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Fixed points and approximately heptic mappings in non-Archimedean normed spaces
Advances in Difference Equations volume 2013, Article number: 209 (2013)
Using the fixed point method, we investigate the stability of the system of additive, quadratic and quartic functional equations with constant coefficients in non-Archimedean 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, 47H10.
The stability problems concerning group homomorphisms was raised by Ulam  in 1940 and affirmatively answered for Banach spaces by Hyers  in the next year. Hyers’ theorem was generalized by Aoki  for additive mappings and by Rassias  for linear mappings by considering an unbounded Cauchy difference. In 1994, a generalization of the Rassias theorem was obtained by Găvruta  by replacing the unbounded Cauchy difference by a general control function.
Let be a generalized metric space. An operator satisfies a Lipschitz condition with the Lipschitz constant L if there exists a constant such that for all . If the Lipschitz constant L is less than 1, 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;
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 [12–15].
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 1.2 Let be a field. A valuation mapping on is a function such that for any we have
and 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 nontrivial, i.e., that there is an such that . The most important examples of non-Archimedean spaces are p-adic numbers.
Definition 1.3 Let X be a linear space over a scalar field with a non-Archimedean nontrivial 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.
It follows from (NA3) that
and so a sequence is Cauchy in X if and only if converges to zero in a non-Archimedean space. It is easy to see that every convergent sequence in a non-Archimedean normed space is Cauchy. If each Cauchy sequence is convergent, then the non-Archimedean normed space is said to be complete and is called a non-Archimedean Banach space.
Khodaei and Rassias  investigated the solution and stability of the n-dimensional additive functional equation such that in the special case ,
for with . The authors proved that the Cauchy equation is equivalent to the above equation.
The functional equation
is related to a symmetric bi-additive function [17, 18]. It is natural that this equation is called a quadratic functional equation. In particular, every solution of quadratic equation (1.2) is said to be a quadratic function. It is well known that a function f between real vector spaces is quadratic if and only if there exists a unique symmetric bi-additive function such that for all x. The bi-additive function is given by . The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof . In , Czerwik proved the Hyers-Ulam stability of equation (1.2). Eshaghi Gordji and Khodaei  obtained the general solution and the Hyers-Ulam stability of the following quadratic functional equation for 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
Ebadian et al.  considered the Hyers-Ulam stability of the systems of additive-quartic functional equations
and the quadratic-cubic functional equations
In this paper, we investigate the Hyers-Ulam stability for the system of additive, quadratic and quartic functional equations
where with . The function given by is a solution of (1.8). In particular, letting , we get a heptic function in one variable given by . The proof of the following proposition is evident, and we omit the details.
Proposition 1.4 Let X and Y be real linear spaces. If a mapping satisfies system (1.8), then for all , and all rational numbers λ, μ, η.
In the rest of this paper, unless otherwise explicitly stated, we assume that X is a non-Archimedean normed space and Y is a non-Archimedean Banach space.
2 Approximation of heptic mappings
In this section, we investigate the Hyers-Ulam stability problem for the system of functional equations (1.8) in non-Archimedean 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 , then there exists a unique heptic mapping satisfying (1.8) and
for all .
Proof Putting and and replacing y, z by 2y, 2z in (2.3), we get
for all . Putting and and replacing x, z by , 2z in (2.4), we get
for all . Putting and and replacing x, y by , in (2.5), we get
for all . Thus
for all . Replacing x, y and z by , and in (2.10), we have
for all . It follows from (2.11) that
for all . From the inequalities (2.12) and (2.13), we obtain
for all .
Let S be the set of all mappings with for all , and let us introduce a generalized metric on S as follows:
for all and . Let and be such that . Then
that is, if , we have . This means that
for all . So, J is a strictly contractive self-mapping on S with the Lipschitz constant L. It follows from (2.14) that . Due to Theorem 1.1, 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 (2.2), (2.3), (2.4) and (2.5) that
for all . It follows from (2.15), (2.16) and (2.17) that T satisfies (1.8). That is, T is heptic.
By Theorem 1.1, we obtain that
which implies the inequality (2.6). Since T is the unique fixed point of J in the set , T is the unique mapping satisfying (2.6). □
Remark 2.2 Let X be a normed space and let Y be a Banach space in Theorem 2.1. Using the direct method, one can show that there exists a unique heptic mapping satisfying (1.8) and
for all , where we assume that
It is easy to show that the approximation in non-Archimedean spaces (inequality (2.6)) is better than the approximation in (Archimedean) normed spaces (inequality (2.18)).
The following corollary in the (Archimedean) normed spaces by the direct method is valid.
Corollary 2.3 Let be fixed and be real numbers such that , and let X, Y be a normed space and a Banach space, respectively. Suppose that a mapping satisfies and
for all . Then there exists a unique heptic mapping satisfying (1.8) and a constant such that
for all .
The following example shows that the previous corollary is not valid in non-Archimedean 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 (we put ). The completion of ℚ with respect to is denoted by and is called a p-adic number field. Note that if , then for each integer n.
Let for a prime number and define by . Then 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.