# Nearly Quadratic Mappings overp-Adic Fields

Authors: M. Eshaghi Gordji; H. Khodaei; Gwang Hui Kim
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We establish some stability results overp-adic fields for the generalized quadratic functional equation ∑k=2n∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1nf(∑i=1,i≠i1,…,in-k+1nxi-∑r=1n-k+1xir)+f(∑i=1nxi)=2n-1∑i=1nf(xi), where n∈ℕ and n≥2.


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## Body

## 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 nx such that x=(a/b)pnx, 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 ℚp, and it is called the p-adic number field. In fact, ℚp is the set of all formal series x=∑k≥nx∞akpk, where |ak|≤p-1 are integers (see, e.g., [2, 3]). Note that if p>2, then |2n|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.Let𝕂 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.LetX 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 {xn}n∈ℕ is Cauchy if and only if {xn+1-xn}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:ℚp→ℝ. They showed that if f:ℚp→ℝ is a continuous function for which there exists a fixed ɛ such that |f(x+y)-f(x)-f(y)|≤ɛ for all x,y∈Qp, then there exists a unique additive function T:ℚp→ℝ such that |f(x)-T(x)|≤ɛ for all x∈ℚp. 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 (1.1)∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1naixi-∑r=1n-k+1airxir)+f(∑i=1naixi)=2n-1a1f(x1)
and proved the generalized Hyers-Ulam stability of the above functional equation. The functional equation(1.2)f(x1+x2)+f(x1-x2)=2f(x1)+2f(x2)
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 inn-variables as follows:(1.3)∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1nxi-∑r=1n-k+1xir)+f(∑i=1nxi)=2n-1∑i=1nf(xi),
where n≥2. Moreover, we investigate the generalized Hyers-Ulam stability of functional equation (1.3) over the p-adic field ℚp.As a special case, ifn=2 in (1.3), then we have the functional equation (1.2). Also, if n=3 in (1.3), we obtain (1.4)∑i1=22∑i2=i1+13f(∑i=1,i≠i1,i23xi-∑r=12xir)+∑i1=23f(∑i=1,i≠i13xi-xi1)+f(∑i=13xi)=22∑i=13f(xi),
that is, (1.5)f(x1-x2-x3)+f(x1-x2+x3)+f(x1+x2-x3)+f(x1+x2+x3)=4f(x1)+4f(x2)+4f(x3).


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

We will use the following lemma.Lemma 2.1.
LetX 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.
Letf satisfy the functional equation (1.3). Setting xi=0 (i=1,…,n) in (1.3), we have
(2.1)∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(0)+f(0)=2n-1∑i=1nf(0),
that is,
(2.2)∑i1=22∑i2=i1+13⋯∑in-1=in-2+1nf(0)+∑i1=23∑i2=i1+14⋯∑in-2=in-3+1nf(0)+⋯+∑i1=2nf(0)+f(0)=2n-1∑i=1nf(0),
or
(2.3)((n-1n-1)+(n-1n-2)+⋯+(n-11)+1)f(0)=2n-1∑i=1nf(0),
but 1+∑j=1n-𝚥(n-𝚥j)=∑j=0n-𝚥(n-𝚥j)=2n-j, and also n>𝚥≥1 so 2n-1(n-1)f(0)=0.
Puttingxi=0 (i=2,…,n-1) in (1.3) and then using f(0)=0, we get
(2.4)f(x1-xn)+((n-21)f(x1-xn)+(n-2n-2)f(x1+xn))+⋯+((n-2n-3)f(x1-xn)+(n-22)f(x1+xn))+((n-2n-2)f(x1-xn)+(n-21)f(x1+xn))+f(x1+xn)=2n-1f(x1)+2n-1f(xn),
that is,
(2.5)(1+∑j=1n-2(n-2j))(f(x1+xn)+f(x1-xn))=2n-1f(x1)+2n-1f(xn),
for all x1,xn∈X, this shows that f satisfies the functional equation (1.2). So the function f is quadratic.
Conversely, suppose thatf 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 onn≥2. It holds on n=2. Assume that it holds on the case where n=t; that is, we have
(2.6)∑k=2t(∑i1=2k∑i2=i1+1k+1⋯∑it-k+1=it-k+1t)f(∑i=1,i≠i1,…,it-k+1txi-∑r=1t-k+1xir)+f(∑i=1txi)=2t-1∑i=1tf(xi)
for all x1,…,xt∈X. It follows from (1.2) that
(2.7)f(∑i=1txi+xt+1)+f(∑i=1txi-xt+1)=2f(∑i=1txi)+2f(xt+1)
for all x1,…,xt+1∈X. Replacing xt by -xt in (2.7), we obtain
(2.8)f(∑i=1t-1xi-xt+xt+1)+f(∑i=1t-1xi-xt-xt+1)=2f(∑i=1t-1xi-xt)+2f(xt+1)
for all x1,…,xt+1∈X. Adding (2.7) to (2.8), we have
(2.9)f(∑i=1t-1xi-xt-xt+1)+f(∑i=1t-1xi-xt+xt+1)+f(∑i=1t-1xi+xt-xt+1)+f(∑i=1t-1xi+xt+xt+1)=2[f(∑i=1t-1xi-xt)+f(∑i=1t-1xi+xt)]+4f(xt+1)
for all x1,…,xt+1∈X. Replacing xt-1 by -xt-1 in (2.9), we get
(2.10)f(∑i=1t-2xi-xt-1-xt-xt+1)+f(∑i=1t-2xi-xt-1-xt+xt+1)+f(∑i=1t-2xi-xt-1+xt-xt+1)+f(∑i=1t-2xi-xt-1+xt+xt+1)=2[f(∑i=1t-2xi-xt-1-xt)+f(∑i=1t-2xi-xt-1+xt)]+4f(xt+1)
for all x1,…,xt+1∈X. Adding (2.9) to (2.10), one gets
(2.11)f(∑i=1t-2xi-xt-1-xt-xt+1)+f(∑i=1t-2xi-xt-1-xt+xt+1)+f(∑i=1t-2xi-xt-1+xt-xt+1)+f(∑i=1t-2xi+xt-1-xt-xt+1)+f(∑i=1t-2xi-xt-1+xt+xt+1)+f(∑i=1t-2xi+xt-1-xt+xt+1)+f(∑i=1t-2xi+xt-1+xt-xt+1)+f(∑i=1t+1xi)=2[f(∑i=1t-2xi-xt-1-xt)+f(∑i=1t-2xi-xt-1+xt)+f(∑i=1t-2xi+xt-1-xt)+f(∑i=1t-2xi+xt-1+xt)]+8f(xt+1)
for all x1,…,xt+1∈X. By using the above method, for xt-2 until x2, we infer that
(2.12)∑k=2t+1(∑i1=2k∑i2=i1+1k+1⋯∑it-k+2=it-k+1+1t+1)f(∑i=1,i≠i1,…,it-k+2t+1xi-∑r=1t-k+2xir)+f(∑i=1t+1xi)=2[∑k=2t(∑i1=2k∑i2=i1+1k+1⋯∑it-k+1=it-k+1t)f(∑i=1,i≠i1,…,it-k+1txi-∑r=1t-k+1xir)+f(∑i=1txi)]+2tf(xt+1)
for all x1,…,xt+1∈X. Now, by the case n=t, we lead to
(2.13)∑k=2t+1(∑i1=2k∑i2=i1+1k+1⋯∑it-k+2=it-k+1+1t+1)f(∑i=1,i≠i1,…,it-k+2t+1xi-∑r=1t-k+2xir)+f(∑i=1t+1xi)=2[2t-1∑i=1tf(xi)]+2tf(xt+1)
for all x1,…,xt+1∈X, so (1.3) holds for n=t+1. This completes the proof of the lemma.Corollary 2.2.
A functionf: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 ℚp. For convenience, we define the difference operator Df for a given function f: (2.14)Df(x1,…,xn):=∑k=2n(∑i1=2k∑i2=i1+1k+1⋯∑in-k+1=in-k+1n)f(∑i=1,i≠i1,…,in-k+1nxi-∑r=1n-k+1xir)+f(∑i=1nxi)-2n-1∑i=1nf(xi).Theorem 2.3.
LetB be a Banach space and let ɛ>0,λ be real numbers. Suppose that a function f:ℚp→B with f(0)=0 satisfies the inequality
(2.15)‖Df(x1,…,xn)‖≤ɛ∑i=1n|xi|pλ
for all x1,…,xn∈ℚp. Then there exists a unique quadratic function Q:ℚp→B such that
(2.16)‖f(x)-Q(x)‖≤{ɛ2n-1-2n-λ-3|x|pλ,p=2,λ>-2;ɛ3.2n-3|x|pλ,p>2;
for all nonzero x∈ℚp.Proof.
Lettingx1=x2=x≠0 and xi=0 (i=3,…,n) in (2.15), we obtain
(2.17)‖f(x)-14f(2x)‖≤ɛ2n-1|x|pλ
for all x∈ℚp. Hence,
(2.18)‖122lf(2lx)-122mf(2mx)‖≤ɛ2n-1∑j=lm-1|2|pλj22j|x|pλ
for all nonnegative integers m and l with m>l and for all x∈ℚp. It follows from (2.18) that the sequence {(1/22m)f(2mx)} is a Cauchy sequence for all x∈ℚp. Since B is complete, the sequence {(1/22m)f(2mx)} converges. Therefore, one can define the function Q:ℚp→B by
(2.19)Q(x):=lim⁡m→∞122mf(2mx)
for all x∈ℚp. It follows from (2.15) and (2.19) that
(2.20)‖DQ(x1,…,xn)‖=lim⁡m→∞122m‖Df(2mx1,…,2mxn)‖≤lim⁡m→∞|2|pλm22m∑i=1nɛ|xi|pλ=0
for all x1,…,xn∈ℚp. So DQ(x1,…,xn)=0. By Lemma 2.1, the function Q:ℚp→B is quadratic.
Taking the limitm→∞ 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:ℚp→B of (1.3).
To prove the aforementioned uniqueness, we assume now that there is another additive functionQ′:ℚp→B which satisfies (1.3) and the inequality (2.16). So
(2.21)‖Q(x)-Q′(x)‖=122m‖Q(2mx)-Q′(2mx)‖≤122m(‖Q(2mx)-f(2mx)‖+‖f(2mx)-Q′(2mx)‖)≤{ɛ22m+λm(2n-2-2n-λ-4)|x|pλ,p=2,λ>-2;ɛ3.22m+n-4|x|pλ,p>2;
which tends to zero as m→∞ for all nonzero x∈ℚp. This proves the uniqueness of Q, completing the proof of uniqueness.The following example shows that the above result is not valid overp-adic fields.Example 2.4.
Letp>2 be a prime number and define f:ℚp→ℚp by f(x)=x2-2x. Since |2n|p=1,
(2.22)|Df(x1,…,xn)|p=|2n∑i=2nxi|p=|∑i=2nxi|p≤∑i=1n|xi|p
for all x1,…,xn∈ℚp. Hence, the conditions of Theorem 2.3 for ɛ=1 and λ=1 hold. However for each n∈ℕ, we have
(2.23)|122(m+1)f(2m+1x)-122mf(2mx)|p=|x|p|2m|p=|x|p
for all x∈ℚp. Hence {(1/22m)f(2mx)} is not convergent for all nonzero x∈ℚp.In the next result, which can be compared with Theorem2.3, we will show that the stability of the functional equation (1.3) in non-Archimedean spaces over p-adic fields.Theorem 2.5.
Letℓ∈{-1,1} be fixed. Let 𝒱 be a non-Archimedean space and 𝒲 be a complete non-Archimedean space over ℚp, where p>2 is a prime number. Suppose that a function f:𝒱→𝒲 satisfies the inequality
(2.24)‖Df(x1,…,xn)‖W≤{ɛ∑i=1n‖xi‖Vλ,λl>2l;ɛ∑i=2n‖x1‖Vλ1‖xi‖Vλi,(λ1+λi)l>2l;ɛmax⁡{‖xi‖Vλ;1≤i≤n},λl>2l;
for all x1,…,xn∈𝒱, where ɛ,λ1,…,λn and λ are nonnegative real numbers. Then, the limit
(2.25)Q(x):=lim⁡m→∞1p2lmf(plmx)
exists for all x∈𝒱 and Q:𝒱→𝒲 is a unique quadratic function satisfying
(2.26)‖f(x)-Q(x)‖W≤{2p1+l+(1-l)λ/2ɛ‖x‖Vλ,p1+l+((1-l)(λ1+λ2)/2)ɛ‖x‖Vλ1+λ2,p1+l+(1-l)λ/2ɛ‖x‖Vλ,
for all x∈𝒱.Proof.
By (2.24),
(2.27)‖Df(x1,…,xn)‖W≤ɛ∑i=1n‖xi‖Vλ
for all x1,…,xn∈𝒱, where λℓ>2ℓ. Putting xi=0 (i=1,…,n) in (2.27) to obtain f(0)=0, setting xi=0 (i=3,…,n) in (2.27), we obtain
(2.28)‖2n-2f(x1+x2)+2n-2f(x1-x2)-2n-1f(x1)-2n-1f(x2)‖W≤ɛ(‖x1‖Vλ+‖x2‖Vλ)
for all x1,x2∈𝒱. So
(2.29)‖f(x1+x2)+f(x1-x2)-2f(x1)-2f(x2)‖W≤ɛ(‖x1‖Vλ+‖x2‖Vλ)
for all x1,x2∈𝒱. Letting x1=x2=x in (2.29), we have
(2.30)‖f(2x)-4f(x)‖W≤2ɛ‖x‖Vλ
for all x∈𝒱. By induction on 𝚥, we will show that for each 𝚥≥2,
(2.31)‖f(𝚥x)-𝚥2f(x)‖W≤2ɛ‖x‖Vλ
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
(2.32)‖f((k+1)x)+f((k-1)x)-2f(kx)-2f(x)‖W≤ɛ(1+|k|pλ)‖x‖Vλ
for all x∈𝒱. It follows from (2.32) and our induction hypothesis that
(2.33)‖f((k+1)x)-(k+1)2f(x)‖W=‖f((k+1)x)+f((k-1)x)-2f(kx)-2f(x)-f((k-1)x)+(k-1)2f(x)-2(f(kx)-k2f(x))‖W≤max⁡{2ɛ‖x‖Vλ,ɛ(1+|k|pλ)‖x‖Vλ}=2ɛ‖x‖Vλ
for all x∈𝒱. This proves (2.31) for each 𝚥≥2. In particular,
(2.34)‖f(px)-p2f(x)‖W≤2ɛ‖x‖Vλ
for all x∈𝒱. So
(2.35)‖f(x)-1p2f(px)‖W≤2p2ɛ‖x‖Vλ,‖f(x)-p2f(xp)‖W≤2pλɛ‖x‖Vλ
for all x∈𝒱. Hence,
(2.36)‖1p2ljf(pljx)-1p2l(j+1)f(pl(j+1)x)‖W≤2p2lj+(1-l)λ/2+1+lpλljɛ‖x‖Vλ
for all x∈𝒱. Since the right side of the above inequality tends to zero as j→∞, {(1/p2ℓm)f(pℓmx)} is a Cauchy sequence in complete non-Archimedean space 𝒲, thus it converges to some function Q(x)=limm→∞(1/p2ℓm)f(pℓmx) for all x∈𝒱. Using (2.35) and induction, one can show that for any m∈ℕ, we have
(2.37)‖f(x)-1p2lmf(plmx)‖W=‖∑j=0m-11p2ljf(pljx)-1p2l(j+1)f(pl(j+1)x)‖W≤max⁡{‖1p2ljf(pljx)-1p2l(j+1)f(pl(j+1)x)‖W;0≤j<m}≤max⁡{2p1+l+(1-l)λ/2+lj(2-λ)ɛ‖x‖Vλ;0≤j<m}
for all x∈𝒱. Letting m→∞ in this inequality, we see that
(2.38)‖f(x)-Q(x)‖W≤2p1+l+(1-l)λ/2ɛ‖x‖Vλ
for all x∈𝒱. Moreover,
(2.39)‖DQ(x1,…,xn)‖W=lim⁡m→∞‖1p2lmDf(plmx1,…,plmxn)‖W≤lim⁡m→∞p2lmpλlm∑i=1nɛ‖xi‖Vλ=0
for all x1,…,xn∈𝒱. So DQ(x1,…,xn)=0. By Lemma 2.1, the function Q:𝒱→𝒲 is quadratic.
Now, letQ′:𝒱→𝒲 be another quadratic function satisfying (1.3) and (2.38). So
(2.40)‖Q(x)-Q′(x)‖W≤p2lmmax⁡{‖Q(plmx)-f(plmx)‖W,‖f(plmx)-Q′(plmx)‖W}≤2p2lm+(1-l)λ/2+1+lpλlmɛ‖x‖Vλ,
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.



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*Source: 285807-2012-01-04.xml*