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Fixed points and approximately octic mappings in non-Archimedean 2-normed spaces
Journal of Inequalities and Applications volume 2012, Article number: 289 (2012)
Using the fixed point method, we investigate the Hyers-Ulam stability of the system of additive-cubic-quartic functional equations with constant coefficients in non-Archimedean 2-normed spaces. Also, we give an example to show that some results in the stability of functional equations in (Archimedean) normed spaces are not valid in non-Archimedean normed spaces.
MSC:39B82, 46S10, 39B52, 47S10, 47H10.
Gähler [1, 2] has introduced the concept of linear 2-normed spaces. Then Gähler and White [3–5] introduced the concept of 2-Banach spaces. In 1999 to 2003, Lewandowska published a series of papers on 2-normed sets and generalized 2-normed spaces [6, 7]. Recently, Park  investigated approximate additive mappings, approximate Jensen mappings and approximate quadratic mappings in 2-Banach spaces. We recall and apply the notions and notes which are given in .
Definition 1.1 Let X be a linear space over ℝ with dim , and let be a function satisfying the following properties:
if and only if x, y are linearly dependent;
for any ;
for all and . Then the function is called a 2-norm on X and the pair is called a linear 2-normed space.
Lemma 1.2 Let be a linear 2-normed space. If and for all , then .
Remark 1.3 Let be a linear 2-normed space. One can show that the conditions (2) and (4) in Definition 1.1 imply that
for all . Hence, the functions are continuous functions of X into ℝ for each fixed .
Definition 1.4 A sequence in a linear 2-normed space X is called a Cauchy sequence if there are two linearly independent points such that
Definition 1.5 A sequence in a linear 2-normed space X is called a convergent sequence if there is an such that
for all . If converges to x, write as and call x the limit of . In this case, we also write .
Lemma 1.6 For a convergent sequence in a linear 2-normed space X,
for all .
Definition 1.7 A linear 2-normed space in which every Cauchy sequence is a convergent sequence is called a 2-Banach space.
Hensel  has introduced a normed space which does not have the Archimedean property. During the last three decades, the 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 . Although many results in the classical normed space theory have a non-Archimedean counterpart, their proofs are different and require a rather new kind of intuition [11–14]. One may note that in each valuation field, every triangle is isosceles and there may be no unit vector in a non-Archimedean normed space; cf. . These facts show that the non-Archimedean framework is of special interest.
Definition 1.8 Let be a field. A valuation mapping on is a function such that for any , we have
and the equality holds if and only if ,
A field endowed with a valuation mapping will be called a valued field. If the condition (iii) in the definition of a valuation mapping is replaced with
then the valuation is said to be non-Archimedean. The condition (iii)′ is called the strict triangle inequality. By (ii), we have . Thus, by induction, it follows from (iii)′ that for each integer n. We always assume in addition that is non-trivial, i.e., there is an such that . The most important examples of non-Archimedean spaces are p-adic numbers.
Example 1.9 Let p be a prime number. For any non-zero rational number such that m and n are coprime to the prime number p, define the p-adic absolute value . Then is a non-Archimedean norm on ℚ. The completion of ℚ with respect to is denoted by and is called the p-adic number field.
Definition 1.10 Let X be a linear space over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean norm (valuation) if it satisfies the following conditions:
(NA1) if and only if ;
(NA2) for all and ;
(NA3) the strong triangle inequality (ultrametric); namely, ().
Then is called a non-Archimedean normed space.
Now, we give the definition of a non-Archimedean 2-normed space which has been introduced in .
Definition 1.11 Let X be a linear space with dim over a scalar field with a non-Archimedean non-trivial valuation . A function is a non-Archimedean 2-norm (valuation) if it satisfies the following conditions:
(NA1) if and only if x, y are linearly dependent;
(NA3) for any ;
for all and . Then is called a non-Archimedean 2-normed space.
It follows from (NA4) that
and so a sequence is Cauchy in X if and only if converges to zero in a non-Archimedean 2-normed space.
The stability problems concerning group homomorphisms were raised by Ulam  in 1940 and affirmatively answered for Banach spaces by Hyers  in the next year. The Hyers theorem was generalized by Rassias  for linear mappings by considering an unbounded Cauchy difference. In 1994, a generalization of the Rassias theorem was obtained by Gǎvruta  by replacing the unbounded Cauchy difference by a general control function.
Let be a generalized metric space. An operator satisfies a Lipschitz condition with a Lipschitz constant L if there exists a constant such that for all . If the Lipschitz constant L is less than one, then the operator T is called a strictly contractive operator. Note that the distinction between the generalized metric and the usual metric is that the range of the former is permitted to include the infinity. We recall the following theorem by Margolis and Diaz.
Suppose that we are given a complete generalized metric space and a strictly contractive mapping with the Lipschitz constant L. Then, for each given , either
or there exists a natural number such that
the sequenceis convergent to a fixed pointof T;
is the unique fixed point of T in;
Khodaei and Rassias  investigated the solution and the Hyers-Ulam stability of an n-dimensional additive functional equation such that in the special case ,
for with . They proved that the Cauchy equation is equivalent to the above equation.
Jun and Kim  introduced the following cubic functional equation:
and they established the general solution and the Hyers-Ulam stability for the functional equation (1.2). They proved that a mapping f between two real vector spaces X and Y is a solution of (1.2) if and only if there exists a unique mapping such that for all . Moreover, C is symmetric for each fixed one variable and is additive for fixed two variables. The mapping C is given by for all . Obviously, the function satisfies the functional equation (1.2), which is called a cubic functional equation. Jun et al.  investigated the solution and the Hyers-Ulam stability for the cubic functional equation
where with .
Lee et al.  considered the following functional equation:
In fact, they proved that a mapping f between two real vector spaces X and Y is a solution of (1.4) if and only if there exists a unique symmetric bi-quadratic mapping such that for all x. The bi-quadratic mapping is given by
Obviously, the function satisfies the functional equation (1.4), which is called a quartic functional equation. Kang  investigated the Hyers-Ulam stability problem for the quartic functional equation
where with .
Recently, Ebadian, Najati and Eshaghi Gordji  considered the Hyers-Ulam stability of the systems of the additive-quartic functional equation
and the quadratic-cubic functional equation
In this paper, we investigate the Hyers-Ulam stability for the system of the additive-cubic-quartic functional equations
where with . The function given by is a solution of (1.6). In particular, letting , we get an octic function in one variable given by . The proof of the following proposition is evident, and we omit the details.
Proposition 1.13 Let X and Y be real linear spaces. If a mapping satisfies (1.6), then for all and all rational numbers .
We mention here the papers [30–35] concerning the Hyers-Ulam stability of the mixed type functional equations, the Hyers-Ulam stability in non-Archimedean Banach spaces and the Hyers-Ulam stability by fixed point methods.
In the rest of this paper, unless otherwise explicitly stated, we will assume that X is a non-Archimedean normed space and Y is a non-Archimedean 2-Banach space.
2 Approximation of octic mappings
In this section, we investigate the Hyers-Ulam stability problem for the system (1.6) in non-Archimedean 2-Banach spaces.
Theorem 2.1 Let be fixed. Let be functions such that
for all and for some ,
for all . If is a mapping such that for all and
for all and , then there exists a unique octic mapping satisfying (1.6) and
for all and .
Proof Putting and and replacing y, z by 2y, 2z in (2.3), we get
for all and . Putting and and replacing x, z by , 2z in (2.4), we get
for all and . Putting and and replacing x, y by , in (2.5), we get
for all and . Thus,
for all and . Replacing x, y and z by , and in (2.10), we have
for all and . It follows from (2.11) that
for all and . From the inequalities (2.12) and (2.13), it follows that
for all and .
Let S be the set of all mappings with for all . And let us introduce a generalized metric on S as follows:
where, as usual, . The proof of the fact that is a complete generalized metric space can be found in . Now, we consider the mapping defined by
for all and . Let such that . Then
that is, if , we have . This means that
for all ; that is, J is a strictly contractive self-mapping on S with the Lipschitz constant L. It follows from (2.14) that
for all and , which implies that . Due to Theorem 1.12, there exists a unique mapping such that T is a fixed point of J, i.e., for all . Also, as , which implies the equality
for all .
It follows from (2.1), (2.3), (2.4) and (2.5) that
for all and ,
for all and ,
for all and . It follows from (2.15), (2.16) and (2.17) that T satisfies (1.6), that is, T is octic.
According to the fixed point alternative, since T is the unique fixed point of J in the set , T is the unique mapping such that
for all and . Using the fixed point alternative, we obtain that
for all , which implies the inequality (2.6). □
Remark 2.2 Let X be a normed space and let Y be a 2-Banach space in Theorem 2.1. Using the fixed point method, one can show that there exists a unique octic mapping satisfying (1.6) and
for all and , and
for all . It is easy to see that the approximation in non-Archimedean 2-normed spaces (inequality (2.6)) is better than the approximation in (Archimedean) 2-normed spaces (inequality (2.18)).
By the direct method, the following corollary is valid in the (Archimedean) Banach spaces.
Corollary 2.3 Let be fixed and θ, p be nonnegative real numbers with , and let X, Y be a normed space and a Banach space, respectively. Suppose that a mapping satisfying and
for all . Then there exists a unique octic mapping satisfying
for all .
The following example shows that the previous corollary is not valid in non-Archi-medean Banach spaces.
Example 2.4 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 and , there exists an integer n such that .
Let p be a prime number. For any nonzero rational number x, there exists a unique integer such that , where a and b are integers not divisible by p. Then defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to is denoted by and is called the p-adic number field. Note that if , then for each integer n.
We consider the following special case of the system (1.6):
Let for a prime number and define by . Then for , and all with , , , we have
But for each natural number n, we have
Hence, for each , is not convergent.
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The authors declare that they have no competing interests.
All authors conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.
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Park, C., Eshaghi Gordji, M., Ghaemi, M.B. et al. Fixed points and approximately octic mappings in non-Archimedean 2-normed spaces. J Inequal Appl 2012, 289 (2012). https://doi.org/10.1186/1029-242X-2012-289
- octic functional equation
- Hyers-Ulam stability
- non-Archimedean 2-normed space
- fixed point method