Stabilities and non-stabilities of the reciprocal-nonic and the reciprocal-decic functional equations

This paper focuses at the various stability results of reciprocal-nonic and reciprocal-decic functional equations in non-Archimedes fields and illustrations of the proper examples for their non-stabilities.


Introduction
The source for the development of stability of functional equations is the question solicited by Ulam [29]. Hyers [11] presented an excellent answer to the question of Ulam. Later, the result of Hyers was generalized and refined further by many great mathematicians like Aoki [1], Th.M. Rassias [14], J.M. Rassias [13] and Gȃvruta [9] in various directions. The progress of the theory of stability of various types of functional equations such as quadratic, cubic, quartic, quintic, sextic, septic, octic, nonic, decic, undecic, duodecic, tredecic, quattordecic have been dealt by many mathematicians and there are lot of interesting and significant results available in the literature.
For the first time, Ravi and the second author [23] achieved various stability results of the following functional equation where φ : R * −→ R is a mapping and R * = R\{0}. The rational function φ(x) = c x is a solution of the functional equation (1.1). The functional equation (1.1) is interconnected with "Reciprocal formula" which will be useful in any electric circuit with couple of parallel resistors [24]. Hence, the equation (1.1) is said to be reciprocal functional equation. The geometrical interpretation of the equation (1.1) is also discussed in [24]. Suitable counter-examples are presented to show the non-stability of the equation (1.1) controlled by the sum of powers of norms and product of powers of norms for singular cases in [18] and [19] respectively.
In this study, we consider the following reciprocal-nonic functional equation The reciprocal-nonic function n(x) = 1 x 9 and the reciprocal-decic function d(x) =

Preliminaries
In this section, we recall the basic facts of non-Archimdean fields.
Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N. We always assume, in addition, that | · | is non-trivial, i.e., there exists an µ 0 ∈ K such that |µ 0 | = 0, 1. A sequence {u n } is Cauchy if and only if {u n+1 − u n } converges to zero in a non-Archimedean field because By a complete non-Archimedean field, we mean that every Cauchy sequence is convergent in the field. In [10], Hensel discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. The most interesting example of non-Archimedean normed spaces is 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. Let p be a prime number. For any non-zero rational number x = p r m n in which m and n are coprime to the prime number p. Consider the p-adic absolute value |x| p = p −r on Q. It is easy to check that | · | is a non-Archimedean norm on Q. The completion of Q with respect to | · | which is denoted by Q p is said to be the p-adic number field. Note that if p > 2, then |2 n | = 1 for all integers n.
Let us presume that throughout this paper, A and B are a non-Archimedean field and a complete non-Archimedean field, respectively. In the sequel, we denote A * = A\{0}, where A is a non-Archimedean field. For the equations (1.4) and (1.5), we define the difference operators ∆ 1 n, for all x, y ∈ A * .

Stability results
In this section, we investigate the various Ulam stabilities of equations (1.4) and (1.5) in non-Archimedean fields.
for all x, y ∈ A * . Suppose that n : A * −→ B is a mapping satisfying the inequality for all x, y ∈ A * . Then, there occurs a distinct reciprocal-nonic mapping N : for all x ∈ A * . Now, by interchanging x into x 3 pk in (3.5) and multiplying the resultant by 1 19683 pk , we arrive at for all x ∈ A * . Using the inequalities (3.1) and (3.5), we see that the sequence Also, for every x ∈ A * and non-negative integers k, we find Letting k → ∞ in the inequality (3.7) and using (3.6), we attain that the inequality (3.3) holds. Once more, by applying the inequalities (3.1), (3.2) and (3.6), for every x, y ∈ A * , we arrive at Hence, the mapping N satisfies (1.4) and so it is reciprocal-nonic mapping. Next, we confirm that N is unique. Let us consider N ′ : A * −→ B be another reciprocalnonic mapping satisfying (3.3). Then for all x ∈ A * . This implies that N is distinct, which concludes the proof.
Corollary 3.5. Let n : A * −→ B be a mapping and let there exist real numbers r, s, q = r + s = −9 and ǫ > 0 such that |∆ 1 n(x, y)| ≤ ǫ |x| r |y| s for all x, y ∈ A * . Then, there exists a unique reciprocal-nonic mapping N : A * −→ B satisfying (1.4) and Proof. Letting ζ(x, y) = ǫ |x| r |y| s , for all x, y ∈ A * in Theorem 3.2, we attain the necessary result.  for all x, y ∈ A * . Suppose that d : A * −→ B is a mapping satisfying the inequality for all x, y ∈ A * . Then, there exists a unique reciprocal-decic mapping D : A * −→ B satisfying (1.5) and |d(x) − D(x)| ≤ max 1 59049 Proof. Letting (x, y) as (x, x) in (3.9), we obtain for all x ∈ A * . Changing x into x 3 pk in (3.11) and multiplying by 1 59049 pk , one finds 1 59049 pk d for all x ∈ A * . From the inequalities (3.8) and (3.12), we conclude that is a Cauchy sequence. By the completeness of B, there exists a mapping D : for all x ∈ A * . The remaining part of the proof is alike Theorem 3.2.
Using Theorem 3.7, we obtain the stability results of equation (1.5) related with the upper bound controlled by a fixed positive constant, sum of powers of norms, product of powers of norms and mixed product-sum of powers of norms via the following corollaries.  Proof. Allowing ξ(x, y) = θ (|x| α + |y| α ), for all x, y ∈ A * in Theorem 3.7, we attain the desired result.
for all x, y ∈ A * . Then, there exists a unique reciprocal-decic mapping D : A * −→ B satisfying (1.5) and Proof. Choosing ξ(x, y) = θ |x| a |y| b , for all x, y ∈ A * in Theorem 3.7, the requisite result is achieved.
Corollary 3.11. Let θ > 0 and α > −10 be real numbers, and d : A * −→ B be a mapping satisfying the functional inequality for all x, y ∈ A * . Then, there exists a unique reciprocal-decic mapping D : A * −→ B satisfying (1.5) and Proof. It is simple to obtain the required result by selecting ξ(x, y) = θ |x| where φ : R * −→ R. Let g : R * −→ R be defined by for all x ∈ R. Let the function g : R * −→ R defined in (4.2) satisfies the functional inequality for all x, y ∈ R * . We show that there do not exist a reciprocal-nonic mapping N : R * −→ R and a constant α > 0 such that for all x ∈ R * . For this, let us first prove that g satisfies (4.3). By computation, we have 19683 19682 k.
In analogous to Example 4.1, we have the following result which acts as a counter-example for the fact that the functional equation (1.5) is not stable for α = −10 in Corollary 3.9.