y is an n -Banach space, the sequence {(1/2j )g(2j x)} converges. So, we can define a mapping A:X[arrow right]Y by [figure omitted; refer to PDF] for all x∈X . Putting l=0 , then passing the limit m[arrow right]∞ in (2.33), and using Lemma 1.6(4), we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X .
Now we show that A is additive. By Lemma 1.6, (2.2), (2.32), and (2.35), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By Lemma 1.6(3), A(2x)=2A(x) for all x∈X . Also, by Lemma 1.6(4), (2.2), (2.3), and (2.35), we get [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . By Lemma 1.6(3), DA(x,y)=0 for all x,y∈X . Hence, the mapping A satisfies (1.1). By [11, Lemma 2.3], the mapping x[arrow right]A(2x)-8A(x) is additive. Therefore, A(2x)=2A(x) implies that the mapping A is additive.
To prove the uniqueness of A , let B:X[arrow right]Y be another additive mapping satisfying (2.4). Fix x∈X . Clearly, A(2l x)=2l A(x) and B(2l x)=2l B(x) for all l∈... . It follows from (2.4) that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , and l∈... . By (2.2), we see that the right-hand side of the above inequality tends to 0 as l[arrow right]∞ . Therefore, ||A(x)-B(x),u2 ,...,un||Y =0 for all u2 ,...,un ∈X . By Lemma 1.6, we can conclude that A(x)=B(x) for all x∈X . So, A=B . This proves the uniqueness of A .
Theorem 2.2.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there is a unique additive mapping A:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Proof.
The proof is similar to the proof of Theorem 2.1.
Corollary 2.3.
Let X be a normed space and Y an n -Banach space. Let θ∈[0,∞),p,r2 ,...,rn ∈(0,∞) such that p...0;1 , and let f:X[arrow right]Y be a mapping with f(0)=0 such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there exists a unique additive mapping A:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [figure omitted; refer to PDF]
Proof.
Define [straight phi](x,y)=θ(||x||Xp +||y||Xp )||u2||Xr2 ...||un||Xrn for all x,y,u2 ,...,un ∈X , and apply Theorems 2.1 and 2.2.
The following example shows that the assumption p...0;1 cannot be omitted in Corollary 2.3.
Example 2.4.
Let X=... be a linear space over ... . Define ||·,·||:X×X[arrow right]... by ||x1 ,x2 ||=|a1b2 -a2b1 | , where xj =aj +bj i∈... , aj ,bj ∈... , j=1,2 ( i=-1 is the imaginary unit). Then, (X,||·,·||) is a 2-normed linear space.
Let [varphi]:...[arrow right]... defined by [figure omitted; refer to PDF]
Consider the function f:...[arrow right]... defined by [figure omitted; refer to PDF] for all x∈... , where α>|k| . Then, f satisfies the functional inequality [figure omitted; refer to PDF] for all x,y,u∈... , but there do not exist an additive mapping A:...[arrow right]... and a constant d>0 such that ||f(x)-A(x),u||...4;d |x||u| for all x,u∈... .
It is clear that |f(x)|...4;α/(α-1) for all x∈... . If |x|+|y|=0 or |x|+|y|...5;1/α for all x,y∈... , then the inequality (2.47) holds. Now suppose that 0<|x|+|y|<1/α . Then, there exists an integer n...5;1 such that [figure omitted; refer to PDF] Hence, αm |kx±y|<1,αm |x±y|<1,αm |x|<1 for all m=0,1,...,n-1 . From the definition of f and (2.48), we obtain that [figure omitted; refer to PDF] Therefore, f satisfies (2.47). Now, we claim that the functional equation (1.1) is not stable for p=1 in Corollary 2.3. Suppose on the contrary that there exist an additive mapping A:...[arrow right]... and a constant d>0 such that ||f(x)-A(x),u||...4;d |x||u| for all x,u∈... . Then, there exists a constant c∈... such that A(x)=cx for all rational numbers x . So, we obtain that [figure omitted; refer to PDF] for all rational numbers x and all u∈... . Let s∈... with s+1>d+|c| . If x is a rational number in (0,α-s ) and u=bi ( b∈... ), then αm x∈(0,1) for all m=0,1,...,s , and we get [figure omitted; refer to PDF] which contradicts (2.50).
Theorem 2.5.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there is a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Proof.
As in the proof of Theorem 2.1, we have [figure omitted; refer to PDF] for all x∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Now, let h:X[arrow right]Y be the mapping defined by h(x):=f(2x)-2f(x) . By (2.55), we have [figure omitted; refer to PDF] for all x∈X . Replacing x by 2j x in (2.56) and dividing both sides of (2.56) by 8j+1 , we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X and all integers j...5;0 . For all integers l,m with 0...4;l<m , we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . So, we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . This shows that the sequence {(1/8j )h(2j x)} is a Cauchy sequence in Y . Since Y is an n -Banach space, the sequence {(1/8j )h(2j x)} converges. So, we can define a mapping C:X[arrow right]Y by [figure omitted; refer to PDF] for all x∈X . Putting l=0 , then passing the limit m[arrow right]∞ in (2.58), and using Lemma 1.6(4), we get [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X .
Now we show that C is cubic. By Lemma 1.6, (2.52), (2.58), and (2.60), we have [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . By Lemma 1.6(3), C(2x)=8C(x) for all x∈X . Also, by Lemma 1.6(4), (2.52), (2.53), and (2.60), we get [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . By Lemma 1.6(3), DC(x,y)=0 for all x,y∈X . Hence the mapping C satisfies (1.1). By [11, Lemma 2.3], the mapping x[arrow right]C(2x)-2C(x) is cubic. Therefore, C(2x)=8C(x) implies that the mapping C is cubic.
To prove the uniqueness of C , let S:X[arrow right]Y be another cubic mapping satisfying (2.54). Fix x∈X . Clearly, C(2l x)=8l A(x) and S(2l x)=8l S(x) for all l∈... . It follows from (2.54) that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , and l∈... . By (2.52), we see that the right-hand side of the above inequality tends to 0 as l[arrow right]∞ . Therefore, ||C(x)-S(x),u2 ,...,un ||Y =0 for all u2 ,...,un ∈X . By Lemma 1.6, we can conclude that C(x)=S(x) for all x∈X . So C=S . This proves the uniqueness of C .
Theorem 2.6.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there is a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Proof.
The proof is similar to the proof of Theorem 2.5.
Corollary 2.7.
Let X be a normed space and Y an n -Banach space. Let θ∈[0,∞),p,r2 ,...,rn ∈(0,∞) such that p...0;3 , and let f:X[arrow right]Y be a mapping with f(0)=0 such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there exists a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where ... is defined as in Corollary 2.3.
Proof.
Define [straight phi](x,y)=θ(||x||Xp +||y||Xp )||u2||Xr2 ...||un||Xrn for all x,y,u2 ,...,un ∈X , and apply Theorems 2.5 and 2.6.
The following example shows that the the generalized Hyers-Ulam stability problem for the case of p=3 was excluded in Corollary 2.7.
Example 2.8.
Let X=... be a linear space over ... , and let ||·,·||:X×X[arrow right]... be defined as in Example 2.4. Then, (X,||·,·||) is a 2-normed linear space.
Let [varphi]:...[arrow right]... be defined by [figure omitted; refer to PDF]
Consider the function f:...[arrow right]... defined by [figure omitted; refer to PDF] for all x∈... , where α>|k| . Then, f satisfies the functional inequality [figure omitted; refer to PDF] for all x,y,u∈... , but there do not exist a cubic mapping C:...[arrow right]... and a constant d>0 such that ||f(x)-C(x),u||...4;d |x|3 |u| for all x,u∈... .
It is clear that |f(x)|...4;α3 /(α3 -1) for all x∈... . If |x|3 +|y|3 =0 or |x|3 +|y|3 ...5;1/α3 for all x,y∈... , then inequality (2.71) holds. Now suppose that 0<|x|3 +|y|3 <1/α3 . Then, there exists an integer n...5;1 such that [figure omitted; refer to PDF] Hence, αm |kx±y|<1,αm |x±y|<1,αm |x|<1 for all m=0,1,...,n-1 . From the definition of f and (2.72), we obtain that [figure omitted; refer to PDF] Therefore, f satisfies (2.71). Now, we claim that the functional equation (1.1) is not stable for p=3 in Corollary 2.7. Suppose on the contrary that there exist a cubic mapping C:...[arrow right]... and a constant d>0 such that ||f(x)-C(x),u||...4;d |x|3 |u| for all x,u∈... . Then, there exists a constant β∈... such that C(x)=βx3 for all rational numbers x . So, we obtain that [figure omitted; refer to PDF] for all rational numbers x and all u∈... . Let s∈... with s+1>d+|β| . If x is a rational number in (0,α-s ) and u=bi ( b∈... ), then αm x∈(0,1) for all m=0,1,...,s , and we get [figure omitted; refer to PDF] which contradicts (2.74).
Theorem 2.9.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there exist a unique additive mapping A:X[arrow right]Y and a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Proof.
By Theorems 2.1 and 2.5, there exist an additive mapping A[variant prime] :X[arrow right]Y and a cubic mapping C[variant prime]:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Hence, [figure omitted; refer to PDF] for all x∈X . So, we obtain (2.78) by letting A(x)=-(1/6)A[variant prime] (x) and C(x)=(1/6)C[variant prime] (x) for all x∈X .
To prove the uniqueness of A and C , let A[variant prime][variant prime] ,C[variant prime][variant prime] :X[arrow right]Y be another additive and cubic mapping satisfying (2.78). Fix x∈X . Let A1 =A-A[variant prime][variant prime] and C1 =C-C[variant prime][variant prime] . So, [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Then (2.76) implies that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X . Thus, C1 =0 . So, it follows from (2.81) that [figure omitted; refer to PDF] for all u2 ,...,un ∈X . Therefore, A1 =0 .
Similarly to Theorem 2.9, one can prove the following result.
Theorem 2.10.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there exist a unique additive mapping A:X[arrow right]Y and a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Proof.
The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems 2.2 and 2.6.
Theorem 2.11.
Let X be a linear space and Y an n -Banach space. Let f:X[arrow right]Y be a mapping with f(0)=0 for which there is a function [straight phi]:Xn+1 [arrow right][0,∞) such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there exist a unique additive mapping A:X[arrow right]Y and a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where [straight phi]...(x,u2 ,...,un ) is defined as in Theorem 2.1.
Proof.
The proof is similar to the proof of Theorem 2.9 and the result follows from Theorems 2.2 and 2.5.
Corollary 2.12.
Let X be a normed space and Y an n -Banach space. Let θ∈[0,∞),r2 ,...,rn ∈(0,∞) , p∈(0,1)∪(1,3)∪(3,∞) , and let f:X[arrow right]Y be a mapping with f(0)=0 such that [figure omitted; refer to PDF] for all x,y,u2 ,...,un ∈X . Then, there exist a unique additive mapping A:X[arrow right]Y and a unique cubic mapping C:X[arrow right]Y such that [figure omitted; refer to PDF] for all x,u2 ,...,un ∈X , where ... is defined as in Corollary 2.3.
Proof.
Define [straight phi](x,y)=θ(||x||Xp +||y||Xp )||u2||Xr2 ...||un||Xrn for all x,y,u2 ,...,un ∈X , and apply Theorems 2.9-2.11.
Remark 2.13.
The generalized Hyers-Ulam stability problem for the cases of p=1 and p=3 was excluded in Corollary 2.12 (see Examples 2.4 and 2.8).
Acknowledgments
The authors would like to thank the Editor Professor Krzysztof Cieplinski and anonymous referees for their valuable comments and suggestions. The first author was supported by the National Natural Science Foundation of China (NNSFC) (grant No. 11171022).
[1] Z. Moszner, "On the stability of functional equations," Aequationes Mathematicae , vol. 77, no. 1-2, pp. 33-88, 2009.
[2] S. M. Ulam A Collection of Mathematical Problems , vol. 8, of Interscience Tracts in Pure and Applied Mathematics, pp. xiii+150, Interscience, New York, NY, USA, 1960.
[3] D. H. Hyers, "On the stability of the linear functional equation," Proceedings of the National Academy of Sciences of the United States of America , vol. 27, pp. 222-224, 1941.
[4] T. Aoki, "On the stability of the linear transformation in Banach spaces," Journal of the Mathematical Society of Japan , vol. 2, pp. 64-66, 1950.
[5] T. M. Rassias, "On the stability of the linear mapping in Banach spaces," Proceedings of the American Mathematical Society , vol. 72, no. 2, pp. 297-300, 1978.
[6] P. Gavruta, "A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings," Journal of Mathematical Analysis and Applications , vol. 184, no. 3, pp. 431-436, 1994.
[7] R. P. Agarwal, B. Xu, W. Zhang, "Stability of functional equations in single variable," Journal of Mathematical Analysis and Applications , vol. 288, no. 2, pp. 852-869, 2003.
[8] A. Najati, G. Z. Eskandani, "Stability of a mixed additive and cubic functional equation in quasi-Banach spaces," Journal of Mathematical Analysis and Applications , vol. 342, no. 2, pp. 1318-1331, 2008.
[9] W.-G. Park, "Approximate additive mappings in 2-Banach spaces and related topics," Journal of Mathematical Analysis and Applications , vol. 376, no. 1, pp. 193-202, 2011.
[10] R. Saadati, Y. J. Cho, J. Vahidi, "The stability of the quartic functional equation in various spaces," Computers & Mathematics with Applications , vol. 60, no. 7, pp. 1994-2002, 2010.
[11] T. Z. Xu, J. M. Rassias, W. X. Xu, "Generalized Hyers-Ulam stability of a general mixed additive-cubic functional equation in quasi-Banach spaces," Acta Mathematica Sinica, English Series , vol. 28, no. 3, pp. 529-560, 2011., [email protected]; [email protected]; [email protected]
[12] T. Z. Xu, J. M. Rassias, W. X. Xu, "Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces," Journal of Mathematical Physics , vol. 51, no. 9, 2010.
[13] S. Gähler, "2-metrische Räume und ihre topologische Struktur," Mathematische Nachrichten , vol. 26, pp. 115-148, 1963.
[14] S. Gähler, "Lineare 2-normierte Räume," Mathematische Nachrichten , vol. 28, pp. 1-43, 1964.
[15] Y. J. Cho, P. C. S. Lin, S. S. Kim, A. Misiak Theory of 2-Inner Product Spaces , pp. xii+330, Nova Science, Huntington, NY, USA, 2001.
[16] A. Misiak, " n -inner product spaces," Mathematische Nachrichten , vol. 140, pp. 299-319, 1989.
[17] X. Y. Chen, M. M. Song, "Characterizations on isometries in linear n -normed spaces," Nonlinear Analysis , vol. 72, no. 3-4, pp. 1895-1901, 2010.
[18] S. Gähler, "Über 2-Banach-Räume," Mathematische Nachrichten , vol. 42, pp. 335-347, 1969.
[19] A. G. White, Jr., "2-Banach spaces," Mathematische Nachrichten , vol. 42, pp. 43-60, 1969.
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Copyright © 2012 Tian Zhou Xu and John Michael Rassias. Tian Zhou Xu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
The objective of the present paper is to determine the generalized Hyers-Ulam stability of the mixed additive-cubic functional equation in n-Banach spaces by the direct method. In addition, we show under some suitable conditions that an approximately mixed additive-cubic function can be approximated by a mixed additive and cubic mapping.
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer