JOURNAL OF MATHEMATICAL PHYSICS 51, 093508 共2010兲
Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces Tian Zhou Xu,1,a兲 John Michael Rassias,2,b兲 and Wan Xin Xu3,c兲 1
Department of Mathematics, School of Science, Beijing Institute of Technology, Beijing 100081, People’s Republic of China 2 Pedagogical Department E.E., Section of Mathematics and Informatics, National and Capodistrian University of Athens, 4, Agamemnonos Str., Aghia Paraskevi, Athens 15342, Greece 3 School of Communication and Information Engineering, University of Electronic Science and Technology of China, Chengdu 611731, People’s Republic of China 共Received 12 June 2010; accepted 16 July 2010; published online 20 September 2010兲
We establish some stability results concerning the general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. In addition, we establish some results of approximately general mixed additive-cubic mappings in non-Archimedean fuzzy normed spaces. The results improve and extend some recent results. © 2010 American Institute of Physics. 关doi:10.1063/1.3482073兴
I. INTRODUCTION
In 1897, Hensel discovered the p-adic numbers as a number theoretical analog of power series in complex analysis. The most important examples of non-Archimedean spaces are p-adic numbers. A key property of p-adic numbers is that they do not satisfy the Archimedean axiom: for all x , y ⬎ 0, there exists an integer n, such that x ⬍ ny. It turned out that non-Archimedean spaces have many nice applications.10,31,36 During the last three decades, theory of non-Archimedean spaces has gained the interest of physicists for their research, in particular, in problems coming from quantum physics, p-adic strings, and superstrings 共cf. Ref. 10兲. Although many results in the classical normed space theory have a non-Archimedean counterpart, but their proofs are essentially different and require an entirely new kind of intuition. One may note that 兩n兩 ⱕ 1 in each valuation field, every triangle is isosceles and there may be no unit vector in a non-Archimedean normed space 共cf. Ref. 10兲. These facts show that the non-Archimedean framework is of special interest. Fuzzy set theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, e.g., population dynamics, chaos control, computer programming, nonlinear dynamical systems, nonlinear operators, statistical convergence, etc. 共cf. Refs. 3, 13, 32, and 33兲. The fuzzy topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The most fascinating application of fuzzy topology in quantum particle physics arises in string and E-infinity theory of EI Naschie 共cf. Refs. 18–22兲. 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. The first stability problem concerning group homomorphisms was raised by Ulam35 in 1940 and affirmatively
a兲
Author to whom correspondence should be addressed. Electronic addresses:
[email protected] and
[email protected]. b兲 Electronic addresses:
[email protected],
[email protected], and
[email protected]. URL: http:// www.primedu.uoa.gr/~jrassias/. c兲 Electronic mail:
[email protected]. 0022-2488/2010/51共9兲/093508/19/$30.00
51, 093508-1
© 2010 American Institute of Physics
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solved by Hyers.9 The result of Hyers was generalized by Aoki1 for approximate additive mappings and by Rassias25 for approximate linear mappings by allowing the Cauchy difference operator CDf共x , y兲 = f共x + y兲 − 关f共x兲 + f共y兲兴 to be controlled by ⑀共储x储 p + 储y储 p兲. In 1994, a generalization of Rassias’ theorem was obtained by Găvruţa,8 who replaced ⑀共储x储 p + 储y储 p兲 by a general control function 共x , y兲. In addition, Rassias27–30,37–39 generalized the Hyers stability result by introducing two weaker conditions controlled by a product of different powers of norms and a mixed productsum of powers of norms, respectively. Recently, several further interesting discussions, modifications, extensions, and generalizations of the original problem of Ulam have been proposed 共see, e.g., Refs. 2, 4–7, 10–17, 23, 24, 26, 30, 34, and 37–40 and the references therein兲. In particular, Mirmostafaee and Moslehian12 introduced a notion of a non-Archimedean fuzzy norm and studied the stability of the Cauchy equation in the context of non-Archimedean fuzzy spaces. They presented an interdisciplinary relation between the theory of fuzzy spaces, the theory of nonArchimedean spaces, and the theory of functional equations. The generalized Hyers–Ulam stability for a mixed additive-cubic functional equation, f共2x + y兲 + f共2x − y兲 = 2f共x + y兲 + 2f共x − y兲 + 2f共2x兲 − 4f共x兲,
共1.1兲
in quasi-Banach spaces has been investigated by Najati and Eskandani.17 Functional equation 共1.1兲 is called mixed additive-cubic functional equation, since the function f共x兲 = ax3 + bx is its solution. Every solution of this mixed additive-cubic functional equation is said to be a mixed additivecubic mapping. In Refs. 37–39, we considered the following general mixed additive-cubic functional equation: f共kx + y兲 + f共kx − y兲 = kf共x + y兲 + kf共x − y兲 + 2f共kx兲 − 2kf共x兲.
共1.2兲
It is easy to show that the function f共x兲 = ax3 + bx satisfies functional equation 共1.2兲. We observe that in case k = 2, Eq. 共1.2兲 yields mixed additive-cubic equation 共1.1兲. Therefore, Eq. 共1.2兲 is a generalized form of the mixed additive-cubic equation. The main purpose of this paper is to establish Ulam–Hyers stability for general mixed additive-cubic functional equation 共1.2兲 in the setting of non-Archimedean fuzzy normed spaces. The achieved results via this paper improve and extend some recent well-known pertinent results. II. PRELIMINARIES
In this section we recall some notations and basic definitions used in this paper. The definition of non-Archimedean fuzzy normed spaces was given in Ref. 12. Definition 2.1: Let K be a field. A non-Archimedean absolute value on K is a function 兩 · 兩 : K → R, such that for any a , b 苸 K we have 共i兲 共ii兲 共iii兲
兩a兩 ⱖ 0 and equality holds if and only if a = 0; 兩ab兩 = 兩a兩兩b兩; 兩a + b兩 ⱕ max兵兩a兩 , 兩b兩其.
The condition 共iii兲 is called the strong triangle inequality. Clearly, 兩1兩 = 兩−1兩 = 1 and 兩n兩 ⱕ 1 for all n 苸 N. We always assume in addition that 兩 · 兩 is non trivial, i.e., that 共iv兲
there is an a0 苸 K, such that 兩a0兩 ⫽ 0 , 1.
The most important examples of non-Archimedean spaces are p-adic numbers. Example 2.2: Let p be a prime number. For any nonzero rational number x, there exists a unique integer nx, such that x = ba pnx, where a and b are integers not divisible by p. Then 兩x兩 p ª p−nx defines a non-Archimedean norm on Q. The completion of Q with respect to the metric d共x , y兲 = 兩x − y兩 p is denoted by Q p which is called the p-adic number field. In fact, Q p is the set of ⬁
all formal series x = 兺 ak pk, where 兩ak兩 ⱕ p − 1 are integers. The addition and multiplication bekⱖnx
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Stability of additive-cubicfunctional equation
⬁
tween any two elements of Q p are defined naturally. The norm 兩 兺 ak pk兩 p = p−nx is a nonkⱖnx
Archimedean norm on Q p and it makes Q p a locally compact field 共see Ref. 31兲. Note that if p ⬎ 2, then 兩2n兩 p = 1 for each integer n but 兩2兩2 ⬍ 1. Now we give the definition of a non-Archimedean fuzzy normed space. Definition 2.3: Let X be a linear space over a non-Archimedean field K. A function N : X ⫻ R → 关0 , 1兴 is said to be a non-Archimedean fuzzy norm on X if for all x , y 苸 X and all s , t 苸 R: 共N1兲 共N2兲 共N3兲 共N4兲 共N5兲
N共x , c兲 = 0 for all c ⱕ 0; x = 0 if and only if N共x , c兲 = 1 for all c ⬎ 0; N共cx , t兲 = N共x , 兩c兩t 兲 if c ⫽ 0; N共x + y , max兵s , t其兲 ⱖ min兵N共x , s兲 , N共y , t兲其; lim N共x , t兲 = 1. t→⬁
A non-Archimedean fuzzy normed space is a pair 共X , N兲, where X is a linear space and N is a non-Archimedean fuzzy norm on X. If 共N4兲 holds then so is 共N6兲 N共x + y , s + t兲 ⱖ min兵N共x , s兲 , N共y , t兲其. Recall that a classical vector space over the complex or real field satisfying 共N1兲–共N5兲 is called a fuzzy normed space in the literature. We repeatedly use the fact N共−x , t兲 = N共x , t兲, x 苸 X, t ⬎ 0, which is deduced from 共N3兲. It is easy to see that 共N4兲 is equivalent to the following condition: 共N7兲 N共x + y,t兲 ⱖ min兵N共x,t兲,N共y,t兲其共x,y 苸 X,t 苸 R兲. Example 2.4: Let 共X , 储 · 储兲 be a non-Archimedean normed space. For all x 苸 X, consider
冦
冧
t , t⬎0 N共x,t兲 = t + 储x储 0, t ⱕ 0.
Then 共X , N兲 is a non-Archimedean fuzzy normed space. Example 2.5: Let 共X , 储 · 储兲 be a non-Archimedean normed space. For all x 苸 X, consider
N共x,t兲 =
再
0, t ⱕ 储x储 1, t ⬎ 储x储.
冎
Then 共X , N兲 is a non-Archimedean fuzzy normed space. Definition 2.6: Let 共X , N兲 be a non-Archimedean fuzzy normed space. Let 兵xn其 be a sequence in X. Then 兵xn其 is said to be convergent if there exists x 苸 X, such that lim N共xn − x , t兲 = 1 for all n→⬁
t ⬎ 0. In that case, x is called the limit of the sequence 兵xn其 and we denote it by lim xn = x. n→⬁
A sequence 兵xn其 in X is said to be a Cauchy sequence if lim N共xn+p − xn , t兲 = 1 for all t ⬎ 0 and n→⬁
p = 1 , 2 , 3 , ¯. Due to the fact that N共xn − xm,t兲 ⱖ min兵N共x j+1 − x j,t兲:m ⱕ j ⱕ n − 1其共n ⬎ m兲, a sequence 兵xn其 is Cauchy if and only if lim N共xn+1 − xn , t兲 = 1 for all t ⬎ 0. n→⬁
It is known that every convergent sequence in a non-Archimedean fuzzy normed space is a Cauchy sequence. If every Cauchy sequence is convergent, then the non-Archimedean fuzzy normed space is called a non-Archimedean fuzzy Banach space.
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III. STABILITY OF THE FUNCTIONAL EQUATION IN NON-ARCHIMEDEAN FUZZY NORMED SPACES
Throughout this section, unless otherwise explicitly stated, we will assume that K is a nonArchimedean field, assume that X is a vector space over K, 共Y , N兲 is a non-Archimedean fuzzy Banach space over K, and 共Z , N⬘兲 is 共an Archimedean or a non-Archimedean fuzzy兲 normed space. We establish the following new stability for general mixed additive-cubic functional equation 共1.2兲 in non-Archimedean fuzzy Banach spaces. For convenience, given f : X → Y, we define the difference operator, Df共x,y兲 = f共kx + y兲 + f共kx − y兲 − kf共x + y兲 − kf共x − y兲 − 2f共kx兲 + 2kf共x兲, for fixed integers k with k ⫽ 0 , ⫾ 1 and for all x , y 苸 X. Theorem 3.1: Let : X ⫻ X → Z be a mapping and for some ␣ ⬎ 0 with 兩2兩 ⬍ ␣, N⬘共共2−1x,2−1y兲,t兲 ⱖ N⬘共共x,y兲, ␣t兲
共3.1兲
for all x , y 苸 X and t ⬎ 0. Let f : X → Y be a mapping with f共0兲 = 0, satisfying condition N共Df共x,y兲,t兲 ⱖ N⬘共共x,y兲,t兲
共3.2兲
for all x , y 苸 X and t ⬎ 0. Then there exists a unique additive mapping A : X → Y, such that N共f共2x兲 − 8f共x兲 − A共x兲,t兲 ⱖ N1共x, ␣兩k3 − k兩t兲
共3.3兲
for all x 苸 X and t ⬎ 0, where
再冉
N1共x,t兲 = min N⬘ 共x,共k + 1兲x兲,
冉
N⬘ 共2x,kx兲,
冊 冉
冊 冉
冊 冉
冊
1 1 1 t ,N⬘ 共x,共k − 1兲x兲, t ,N⬘ 共2x,x兲, t , 兩2兩 兩2兩 兩2兩
冊 冉
冊
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,x兲, t ,N⬘ 共0,kx兲, t ,N⬘共共x,共2k + 1兲x兲,t兲, 兩2兩 兩2兩 兩2k兩
N⬘共共x,共2k − 1兲x兲,t兲,N⬘共共3x,x兲,t兲,N⬘共共x,x兲,t兲,N⬘共共0,共k + 1兲x兲,兩k − 1兩t兲,
冉
N⬘共共0,共k − 1兲x兲,兩k − 1兩t兲,N⬘ 共0,2kx兲,
冉 冉 冉冉 冉 冉
N⬘ 共2x,2x兲,
冊
冉
冊 冉
兩k − 1兩 1 1 t ,N⬘ 共0,kx兲, t 2 t ,N⬘共共2x,2kx兲,t兲,N⬘ 共0,共3k − 1兲x兲, 兩k 兩 兩k兩 兩8k + 8兩
冊 冊 冊 冉 冊 冉冉 冊 冊 冉冉 冊 冊 冉冉 冊 冊 冊 冊 冉冉 冊 冊 冊 冉冉 冊冎
N⬘ 共0,2共k − 1兲x兲, N⬘
冊
兩k − 1兩 t ,N⬘共共x,3kx兲,t兲,N⬘共共x,kx兲,t兲, 兩k兩
冊
兩k − 1兩 1 1 x 共2k + 1兲x , t ,N⬘ , t , 2 t ,N⬘ 共0,2kx兲, 兩k 兩 兩k + 1兩 2 2 兩8k兩
1 x kx 1 x 共2k − 1兲x x 3kx 1 , t ,N⬘ , , , t ,N⬘ , , t , 兩8兩 2 2 兩8k兩 2 2 2 2 兩8兩
N⬘ 共x,x兲,
1 共3k − 1兲x 兩k − 1兩 共k + 1兲x 兩k − 1兩 t ,N⬘ 0, t ,N⬘ 0, t , , , 兩8k2兩 2 兩8k兩 2 兩8k兩
N⬘ 共0,共k − 1兲x兲,
兩k − 1兩 t 兩8k2兩
.
Proof: Letting x = 0 in 共3.2兲, we get N共f共y兲 + f共− y兲,t兲 ⱖ N⬘共共0,y兲,兩k − 1兩t兲 for all y 苸 X and t ⬎ 0. Putting y = x in 共3.2兲, we have
共3.4兲
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N共f共共k + 1兲x兲 + f共共k − 1兲x兲 − kf共2x兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ N⬘共共x,x兲,t兲
共3.5兲
for all x 苸 X and t ⬎ 0. Hence N共f共2共k + 1兲x兲 + f共2共k − 1兲x兲 − kf共4x兲 − 2f共2kx兲 + 2kf共2x兲,t兲 ⱖ N⬘共共2x,2x兲,t兲
共3.6兲
for all x 苸 X and t ⬎ 0. Letting y = kx in 共3.2兲, we get N共f共2kx兲 − kf共共k + 1兲x兲 − kf共− 共k − 1兲x兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ N⬘共共x,kx兲,t兲
共3.7兲
for all x 苸 X and t ⬎ 0. Letting y = 共k + 1兲x in 共3.2兲, we have N共f共共2k + 1兲x兲 + f共− x兲 − kf共共k + 2兲x兲 − kf共− kx兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ N⬘共共x,共k + 1兲x兲,t兲 共3.8兲 for all x 苸 X and t ⬎ 0. Letting y = 共k − 1兲x in 共3.2兲, we have N共f共共2k − 1兲x兲 − 共k + 2兲f共kx兲 − kf共− 共k − 2兲x兲 + 共2k + 1兲f共x兲,t兲 ⱖ N⬘共共x,共k − 1兲x兲,t兲 共3.9兲 for all x 苸 X and t ⬎ 0. Replacing x and y by 2x and x in 共3.2兲, respectively, we get N共f共共2k + 1兲x兲 + f共共2k − 1兲x兲 − 2f共2kx兲 − kf共3x兲 + 2kf共2x兲 − kf共x兲,t兲 ⱖ N⬘共共2x,x兲,t兲 共3.10兲 for all x 苸 X and t ⬎ 0. Replacing x and y by 3x and x in 共3.2兲, respectively, we get N共f共共3k + 1兲x兲 + f共共3k − 1兲x兲 − 2f共3kx兲 − kf共4x兲 − kf共2x兲 + 2kf共3x兲,t兲 ⱖ N⬘共共3x,x兲,t兲 共3.11兲 for all x 苸 X and t ⬎ 0. Replacing x and y by 2x and kx in 共3.2兲, respectively, we have N储共f共3kx兲 + f共kx兲 − kf共共k + 2兲x兲 − kf共− 共k − 2兲x兲 − 2f共2kx兲 + 2kf共2x兲,t兲储 ⱕ N⬘共共2x,kx兲,t兲 共3.12兲 for all x 苸 X and t ⬎ 0. Setting y = 共2k + 1兲x in 共3.2兲, we have N共f共共3k + 1兲x兲 + f共− 共k + 1兲x兲 − kf共2共k + 1兲x兲 − kf共− 2kx兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ N⬘共共x,共2k + 1兲x兲,t兲
共3.13兲
for all x 苸 X and t ⬎ 0. Letting y = 共2k − 1兲x in 共3.2兲, we have N共f共共3k − 1兲x兲 + f共− 共k − 1兲x兲 − kf共− 2共k − 1兲x兲 − kf共2kx兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ N⬘共共x,共2k − 1兲x兲,t兲
共3.14兲
for all x 苸 X and t ⬎ 0. Letting y = 3kx in 共3.2兲, we have N共f共4kx兲 + f共− 2kx兲 − kf共共3k + 1兲x兲 − kf共− 共3k − 1兲x兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ N⬘共共x,3kx兲,t兲 共3.15兲 for all x 苸 X and t ⬎ 0. By 共3.4兲, 共3.5兲, 共3.11兲, 共3.13兲, and 共3.14兲, we get
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N共kf共2共k + 1兲x兲 + kf共− 2共k − 1兲x兲 + 6f共kx兲 − 2f共3kx兲 − kf共4x兲 + 2kf共3x兲 − 6kf共x兲,t兲
再
ⱖ min N⬘共共x,共2k + 1兲x兲,t兲,N⬘共共x,共2k − 1兲x兲,t兲,N⬘共共3x,x兲,t兲,N⬘共共x,x兲,t兲,
冉
N⬘共共0,共k + 1兲x兲,兩k − 1兩t兲,N⬘共共0,共k − 1兲x兲,兩k − 1兩t兲,N⬘ 共0,2kx兲,
兩k − 1兩 t 兩k兩
冊冎 共3.16兲
for all x 苸 X and t ⬎ 0. By 共3.4兲, 共3.8兲, and 共3.9兲, we have N共f共共2k + 1兲x兲 + f共共2k − 1兲x兲 − kf共共k + 2兲x兲 − kf共− 共k − 2兲x兲 − 4f共kx兲 + 4kf共x兲,t兲
再
ⱖ min N⬘共共x,共k + 1兲x兲,t兲,N⬘共共x,共k − 1兲x兲,t兲,N⬘共共0,x兲,兩k − 1兩t兲,
冉
N⬘ 共0,kx兲,
兩k − 1兩 t 兩k兩
冊冎
共3.17兲
for all x 苸 X and t ⬎ 0. It follows from 共3.10兲 and 共3.17兲 that N共kf共共k + 2兲x兲 + kf共− 共k − 2兲x兲 − 2f共2kx兲 + 4f共kx兲 − kf共3x兲 + 2kf共2x兲 − 5kf共x兲,t兲
再
ⱖ min N⬘共共x,共k + 1兲x兲,t兲,N⬘共共x,共k − 1兲x兲,t兲,N⬘共共2x,x兲,t兲,N⬘共共0,x兲,兩k − 1兩t兲,
冉
N⬘ 共0,kx兲,
兩k − 1兩 t 兩k兩
冊冎
共3.18兲
for all x 苸 X and t ⬎ 0. By 共3.12兲 and 共3.18兲, we have N共f共3kx兲 − 4f共2kx兲 + 5f共kx兲 − kf共3x兲 + 4kf共2x兲 − 5kf共x兲,t兲
再
ⱖ min N⬘共共x,共k + 1兲x兲,t兲,N⬘共共x,共k − 1兲x兲,t兲,N⬘共共2x,x兲,t兲,
冉
N⬘共共2x,kx兲,t兲,N⬘共共0,x兲,兩k − 1兩t兲,N⬘ 共0,kx兲,
兩k − 1兩 t 兩k兩
冊冎
共3.19兲
for all x 苸 X and t ⬎ 0. By 共3.4兲 and 共3.13兲–共3.15兲, we have N共kf共− 共k + 1兲x兲 − kf共− 共k − 1兲x兲 − k2 f共2共k + 1兲x兲 + k2 f共− 2共k − 1兲x兲 + k2 f共2kx兲 − 共k2 − 1兲f共− 2kx兲
再冉
+ f共4kx兲 − 2f共kx兲 + 2kf共x兲,t兲 ⱖ min N⬘ 共x,共2k + 1兲x兲,
冉
N⬘ 共x,共2k − 1兲x兲,
冊
冉
冊
1 t , 兩k兩
兩k − 1兩 1 t ,N⬘共共x,3kx兲,t兲,N⬘ 共0,共3k − 1兲x兲, t 兩k兩 兩k兩
冊冎
共3.20兲
for all x 苸 X and t ⬎ 0. It follows from 共3.4兲, 共3.6兲, 共3.7兲, and 共3.20兲 that N共f共4kx兲 − 2f共2kx兲 − k3 f共4x兲 + 2k3 f共2x兲,t兲
再冉
ⱖ min N⬘ 共x,共2k + 1兲x兲,
冉
N⬘ 共2x,2x兲,
冊 冉
冊 冉
冊
1 1 t ,N⬘ 共x,共2k − 1兲x兲, t ,N⬘共共x,3kx兲,t兲,N⬘共共x,kx兲,t兲, 兩k兩 兩k兩
冊 冉
冊
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,共3k − 1兲x兲, t ,N⬘ 共0,共k + 1兲x兲, t , 兩k2兩 兩k兩 兩k兩
093508-7
J. Math. Phys. 51, 093508 共2010兲
Stability of additive-cubicfunctional equation
冉
N⬘ 共0,2共k − 1兲x兲,
冊 冉
兩k − 1兩 1 t 2 t ,N⬘ 共0,2kx兲, 兩k 兩 兩k + 1兩
冊冎
共3.21兲
for all x 苸 X and t ⬎ 0. Thus, N共f共2kx兲 − 2f共kx兲 − k3 f共2x兲 + 2k3 f共x兲,t兲
再 冉冉
冊 冊 冉冉 冉冉 冊冊 冉 冊 冉冉 冉冉 冊 冊 冉
ⱖ min N⬘ N⬘
冊 冊 冉冉 冊冊 冊 冊 冊 冉 冊冎
x 共2k + 1兲x 1 x 共2k − 1兲x 1 x 3kx , t ,N⬘ , , t ,N⬘ , ,t , , 2 2 兩k兩 2 2 兩k兩 2 2
1 共3k − 1兲x 兩k − 1兩 x kx , t , ,t ,N⬘ 共x,x兲, 2 t ,N⬘ 0, , 2 2 兩k 兩 2 兩k兩
N⬘ 0,
兩k − 1兩 共k + 1兲x 兩k − 1兩 1 t ,N⬘ 共0,共k − 1兲x兲, 2 t ,N⬘ 共0,kx兲, t , 2 兩k兩 兩k 兩 兩k + 1兩
共3.22兲 for all x 苸 X and t ⬎ 0. By 共3.7兲, we have N共f共4kx兲 − kf共2共k + 1兲x兲 − kf共− 2共k − 1兲x兲 − 2f共2kx兲 + 2kf共2x兲,t兲 ⱖ N⬘共共2x,2kx兲,t兲 共3.23兲 for all x 苸 X and t ⬎ 0. From 共3.21兲 and 共3.23兲, we have N共kf共2共k + 1兲x兲 + kf共− 2共k − 1兲x兲 − k3 f共4x兲 + 共2k3 − 2k兲f共2x兲,t兲
再冉
冊 冉 冉 冊 冊 冉
ⱖ min N⬘ 共x,共2k + 1兲x兲,
N⬘共共x,kx兲,t兲,N⬘ 共2x,2x兲,
冉
N⬘ 共0,共k + 1兲x兲,
冊
1 1 t ,N⬘ 共x,共2k − 1兲x兲, t ,N⬘共共x,3kx兲,t兲, 兩k兩 兩k兩
冉 冊
冊
兩k − 1兩 1 t , 2 t ,N⬘共共2x,2kx兲,t兲,N⬘ 共0,共3k − 1兲x兲, 兩k 兩 兩k兩
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,2共k − 1兲x兲, 2 t ,N⬘共共0,2kx兲兲, 兩k兩 兩k 兩 兩k + 1兩
冎
共3.24兲 for all x 苸 X and t ⬎ 0. Also, from 共3.16兲 and 共3.24兲, we get N共2f共3kx兲 − 6f共kx兲 + 共k − k3兲f共4x兲 − 2kf共3x兲 + 共2k3 − 2k兲f共2x兲 + 6kf共x兲,t兲
再
ⱖ min N⬘共共x,共2k + 1兲x兲,t兲,N⬘共共x,共2k − 1兲x兲,t兲,N⬘共共3x,x兲,t兲,N⬘共共x,x兲,t兲,
冉
N⬘共共0,共k + 1兲x兲,兩k − 1兩t兲,N⬘共共0,共k − 1兲x兲,兩k − 1兩t兲,N⬘ 共0,2kx兲,
冉 冊 冉
N⬘共共x,3kx兲,t兲,N⬘共共x,kx兲,t兲,N⬘ 共2x,2x兲,
冉
N⬘ 共0,共3k − 1兲x兲,
冊
冊
兩k − 1兩 t , 兩k兩
1 t ,N⬘共共2x,2kx兲,t兲, 兩k2兩
冊 冉
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,2共k − 1兲x兲, 2 t ,N⬘ 共0,2kx兲, 兩k兩 兩k 兩 兩k + 1兩
冊冎 共3.25兲
for all x 苸 X and t ⬎ 0. On the other hand, it follows from 共3.19兲 and 共3.25兲 that
093508-8
J. Math. Phys. 51, 093508 共2010兲
Xu, Rassias, and Xu
N共8f共2kx兲 − 16f共kx兲 + 共k − k3兲f共4x兲 + 共2k3 − 10k兲f共2x兲 + 16kf共x兲,t兲
再冉
ⱖ min N⬘ 共x,共k + 1兲x兲,
冉
N⬘ 共2x,kx兲,
冊 冉
冊 冉
冊 冉
冊
1 1 1 t ,N⬘ 共x,共k − 1兲x兲, t ,N⬘ 共2x,x兲, t , 兩2兩 兩2兩 兩2兩
冊 冉
冊
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,x兲, t ,N⬘ 共0,kx兲, t ,N⬘共共x,共2k + 1兲x兲,t兲, 兩2兩 兩2兩 兩2k兩
N⬘共共x,共2k − 1兲x兲,t兲,N⬘共共3x,x兲,t兲,N⬘共共x,x兲,t兲,N⬘共共0,共k + 1兲x兲,兩k − 1兩t兲,
冉
N⬘共共0,共k − 1兲x兲,兩k − 1兩t兲,N⬘ 共0,2kx兲,
冉 冉
N⬘ 共2x,2x兲,
冊
冊
兩k − 1兩 t ,N⬘共共x,3kx兲,t兲,N⬘共共x,kx兲,t兲, 兩k兩
冉
冊
兩k − 1兩 1 t , 2 t ,N⬘共共2x,2kx兲,t兲,N⬘ 共0,共3k − 1兲x兲, 兩k 兩 兩k兩
N⬘ 共0,2共k − 1兲x兲,
冊 冉
兩k − 1兩 1 t 2 t ,N⬘ 共0,2kx兲, 兩k 兩 兩k + 1兩
冊冎
共3.26兲
for all x 苸 X. Hence by 共3.22兲 and 共3.26兲, we get N共共k3 − k兲关f共4x兲 − 10f共2x兲 + 16f共x兲兴,t兲
再冉
ⱖ min N⬘ 共x,共k + 1兲x兲,
冉
N⬘ 共2x,kx兲,
冊 冉
冊 冉
冊 冉
冊
1 1 1 t ,N⬘ 共x,共k − 1兲x兲, t ,N⬘ 共2x,x兲, t , 兩2兩 兩2兩 兩2兩
冊 冉
冊
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,x兲, t ,N⬘ 共0,kx兲, t ,N⬘共共x,共2k + 1兲x兲,t兲, 兩2兩 兩2兩 兩2k兩
N⬘共共x,共2k − 1兲x兲,t兲,N⬘共共3x,x兲,t兲,N⬘共共x,x兲,t兲,N⬘共共0,共k + 1兲x兲,兩k − 1兩t兲,
冉
N⬘共共0,共k − 1兲x兲,兩k − 1兩t兲,N⬘ 共0,2kx兲,
冉 冉 冉冉 冉冉
N⬘ 共2x,2x兲,
冊
冉
冊 冉
冊
兩k − 1兩 1 1 t ,N⬘共共2x,2kx兲,t兲,N⬘ 共0,共3k − 1兲x兲, t ,N⬘ 共0,kx兲, t , 兩k2兩 兩k兩 兩8k + 8兩
冊 冊 冊 冉 冊 冉冉 冊 冊 冉冉 冊 冊 冉冉 冊 冊 冉 冊 冊 冊 冉冉 冊 冊 冉 冊冎
N⬘ 共0,2共k − 1兲x兲, N⬘
冊
兩k − 1兩 t ,N⬘共共x,3kx兲,t兲,N⬘共共x,kx兲,t兲, 兩k兩
兩k − 1兩 1 1 x 共2k + 1兲x t ,N⬘ , t , , 2 t ,N⬘ 共0,2kx兲, 兩k 兩 兩k + 1兩 2 2 兩8k兩
1 1 x kx 1 x 共2k − 1兲x x 3kx 1 , t ,N⬘ , , , t ,N⬘ , , t ,N⬘ 共x,x兲, 2 t , 兩8兩 2 2 兩8k兩 2 2 2 2 兩8兩 兩8k 兩
N⬘ 0,
兩k − 1兩 共3k − 1兲x 兩k − 1兩 共k + 1兲x 兩k − 1兩 t ,N⬘ 0, t ,N⬘ 共0,共k − 1兲x兲, t , , 2 兩8k兩 2 兩8k兩 兩8k2兩
.
共3.27兲
Therefore, N共f共4x兲 − 10f共2x兲 + 16f共x兲,t兲 ⱖ N1共x,兩k3 − k兩t兲 for all x 苸 X and t ⬎ 0, where
共3.28兲
093508-9
J. Math. Phys. 51, 093508 共2010兲
Stability of additive-cubicfunctional equation
再冉
N1共x,t兲 = min N⬘ 共x,共k + 1兲x兲,
冉
N⬘ 共2x,kx兲,
冊 冉
冊 冉
冊 冉
冊
1 1 1 t ,N⬘ 共x,共k − 1兲x兲, t ,N⬘ 共2x,x兲, t , 兩2兩 兩2兩 兩2兩
冊 冉
冊
兩k − 1兩 兩k − 1兩 1 t ,N⬘ 共0,x兲, t ,N⬘ 共0,kx兲, t ,N⬘共共x,共2k + 1兲x兲,t兲, 兩2兩 兩2兩 兩2k兩
N⬘共共x,共2k − 1兲x兲,t兲,N⬘共共3x,x兲,t兲,N⬘共共x,x兲,t兲,N⬘共共0,共k + 1兲x兲,兩k − 1兩t兲,
冉
N⬘共共0,共k − 1兲x兲,兩k − 1兩t兲,N⬘ 共0,2kx兲,
冉 冉 冉冉 冉 冉
N⬘ 共2x,2x兲,
冊
冉
冊 冉
兩k − 1兩 1 1 t ,N⬘ 共0,kx兲, t 2 t ,N⬘共共2x,2kx兲,t兲,N⬘ 共0,共3k − 1兲x兲, 兩k 兩 兩k兩 兩8k + 8兩
冊 冊 冊 冉 冊 冉冉 冊 冊 冉冉 冊 冊 冉冉 冊 冊 冊 冊 冉冉 冊 冊 冊 冉冉 冊冎
N⬘ 共0,2共k − 1兲x兲, N⬘
冊
兩k − 1兩 t ,N⬘共共x,3kx兲,t兲,N⬘共共x,kx兲,t兲, 兩k兩
冊
兩k − 1兩 1 1 x 共2k + 1兲x , t ,N⬘ 共0,2kx兲, t ,N⬘ , t , 兩k2兩 兩k + 1兩 2 2 兩8k兩
1 x kx 1 x 共2k − 1兲x x 3kx 1 , t ,N⬘ , , , t ,N⬘ , , t , 兩8兩 2 2 兩8k兩 2 2 2 2 兩8兩
N⬘ 共x,x兲,
1 共3k − 1兲x 兩k − 1兩 共k + 1兲x 兩k − 1兩 t ,N⬘ 0, t , , , 2 t ,N⬘ 0, 兩8k 兩 2 兩8k兩 2 兩8k兩
N⬘ 共0,共k − 1兲x兲,
兩k − 1兩 t 兩8k2兩
.
共3.29兲
Let g : X → Y is the mapping defined by g共x兲 ª f共2x兲 − 8f共x兲 for all x 苸 X. From 共3.28兲, we have N共g共2x兲 − 2g共x兲,t兲 ⱖ N1共x,兩k3 − k兩t兲
共3.30兲
for all x 苸 X and t ⬎ 0. Replacing x by 2−n−1x in 共3.30兲 and using inequality 共3.1兲, we get N共g共2−nx兲 − 2g共2−共n+1兲x兲,t兲 ⱖ N1共x, ␣n+1兩k3 − k兩t兲
共3.31兲
for all x 苸 X and t ⬎ 0. Hence
冉
N共2ng共2−nx兲 − 2n+1g共2−共n+1兲x兲,t兲 ⱖ N1 x,
␣n+1 3 兩k − k兩t 兩2兩n
冊
共3.32兲
for all x 苸 X, t ⬎ 0 and all non-negative integers n. n+1 Since limn→⬁ N1共x , ␣兩2兩n 兩k3 − k兩t兲 = 1, inequality 共3.32兲 shows that 兵2ng共2−nx兲其 is a Cauchy sequence in the non-Archimedean fuzzy Banach space 共Y , N兲 for each x 苸 X. Hence we can define the mapping A : X → Y by A共x兲 ª lim 2ng共2−nx兲
共3.33兲
lim N共2ng共2−nx兲 − A共x兲,t兲 = 1
共3.34兲
n→⬁
for all x 苸 X. Thus,
n→⬁
for all x 苸 X and t ⬎ 0. For each n ⱖ 1, x 苸 X, and t ⬎ 0,
093508-10
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Xu, Rassias, and Xu
冉兺
n−1
n
−n
N共g共x兲 − 2 g共2 x兲,t兲 = N
关2ig共2−ix兲 − 2i+1g共2−共i+1兲x兲兴,t
i=0
冊
n−1
ⱖ min 艛 兵N共2ig共2−ix兲 − 2i+1g共2−共i+1兲x兲,t兲其 i=0
ⱖ N1共x, ␣兩k3 − k兩t兲.
共3.35兲
It follows from 共3.34兲 and 共3.35兲 that for each x 苸 X, t ⬎ 0, and large enough n, we have N共g共x兲 − A共x兲,t兲 ⱖ min兵N共g共x兲 − 2ng共2−nx兲,t兲,N共2ng共2−nx兲 − A共x兲,t兲其 ⱖ N1共x, ␣兩k3 − k兩t兲. 共3.36兲 This proves 共3.3兲. Now, we will show that A is an additive mapping. It follows from 共3.34兲 that lim N共2ng共2−n+1x兲 − A共2x兲,t兲 = 1,
lim N共A共x兲 − 2n−1g共2−n+1x兲,t兲 = 1
n→⬁
n→⬁
for all x 苸 X and t ⬎ 0. Therefore, N共A共2x兲 − 2A共x兲,t兲 = N共A共2x兲 − 2ng共2−n+1x兲 + 2ng共2−n+1x兲 − 2A共x兲,t兲 ⱖ min兵N共A共2x兲 − 2ng共2−n+1x兲,t兲,N共2ng共2−n+1x兲 − 2A共x兲,t兲其
再
冉
= min N共A共2x兲 − 2ng共2−n+1x兲,t兲,N 2n−1g共2−n+1x兲 − A共x兲,
t 兩2兩
冊冎
→ 1共as n → ⬁兲, and so 共N2兲 implies that 共3.37兲
A共2x兲 = 2A共x兲
for all x 苸 X. Replacing x , y by 2−nx , 2−ny, respectively, in 共3.2兲 and using 共N3兲, we have
冉
N共2nDf共2−nx,2−ny兲,t兲 ⱖ N⬘ 共2−nx,2−ny兲,
t 兩2n兩
冊
for all x , y 苸 X and t ⬎ 0. On the other hand, it can be easily verified that Dg共x,y兲 = Df共2x,2y兲 − 8Df共x,y兲 for all x , y 苸 X. Hence N共DA共x,y兲,t兲 = N共A共kx + y兲 + A共kx − y兲 − kA共x + y兲 − kA共x − y兲 − 2A共kx兲 + 2kA共x兲,t兲 = N共关A共kx + y兲 − 2ng共2−n共kx + y兲兲兴 + 关A共kx − y兲 − 2ng共2−n共kx − y兲兲兴 − k关A共x + y兲 − 2ng共2−n共x + y兲兲兴 − k关A共x − y兲 − 2ng共2−n共x − y兲兲兴 − 2关A共kx兲 − 2ng共2−nkx兲兴 + 2k关A共x兲 − 2ng共2−nx兲兴 + 2n关Df共2−n+1x,2−n+1y兲 − 8Df共2−nx,2−ny兲兴,t兲
再
ⱖ min N共A共kx + y兲 − 2ng共2−n共kx + y兲兲,t兲,N共A共kx − y兲 − 2ng共2−n共kx − y兲兲,t兲,
冉
N A共x + y兲 − 2ng共2−n共x + y兲兲,
冊冉
冊
t t ,N A共x − y兲 − 2ng共2−n共x − y兲兲, , 兩k兩 兩k兩
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冉 冉
J. Math. Phys. 51, 093508 共2010兲
Stability of additive-cubicfunctional equation
N A共kx兲 − 2ng共2−nkx兲, N⬘ 共x,y兲,
冊冉
冊
t t ,N A共x兲 − 2ng共2−nx兲, , 兩2兩 兩2k兩
冊 冉
␣n−1t ␣ nt ,N 共x,y兲, ⬘ 兩2n兩 兩2n+3兩
冊冎
for all x , y 苸 X and t ⬎ 0. The first six terms on the right hand side of the above inequality tend to 1 as n → ⬁ by 共3.34兲 and the seventh and eighth terms tend to 1 as n → ⬁ by 兩2兩 ⬍ ␣ and 共N5兲. Therefore, N共DA共x , y兲 , t兲 = 1 for all t ⬎ 0. By 共N2兲, we infer that A共kx + y兲 + A共kx − y兲 − kA共x + y兲 − kA共x − y兲 − 2A共kx兲 + 2kA共x兲 = 0 for all x , y 苸 X, and so by Ref. 39, Lemma 3.1, we see that the mapping x → A共2x兲 − 8A共x兲 is additive. Equation 共3.37兲 implies that the mapping A is additive. To prove the uniqueness of the mapping A, let B : X → Y be another additive mapping, such that N共f共2x兲 − 8f共x兲 − B共x兲 , t兲 ⱖ N1共x , ␣兩k3 − k兩t兲. Then for each x 苸 X and all t ⬎ 0, we have N共A共x兲 − B共x兲,t兲 = N共A共x兲 − f共2x兲 + 8f共x兲 + f共2x兲 − 8f共x兲 − B共x兲,t兲 ⱖ min兵N共A共x兲 − f共2x兲 + 8f共x兲,t兲,N共f共2x兲 − 8f共x兲 − B共x兲,t兲其 ⱖ N1共x, ␣兩k3 − k兩t兲. Hence by the above inequality, 共3.1兲, 共N3兲, and the additivity of A and B, we get
冉
N共A共x兲 − B共x兲,t兲 = N A共2−nx兲 − B共2−nx兲,
冊 冉
冊 冉
␣ 3 ␣n+1 3 t −n 兩k 兩k − k兩t ⱖ N 2 x, − k兩t ⱖ N x, 1 1 兩2n兩 兩2n兩 兩2n兩
冊
␣ n 兲 = ⬁. Thus, the right hand side of the above for all x 苸 X, t ⬎ 0, and n 苸 N. Since 兩2兩 ⬍ ␣, limn→⬁共 兩2兩 inequality tends to 1 as n → ⬁. So A共x兲 = B共x兲 for all x 苸 X. This completes the proof. 䊏 Similar to Theorem 3.1, one can prove the following result. Theorem 3.2: Let : X ⫻ X → Z be a mapping and for some  ⬎ 0 with 兩8兩 ⬍ ,
N⬘共共2−1x,2−1y兲,t兲 ⱖ N⬘共共x,y兲, t兲
共3.38兲
for all x , y 苸 X and t ⬎ 0. Let f : X → Y be a mapping with f共0兲 = 0, satisfies condition N共Df共x,y兲,t兲 ⱖ N⬘共共x,y兲,t兲
共3.39兲
for all x , y 苸 X and t ⬎ 0. Then there exists a unique cubic mapping C : X → Y, such that N共f共2x兲 − 2f共x兲 − C共x兲,t兲 ⱖ N1共x, 兩k3 − k兩t兲
共3.40兲
for all x 苸 X and t ⬎ 0, where N1共x , t兲 is defined as in Theorem 3.1. Proof: Similar to the proof of Theorem 3.1, let h : X → Y be the mapping defined by h共x兲 ª f共2x兲 − 2f共x兲 for all x 苸 X. From 共3.28兲, we have N共h共2x兲 − 8h共x兲,t兲 ⱖ N1共x,兩k3 − k兩t兲
共3.41兲
for all x 苸 X and t ⬎ 0. Replacing x by 2−n−1x in 共3.41兲 and using inequality 共3.38兲, we get N共h共2−nx兲 − 8h共2−共n+1兲x兲,t兲 ⱖ N1共x, n+1兩k3 − k兩t兲
共3.42兲
for all x 苸 X and t ⬎ 0. Hence
冉
N共8nh共2−nx兲 − 8n+1h共2−共n+1兲x兲,t兲 ⱖ N1 x, for all x 苸 X, t ⬎ 0, and all non-negative integers n.
n+1 3 兩k − k兩t 兩8兩n
冊
共3.43兲
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Xu, Rassias, and Xu
From 兩8兩 ⬍ , we conclude that limn→⬁ N1共x , 兩8兩n 兩k3 − k兩t兲 = 1. Then inequality 共3.43兲 shows that 兵8nh共2−nx兲其 is a Cauchy sequence in the non-Archimedean fuzzy Banach space 共Y , N兲 for each x 苸 X. Hence we can define the mapping C : X → Y by n+1
C共x兲 ª lim 8nh共2−nx兲
共3.44兲
lim N共8nh共2−nx兲 − C共x兲,t兲 = 1
共3.45兲
n→⬁
for all x 苸 X. Thus,
n→⬁
for all x 苸 X and t ⬎ 0. For each n ⱖ 1, x 苸 X, and t ⬎ 0,
冉兺
n−1
n
−n
N共h共x兲 − 8 h共2 x兲,t兲 = N
关8ih共2−ix兲 − 8i+1h共2−共i+1兲x兲兴,t
i=0
冊
n−1
ⱖ min 艛 兵N共8ih共2−ix兲 − 8i+1h共2−共i+1兲x兲,t兲其 i=0
ⱖ N1共x, 兩k3 − k兩t兲.
共3.46兲
It follows from 共3.45兲 and 共3.46兲 that for each x 苸 X, t ⬎ 0, and large enough n, we have N共h共x兲 − C共x兲,t兲 ⱖ min兵N共h共x兲 − 8nh共2−nx兲,t兲,N共8nh共2−nx兲 − C共x兲,t兲其 ⱖ N1共x, 兩k3 − k兩t兲. 共3.47兲 This proves 共3.40兲. The rest of the proof is similar to the proof of Theorem 3.1. Theorem 3.3: Let : X ⫻ X → Z be a mapping and for some ␣ ⬎ 0 with 兩2兩 ⬍ ␣, N⬘共共2−1x,2−1y兲,t兲 ⱖ N⬘共共x,y兲, ␣t兲
䊏
共3.48兲
for all x , y 苸 X and t ⬎ 0. Let f : X → Y be a mapping with f共0兲 = 0, satisfies condition N共Df共x,y兲,t兲 ⱖ N⬘共共x,y兲,t兲
共3.49兲
for all x , y 苸 X and t ⬎ 0. Then there exist an additive mapping A : X → Y and a cubic mapping C : X → Y, such that N共f共x兲 − A共x兲 − C共x兲,t兲 ⱖ N1共x, ␣兩6兩兩k3 − k兩t兲
共3.50兲
for all x 苸 X and t ⬎ 0, where N1共x , t兲 is defined as in Theorem 3.1. Proof: Clearly 兩8兩 ⱕ 兩2兩 ⬍ ␣. By Theorems 3.1 and 3.2, there exist a unique additive mapping A1 : X → Y and a unique cubic mapping C1 : X → Y, such that N共f共2x兲 − 8f共x兲 − A1共x兲,t兲 ⱖ N1共x, ␣兩k3 − k兩t兲
共3.51兲
N共f共2x兲 − 2f共x兲 − C1共x兲,t兲 ⱖ N1共x, ␣兩k3 − k兩t兲
共3.52兲
and
for all x 苸 X and t ⬎ 0, where N1共x , t兲 is defined as in Theorem 3.1. Therefore, from 共3.51兲 and 共3.52兲, we get
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Stability of additive-cubicfunctional equation
冉
1 1 N f共x兲 + A1共x兲 − C1共x兲,t 6 6 =N
冉
冊
1 1 关f共2x兲 − 2f共x兲 − C1共x兲兴 − 关f共2x兲 − 8f共x兲 − A1共x兲兴,t 6 6
再冉
ⱖ min N
冊冉
冊
1 1 关f共2x兲 − 2f共x兲 − C1共x兲兴,t ,N 关f共2x兲 − 8f共x兲 − A1共x兲兴,t 6 6
冊冎
= min兵N共f共2x兲 − 2f共x兲 − C1共x兲,兩6兩t兲,N共f共2x兲 − 8f共x兲 − A1共x兲,兩6兩t兲其 ⱖ N1共x, ␣兩6兩兩k3 − k兩t兲 for all x 苸 X and t ⬎ 0. Letting 共3.53兲 that
共3.53兲
A共x兲 = − 61 A1共x兲
and
C共x兲 = 61 C1共x兲
for all x 苸 X, it follows from
N共f共x兲 − A共x兲 − C共x兲,t兲 ⱖ N1共x, ␣兩6兩兩k3 − k兩t兲 for all x 苸 X and t ⬎ 0. This completes the proof.
䊏
IV. APPLICATIONS OF FUZZY STABILITY
In this section, we investigate applications of fuzzy stability to the stability of general mixed additive-cubic functional equation in non-Archimedean normed spaces. Theorem 4.1: Let K be a non-Archimedean field, X be a linear space over K, 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ 共x,y兲 for all x , y 苸 X, where : X ⫻ X → 关0 , ⬁兲. Suppose that
共2−1x,2−1y兲 ⱕ
1 共x,y兲 ␣
for all x , y 苸 X, where ␣ is a positive real number with 兩2兩 ⬍ ␣. Then there exists a unique additive mapping A : X → Y, such that 储f共2x兲 − 8f共x兲 − A共x兲储Y ⱕ for all x 苸 X, where M共x兲 =
1 M共x兲 ␣
再
兩2兩 1 max 兩2兩共x,共k + 1兲x兲,兩2兩共x,共k − 1兲x兲,兩2兩共2x,x兲,兩2兩共2x,kx兲, 共0,x兲, 兩k3 − k兩 兩k − 1兩 兩2k兩 1 共0,kx兲, 共x,共2k + 1兲x兲, 共x,共2k − 1兲x兲, 共3x,x兲, 共x,x兲, 共0,共k + 1兲x兲, 兩k − 1兩 兩k − 1兩 兩k兩 1 共0,共k − 1兲x兲, 共0,2kx兲, 共x,3kx兲, 共x,kx兲,兩k2兩共2x,2x兲, 共2x,2kx兲, 兩k − 1兩 兩k − 1兩 兩k2兩 兩k兩 共0,共3k − 1兲x兲,兩8k + 8兩共0,kx兲, 共0,2共k − 1兲x兲,兩k + 1兩共0,2kx兲, 兩k − 1兩 兩k − 1兩 兩8k兩
冉
冊 冉 冊 冉 冊 冉 冊 冊 冉 冊 冎
x 共2k + 1兲x x 共2k − 1兲x x 3kx x kx ,兩8k兩 , ,兩8兩 , ,兩8兩 , , ,兩8k2兩共x,x兲, 2 2 2 2 2 2 2 2
冉
兩8k兩 兩8k兩 共3k − 1兲x 共k + 1兲x 兩8k2兩 0, , 0, , 共0,共k − 1兲x兲 . 兩k − 1兩 2 兩k − 1兩 2 兩k − 1兩
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Xu, Rassias, and Xu
Proof: Let Z = R with the fuzzy norm,
冦
t , t ⬎ 0, N⬘共x,t兲 = t + 兩x兩 t ⱕ 0, 0,
x苸R x 苸 R,
冧
and define
冦
t , t ⬎ 0, N共y,t兲 = t + 储y储Y 0, t ⱕ 0,
y苸Y y 苸 Y.
冧
Then N is a non-Archimedean fuzzy norm on Y and N⬘ is a fuzzy norm on R. The result follows from Theorem 3.1. 䊏 Corollary 4.2: Let K be a non-Archimedean field, 共X , 储 · 储X兲 be a non-Archimedean normed space over K, and 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K. Let ␦ ⬎ 0, 0 ⱕ r ⬍ 1, 兩2兩 ⬍ 1, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ ␦共储x储Xr + 储y储Xr兲 for all x , y 苸 X. Then there exists a unique additive mapping A : X → Y, such that 储f共2x兲 − 8f共x兲 − A共x兲储Y ⱕ
␦储x储Xr 兩k − k兩兩2兩2r 3
再
max 2,
1 兩k − 1兩
冎
for all x 苸 X. Proof: Let : X ⫻ X → 关0 , ⬁兲 be defined by 共x , y兲 = ␦共储x储Xr + 储y储Xr兲 for all x , y 苸 X. Then the 䊏 corollary is followed from Theorem 4.1 by ␣ = 兩2兩r. Corollary 4.3: Let K be a non-Archimedean field, 共X , 储 · 储X兲 be a non-Archimedean normed space over K, and 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K. Let ␦ ⬎ 0, 兩2兩 ⬍ 1, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ ␦关储x储Xr储y储Xs + 共储x储Xr+s + 储y储Xr+s兲兴
共x,y 苸 X兲
where r , s be non-negative real numbers, such that ª r + s ⬍ 1. Then there exists a unique additive mapping A : X → Y, such that 储f共2x兲 − 8f共x兲 − A共x兲储Y ⱕ
␦储x储X 兩k − k兩兩2兩2 3
再
max 3,
1 兩k − 1兩
冎
for all x 苸 X. Proof: Let : X ⫻ X → 关0 , ⬁兲 be defined by 共x , y兲 = ␦关储x储Xr储y储Xs + 共储x储Xr+s + 储y储Xr+s兲兴 for all x , y 苸 X. Then the corollary is followed from Theorem 4.1 by ␣ = 兩2兩. This mixed product-sum sta䊏 bility function was introduced by Rassias30 共in 2008兲. Ulam–Gavruta–Rassias stability product of powers of norm, introduced by Rassias28,29 共in 1982 and 1989兲. See also Refs. 37–39. The following example shows that the assumption 兩2兩 ⬍ 1 cannot be omitted in Corollaries 4.2 and 4.3. This example is a modification of the example of Ref. 11. Example 4.4: Let p ⬎ 2 be a prime number and f : Q p → Q p be defined by f共x兲 = 2. By Example 2.2, 兩2n兩 p = 1 for all n 苸 Z. Then for ⬎ 0, 兩Df共x,y兲兩 p = 兩0兩 p = 0 ⬍ However,
共x,y 苸 Q p兲.
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Stability of additive-cubicfunctional equation
兩2ng共2−nx兲 − 2n+1g共2−共n+1兲x兲兩 p = 兩2n+1兩 p兩7兩 p = 兩7兩 p for all x 苸 Q p and n 苸 N. Hence 兵2ng共2−nx兲其 is not a Cauchy sequence, where g共x兲 = f共2x兲 − 8f共x兲 共see the proof of Theorem 3.1兲. Theorem 4.5: Let K be a non-Archimedean field, X be a linear space over K, 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ 共x,y兲 for all x , y 苸 X, where : X ⫻ X → 关0 , ⬁兲. Suppose that
共2−1x,2−1y兲 ⱕ
1 共x,y兲 
for all x , y 苸 X, where  is a positive real number with 兩8兩 ⬍ . Then there exists a unique cubic mapping C : X → Y, such that 储f共2x兲 − 2f共x兲 − C共x兲储Y ⱕ
1 M共x兲 
for all x 苸 X, where M共x兲 is defined as in Theorem 4.1. Proof: Similar to the proof of Theorem 4.1. The result follows from Theorem 3.2. 䊏 Corollary 4.6: Let K be a non-Archimedean field, 共X , 储 · 储X兲 be a non-Archimedean normed space over K, and 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K. Let ␦ ⬎ 0, 0 ⱕ r ⬍ 3, 兩2兩 ⬍ 1, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ ␦共储x储Xr + 储y储Xr兲 for all x , y 苸 X. Then there exists a unique cubic mapping C : X → Y, such that 储f共2x兲 − 2f共x兲 − C共x兲储Y ⱕ
␦储x储Xr 兩k − k兩兩2兩2r 3
再
max 2,
1 兩k − 1兩
冎
for all x 苸 X. Proof: Let : X ⫻ X → 关0 , ⬁兲 be defined by 共x , y兲 = ␦共储x储Xr + 储y储Xr兲 for all x , y 苸 X. Then the 䊏 corollary is followed from Theorem 4.5 by replacing  = 兩2兩r. Corollary 4.7: Let K be a non-Archimedean field, 共X , 储 · 储X兲 be a non-Archimedean normed space over K, and 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K. Let ␦ ⬎ 0, 兩2兩 ⬍ 1, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ ␦关储x储Xr储y储Xs + 共储x储Xr+s + 储y储Xr+s兲兴
共x,y 苸 X兲
where r , s be non-negative real numbers, such that ª r + s ⬍ 3. Then there exists a unique cubic mapping C : X → Y, such that 储f共2x兲 − 2f共x兲 − C共x兲储Y ⱕ
␦储x储X 兩k − k兩兩2兩 3
再
2 max
3,
1 兩k − 1兩
冎
for all x 苸 X. Proof: Let : X ⫻ X → 关0 , ⬁兲 be defined by 共x , y兲 = ␦关储x储Xr储y储Xs + 共储x储Xr+s + 储y储Xr+s兲兴 for all x , y 苸 X. Since 兩2兩 ⬎ 兩8兩, we see that all conditions of Theorem 4.5 hold for  = 兩2兩. Then the corollary is followed from Theorem 4.5. 䊏 The following example shows that the assumption 兩2兩 ⬍ 1 cannot be omitted in Corollaries 4.6 and 4.7. Example 4.8: Let p ⬎ 2 be a prime number and f : Q p → Q p be defined by f共x兲 = 2. By Example 2.2, 兩2n兩 p = 1 for all n 苸 Z. Then for ⬎ 0,
093508-16
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Xu, Rassias, and Xu
兩Df共x,y兲兩 p = 兩0兩 p = 0 ⬍ 共x,y 苸 Q p兲. However, 兩8nh共2−nx兲 − 8n+1h共2−共n+1兲x兲兩 p = 兩23n+1兩 p兩7兩 p = 兩7兩 p for all x 苸 Q p and n 苸 N. Therefore, 兵8nh共2−nx兲其 is not a Cauchy sequence, where h共x兲 = f共2x兲 − 2f共x兲 共see the proof of Theorem 3.2兲. Theorem 4.9: Let K be a non-Archimedean field, X be a linear space over K, 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ 共x,y兲 for all x , y 苸 X, where : X ⫻ X → 关0 , ⬁兲. Suppose that
共2−1x,2−1y兲 ⱕ
1 共x,y兲 ␣
for all x , y 苸 X, where ␣ is a positive real number with 兩2兩 ⬍ ␣. Then there exist an additive mapping A : X → Y and a cubic mapping C : X → Y, such that 储f共x兲 − A共x兲 − C共x兲储Y ⱕ
1 M共x兲 兩6兩␣
for all x 苸 X, where M共x兲 is defined as in Theorem 4.1. Proof: Similar to the proof of Theorem 4.1. The result follows from Theorem 3.3. 䊏 The following results are due to Xu et al. 共Ref. 38, see also Ref. 39兲. Corollary 4.10: Let 共X , 储 · 储X兲 be a normed space, 共Y , 储 · 储Y 兲 be a Banach space, and let ␦ , r be non-negative real numbers, such that r 苸 共0 , 1兲 艛 共1 , 3兲 艛 共3 , ⬁兲. Let f : X → Y be a mapping with f共0兲 = 0, satisfying the inequality 储Df共x,y兲储Y ⱕ ␦共储x储Xr + 储y储Xr兲 for all x , y 苸 X. Then there exist a unique additive mapping A : X → Y and a unique cubic mapping C : X → Y, such that
储f共x兲 − A共x兲 − C共x兲储Y ⱕ
冦
768k2␦储x储Xr , 共k3 − k兲共2 − 2r兲
r 苸 共0,1兲
768 · 3r−1kr+1␦储x储Xr , r 苸 共3,⬁兲 共k3 − k兲共2r − 8兲
冉 冊 冉 冊
ln 3 768 · 3r−1kr+1␦储x储Xr , r 苸 1, 3 r ln 2 共k − k兲共2 − 2兲
ln 3 768 · 3r−1kr+1␦储x储Xr ,3 , , r苸 3 r ln 2 共k − k兲共8 − 2 兲
冧
for all x 苸 X. Corollary 4.11: Let 共X , 储 · 储X兲 be a normed space, 共Y , 储 · 储Y 兲 be a Banach space and let ␦, r, and s be non-negative real numbers, such that ª r + s 苸 共0 , 1兲 艛 共1 , 3兲 艛 共3 , ⬁兲. Let f : X → Y be a mapping with f共0兲 = 0, satisfying the inequality 储Df共x,y兲储Y ⱕ ␦关储x储Xr储y储Xs + 共储x储Xr+s + 储y储Xr+s兲兴 for all x , y 苸 X. Then there exist a unique additive mapping A : X → Y and a unique cubic mapping C : X → Y, such that
093508-17
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Stability of additive-cubicfunctional equation
储f共x兲 − A共x兲 − C共x兲储Y ⱕ
冦
768 · 3k2␦储x储X , 苸 共0,1兲; 2共k3 − k兲共2 − 2p兲 768 · 3k+1␦储x储X , 苸 共3,⬁兲 2共k3 − k兲共2 − 8兲
冉 冊 冉 冊
ln 3 768 · 3k+1␦储x储X , 苸 1, 3 ln 2 2共k − k兲共2 − 2兲
ln 3 768 · 3k+1␦储x储X 苸 ,3 , 3 , ln 2 2共k − k兲共8 − 2 兲
冧
for all x 苸 X. Note that A共x兲 = − 61 limn→⬁ 2nj关f共2−nj+1x兲 − 8f共2−njx兲兴 and C共x兲 = 61 limn→⬁ 8nj关f共2−nj+1x兲 − 2f共2−njx兲兴 for x 苸 X, where j 苸 兵1 , −1其 共Ref. 38, see also Ref. 39兲. The following example shows that Corollaries 4.10 and 4.11 are not true in non-Archimedean normed spaces. Example 4.12: Let p ⬎ 2 be a prime number and f : Q p → Q p be defined by f共x兲 = 2 for all x 苸 Q p. Since 兩2n兩 p = 1 for all n 苸 Z, then for ⬎ 0, 兩Df共x,y兲兩 p = 兩0兩 p = 0 ⬍ for all x , y 苸 Q p. However, 兵2nj关f共2−nj+1x兲 − 8f共2−njx兲兴其 and 兵8nj关f共2−nj+1x兲 − 2f共2−njx兲兴其 are not Cauchy sequences for j = 1 or ⫺1. In fact, by using the fact that 兩2n兩 p = 1共n 苸 Z兲, we have 兩2nj关f共2−nj+1x兲 − 8f共2−njx兲兴 − 2共n+1兲j关f共2−共n+1兲j+1x兲 − 8f共2−共n+1兲jx兲兴兩 p = 兩7兩 p and 兩8nj关f共2−nj+1x兲 − 2f共2−njx兲兴 − 8共n+1兲j关f共2−共n+1兲j+1x兲 − 2f共2−共n+1兲jx兲兴兩 p = 兩7兩 p for all x 苸 Q p and n 苸 N. Hence the sequences 兵2nj关f共2−nj+1x兲 − 8f共2−njx兲兴其 and 兵8nj关f共2−nj+1x兲 − 2f共2−njx兲兴其 are not converge in Q p. However, we have the following version of Corollaries 4.10 and 4.11 for non-Archimedean normed spaces. Corollary 4.13: Let K be a non-Archimedean field, 共X , 储 · 储X兲 be a non-Archimedean normed space over K, and 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K. Let ␦ ⬎ 0, 0 ⱕ r ⬍ 1, 兩2兩 ⬍ 1, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ ␦共储x储Xr + 储y储Xr兲 for all x , y 苸 X. Then there exist an additive mapping A : X → Y and a cubic mapping C : X → Y, such that 储f共x兲 − A共x兲 − C共x兲储Y ⱕ
␦储x储Xr 兩6兩兩k3 − k兩兩2兩2r
再
max 2,
1 兩k − 1兩
冎
for all x 苸 X. Proof: Similar to the proof of Corollary 4.2. The result follows from Theorem 4.9. 䊏 Corollary 4.14: Let K be a non-Archimedean field, 共X , 储 · 储X兲 be a non-Archimedean normed space over K, and 共Y , 储 · 储Y 兲 be a complete non-Archimedean normed space over K. Let ␦ ⬎ 0, 兩2兩 ⬍ 1, and f : X → Y with f共0兲 = 0, such that 储Df共x,y兲储Y ⱕ ␦关储x储Xr储y储Xs + 共储x储Xr+s + 储y储Xr+s兲兴
共x,y 苸 X兲
where r , s be non-negative real numbers, such that ª r + s ⬍ 1. Then there exist an additive mapping A : X → Y and a cubic mapping C : X → Y, such that
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储f共x兲 − A共x兲 − C共x兲储Y ⱕ
␦储x储X 兩6兩兩k − k兩兩2兩 3
再
2 max
3,
1 兩k − 1兩
冎
for all x 苸 X. Proof: Similar to the proof of Corollary 4.3. The result follows from Theorem 4.9. 䊏 Remark: Example 4.12 shows that the assumption 兩2兩 ⬍ 1 cannot be omitted in Corollaries 4.13 and 4.14. ACKNOWLEDGMENTS
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