AAAAbstract and Applied Analysis1687-04091085-3375Hindawi Publishing Corporation28580710.1155/2012/285807285807Research ArticleNearly Quadratic Mappings over p-Adic FieldsGordjiM. Eshaghi1KhodaeiH.1KimGwang Hui2RassiasJohn1Department of MathematicsSemnan UniversityP.O. Box 35195-363, SemnanIransemnan.ac.ir2Department of MathematicsKangnam University, Yongin, Gyeonggi 446-702Republic of Koreakangnam.ac.kr201241201220123010201120112011211120112012Copyright © 2012 M. Eshaghi Gordji et al.This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We establish some stability results over p-adic fields for the generalized quadratic functional equation k=2ni1=2ki2=i1+1k+1in-k+1=in-k+1nf(i=1,ii1,,in-k+1nxi-r=1n-k+1xir)+f(i=1nxi)=2n-1i=1nf(xi), where n and n2.

1. Introduction and Preliminaries

In 1899, Hensel  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 . The completion of with respect to the metric d(x,y)=|x-y|p is denoted by p, and it is called the p-adic number field. In fact, p is the set of all formal series x=knxakpk, 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 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,y𝕂. The associated field 𝕂 is referred to as a non-Archimedean field. Clearly, |1|=|-1|=1 and |n|1 for all n1. 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 z𝕂 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+ymax{x,y} for all x,yX. Then, (X,·) is called a non-Archimedean space. In any such a space, a sequence {xn}n is Cauchy if and only if {xn+1-xn}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  concerning the stability of group homomorphisms, which was affirmatively answered for Banach spaces by Hyers . Subsequently, the result of Hyers was generalized by Aoki  for additive mappings and by Rassias  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  considered the Cauchy difference controlled by a product of different powers of norm. The above results have been generalized by Forti  and Găvruţa  who permitted the Cauchy difference to become arbitrary unbounded (see also ). Arriola and Beyer  investigated stability of approximate additive functions f:p. They showed that if f:p is a continuous function for which there exists a fixed ɛ such that |f(x+y)-f(x)-f(y)|ɛ for all x,yQp, then there exists a unique additive function T:p such that |f(x)-T(x)|ɛ for all xp. For more details about the results concerning such problems, the reader is referred to .

Recently, Khodaei and Rassias  introduced the generalized additive functional equation k=2n(i1=2ki2=i1+1k+1in-k+1=in-k+1n)f(i=1,ii1,,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 equationf(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 in n-variables as follows:k=2n(i1=2ki2=i1+1k+1in-k+1=in-k+1n)f(i=1,ii1,,in-k+1nxi-r=1n-k+1xir)+f(i=1nxi)=2n-1i=1nf(xi), where n2. Moreover, we investigate the generalized Hyers-Ulam stability of functional equation (1.3) over the p-adic field 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 i1=22i2=i1+13f(i=1,ii1,i23xi-r=12xir)+i1=23f(i=1,ii13xi-xi1)+f(i=13xi)=22i=13f(xi), that is, 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 (<xref ref-type="disp-formula" rid="EEq1.2">1.3</xref>) over <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M78"><mml:mrow><mml:mi>p</mml:mi></mml:mrow></mml:math></inline-formula>-Adic Fields

We will use the following lemma.

Lemma 2.1.

Let X and Y be real vector spaces. A function f:XY 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 xi=0 (i=1,,n) in (1.3), we have k=2n(i1=2ki2=i1+1k+1in-k+1=in-k+1n)f(0)+f(0)=2n-1i=1nf(0), that is, i1=22i2=i1+13in-1=in-2+1nf(0)+i1=23i2=i1+14in-2=in-3+1nf(0)++i1=2nf(0)+f(0)=2n-1i=1nf(0), or ((n-1n-1)+(n-1n-2)++(n-11)+1)f(0)=2n-1i=1nf(0), but 1+j=1n-𝚥(n-𝚥j)=j=0n-𝚥(n-𝚥j)=2n-j, and also n>𝚥1 so 2n-1(n-1)f(0)=0.

Putting xi=0 (i=2,,n-1) in (1.3) and then using f(0)=0, we get 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, (1+j=1n-2(n-2j))(f(x1+xn)+f(x1-xn))=2n-1f(x1)+2n-1f(xn), for all x1,xnX, 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 n2. It holds on n=2. Assume that it holds on the case where n=t; that is, we have k=2t(i1=2ki2=i1+1k+1it-k+1=it-k+1t)f(i=1,ii1,,it-k+1txi-r=1t-k+1xir)+f(i=1txi)=2t-1i=1tf(xi) for all x1,,xtX. It follows from (1.2) that f(i=1txi+xt+1)+f(i=1txi-xt+1)=2f(i=1txi)+2f(xt+1) for all x1,,xt+1X. Replacing xt by -xt in (2.7), we obtain 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+1X. Adding (2.7) to (2.8), we have 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+1X. Replacing xt-1 by -xt-1 in (2.9), we get 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+1X. Adding (2.9) to (2.10), one gets 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+1X. By using the above method, for xt-2 until x2, we infer that k=2t+1(i1=2ki2=i1+1k+1it-k+2=it-k+1+1t+1)f(i=1,ii1,,it-k+2t+1xi-r=1t-k+2xir)+f(i=1t+1xi)=2[k=2t(i1=2ki2=i1+1k+1it-k+1=it-k+1t)f(i=1,ii1,,it-k+1txi-r=1t-k+1xir)+f(i=1txi)]+2tf(xt+1) for all x1,,xt+1X. Now, by the case n=t, we lead to k=2t+1(i1=2ki2=i1+1k+1it-k+2=it-k+1+1t+1)f(i=1,ii1,,it-k+2t+1xi-r=1t-k+2xir)+f(i=1t+1xi)=2[2t-1i=1tf(xi)]+2tf(xt+1) for all x1,,xt+1X, so (1.3) holds for n=t+1. This completes the proof of the lemma.

Corollary 2.2.

A function f:XY satisfies the functional equation (1.3) if and only if there exists a symmetric biadditive function B1:X×XY such that f(x)=B1(x,x) for all xX.

Now, we investigate the stability of the functional equation (1.3) from a Banach space B into p-adic field p. For convenience, we define the difference operator Df for a given function f: Df(x1,,xn):=k=2n(i1=2ki2=i1+1k+1in-k+1=in-k+1n)f(i=1,ii1,,in-k+1nxi-r=1n-k+1xir)+f(i=1nxi)-2n-1i=1nf(xi).

Theorem 2.3.

Let B be a Banach space and let ɛ>0,  λ be real numbers. Suppose that a function f:pB with f(0)=0 satisfies the inequality Df(x1,,xn)ɛi=1n|xi|pλ for all x1,,xnp. Then there exists a unique quadratic function Q:pB such that 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 xp.

Proof.

Letting x1=x2=x0 and xi=0 (i=3,,n) in (2.15), we obtain f(x)-14f(2x)ɛ2n-1|x|pλ for all xp. Hence, 122lf(2lx)-122mf(2mx)ɛ2n-1j=lm-1|2|pλj22j|x|pλ for all nonnegative integers m and l with m>l and for all xp. It follows from (2.18) that the sequence {(1/22m)f(2mx)} is a Cauchy sequence for all xp. Since B is complete, the sequence {(1/22m)f(2mx)} converges. Therefore, one can define the function Q:pB by Q(x):=limm122mf(2mx) for all xp. It follows from (2.15) and (2.19) that DQ(x1,,xn)=limm122mDf(2mx1,,2mxn)limm|2|pλm22mi=1nɛ|xi|pλ=0 for all x1,,xnp. So DQ(x1,,xn)=0. By Lemma 2.1, the function Q:pB 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:pB of (1.3).

To prove the aforementioned uniqueness, we assume now that there is another additive function Q:pB which satisfies (1.3) and the inequality (2.16). So Q(x)-Q(x)=122mQ(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 xp. 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:pp by f(x)=x2-2x. Since |2n|p=1, |Df(x1,,xn)|p=|2ni=2nxi|p=|i=2nxi|pi=1n|xi|p for all x1,,xnp. Hence, the conditions of Theorem 2.3 for ɛ=1 and λ=1 hold. However for each n, we have |122(m+1)f(2m+1x)-122mf(2mx)|p=|x|p|2m|p=|x|p for all xp. Hence {(1/22m)f(2mx)} is not convergent for all nonzero xp.

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 {-1,1} be fixed. Let 𝒱 be a non-Archimedean space and 𝒲 be a complete non-Archimedean space over p, where p>2 is a prime number. Suppose that a function f:𝒱𝒲 satisfies the inequality Df(x1,,xn)W{ɛi=1nxiVλ,λl>2l;ɛi=2nx1Vλ1xiVλi,(λ1+λi)l>2l;ɛmax{xiVλ;1in},λl>2l; for all x1,,xn𝒱, where ɛ,λ1,,λn and λ are nonnegative real numbers. Then, the limit Q(x):=limm1p2lmf(plmx) exists for all x𝒱 and Q:𝒱𝒲 is a unique quadratic function satisfying f(x)-Q(x)W{2p1+l+(1-l)λ/2ɛxVλ,p1+l+((1-l)(λ1+λ2)/2)ɛxVλ1+λ2,p1+l+(1-l)λ/2ɛxVλ, for all x𝒱.

Proof.

By (2.24), Df(x1,,xn)Wɛi=1nxiVλ for all x1,,xn𝒱, where λ>2. 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 2n-2f(x1+x2)+2n-2f(x1-x2)-2n-1f(x1)-2n-1f(x2)Wɛ(x1Vλ+x2Vλ) for all x1,x2𝒱. So f(x1+x2)+f(x1-x2)-2f(x1)-2f(x2)Wɛ(x1Vλ+x2Vλ) for all x1,x2𝒱. Letting x1=x2=x in (2.29), we have f(2x)-4f(x)W2ɛxVλ for all x𝒱. By induction on 𝚥, we will show that for each 𝚥2, f(𝚥x)-𝚥2f(x)W2ɛxVλ for all x𝒱. 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)Wɛ(1+|k|pλ)xVλ for all x𝒱. It follows from (2.32) and our induction hypothesis that 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))Wmax{2ɛxVλ,ɛ(1+|k|pλ)xVλ}=2ɛxVλ for all x𝒱. This proves (2.31) for each 𝚥2. In particular, f(px)-p2f(x)W2ɛxVλ for all x𝒱. So f(x)-1p2f(px)W2p2ɛxVλ,f(x)-p2f(xp)W2pλɛxVλ for all x𝒱. Hence, 1p2ljf(pljx)-1p2l(j+1)f(pl(j+1)x)W2p2lj+(1-l)λ/2+1+lpλljɛxVλ for all x𝒱. Since the right side of the above inequality tends to zero as j, {(1/p2m)f(pmx)} is a Cauchy sequence in complete non-Archimedean space 𝒲, thus it converges to some function Q(x)=limm(1/p2m)f(pmx) for all x𝒱. Using (2.35) and induction, one can show that for any m, we have f(x)-1p2lmf(plmx)W=j=0m-11p2ljf(pljx)-1p2l(j+1)f(pl(j+1)x)Wmax{1p2ljf(pljx)-1p2l(j+1)f(pl(j+1)x)W;0j<m}max{2p1+l+(1-l)λ/2+lj(2-λ)ɛxVλ;0j<m} for all x𝒱. Letting m in this inequality, we see that f(x)-Q(x)W2p1+l+(1-l)λ/2ɛxVλ for all x𝒱. Moreover, DQ(x1,,xn)W=limm1p2lmDf(plmx1,,plmxn)Wlimmp2lmpλlmi=1nɛxiVλ=0 for all x1,,xn𝒱. So DQ(x1,,xn)=0. By Lemma 2.1, the function Q:𝒱𝒲 is quadratic.

Now, let Q:𝒱𝒲 be another quadratic function satisfying (1.3) and (2.38). So Q(x)-Q(x)Wp2lmmax{Q(plmx)-f(plmx)W,f(plmx)-Q(plmx)W}2p2lm+(1-l)λ/2+1+lpλlmɛxVλ, which tends to zero as m for all x𝒱. 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).