# Hyers-Ulam Stability of Functional Equation Deriving from Quadratic Mapping in Non-Archimedeann,β-Normed Spaces

Authors: Nazek Alessa; K. Tamilvanan; K. Loganathan; K. Kalai Selvi
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this work, we have to introduce a generalized quadratic functional equation and derive its solution. The main objective of this work is to investigate the Hyers-Ulam stability of quadratic functional equation in non-Archimedeann,β-normed spaces.


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## Body

## 1. Introduction

The theory of functional equation is one of the most interesting topics in the field of Mathematics. It deals with the search of functions which satisfies a given equation. A functional equation is like a regular algebraic equation; though instead of unknown elements in some set, we are interested in finding a function satisfying our equation.The concept of stability of a functional equation arises when one replaces a functional equation by an inequality which acts as a perturbation of the equation. The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940 and affirmatively solved by Hyers [2]. Aoki was the first author who has generalized the theorem of Hyers (see [3]).Moreover, Gavruta [4], Rassias [5], and Bourgin [6] have considered the stability problem with unbounded Cauchy difference (see also [7]). On the other hand, Rassias [8–13] considered the Cauchy difference controlled by a product of different powers of norm. This stability phenomenon is called the Ulam-Gavruta-Rassias stability (see also [14]).The Hyers-Ulam stability issue for the quadratic functional equation was cleared by Skof [15]. In [16], Czerwik demonstrated the Hyers-Ulam-Rassias stability of the quadratic functional equation. Afterward, Jung [17] has summed up the outcomes gotten by Skof and Czerwik.The Hyers-Ulam stability problem for the quadratic functional equation was solved by Skof [15]. In [16], Czerwik proved the Hyers-Ulam-Rassias stability of quadratic functional equation. Later, Jung [17] has generalized the results obtained by Skof and Czerwik.The first work on the Hyers-Ulam stability of functional equations in complete non-Archimedean normed spaces (some particular cases were considered earlier; see [18] for details) is [19]. After it, a lot of articles (see, for instance, [20] and the references given there) on the stability of other equations in such spaces were published. In [21], the stability of the additive Cauchy equation in non-Archimedean fuzzy normed spaces under the strongest t-norm TM Rassias has been established.In 1897, Hensel [22] has introduced a normed space which does not have the Archimedean property. It turned out that non-Archimedean spaces have many nice applications [23–26].Initially, Liu [27] introduced the notions of n,β-normed space and non-Archimedean n,β-normed space. Then, they investigated Hyers-Ulam stability of the Cauchy functional equation and the Jensen functional equation in non-Archimedean n,β-normed spaces and that of the pexiderized Cauchy functional equation in n,β-normed spaces. During the most recent many years, a few stability issues of functional equations have been researched by various mathematicians (see [28–40]).The objective of this work is to introduce that a generalized quadratic functional equation is(1)∑1≤i<j≤lϕvi+vj+∑1≤i<j≤lϕvi−vj=2l−1∑1≤i≤lϕvi,where l≥2 and examine the Hyers-Ulam stability of the above mentioned equation in non-Archimedean n,β-normed spaces.


## 2. Preliminaries

Now, we recall some notions and results which will be used.Throughout this paper, letℕ denote the set of nonnegative integers and i,t,p,n∈ℕ, and let n≥2 be fixed.Definition 1 (see [27]).
LetE be a linear space over ℝ with dim E≥n, n∈ℕ, and 0<β≤1, let a mapping ·,⋯,·β:En⟶ℝ satisfies the following conditions:
(i)
m1,⋯,mnβ=0 if and only if m1,⋯,mn are linearly dependent(ii)
m1,⋯,mnβ is invariant under permutations of m1,⋯,mn(iii)
λm1,⋯,mnβ=λβm1,⋯,mnβ(iv)
m1,⋯,mn−1,u+vβ≤m1,⋯,mn−1,uβ+m1,⋯,mn−1,vβfor all m1,⋯,mn∈E, and λ∈ℝ.
Then,·,⋯,·β is called as n,β-norm on E, and E,·,⋯,·β is called a linear n,β-normed space or n,β-normed space.
We remark that the notion of a linearn,β-normed space is a summed up of a linear n-normed space β=1 and of a β-normed space n=1.Definition 2 (see [27]).
LetE be a real vector space with dim E≥n over a scalar field K with a non-Archimedean nontrivial valuation ·, where n∈ℤ+ and a constant β with 0<β≤1. A real-valued function ∙,⋯,·β:En⟶ℝ is called an n,β-norm on E if the upcoming conditions true:
(a)
m1,⋯,mnβ=0 if and only if m1,⋯,mn are linearly dependent(b)
m1,⋯,mnβ is invariant under permutations of m1,⋯,mn(c)
λm1,⋯,mnβ=λβm1,⋯,mnβ(d)
m0+m1,⋯,mnβ≤maxm0,m2,⋯,mnβ,m1,m2⋯,mnβfor all λ∈K and m0,m1,⋯,mn∈E. Then, E,·,⋯,·β is called as non-Archimedean n,β-normed space.Example 3.
Letp be a prime number. For any nonzero rational number x=a/bpr such that a and b are coprime to the prime number p, define the p-adic absolute value xp≔p−r. Then, ·p is a non-Archimedean norm on ℚ. The completion of ℚ with respect to ·p is denoted by ℚp and is called the p-adic number field.Note that ifp>3; then, 2n=1 in for each integer n.Remark 4 (see [27]).
A sequencevm in a non-Archimedean n,β-normed space E is a Cauchy sequence if and only if vm+1−vm converges to zero.Lemma 5 (see [27]).
For a convergent sequencevp in a linear n,β-normed space E,
(2)limp→∞vp,w1,⋯,wn−1β=limp⟶∞vp,w1,⋯,wn−1β,for all w1,⋯,wn−1∈E.Lemma 6 (see [27]).
LetE,·,⋯,·β be a linear n,β-normed space, n≥2, 0<β≤1. If v1∈E and v1,w1,⋯,wn−1β=0 for all w1,⋯,wn−1∈E, then v1=0.


## 3. General Solution

Here, the authors discussed the general solution of the equation (1). Consider E and F are real vector spaces.Theorem 7.
Ifϕ:E⟶F be a mapping satisfies the functional equation (1) for all v1,v2,⋯,vl∈E, then the mapping ϕ:E⟶F is quadratic, that is ϕ satisfies the equality
(3)ϕv1+v2+ϕv1−v2=2ϕv1+2ϕv2,for all v1,v2∈E.Proof.
Suppose that the mappingϕ:E⟶F satisfies the functional equation (1). Replacing v1,v2,⋯,vl by 0,0,⋯,0 in (1), we get ϕ0=0. Now, setting v1,v2,⋯,vl by v,0,⋯,0 in (1), we get ϕ−v=ϕv for all v∈E. Therefore, the function ϕ is an even function. Substituting v1,v2,⋯,vl by v,v,0,⋯,0 and v,v,v,0,⋯,0 in (1), we obtain ϕ2v=22ϕv, ϕ3v=32ϕv, and so on, for every v∈E. In general, for any nonnegative integer n>0, we get ϕnv=n2ϕv for all v∈E. Next, replacing v1,v2,⋯,vl by v1,v2,0,⋯,0 in (1), we get our desired outcome.


## 4. Stability Results

Here, we consider2≠1 and examine the Hyers-Ulam stability of the functional equation (1).Let us assumeE and F are non-Archimedean β1-normed space and complete non-Archimedean n,β-normed space, respectively, where n≥2, 0<β,β1≤1.Define a mappingΔϕ:E⟶F by
(4)Δϕv1,v2,⋯,vl=∑1≤i<j≤lϕvi+vj+∑1≤i<j≤lϕvi−vj−2l−1∑1≤i≤lϕvi,for all v1,v2,⋯,vl∈E.Theorem 8.
Letτ∈0,∞ and s∈0,∞ with sβ1>β, and let χ:F×F×⋯×F⏟n−1⟶0,∞ be a function. Suppose that ϕ:E⟶F be a mapping satisfies
(5)Δϕv1,v2,⋯,vl,w1,⋯,wn−1β≤τ∑1≤j≤lvjβ1sχm1,⋯,mn−1,for all v1,v2,⋯,vl and m1,⋯,mn−1∈F. Then, there exists a unique quadratic mapping Q2:E⟶F satisfies
(6)ϕv−Q2v,w1,⋯,wn−1β≤2τ2−2βvβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F.Proof.
Replacingv1,v2,⋯,vl by v,v,0,⋯,0 in (5), we obtain
(7)ϕ2v−22ϕv,w1,⋯,wn−1β≤2τvβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. Dividing both sides by 22β, we have
(8)ϕ2v22−ϕv,w1,⋯,wn−1β≤22−2βτvβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. Replacing v by 2pv in (8), we attain
(9)ϕ2p+1v22p+1−ϕ2pv22p,m1,⋯,mn−1β≤22−2p+1βτ2pvβ1sχm1,⋯,mn−1≤22−2β2sβ1−2βpτvβ1sχm1,⋯,mn−1.
Assβ1>β and 2≠1, we obtain that
(10)limp⟶∞ϕ2p+1v22p+1−ϕ2pv22p,m1,⋯,mn−1β=0,for all v∈E,m1,⋯,mn−1∈F. From Remark 4, we conclude that the sequence ϕ2pv/22p is a Cauchy sequence in F. Since F is complete space, we can define Q2:E⟶F by
(11)Q2v=limp⟶∞ϕ2pv22p,for all v∈E. Next, our aim is to prove that the function Q2 is quadratic. From (5), (11) and Lemma 5 that
(12)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β=limp⟶∞2−2pβΔϕ2pv1,2pv2,⋯,2pvl,m1,⋯,mn−1β≤limp⟶∞2−2pβτ∑1≤j≤l2pvjβ1sχm1,⋯,mn−1≤limp⟶∞τ2sβ1−2βp∑1≤j≤lvjβ1sχm1,⋯,mn−1,for all v1,v2,⋯,vl∈E and m1,⋯,mn−1∈F. As sβ1>β and 2≠1, we obtain
(13)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β=0,for all v1,v2,⋯,vl∈E,m1,⋯,mn−1∈F. By Lemma 6, we have
(14)ΔQ2v1,v2,⋯,vl=0,for all v1,v2,⋯,vl∈E. Therefore, the function Q2 is quadratic. Switching v through 2v in (7) and divide by 22β, we attain
(15)ϕ22v24−ϕ2v22,m1,⋯,mn−1β≤2τ2−4β2vβ1sχm1,⋯,mn−1.
Thus, by (7) and (15), we reach
(16)ϕv−ϕ22v24,m1,⋯,mn−1β≤maxϕ2v22−ϕv,m1,⋯,mn−1β,ϕ22v24−ϕ2v22,m1,⋯,mn−1β≤max2τ2−2βvβ1sχm1,⋯,mn−1,2τ2−4βvβ1sχm1,⋯,mn−1.
Assβ1>β and 2≠1, we obtain
(17)ϕv−ϕ22v24,m1,⋯,mn−1β≤2−2β2τvβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. By induction on p, we can conclude that
(18)ϕv−ϕ2pv22p,m1,⋯,mn−1β≤22−2βτvβ1sχm1,⋯,mn−1,for all p∈ℕ, v∈E, and m1,⋯,mn−1∈F. Replacing v by 2v in (18) and dividing both sides by 22β, we have
(19)ϕ2v22−ϕ2p+1v22p+1,m1,⋯,mn−1β≤22−4βτ2vβ1sχm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and p∈ℕ. It follows from (7) and (19) that
(20)ϕv−ϕ2p+1v22p+1,m1,⋯,mn−1β≤22−2βτvβ1sχm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and p∈ℕ. Passing the limit as p tends to ∞ in inequality (18), we can get (6). Next, we want to prove that the function Q2 is unique. Let Q2′ be an another quadratic mapping which satisfies (6),
(21)Q2v−Q2′v,m1,⋯,mn−1β=2−2pβQ22pv−Q2′2pv,m1,⋯,mn−1β≤2−2pβmaxQ22pv−ϕ2pv,m1,⋯,mn−1β,ϕ2pv−Q2′2pv,m1,⋯,mn−1β≤22−2pβ2−2βτ2pvβ1sχm1,⋯,mn−1≤2τ2sβ1−2βp2−2βvβ1sχm1,⋯,mn−1.
Taking the limit asp tends to ∞ in the last inequality, we obtain that
(22)Q2v−Q2′v,m1,⋯,mn−1β=0,for all v∈E,m1,⋯,mn−1∈F. By Lemma 6, we conclude that Q2v=Q2′v for all v∈E. Hence, Q2 is the unique quadratic mapping which satisfies (6).Theorem 9.
Letτ∈0,∞, s∈0,∞ with sβ1<β. Let χ:F×F×⋯×F⏟n−1⟶0,∞ be a function. Suppose that a mapping ϕ:E⟶F satisfies
(23)Δϕv1,v2,⋯,vl,m1,⋯,mn−1β≤τ∑1≤j≤lvjβ1sχm1,⋯,mn−1,for all v1,v2,⋯,vl and m1,⋯,mn−1∈F. Then, there exists a unique quadratic mapping Q2:E⟶F satisfies
(24)ϕv−Q2v,m1,⋯,mn−1β≤2τ2−sβ1vβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F.Proof.
Switchingv1,v2,⋯,vl by v,v,0,⋯,0 in (23), we have
(25)ϕ2v−22ϕv,m1,⋯,mn−1β≤2τvβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. Interchanging v by v/2 in (25), we obtain
(26)ϕv−22ϕv2,m1,⋯,mn−1β≤2τ2−sβ1vβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. Replacing v by v/2p in (26), we reach
(27)22pϕv2p−22p+1ϕv2p+1,m1,⋯,mn−1β≤222pβτ2−p+1sβ1vβ1sχm1,⋯,mn−1≤22−sβ122β−sβ1pτvβ1sχm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. As sβ1<β and 2≠1, we have
(28)limp⟶∞22p+1ϕv2p+1−22pϕv2p,m1,⋯,mn−1β=0,for all v∈E and m1,⋯,mn−1∈F. From Remark 4, we conclude that the sequence 22pϕv/2p is a Cauchy sequence in F. As F is complete space. We can define Q2:E⟶F by
(29)Q2v=limp⟶∞22pϕv2p,for all v∈E. Next, our aim is to prove that the function Q2 is quadratic. From (23), (29) and Lemma 5 that
(30)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β=limp⟶∞22pβΔϕv12p,v22p,⋯,vl2p,m1,⋯,mn−1β≤limp⟶∞22pβτ∑1≤j≤lvj2pβ1sχm1,⋯,mn−1≤limp⟶∞τ22β−sβ1p∑1≤j≤lvjβ1sχm1,⋯,mn−1,for all v1,v2,⋯,vl∈E and m1,⋯,mn−1∈F. Since sβ1<β and 2≠1, we have
(31)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β=0,for all v1,v2,⋯,vl∈E,m1,m2,⋯,mn−1∈F. By Lemma 6, we obtain
(32)ΔQ2v1,v2,⋯,vl=0,for all v1,v2,⋯,vl∈E. Therefore, the function Q2 is quadratic. Switching v through v/2 in (25) and multiplying by ∣22β∣, we reach
(33)24ϕv22−22ϕv2,m1,⋯,mn−1β≤2τ22βv22β1sχm1,⋯,mn−1.
Thus, by (25) and (33), we obtain
(34)ϕv−24ϕv22,m1,⋯,mn−1β≤max22ϕv2−ϕv,m1,⋯,mn−1β,24ϕv22−22ϕv2,m1,⋯,mn−1β≤max2τv2β1sχm1,⋯,mn−1,2τ22βv22β1sχm1,⋯,mn−1.
Sincesβ1<β and 2≠1, we attain
(35)ϕv−24ϕv22,m1,⋯,mn−1β≤2−sβ12τvβ1sχm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. By induction on p, we can conclude that
(36)ϕv−22pϕv2p,m1,⋯,mn−1β≤22−sβ1τvβ1sχm1,⋯,mn−1,for all p∈ℕ, v∈E and m1,⋯,mn−1∈F. Replacing v by v/2 in (36) and multiplying both sides by ∣22β∣, we get.
(37)22ϕv2−22p+1ϕv2p+1,m1,⋯,mn−1β≤222β2−2sβ1τvβ1sχm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F. It follows from (25) and (37) that
(38)ϕv−22p+1ϕv2p+1,m1,⋯,mn−1β≤22−sβ1τvβ1sχm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and p∈ℕ. Passing the limit as p tends to ∞ in (36), we can get (24). Finally, we want to prove that the function Q2 is unique. Consider an another quadratic mapping Q2′ satisfying (24),
(39)Q2v−Q2′v,m1,⋯,mn−1β=22pβQ2v2p−Q2′v2p,m1,⋯,mn−1β≤22pβmaxQ2v2p−ϕv2p,m1,⋯,mn−1β,ϕv2p−Q2′v2p,m1,⋯,mn−1β≤222pβ2−sβ1τv2pβ1sχm1,⋯,mn−1≤2τ22β−sβ1p2−sβ1vβ1sχm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. Taking the limit as p tends to ∞, we obtain that
(40)Q2v−Q2′v,m1,⋯,mn−1β=0,for all v∈E,m1,⋯,mn−1∈F. By Lemma 6, we conclude that Q2v=Q2′v for all v∈E. So that the function Q2 is the unique quadratic function. Hence, the proof is completed.We obtain the following results of theorem with a generalized control function when the domainE is a vector space and codomain F be a complete non-Archimendean n,β-normed space, where n≥2 and 0<β≤1.Theorem 10.
Letψ:El⟶0,∞ be a function such that
(41)limp⟶∞122pβψ2pv1,2pv2,⋯,2pvl=0,for all v1,v2,⋯,vl∈E, and let χ:F×F×⋯×F⏟n−1⟶0,∞ be a function. Suppose ϕ:E→F be a mapping which satisfies
(42)Δϕv1,v2,⋯,vl,m1,⋯,mn−1β≤ψv1,v2,⋯,vlχm1,⋯,mn−1,for all v1,v2,⋯,vl∈E and m1,⋯,mn−1∈F. Then, there exists a quadratic mapping Q2:E⟶F such that
(43)ϕv−Q2v,m1,⋯,mn−1β≤ψ~vχm1,⋯,mn−1,where
(44)ψ~v=limp⟶∞max2−2iβψ2i−1v,2i−1v,0,⋯,0:1≤i≤p,for all v∈E,m1,⋯,mn−1∈F. Moreover, if
(45)limt⟶∞limp⟶∞max2−2iβψ2i−1v,2i−1v,0,⋯,0:1+t≤i≤p+t=0,for all v∈E. Then, the unique mapping Q2 is quadratic which satisfies the inequality (43).Proof.
Settingv1,v2,⋯,vl by v,v,0,⋯,0 in (42) and dividing both sides by 22β, we have
(46)ϕ2v22−ϕv,m1,⋯,mn−1β≤2−2βψv,v,0,⋯,0χm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. Replacing v by 2iv in (46) and dividing both sides by ∣22iβ∣, we obtain
(47)ϕ2i+1v22i+1−ϕ2iv22i,m1,⋯,mn−1β≤2−2iβ2−2βψ2iv,2iv,0,⋯,0χm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and i∈ℕ. Taking the limit as i tends to ∞ and considering (41), we attain
(48)limi⟶∞ϕ2i+1v22i+1−ϕ2iv22i,m1,⋯,mn−1β=0,for all v∈E and m1,⋯,mn−1∈F. Utilizing Remark 4, it is clear that the sequence ϕ2mv/22m is a Cauchy sequence. As F is a complete space. We can define the mapping Q2:E⟶F by
(49)limi⟶∞ϕ2i+1v22i+1−ϕ2iv22i,m1,⋯,mn−1β=0,for all v∈E,m1,⋯,mn−1∈F. Next, we want to prove that the function Q2 is quadratic. So,
(50)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β≤2−2pβΔQ22pv1,2pv2,⋯,2pvl,m1,⋯,mn−1β≤2−2pβψ2pv,2pv,0,⋯,0χm1,⋯,mn−1,for all v1,v2,⋯,vl∈E and m1,⋯,mn−1∈F. Taking the limit as p⟶∞ and considering (41), we arrive
(51)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β=0,for all v∈E,m1,⋯,mn−1∈F. Using Lemma 6, we conclude that Q2 is quadratic. Switching v through 2v in (46) and dividing by 22β, we obtain
(52)ϕ22v24−ϕ2v22,m1,⋯,mn−1β≤2−4βψ2v,2v,0,⋯,0χm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. Considering (46), we receive
(53)ϕv−ϕ22v24,m1,⋯,mn−1β≤max2−2βψv,v,0,⋯,0,2−4βψ2v,2v,0,⋯,0χm1,⋯,mn−1.
By induction onp, we reach
(54)ϕv−ϕ2pv22p,m1,⋯,mn−1β≤maxψ2t−1v,2t−1v,0,⋯,022tβ:1≤t≤pχm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. Replacing v by 2v in (54) and dividing both sides by ∣22β∣, we have
(55)ϕ2v22−ϕ2m+1v22m+1,m1,⋯,mn−1β≤maxψ2tv,2tv,0,⋯,022t+1β:1≤t≤pχm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and p∈ℕ, which together with (46) implies
(56)ϕv−ϕ2p+1v22p+1,m1,⋯,mn−1β≤maxψv,v,0,⋯,022β,ψ2tv,2tv,0,⋯,022t+1β:1≤t≤pχm1,⋯,mn−1≤max2−2t+1βψ2tv,2tv,0,⋯,0:0≤t≤pχm1,⋯,mn−1≤max2−2tβψ2t−1v,2t−1v,0,⋯,0:1≤t≤p+1χm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and p∈ℕ. Passing the limit as p tends to ∞ in (54), we get (43). Finally, we want to show that the quadratic function Q2 is unique. Consider an another quadratic function Q2′ satisfying (43). Hence,
(57)limp⟶∞2−2tβψ~2tv=limt⟶∞2−2tβlimp⟶∞max2−2iβψ2i+t−1v,2i+t−1v,0,⋯,0:1≤i≤p≤limt⟶∞limp⟶∞max2−2iβψ2i−1v,2i−1v,0,⋯,0:1+t≤i≤p+t,for all v∈E and m1,⋯,mn−1∈F. It follows from (45) that
(58)Q2v−Q2′v,m1,⋯,mn−1β≤limt⟶∞2−2tβQ22tv−Q2′2tv,m1,⋯,mn−1β≤limt⟶∞2−2tβmaxQ22tv−ϕ2tv,m1,⋯,mn−1β,ϕ2tv−Q2′2tv,m1,⋯,mn−1β≤limt⟶∞2−2tβψ~2tvχm1,⋯,mn−1=0.
From Lemma6, we conclude that the quadratic function Q2 is unique. This ends the proof of the theorem.Theorem 11.
Letψ:El→0,∞ be a mapping such that
(59)limp→∞22pβψv12p,v22p,⋯,vl2p=0,for all v1,v2,⋯,vl∈E, and let χ:F×F×⋯×F⏟n−1→0,∞ be a function. Suppose that ϕ:E→F be a mapping satisfies
(60)Δϕv1,v2,⋯,vl,m1,⋯,mn−1β≤ψv1,v2,⋯,vlχm1,⋯,mn−1,for all v1,v2,⋯,vl∈E and m1,⋯,mn−1∈F. Then, there exists a quadratic mapping Q2:E→F such that
(61)ϕv−Q2v,m1,⋯,mn−1β≤ψ~vχm1,⋯,mn−1,where
(62)ψ~v=limp→∞max22i−1βψ2−iv,2−iv,0,⋯,0:1≤i≤p,for all v∈E and m1,⋯,mn−1∈F. Moreover, if
(63)limt→∞limp→∞max22i−1βψ2−iv,2−iv,0,⋯,0:1+t≤i≤p+t=0,for all v∈E, then the quadratic mapping Q2 is unique which satisfies (61).Proof.
Switchingv1,v2,⋯,vl by v,v,0,⋯,0 in (60), we have
(64)ϕ2v−22ϕv,m1,⋯,mn−1β≤ψv,v,0,⋯,0χm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. Interchanging v by v/2 in (64), we obtain
(65)ϕv−22ϕv2,m1,⋯,mn−1β≤ψv2,v2,0,⋯,0χm1,⋯,mn−1,for all v∈E,m1,⋯,mn−1∈F. Replacing v by v/2i−1 in (65) and multiplying both sides by 22i−1β, we obtain
(66)22iϕv2i−22i−1ϕv2i−1,m1,⋯,mn−1β≤22i−1βψv2i,v2i,0,⋯,0χm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and i∈ℕ. Taking the limit as i tends to ∞ and considering (59), we attain
(67)limi→∞∥22iϕv2i−22i−1ϕv2i−1,m1,⋯,mn−1∥β=0,for all v∈E,m1,⋯,mn−1∈F. From Remark 4, we can conclude that the sequence 22mϕv/2m is a Cauchy sequence. We know that F is a complete space; we define a mapping Q2:E→F by
(68)limi→∞22iϕv2i−22i−1ϕv2i−1,m1,⋯,mn−1β=0,for all v∈E,m1,⋯,mn−1∈F. Next, we prove that Q2 is quadratic. So,
(69)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β≤22pβΔQ2v12p,v22p,⋯,vl2p,m1,⋯,mn−1β≤22pβψv2p,v2p,0,⋯,0χm1,⋯,mn−1,for all v1,v2,⋯,vl∈E and m1,⋯,mn−1∈F. Taking the limit as p→∞ and considering (59), we arrive
(70)ΔQ2v1,v2,⋯,vl,m1,⋯,mn−1β=0,for all v∈E and m1,⋯,mn−1∈F. By Lemma 6, we know that Q2 is quadratic. Replacing v by v/2 in (65) and multiplying both sides by 22β, we obtain
(71)24ϕv22−22ϕv2,m1,⋯,mn−1β≤22βψv22,v22,0,⋯,0χm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. Considering (65), we receive
(72)ϕv−24ϕv22,m1,⋯,mn−1β≤maxψv2,v2,0,⋯,0,22βψv22,v22,0,⋯,0χm1,⋯,mn−1.
By induction onp, we reach
(73)ϕv−22pϕv2p,m1,⋯,mn−1β≤max22t−1βψ2−tv,2−tv,0,⋯,0:1≤t≤pχm1,⋯,mn−1,for all v∈E and m1,⋯,mn−1∈F. Replacing v by v/2 in (73) and multiplying both sides by ∣22β∣, we have
(74)∥22ϕv2−22m+1ϕv2m+1,m1,⋯,mn−1∥β≤max22t−1βψv2t,v2t,0,⋯,0:1≤t≤pχm1,⋯,mn−1,which together with (65) implies
(75)∥ϕv−22p+1ϕv2p+1,m1,⋯,mn−1∥β≤maxψv2,v2,0,⋯,0,22t−1βψv2t,v2t,0,⋯,0:1≤t≤pχm1,⋯,mn−1≤max22t−1βψv2t,v2t,0,⋯,0:1≤t≤p+1χm1,⋯,mn−1,for all v∈E, m1,⋯,mn−1∈F, and p∈ℕ. Passing the limit as p tends to ∞ in (73), we get (61). At the end, we show that the quadratic function Q2 is unique. Consider an another quadratic mapping Q2′ which satisfies (61). Therefore,
(76)limp→∞22tβψ~v2t=limt→∞22tβlimp→∞max22i−1βψv2i+t,v2i+t,0,⋯,0:1≤i≤p≤limt→∞limp→∞max22i−1βψv2i,v2i,0,⋯,0:1+t≤i≤p+t,for all v∈E. It follows from (63) that
(77)Q2v−Q2′v,m1,⋯,mn−1β≤limt→∞22tβQ2v2t−Q2′v2t,m1,⋯,mn−1β≤limt→∞22tβmaxQ2v2t−ϕv2t,m1,⋯,mn−1β,ϕ2tv−Q2′2tv,m1,⋯,mn−1β≤limt→∞22t−1βψ~v2tχm1,⋯,mn−1=0.
By Lemma6, we conclude that the quadratic function Q2 is unique. This ends the proof of the theorem.


## 5. Conclusion

In this work, we introduced a new finite dimensional quadratic functional equation (1) and obtained its solution to show which satisfies the quadratic properties. Mainly, investigated Hyers-Ulam stability of considered domain E is a non-Archimedean β-normed space in Theorem 8 and Theorem 9 and examined by considered the domain E is a vector space in Theorem 10 and Theorem 11. The results obtained in this article may be further generalized to be in non-Archimedean quasin,β-Banach spaces. This could be a potential future work.



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*Source: 9953214-2021-05-04.xml*