Fixed points and approximately heptic mappings in non-Archimedean normed spaces
© Park et al.; licensee Springer 2013
Received: 3 December 2012
Accepted: 23 May 2013
Published: 12 July 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.
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.
and equality holds if and only if ,
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.
(NA1) if and only if ;
(NA2) for all and ;
Then is called a non-Archimedean normed space.
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.
for with . The authors proved that the Cauchy equation is equivalent to the above equation.
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.
for all .
for all .
for all .
for all . It follows from (2.15), (2.16) and (2.17) that T satisfies (1.8). That is, T is heptic.
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). □
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.
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.
Hence, for each , is not convergent.
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