Nearly Quadratic Mappings over 𝑝-Adic Fields

Abstract

We establish some stability results over 𝑝-adic fields for the generalized quadratic functional equation ∑^𝑛_𝑘=2∑^𝑘_𝑖1=2∑_𝑖^𝑘+1_2=𝑖1+1⋯∑^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓(∑^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑥_𝑖−∑^{𝑛−𝑘+1}_𝑟=1𝑥_𝑖𝑟)+𝑓(∑^𝑛_𝑖=1𝑥_𝑖)=2^𝑛−1∑^𝑛_𝑖=1𝑓(𝑥_𝑖), where 𝑛∈ℕ and 𝑛≥2.

1. Introduction and Preliminaries

In 1899, Hensel [1] discovered the 𝑝-adic numbers as a number of 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 such that 𝑥=(𝑎/𝑏)𝑝^𝑛𝑥, where 𝑎 and 𝑏 are integers not divisible by 𝑝. Then, 𝑝-adic absolute value |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, and it is called the 𝑝-adic number field. In fact, ℚ_𝑝 is the set of all formal series 𝑥=∑^∞_{𝑘≥𝑛𝑥}𝑎_𝑘𝑝^𝑘, where |𝑎_𝑘|≤𝑝−1 are integers (see, e.g., [2, 3]). Note that if 𝑝>2, then |2^𝑛|_𝑝=1 for each integer 𝑛.

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 [4, 5]. A key property of 𝑝-adic numbers is that they do not satisfy the Archimedean axiom: For 𝑥,𝑦>0, there exists 𝑛∈ℕ such that 𝑥<𝑛𝑦.

Let 𝕂 denote a field and function (valuation absolute) |⋅| from 𝕂 into [0,∞). A non-Archimedean valuation is a function |⋅| that satisfies the strong triangle inequality; namely, |𝑥+𝑦|≤max{|𝑥|,|𝑦|}≤|𝑥|+|𝑦| for all 𝑥,𝑦∈𝕂. The associated field 𝕂 is referred to as a non-Archimedean field. Clearly, |1|=|−1|=1 and |𝑛|≤1 for all 𝑛≥1. A trivial example of a non-Archimedean valuation is the function |⋅| taking everything except 0 into 1 and |0|=0. We always assume in addition that |⋅| is nontrivial, that is, there is a 𝑧∈𝕂 such that |𝑧|≠0,1.

Let 𝑋 be a linear space over a field 𝕂 with a non-Archimedean nontrivial valuation |⋅|. A function ‖⋅‖∶𝑋→[0,∞) is said to be a non-Archimedean norm if it is a norm over 𝕂 with the strong triangle inequality (ultrametric); namely, ‖𝑥+𝑦‖≤max{‖𝑥‖,‖𝑦‖} for all 𝑥,𝑦∈𝑋. Then, (𝑋,‖⋅‖) is called a non-Archimedean space. In any such a space, a sequence {𝑥_𝑛}_𝑛∈ℕ is Cauchy if and only if {𝑥_𝑛+1−𝑥_𝑛}_𝑛∈ℕ converges to zero. By a complete non-Archimedean space, we mean one in which every Cauchy sequence is convergent.

The study of stability problems for functional equations is related to a question of Ulam [6] concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers [7]. Subsequently, the result of Hyers was generalized by Aoki [8] for additive mappings and by Rassias [9] for linear mappings by considering an unbounded Cauchy difference. The paper by Rassias has provided a lot of influences in the development of what we now call the generalized Hyers-Ulam stability or Hyers-Ulam-Rassias stability of functional equations. Rassias [10] considered the Cauchy difference controlled by a product of different powers of norm. The above results have been generalized by Forti [11] and Găvruţa [12] who permitted the Cauchy difference to become arbitrary unbounded (see also [13–22]). Arriola and Beyer [23] investigated stability of approximate additive functions 𝑓∶ℚ_𝑝→ℝ. They showed that if 𝑓∶ℚ_𝑝→ℝ is a continuous function for which there exists a fixed 𝜀 such that |𝑓(𝑥+𝑦)−𝑓(𝑥)−𝑓(𝑦)|≤𝜀 for all 𝑥,𝑦∈𝑄_𝑝, then there exists a unique additive function 𝑇∶ℚ_𝑝→ℝ such that |𝑓(𝑥)−𝑇(𝑥)|≤𝜀 for all 𝑥∈ℚ_𝑝. For more details about the results concerning such problems, the reader is referred to [24–45].

Recently, Khodaei and Rassias [46] introduced the generalized additive functional equation ^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑎_𝑖𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑎_𝑖𝑟𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑎_𝑖𝑥_𝑖=2^𝑛−1𝑎₁𝑓𝑥₁ (1.1) and proved the generalized Hyers-Ulam stability of the above functional equation. The functional equation𝑓𝑥₁+𝑥₂+𝑓𝑥₁−𝑥₂=2𝑓𝑥₁+2𝑓𝑥₂ (1.2) is related to symmetric biadditive function and is called a quadratic functional equation [47, 48]. Every solution of the quadratic equation (1.2) is said to be a quadratic function.

Now, we introduce the generalized quadratic functional equation in 𝑛-variables as follows:^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑥_𝑖=2^𝑛𝑛−1_𝑖=1𝑓𝑥_𝑖, (1.3) where 𝑛≥2. Moreover, we investigate the generalized Hyers-Ulam stability of functional equation (1.3) over the 𝑝-adic field ℚ_𝑝.

As a special case, if 𝑛=2 in (1.3), then we have the functional equation (1.2). Also, if 𝑛=3 in (1.3), we obtain ²_𝑖1³_{=2𝑖2=𝑖1+1}𝑓³_{𝑖=1,𝑖≠𝑖1,𝑖2}𝑥_𝑖−²_𝑟=1𝑥_𝑖𝑟+³_𝑖1=2𝑓³_{𝑖=1,𝑖≠𝑖1}𝑥_𝑖−𝑥_𝑖1+𝑓³_𝑖=1𝑥_𝑖=2²³_𝑖=1𝑓𝑥_𝑖, (1.4) that is, 𝑓𝑥₁−𝑥₂−𝑥₃+𝑓𝑥₁−𝑥₂+𝑥₃+𝑓𝑥₁+𝑥₂−𝑥₃+𝑓𝑥₁+𝑥₂+𝑥₃=4𝑓𝑥₁+4𝑓𝑥₂+4𝑓𝑥₃. (1.5)

2. Stability of Quadratic Functional Equation (1.3) over 𝑝-Adic Fields

We will use the following lemma.

Lemma 2.1. Let 𝑋 and 𝑌 be real vector spaces. A function 𝑓∶𝑋→𝑌 satisfies the functional equation (1.3) if and only if the function 𝑓 is quadratic.

Proof. Let 𝑓 satisfy the functional equation (1.3). Setting 𝑥_𝑖=0 (𝑖=1,…,𝑛) in (1.3), we have ^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓(0)+𝑓(0)=2^𝑛𝑛−1_𝑖=1𝑓(0), (2.1) that is, ²_𝑖1³_{=2𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−1=𝑖𝑛−2+1}𝑓(0)+³_𝑖1⁴_{=2𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−2=𝑖𝑛−3+1}𝑓(0)+⋯+^𝑛_𝑖1=2𝑓(0)+𝑓(0)=2^𝑛𝑛−1_𝑖=1𝑓(0), (2.2) or +1𝑓(0)𝑛−1𝑛−1𝑛−1𝑛−2+⋯+𝑛−1+1=2^𝑛𝑛−1_𝑖=1𝑓(0), (2.3) but 1+∑^𝑛−𝚥_𝑗=1𝑗^𝑛−𝚥=∑^𝑛−𝚥_𝑗=0𝑗^𝑛−𝚥=2^𝑛−𝑗, and also 𝑛>𝚥≥1 so 2^𝑛−1(𝑛−1)𝑓(0)=0.
Putting 𝑥_𝑖=0 (𝑖=2,…,𝑛−1) in (1.3) and then using 𝑓(0)=0, we get 𝑓𝑥₁−𝑥_𝑛+1𝑓𝑥𝑛−2₁−𝑥_𝑛+𝑓𝑥𝑛−2𝑛−2¹+𝑥_𝑛+𝑓𝑥+⋯𝑛−2𝑛−3¹−𝑥_𝑛+2𝑓𝑥𝑛−2₁+𝑥_𝑛+𝑓𝑥𝑛−2𝑛−2¹−𝑥_𝑛+1𝑓𝑥𝑛−2₁+𝑥_𝑛+𝑓𝑥₁+𝑥_𝑛=2^𝑛−1𝑓𝑥₁+2^𝑛−1𝑓𝑥_𝑛, (2.4) that is, 1+^𝑛−2_𝑗=1𝑗𝑓𝑥𝑛−2₁+𝑥_𝑛+𝑓𝑥₁−𝑥_𝑛=2^𝑛−1𝑓𝑥₁+2^𝑛−1𝑓𝑥_𝑛, (2.5) for all 𝑥₁,𝑥_𝑛∈𝑋, this shows that 𝑓 satisfies the functional equation (1.2). So the function 𝑓 is quadratic.
Conversely, suppose that 𝑓 is quadratic, thus 𝑓 satisfies the functional equation (1.2). Hence, we have 𝑓(0)=0 and 𝑓 is even.
We are going to prove our assumption by induction on 𝑛≥2. It holds on 𝑛=2. Assume that it holds on the case where 𝑛=𝑡; that is, we have ^𝑡_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑡_{𝑖𝑡−𝑘+1=𝑖𝑡−𝑘+1}𝑓^𝑡_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑡−𝑘+1}𝑥_𝑖−^{𝑡−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑡_𝑖=1𝑥_𝑖=2^𝑡𝑡−1_𝑖=1𝑓𝑥_𝑖 (2.6) for all 𝑥₁,…,𝑥_𝑡∈𝑋. It follows from (1.2) that 𝑓^𝑡_𝑖=1𝑥_𝑖+𝑥_𝑡+1+𝑓^𝑡_𝑖=1𝑥_𝑖−𝑥_𝑡+1=2𝑓^𝑡_𝑖=1𝑥_𝑖+2𝑓𝑥_𝑡+1 (2.7) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋. Replacing 𝑥_𝑡 by −𝑥_𝑡 in (2.7), we obtain 𝑓^𝑡−1_𝑖=1𝑥_𝑖−𝑥_𝑡+𝑥_𝑡+1+𝑓^𝑡−1_𝑖=1𝑥_𝑖−𝑥_𝑡−𝑥_𝑡+1=2𝑓^𝑡−1_𝑖=1𝑥_𝑖−𝑥_𝑡+2𝑓𝑥_𝑡+1 (2.8) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋. Adding (2.7) to (2.8), we have 𝑓^𝑡−1_𝑖=1𝑥_𝑖−𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−1_𝑖=1𝑥_𝑖−𝑥_𝑡+𝑥_𝑡+1+𝑓^𝑡−1_𝑖=1𝑥_𝑖+𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−1_𝑖=1𝑥_𝑖+𝑥_𝑡+𝑥_𝑡+1=2𝑓^𝑡−1_𝑖=1𝑥_𝑖−𝑥_𝑡+𝑓^𝑡−1_𝑖=1𝑥_𝑖+𝑥_𝑡+4𝑓𝑥_𝑡+1 (2.9) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋. Replacing 𝑥_𝑡−1 by −𝑥_𝑡−1 in (2.9), we get 𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1−𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1−𝑥_𝑡+𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1+𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1+𝑥_𝑡+𝑥_𝑡+1=2𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1−𝑥_𝑡+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1+𝑥_𝑡+4𝑓𝑥_𝑡+1 (2.10) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋. Adding (2.9) to (2.10), one gets 𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1−𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1−𝑥_𝑡+𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1+𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖+𝑥_𝑡−1−𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1+𝑥_𝑡+𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖+𝑥_𝑡−1−𝑥_𝑡+𝑥_𝑡+1+𝑓^𝑡−2_𝑖=1𝑥_𝑖+𝑥_𝑡−1+𝑥_𝑡−𝑥_𝑡+1+𝑓^𝑡+1_𝑖=1𝑥_𝑖=2𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1−𝑥_𝑡+𝑓^𝑡−2_𝑖=1𝑥_𝑖−𝑥_𝑡−1+𝑥_𝑡+𝑓^𝑡−2_𝑖=1𝑥_𝑖+𝑥_𝑡−1−𝑥_𝑡+𝑓^𝑡−2_𝑖=1𝑥_𝑖+𝑥_𝑡−1+𝑥_𝑡+8𝑓𝑥_𝑡+1 (2.11) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋. By using the above method, for 𝑥_𝑡−2 until 𝑥₂, we infer that ^𝑡+1_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑡+1_{𝑖𝑡−𝑘+2=𝑖𝑡−𝑘+1+1}𝑓^𝑡+1_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑡−𝑘+2}𝑥_𝑖−^{𝑡−𝑘+2}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑡+1_𝑖=1𝑥_𝑖=2^𝑡_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑡_{𝑖𝑡−𝑘+1=𝑖𝑡−𝑘+1}𝑓^𝑡_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑡−𝑘+1}𝑥_𝑖−^{𝑡−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑡_𝑖=1𝑥_𝑖+2^𝑡𝑓𝑥_𝑡+1 (2.12) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋. Now, by the case 𝑛=𝑡, we lead to ^𝑡+1_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑡+1_{𝑖𝑡−𝑘+2=𝑖𝑡−𝑘+1+1}𝑓^𝑡+1_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑡−𝑘+2}𝑥_𝑖−^{𝑡−𝑘+2}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑡+1_𝑖=1𝑥_𝑖=22^𝑡𝑡−1_𝑖=1𝑓𝑥_𝑖+2^𝑡𝑓𝑥_𝑡+1 (2.13) for all 𝑥₁,…,𝑥_𝑡+1∈𝑋, so (1.3) holds for 𝑛=𝑡+1. This completes the proof of the lemma.

Corollary 2.2. A function 𝑓∶𝑋→𝑌 satisfies the functional equation (1.3) if and only if there exists a symmetric biadditive function 𝐵₁∶𝑋×𝑋→𝑌 such that 𝑓(𝑥)=𝐵₁(𝑥,𝑥) for all 𝑥∈𝑋.

Now, we investigate the stability of the functional equation (1.3) from a Banach space 𝐵 into 𝑝-adic field ℚ_𝑝. For convenience, we define the difference operator 𝐷_𝑓 for a given function 𝑓: 𝐷_𝑓𝑥₁,…,𝑥_𝑛∶=^𝑛_𝑘=2^𝑘_𝑖1=2^𝑘+1_{𝑖2=𝑖1+1}⋯^𝑛_{𝑖𝑛−𝑘+1=𝑖𝑛−𝑘+1}𝑓^𝑛_{𝑖=1,𝑖≠𝑖1,…,𝑖𝑛−𝑘+1}𝑥_𝑖−^{𝑛−𝑘+1}_𝑟=1𝑥_𝑖𝑟+𝑓^𝑛_𝑖=1𝑥_𝑖−2^𝑛𝑛−1_𝑖=1𝑓𝑥_𝑖. (2.14)

Theorem 2.3. Let 𝐵 be a Banach space and let 𝜀>0,𝜆 be real numbers. Suppose that a function 𝑓∶ℚ_𝑝→𝐵 with 𝑓(0)=0 satisfies the inequality 𝐷_𝑓𝑥₁,…,𝑥_𝑛≤𝜀^𝑛_𝑖=1𝑥_𝑖^𝜆_𝑝 (2.15) for all 𝑥₁,…,𝑥_𝑛∈ℚ_𝑝. Then there exists a unique quadratic function 𝑄∶ℚ_𝑝→𝐵 such that 𝑓(𝑥)(𝑥≤𝜀−𝑄)2^𝑛−1−2^{𝑛−𝜆−3}|𝑥|^𝜆_𝑝𝜀,𝑝=2,𝜆>−2;3.2^𝑛−3|𝑥|^𝜆_𝑝,𝑝>2; (2.16) for all nonzero 𝑥∈ℚ_𝑝.

Proof. Letting 𝑥₁=𝑥₂=𝑥≠0 and 𝑥_𝑖=0 (𝑖=3,…,𝑛) in (2.15), we obtain 𝑓(𝑥)−14𝑓(≤𝜀2𝑥)2^𝑛−1|𝑥|^𝜆_𝑝 (2.17) for all 𝑥∈ℚ_𝑝. Hence, 12^2𝑙𝑓2^𝑙𝑥−12^2𝑚𝑓2^𝑚𝑥≤𝜀2^{𝑛−1𝑚−1}_𝑗=𝑙|2|_𝑝^𝜆𝑗2^2𝑗|𝑥|^𝜆_𝑝 (2.18) for all nonnegative integers 𝑚 and 𝑙 with 𝑚>𝑙 and for all 𝑥∈ℚ_𝑝. It follows from (2.18) that the sequence {(1/2^2𝑚)𝑓(2^𝑚𝑥)} is a Cauchy sequence for all 𝑥∈ℚ_𝑝. Since 𝐵 is complete, the sequence {(1/2^2𝑚)𝑓(2^𝑚𝑥)} converges. Therefore, one can define the function 𝑄∶ℚ_𝑝→𝐵 by 𝑄(𝑥)∶=lim_𝑚→∞12^2𝑚𝑓2^𝑚𝑥 (2.19) for all 𝑥∈ℚ_𝑝. It follows from (2.15) and (2.19) that 𝐷_𝑄𝑥₁,…,𝑥_𝑛=lim_𝑚→∞12^2𝑚𝐷_𝑓2^𝑚𝑥₁,…,2^𝑚𝑥_𝑛≤lim_𝑚→∞|2|_𝑝^𝜆𝑚2^𝑛2𝑚_𝑖=1𝜀𝑥_𝑖^𝜆_𝑝=0 (2.20) for all 𝑥₁,…,𝑥_𝑛∈ℚ_𝑝. So 𝐷_𝑄(𝑥₁,…,𝑥_𝑛)=0. By Lemma 2.1, the function 𝑄∶ℚ_𝑝→𝐵 is quadratic.
Taking the limit 𝑚→∞ in (2.18) with 𝑙=0, we find that the function 𝑄 is quadratic function satisfying the inequality (2.16) near the approximate function 𝑓∶ℚ_𝑝→𝐵 of (1.3).
To prove the aforementioned uniqueness, we assume now that there is another additive function 𝑄′∶ℚ_𝑝→𝐵 which satisfies (1.3) and the inequality (2.16). So 𝑄(𝑥)−𝑄^′(𝑥=1)2^2𝑚𝑄2^𝑚𝑥−𝑄′2^𝑚𝑥≤12^2𝑚𝑄2^𝑚𝑥−𝑓2^𝑚𝑥+𝑓2^𝑚𝑥−𝑄^′2^𝑚𝑥≤𝜀2^{2𝑚+𝜆𝑚}2^𝑛−2−2^{𝑛−𝜆−4}|𝑥|^𝜆_𝑝𝜀,𝑝=2,𝜆>−2;3.2^{2𝑚+𝑛−4}|𝑥|^𝜆_𝑝,𝑝>2; (2.21) which tends to zero as 𝑚→∞ for all nonzero 𝑥∈ℚ_𝑝. This proves the uniqueness of 𝑄, completing the proof of uniqueness.

The following example shows that the above result is not valid over 𝑝-adic fields.

Example 2.4. Let 𝑝>2 be a prime number and define 𝑓∶ℚ_𝑝→ℚ_𝑝 by 𝑓(𝑥)=𝑥²−2𝑥. Since |2^𝑛|_𝑝=1, 𝐷_𝑓(𝑥₁,…,𝑥_𝑛)_𝑝=2^𝑛𝑛_𝑖=2𝑥_𝑖𝑝=^𝑛_𝑖=2𝑥_𝑖𝑝≤^𝑛_𝑖=1𝑥_𝑖𝑝 (2.22) for all 𝑥₁,…,𝑥_𝑛∈ℚ_𝑝. Hence, the conditions of Theorem 2.3 for 𝜀=1 and 𝜆=1 hold. However for each 𝑛∈ℕ, we have 12^2(𝑚+1)𝑓(2^𝑚+11𝑥)−2^2𝑚𝑓(2^𝑚𝑥)_𝑝=|𝑥|_𝑝|2^𝑚|_𝑝=|𝑥|_𝑝 (2.23) for all 𝑥∈ℚ_𝑝. Hence {(1/2^2𝑚)𝑓(2^𝑚𝑥)} is not convergent for all nonzero 𝑥∈ℚ_𝑝.

In the next result, which can be compared with Theorem 2.3, we will show that the stability of the functional equation (1.3) in non-Archimedean spaces over 𝑝-adic fields.

Theorem 2.5. Let ℓ∈{−1,1} be fixed. Let 𝒱 be a non-Archimedean space and 𝒲 be a complete non-Archimedean space over ℚ_𝑝, where 𝑝>2 is a prime number. Suppose that a function 𝑓∶𝒱→𝒲 satisfies the inequality 𝐷_𝑓(𝑥₁,…,𝑥_𝑛)_𝒲≤𝜀^𝑛_𝑖=1𝑥_𝑖^𝜆_𝒱𝜀,𝜆ℓ>2ℓ;_𝑛𝑖=2𝑥₁^𝜆1_𝒱𝑥_𝑖^𝜆𝑖_𝒱,𝜆₁+𝜆_𝑖𝑥ℓ>2ℓ;𝜀max_𝑖^𝜆_𝒱;1≤𝑖≤𝑛,𝜆ℓ>2ℓ; (2.24) for all 𝑥₁,…,𝑥_𝑛∈𝒱, where 𝜀,𝜆₁,…,𝜆_𝑛 and 𝜆 are nonnegative real numbers. Then, the limit 𝑄(𝑥)∶=lim_𝑚→∞1𝑝^2ℓ𝑚𝑓𝑝^ℓ𝑚𝑥 (2.25) exists for all 𝑥∈𝒱 and 𝑄∶𝒱→𝒲 is a unique quadratic function satisfying 𝑓(𝑥)−𝑄(𝑥)_𝒲≤2𝑝^{1+ℓ+(1−ℓ)𝜆/2}𝜀‖𝑥‖^𝜆_𝒱,𝑝^{1+ℓ+((1−ℓ)(𝜆1+𝜆2)/2)}𝜀‖𝑥‖^𝜆1+𝜆2_𝒱,𝑝^{1+ℓ+(1−ℓ)𝜆/2}𝜀‖𝑥‖^𝜆_𝒱, (2.26) for all 𝑥∈𝒱.

Proof. By (2.24), 𝐷_𝑓(𝑥₁,…,𝑥_𝑛)_𝒲≤𝜀^𝑛_𝑖=1𝑥_𝑖^𝜆_𝒱 (2.27) for all 𝑥₁,…,𝑥_𝑛∈𝒱, where 𝜆ℓ>2ℓ. Putting 𝑥_𝑖=0 (𝑖=1,…,𝑛) in (2.27) to obtain 𝑓(0)=0, setting 𝑥_𝑖=0 (𝑖=3,…,𝑛) in (2.27), we obtain 2^𝑛−2𝑓(𝑥₁+𝑥₂)+2^𝑛−2𝑓(𝑥₁−𝑥₂)−2^𝑛−1𝑓(𝑥₁)−2^𝑛−1𝑓(𝑥₂)_𝒲≤𝜀𝑥₁^𝜆_𝒱+𝑥₂^𝜆_𝒱 (2.28) for all 𝑥₁,𝑥₂∈𝒱. So 𝑓(𝑥₁+𝑥₂)+𝑓(𝑥₁−𝑥₂)−2𝑓(𝑥₁)−2𝑓(𝑥₂)_𝒲≤𝜀𝑥₁^𝜆_𝒱+𝑥₂^𝜆_𝒱 (2.29) for all 𝑥₁,𝑥₂∈𝒱. Letting 𝑥₁=𝑥₂=𝑥 in (2.29), we have 𝑓(2𝑥)−4𝑓(𝑥)_𝒲≤2𝜀‖𝑥‖^𝜆_𝒱 (2.30) for all 𝑥∈𝒱. By induction on 𝚥, we will show that for each 𝚥≥2, 𝑓(𝚥𝑥)−𝚥²𝑓(𝑥)_𝒲≤2𝜀‖𝑥‖^𝜆_𝒱 (2.31) for all 𝑥∈𝒱. It holds on 𝚥=2; see (2.30). Let (2.31) hold for 𝚥=2,…,𝑘. Replacing 𝑥₁ and 𝑥₂ by 𝑘𝑥 and 𝑥 in (2.29), respectively, we get 𝑓((𝑘+1)𝑥)+𝑓((𝑘−1)𝑥)−2𝑓(𝑘𝑥)−2𝑓(𝑥)_𝒲≤𝜀1+|𝑘|^𝜆_𝑝‖𝑥‖^𝜆_𝒱 (2.32) for all 𝑥∈𝒱. It follows from (2.32) and our induction hypothesis that 𝑓((𝑘+1)𝑥)−(𝑘+1)²𝑓(𝑥)_𝒲=𝑓)𝑥))𝑥)()(𝑥)()((𝑘+1+𝑓((𝑘−1−2𝑓𝑘𝑥−2𝑓−𝑓((𝑘−1)𝑥)+𝑘−1²𝑓(𝑥)−2𝑓(𝑘𝑥)−𝑘²𝑓(𝑥)_𝒲≤max2𝜀‖𝑥‖^𝜆_𝒱,𝜀1+|𝑘|^𝜆_𝑝‖𝑥‖^𝜆_𝒱=2𝜀‖𝑥‖^𝜆_𝒱 (2.33) for all 𝑥∈𝒱. This proves (2.31) for each 𝚥≥2. In particular, 𝑓(𝑝𝑥)−𝑝²𝑓(𝑥)_𝒲≤2𝜀‖𝑥‖^𝜆_𝒱 (2.34) for all 𝑥∈𝒱. So 1𝑓(𝑥)−𝑝²𝑓(𝑝𝑥)_𝒲≤2𝑝²𝜀‖𝑥‖^𝜆_𝒱,𝑓(𝑥)−𝑝²𝑓𝑥𝑝_𝒲≤2𝑝^𝜆𝜀‖𝑥‖^𝜆_𝒱 (2.35) for all 𝑥∈𝒱. Hence, 1𝑝^2ℓ𝑗𝑓𝑝^ℓ𝑗𝑥−1𝑝^2ℓ(𝑗+1)𝑓𝑝^ℓ(𝑗+1)𝑥_𝒲≤2𝑝^{2ℓ𝑗+(1−ℓ)𝜆/2+1+ℓ}𝑝^𝜆ℓ𝑗𝜀‖𝑥‖^𝜆_𝒱 (2.36) for all 𝑥∈𝒱. Since the right side of the above inequality tends to zero as 𝑗→∞, {(1/𝑝^2ℓ𝑚)𝑓(𝑝^ℓ𝑚𝑥)} is a Cauchy sequence in complete non-Archimedean space 𝒲, thus it converges to some function 𝑄(𝑥)=lim_𝑚→∞(1/𝑝^2ℓ𝑚)𝑓(𝑝^ℓ𝑚𝑥) for all 𝑥∈𝒱. Using (2.35) and induction, one can show that for any 𝑚∈ℕ, we have 1𝑓(𝑥)−𝑝^2ℓ𝑚𝑓(𝑝^ℓ𝑚𝑥)_𝒲=^𝑚−1_𝑗=01𝑝^2ℓ𝑗𝑓(𝑝^ℓ𝑗1𝑥)−𝑝^2ℓ(𝑗+1)𝑓(𝑝^ℓ(𝑗+1)𝑥)_𝒲1≤max𝑝^2ℓ𝑗𝑓(𝑝^ℓ𝑗1𝑥)−𝑝^2ℓ(𝑗+1)𝑓𝑝^ℓ(𝑗+1)𝑥_𝒲;0≤𝑗<𝑚≤max2𝑝^{1+ℓ+(1−ℓ)𝜆/2+ℓ𝑗(2−𝜆)}𝜀‖𝑥‖^𝜆_𝒱;0≤𝑗<𝑚 (2.37) for all 𝑥∈𝒱. Letting 𝑚→∞ in this inequality, we see that 𝑓(𝑥)−𝑄(𝑥)_𝒲≤2𝑝^{1+ℓ+(1−ℓ)𝜆/2}𝜀‖𝑥‖^𝜆_𝒱 (2.38) for all 𝑥∈𝒱. Moreover, 𝐷_𝑄(𝑥₁,…,𝑥_𝑛)_𝒲=lim_𝑚→∞1𝑝^2ℓ𝑚𝐷_𝑓(𝑝^ℓ𝑚𝑥₁,…,𝑝^ℓ𝑚𝑥_𝑛)_𝒲≤lim_𝑚→∞𝑝^2ℓ𝑚𝑝^{𝑛𝜆ℓ𝑚}_𝑖=1𝜀𝑥_𝑖^𝜆_𝒱=0 (2.39) for all 𝑥₁,…,𝑥_𝑛∈𝒱. So 𝐷_𝑄(𝑥₁,…,𝑥_𝑛)=0. By Lemma 2.1, the function 𝑄∶𝒱→𝒲 is quadratic.
Now, let 𝑄^′∶𝒱→𝒲 be another quadratic function satisfying (1.3) and (2.38). So 𝑄(𝑥)−𝑄^′(𝑥)_𝒲≤𝑝^2ℓ𝑚max𝑄𝑝^ℓ𝑚𝑥−𝑓𝑝^ℓ𝑚𝑥_𝒲,𝑓𝑝^ℓ𝑚𝑥−𝑄′𝑝^ℓ𝑚𝑥_𝒲≤2𝑝^{2ℓ𝑚+(1−ℓ)𝜆/2+1+ℓ}𝑝^𝜆ℓ𝑚𝜀‖𝑥‖^𝜆_𝒱, (2.40) which tends to zero as 𝑚→∞ for all 𝑥∈𝒱. This proves the uniqueness of 𝑄.
The rest of the proof is similar to the above proof, hence it is omitted.

Acknowledgments

The third author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant number: 2011-0005197).