The Jensen functional equation in non-Archimedean normed spaces

We investigate the Hyers–Ulam–Rassias stability of the Jensen functional equation in non-Archimedean normed spaces and study its asymptotic behavior in two directions: bounded and unbounded Jensen differences. In particular, we show that a mapping f between non-Archimedean spaces with f(0) = 0 is additive if and only if ‖f( + y 2 ) − f(x) + f(y) 2 ‖ → 0 as max {‖x‖, ‖y‖} → ∞ .


Introduction and preliminaries
The history of the stability theory of functional equations started with a problem concerning group homomorphisms posed by S.M. Ulam [30] in 1940 and its solution given by H.D. Hyers [7] in 1941. Hyers' theorem was generalized by T. Aoki [1] for additive mappings and by Th.M. Rassias [24] for linear mappings by considering an unbounded Cauchy difference. The paper [24] of Th.M. Rassias has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. During the last decades many stability problems for various functional equations have been studied by numerous mathematicians. We refer the reader to [4,8,13,25,26] and references therein. The first result on the stability of the classical Jensen equation f ( x+y 2 ) = f (x)+f (y) 2 was given by Z. Kominek [16]. The first author, who investigated the stability problem on a restricted domain was F. Skof [29]. The stability of the Jensen equation and its generalizations were studied by numerous researchers, cf. [5,12,17,22] By a non-Archimedean field we mean a field K equipped with a function (valuation) | · | from K into [0, ∞) such that |r| = 0 if and only if r = 0, |rs| = |r| |s|, and |r + s| ≤ max {|r|, |s|} for all r, s ∈ K . Clearly |1| = | − 1| = 1 and |n| ≤ 1 for all n ∈ N.
Let X be a vector space over a field K with a non-Archimedean nontrivial valuation | · |. A function · : X → [0, ∞) is called a non-Archimedean norm if it satisfies the following conditions: (iii) the strong triangle inequality (ultrametric); namely, Then (X, · ) is called a non-Archimedean normed space; cf. [28,10,18]. Due to the fact that a sequence {x n } is Cauchy if and only if {x n+1 − x n } converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space we mean one in which every Cauchy sequence is convergent. Theory of non-Archimedean normed spaces is not trivial, for instance there may not be any unit vector. Although many results in classical normed space theory have a non-Archimedean counterpart, but their proofs are essentially different and require an entirely new kind of intuition, cf. [20,21,23].
In 1897, Hensel [6] discovered the p-adic numbers as a number theoretical analogue of power series in complex analysis. Fix a prime number p. For any nonzero rational number x, there exists a unique integer n x ∈ Z such that x = a b p nx , where a and b are integers not divisible by p. Then |x| p := p −nx defines a non-Archimedean norm on Q . The completion of Q with respect to the metric d(x, y) = |x − y| p is denoted by Q p , which is called the p-adic number field; cf. [27,3]. During the last three decades p-adic numbers have gained the interest of physicists for their research in particular in problems coming from quantum physics, p-adic strings and superstrings; cf. [15].
In [2], the authors investigated stability of approximate additive mappings f : Q p → R. The stability of the Cauchy equation in normed spaces over fields with valuation was studied in [14]. In [20], the stability of Cauchy and quadratic functional equations were investigated in the context of non-Archimedean normed spaces. In this paper, using some ideas from [9,11,12,19] we establish the Hyers-Ulam-Rassias stability of the Jensen functional equation in the setting of non-Archimedean normed spaces and study its asymptotic behavior in two directions: bounded and unbounded Jensen differences.

Stability of the Jensen equation
In this section, we prove the stability of the Jensen functional equation. Throughout this section we assume that X is a non-Archimedean normed space and Y is a non-Archimedean Banach space over a non-Archimedean field K with |3| < 1.
Then there exists a unique Jensen mapping T : . Replace x and y by x and −x 3 in (2.1), respectively, to get Replace x and y by x 3 and −x 3 in (2.1), respectively, to obtain 3 n in (2.5) and multiple the obtained inequality with |3| n to get The right hand side tends to zero as n → ∞, so the sequence Hence (2.6) holds for all positive integer n. Letting n approach to infinity in (2.6) we get Replacing x and y by x 3 n and y 3 n , respectively, in (2.1) we get Taking the limit as n → ∞ we obtain If T is another Jensen mapping satisfying (2.2), then Therefore T = T . This completes the proof of the uniqueness of T .
Then there exists a unique additive mapping T : X → Y such that Proof. It follows from Theorem 2.1 that there is a unique Jensen mapping T satisfying (2.2). Since f (0) = 0 , we have T (0) = 0 . Hence T is clearly additive and satisfies (2.7).

Asymptotic aspect of a bounded Jensen difference
In this section, we deal with the asymptotic behavior of the Jensen functional equation. Throughout this section we assume that X is a non-Archimedean normed space over K with { x : x ∈ X } = {|r| : r ∈ K} and Y is a non-Archimedean Banach space over a non-Archimedean field K with |3| < 1 . Utilizing the strategy of Theorem 3 of [12] we get the following result.
for all x, y ∈ X with max { x , y } ≥ |β|. Then there exists a unique additive mapping T : X → Y such that Proof. Assume that max { x , y } < |β|. For x = y = 0 take z ∈ X to be an element of X with z = |β|. Without loss of generality, assume that y ≤ x < |β|. Let γ ∈ K with |γ| = x . Set z := x + β 3 n γ x for large enough n such that x = 0 or y = 0, It follows from (3.1) and (3.2), we get for all x, y ∈ X . Now the result is deduced from Corollary 2.2.
Proof. If f is additive, then (3.3) evidently holds. Conversely, use the limit (3.3) to get for each n ∈ N a real number β n > |β n | (replace β n by a real number of the form k(n)β n where k(n) is an integer, if necessary, to get k(n)β n > |β n | ≥ |k(n)β n |, since |k(n)| ≤ 1) such that Next use Theorem 3.1 to conclude a unique additive mapping T n such that for all x ∈ X . Thus f (x) − T 1 (x) ≤ 1 and f (x) − T n (x) ≤ 1/n ≤ 1 for each n. By the uniqueness of T 1 we conclude that T n = T 1 for all n. Tending n to infinity in (3.4) we deduce that f = T 1 is additive.

Asymptotic aspect of an unbounded Jensen difference
In this section, we deal with the asymptotic behavior of an unbounded Jensen difference. Throughout this section we assume that Let X is a non-Archimedean normed space and Y is a non-Archimedean Banach space over a non-Archimedean field K with |2| < 1.
for all x, y ∈ X with max { x , y } ≥ M . Then there exists an additive mapping T : X → Y such that for all x ∈ X with x ≥ M . Furthermore, T is independent of given positive numbers α and M .
Using (4.3) and the fact that x/2 n ≥ M for large enough n, we can follow the same argument as in the proof of Theorem 2.1 to get a mapping τ : Given any x ∈ X with 0 < x < M , let k = k(x) denotes the least positive integer such that x/2 k || ≥ M and define x ≥ M Now we show that To see this, take any x ∈ X with 0 < x < M . Let k be the least positive integer satisfying x/2 k−1 ≥ M . Then k − 1 is the least positive integer satisfying By (4.4) and the definition of T , we therefore conclude that for all x ∈ X . Let x ∈ X\ {0}. There is a positive integer k 0 such that 2 −k0 x ≥ M . We have Trivially T (0) = 0 = lim n→∞ 2 n f ( 0 2 n ). Hence (4.5) holds for all x ∈ X . If x = 0 or y = 0 , by taking (4.6) into account, we get T ( x+y 2 ) = 1 2 (T (x) + T (y)). So we may assume that x = 0 and y = 0. Choose n ∈ N to be large enough such that Utilizing (4.1) we obtain Letting n approach to infinity we get Hence T is additive. Suppose that T is another additive mapping satisfying (4.2) with α and M replaced by α and M , respectively. For x ∈ X choose n ∈ N large enough so that 2 −n x ≥ max {M, M } . Then Hence T (x) = T (x).
Let f : X → Y be a mapping. Following [9] (i) f is called p-asymptotic close to an additive mapping T if lim x →∞ f (x)−T (x) x p = 0.
(ii) f is said to satisfy p-asymptotically the Jensen equation if for each α > 0 there exists M > 0 such that for all x, y ∈ X with max { x , y } ≥ M .
Applying Theorem 4.1 and the uniqueness of obtained additive mapping we infer the following corollary.