# Ulam-Hyers Stability Results ofλ-Quadratic Functional Equation with Three Variables in Non-Archimedean Banach Space and Non-Archimedean Random Normed Space

Authors: Ly Van An; Kandhasamy Tamilvanan; R. Udhayakumar; Masho Jima Kabeto; Ly Van Ngoc
Journal: Journal of Function Spaces
 (2022)

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## Abstract

In this paper, we introduce theλ-quadratic functional equation with three variables and obtain its general solution. The main aim of this work is to examine the Ulam-Hyers stability of this functional equation in non-Archimedean Banach space by using direct and fixed point techniques and examine the stability results in non-Archimedean random normed space.


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## Body

## 1. Introduction

One of the most important areas of research in mathematics is the investigation of stability issues for functional equations, which has its origins in concerns of applied mathematics. The first question about the stability of homomorphisms was given by Ulam [1] as follows.Given a groupM,∗, a metric group M′,· with the metric d, and a function ϕ from B and B′, does there exists δ>0 satisfying
(1)dϕu∗v,ϕu·ϕv≤δ,for all u,v∈B, then there exists a homomorphism h:B⟶B′ such that
(2)dϕu,hu≤ε,for all u∈B?Ulam’s question on Banach spaces was partially answered affirmatively by Hyers [2]. By assuming an infinite Cauchy difference, Aoki [3] expanded Hyers’ and Rassias’ theorems for additive and linear mappings, respectively. Using the same method as Rassias [4], Gajda [5] discovered a positive solution to the question p>1. Rassias and Šemrl [6], as well as Gajda [5], have proved that a Rassias’ type theorem cannot be formed for p=1.The functional equation(3)ϕu+v=ϕu+ϕvis known as the Cauchy additive equation.Since the functionϕu=u is the solution of the functional equation (3), every solution of the additive functional equation (3) is called as an additive function. Every solution of the functional equation (3), in particular, is called as an additive function.The functional equation(4)ϕu+v+ϕu−v=2ϕu+2ϕvis known as the quadratic functional equation.Since the functionϕu=u2 is the solution of the functional equation (4), each solution of the functional equation (4) is called as a quadratic function. Every solution of functional equation (4), in particular, is called as a quadratic mapping.Skof [7] established the stability of the quadratic functional equation for the function f between normed space and complete normed space. The authors [8–14] recently examined the Ulam-Hyers stability results for the following α-functional equation
(5)2fx−2fy=fx+y+α−2fαx−y,in non-Archimedean Banach spaces.The Skof theorem still applies when the relevant domainB is replaced by an Abelian group, according to Cholewa [15]. See [15–21] for other functional equations. A survey of the Ulam-Hyers stability results of functional equations was conducted by Brillouët-Belluot [22]. Park and Kim [11] demonstrated the Ulam-Hyers stability of quadratic α-functional equation.In this paper, the authors present a newλ-quadratic functional equation with three variables as
(6)2ξϑ1+ϑ22+2ξϑ3=ξϑ1+ϑ22+ϑ3+λ−2ξλϑ1+ϑ22−ϑ3,where λ is a fixed non-Archimedean number with λ−2≠3, and its general solution was obtained. The motivation behind this study is to investigate the Ulam-Hyers stability results for the above functional equation (6) in non-Archimedean Banach space by using direct and fixed point methods and non-Archimedean random normed space.The following is the structure of this paper: in Section2, we recall some fundamental notions and definitions, in Section 3, we look at the general solution of the equation (6), where V and W are two vector spaces. We investigate the Ulam-Hyers stability in non-Archimedean Banach space by using fixed point method and direct method in Sections 4 and 5, where V is a non-Archimedean normed space, W is a non-Archimedean Banach space, and 2≠1 is a non-Archimedean Banach space. In Section 6, we recall some fundamental notions and results and investigate the Ulam-Hyers stability in non-Archimedean random normed space.


## 2. Preliminaries

To reach our major results, we use certain fundamental notations in [8, 10, 11].A map·:K⟶0,∞ is a valuation such that zero is the only one element having the zero valuation, k1k2=k1k2, and the inequality of the triangle holds true, that is, k1+k2≤k1+k2, for all k1,k2∈K.We call a fieldK valued if K holds a valuation. Examples of valuations include the typical absolute values of ℝ and ℂ.Consider a valuation that satisfies a criterion that is stronger than the triangle inequality. A· is called a non-Archimedean valuation if the triangle inequality is replaced by k1+k2≤maxk1,k2, for all k1,k2∈K, and a field is called a non-Archimedean field. Evidently, −1=1=1 and n are greater than or equal to 1, for all n in ℕ. The map · takes everything except 0 for 1, and 0=0 is a basic example of a non-Archimedean valuation.Definition 1.
LetV be a linear space over K with ·. A mapping ·:V⟶0,∞ is known as a non-Archimedean norm if it satisfies
(i)
v=0 if and only if v=0.(ii)
rv=rv, v∈V, and r∈K.(iii)
the strong triangle inequality.(7)v1+v2≤maxv1,v2,v1,v2∈V.
Then,V,· is called a non-Archimedean normed space. Every Cauchy sequence converges in a complete non-Archimedean normed space, which we call a complete non-Archimedean normed space.Definition 2.
LetV be a non-Archimedean normed space and a sequence vp in V. Then,
(1)
a sequencevpp=1∞ in V is a Cauchy sequence if vp+1−vpp=1∞ converges to 0.(2)
vp is called convergent if, for any ε>0, there is an integer p>0 in ℕ and v∈V satisfies
(8)∥vp−v∥≤εforallp≥ℕ,for every p,q≥ℕ. Then, we called as v is a limit of the sequence vp and denoted by limp⟶∞vp=v.(3)
if every Cauchy sequence in a non-Archimedean normed spaceV converges, it is called a non-Archimedean Banach space.Theorem 3 (alternative fixed point theorem).
LetV,d be a generalized complete metric space and a strictly contractive mapping M:V⟶V with Lipschitz constant 0<L<1. Then, for all v1∈V, either
(9)dMmv1,Mm+1v1=∞,m≥m0,
or there exists a positive integerm0 such that
(i)
dMmv1,Mm+1v1<∞,m≥m0.(ii)
the sequenceMmv1m∈ℕ converges to a fixed point v1∗ of M.(iii)
v1∗ is the unique fixed point of M in V∗=v2∈VdMm0v1,v2<∞.(iv)
dv2,v1∗≤1/1−LdMv2,v2, for all v2∈V∗.


## 3. Solution

Lemma 4.
If a mappingξ:V⟶W satisfies the functional equation (6) for all ϑ1,ϑ2,ϑ3∈V, then the function ξ is quadratic.Proof.
A mappingξ:V⟶W satisfies the functional equation (6). Replacing ϑ1,ϑ2,ϑ3 by 0,0,0 in (6), we obtain
(10)3ξ0=λ−2ξ0.
This implies thatξ0=0. Replacing ϑ1,ϑ2,ϑ3 by ϑ,ϑ,0 in (6), we obtain
(11)ξϑ=λ−2ξλϑ,and so
(12)ξλv=λ2ξϑ,for all ϑ∈V. Thus, equation (6) is reduced as
(13)2ξϑ1+ϑ22+2ξϑ3=ξϑ1+ϑ22+ϑ3+ξϑ1+ϑ22−ϑ3,
for allϑ1,ϑ2,ϑ3∈V. Now, replacing ϑ1=ϑ2=ϑ3=ϑ in (13), we get
(14)ξ2ϑ=22ξϑ,for all ϑ∈V. Again, replacing ϑ by 2ϑ in (14), we have
(15)ξ22ϑ=24ξϑ,
for allϑ∈V. From equalities (14) and (15), we can conclude that for any integer p>0, we get
(16)ξ2pϑ=22pξϑ,for all ϑ∈V. Now, replacing ϑ1,ϑ2,ϑ3 by ϑ1,ϑ1,ϑ2 in (13), we reach (3) for all ϑ1,ϑ2∈V. Hence, the function ξ is quadratic.For our notational simplicity, we use the following abbreviation:(17)Δξϑ1,ϑ2,ϑ3=2ξϑ1+ϑ22+2ξϑ3−ξϑ1+ϑ22+ϑ3−λ−2ξλϑ1+ϑ22−ϑ3.


## 4. Stability of (6) in Non-Archimedean Banach Space: Direct Method

Theorem 5.
Letρ:V3⟶0,∞ be a mapping and a mapping ξ:V⟶W such that ξ0=0 and
(18)limj⟶∞22jρ2−jϑ1,2−jϑ2,2−jϑ3=0,(19)∥Δξϑ1,ϑ2,ϑ3∥≤ρϑ1,ϑ2,ϑ3,for all ϑ1,ϑ2,ϑ3∈V. Then, there exists a unique quadratic mapping Q:V⟶W satisfying
(20)∥ξϑ−Qϑ∥≤supj∈ℕ22j−1ρϑ12j,ϑ22j,ϑ32j,for all ϑ∈V.Proof.
Settingϑ1=ϑ2=ϑ3=ϑ in (19), we have
(21)ξ2ϑ−22ξϑ≤ρϑ,ϑ,ϑ,ϑ∈V.
Thus, from inequality (21), it implies that
(22)ξϑ−22ξϑ2≤ρϑ2,ϑ2,ϑ2,for all ϑ∈V. Replacing ϑ by ϑ/2 in (22), we obtain
(23)22ξϑ2−24ξϑ22≤22ρϑ22,ϑ22,ϑ22,for all ϑ∈V. Hence,
(24)22lξϑ2l−22mξϑ2m≤max22lξϑ2lv−22l+1ξϑ2l+1∥,⋯,∥22m−1ξϑ2m−1−22mξϑ2m≤max22l∥ξϑ2l−22ξϑ2l+1,⋯,22m−1ξϑ2m−1−22ξϑ2m∥≤supj∈l,l+1,⋯22jρϑ2j+1,ϑ2j+1,ϑ2j+1,for all m>l>0 and all ϑ∈V. From inequality (24), the sequence 22nξϑ/2n is a Cauchy sequence for all ϑ∈V. Since W is complete, thus the sequence 22nξϑ/2n is convergent. Now, we can define a mapping Q:V⟶W by
(25)Qϑ≔liml⟶∞22lξϑ2l,ϑ∈V.
Takingl=0 and passing the limit m⟶∞ in (24), we obtain (20). From inequalities (18) and (19), we have
(26)∥ΔQϑ1,ϑ2,ϑ3∥=limj⟶∞22j∥Δξ2−jϑ1,2−jϑ2,2−jϑ3∥≤limj⟶∞22jρ2−jϑ1,2−jϑ2,2−jϑ3=0.
From above, we conclude thatΔQϑ1,ϑ2,ϑ3=0 for all ϑ1,ϑ2,ϑ3∈V. By using Lemma 4, the function Q is quadratic. Consider another quadratic mapping T:V⟶W satisfying (20). Then, we have
(27)∥Qϑ−Tϑ∥=∥22qQϑ2q−22qTϑ2q∥≤max22qQϑ2q−22qξϑ2q∥,∥22qTϑ2q−22qξϑ2q≤supj∈ℕ22q+j−1ρϑ2j+1,ϑ2j+1,ϑ2j+1⟶0asq⟶∞,for all ϑ∈V. Thus, we can conclude that Tϑ=Qϑ,ϑ∈V. Hence, the function Q is unique. Thus, the unique quadratic mapping Q:V⟶W satisfies (20). Hence, the proof of the theorem is now completed.Theorem 6.
Letρ:V3⟶0,∞ be a mapping and a mapping ξ:V⟶W such that ξ0=0 and
(28)limj⟶∞122jρ2j−1ϑ1,2j−1ϑ2,2j−1ϑ3=0,and (19) for all ϑ1,ϑ2,ϑ3∈V. Then, there exists a unique quadratic mapping Q:V⟶W satisfying
(29)∥ξϑ−Qϑ∥≤Supj∈ℕ122j−1ρ2j−1ϑ1,2j−1ϑ2,2j−1ϑ3,for all ϑ∈V.Proof.
Settingϑ1=ϑ2=ϑ3=ϑ in (19), we have
(30)ξ2ϑ−22ξϑ≤ρϑ,ϑ,ϑ,ϑ∈V.
From inequality (30), we obtain
(31)ξϑ−122ξ2ϑ≤122ρϑ,ϑ,ϑ,ϑ∈V.
Replacingϑ by 2ϑ in (31), we get
(32)ξ2ϑ22−124ξ22ϑ≤1222ρ2ϑ,2ϑ,2ϑ,for all ϑ∈V. Hence,
(33)122lξ2lϑ−122mξ2mϑ≤max122lξ2lϑ−122l+1ξ2l+1ϑ∥,⋯,∥122m−1ξ2m−1ϑ−122mξ2mϑ≤max122lξ2lϑ−122m−1ξ2l+1ϑ,⋯,122m−1ξ2m−1ϑ−122ξ2mϑ≤supj∈l,l+1,⋯122j+1ρ2jϑ1,2jϑ2,2jϑ3,for all m>l>0 and all ϑ∈V. From inequality (33), the sequence 1/22nξ2nϑ is a Cauchy sequence for all ϑ∈V. Since W is complete, the sequence 1/22nξ2nϑ is convergent. Now, we can define a mapping Q:V⟶W by
(34)Qϑ≔limn⟶∞122nξ2nϑ,ϑ∈V.
The remaining proof is the same as the proof of Theorem5.Corollary 7.
Letξ:V⟶W be a mapping such that ξϑ=0 and
(35)∥Δϑ1,ϑ2,ϑ3∥≤θ∥ϑ1∥r+∥ϑ2∥r+∥ϑ3∥r,for all ϑ1,ϑ2,ϑ3∈V, where r and θ are in ℝ+ with r<2. Then, there exists a unique quadratic mapping Q:V⟶W satisfying
(36)∥ξϑ−Qϑ∥≤3θ2r∥ϑ∥r,for all ϑ∈V.Corollary 8.
Letξ:V⟶W be a mapping such that ξϑ=0 and
(37)Δϑ1,ϑ2,ϑ3≤θϑ1r+ϑ2r+ϑ3r,for all ϑ1,ϑ2,ϑ3∈V, where r and θ are in ℝ+ with r>2. Then, there exists a unique quadratic mapping Q:V⟶W satisfying
(38)ξϑ−Qϑ≤3θ22ϑr,for all ϑ∈V.


## 5. Stability of (6) in Non-Archimedean Banach Space: Fixed Point Method

Theorem 9.
Letρ:V3⟶0,∞ be a mapping such that there exists L<1 with
(39)ρ2−1ϑ1,2−1ϑ2,2−1ϑ3≤L4ρ2−1ϑ1,2−1ϑ2,2−1ϑ3,for all ϑ1,ϑ2,ϑ3∈V. If a mapping ξ:V⟶W such that ξ0=0 and (19) for all ϑ1,ϑ2,ϑ3∈V, then there exists a unique quadratic mapping Q:V⟶W satisfying
(40)ξϑ−Qϑ≤L221−Lρϑ,ϑ,ϑ,for all ϑ∈V.Proof.
Settingϑ1=ϑ2=ϑ3=ϑ in (19), we obtain
(41)ξ2ϑ−4ξϑ≤ρϑ,ϑ,ϑ,for all ϑ∈V. Consider
(42)S≔q:V⟶W,q0=0,and the generalized metric d defined by
(43)dp,q≔infε∈ℝ:pϑ−qϑ≤ερϑ,ϑ,ϑ,∀ϑ∈V,here, as usual, infξ=+∞. Clearly, S,q is complete (see [23]). Next, consider a mapping J:S⟶S defined by
(44)Jpϑ≔22pϑ2,ϑ∈V
For allp,q∈S such that dp,q=ε, then
(45)pϑ−qϑ≤ερϑ,ϑ,ϑ,for all ϑ∈V. Hence,
(46)Jpϑ−Jqξϑ=22p2−1ϑ−22qξ2−1ϑ≤22ερ2−1ϑ,2−1ϑ,2−1v≤22εL22ρϑ,ϑ,ϑ≤22Lερϑ,ϑ,ϑ,for all ϑ∈V. Thus,
(47)dp,q=ε⇒dJp,Jq≤Lε.
This concludes that(48)dJp,Jq≤Ldp,q,p,q∈S.
From inequality (41),
(49)ξϑ−22ξϑ2≤ρ2−1ϑ,2−1ϑ,2−1ϑ≤L22ρϑ,ϑ,ϑ,ϑ∈V.
Therefore,(50)dξ,Jξ≤122L,ϑ∈V.
By using Theorem3, there exists a mapping Q:V⟶W satisfying the following conditions:
(1)
Q is a fixed point of J, i.e.,(51)Qϑ=22Q2−1ϑ∀ϑ∈V.
In the set below, the functionQ is the unique fixed point J.
(52)M=p∈S:dξ,p<∞.
This proves that the uniqueness of the functionQ satisfies (51) such that there exists ε∈0,∞ such that
(53)ξϑ−Qϑ≤ερϑ,ϑ,ϑ,ϑ∈V.(2)
dJlξ,Q tends to 0 as taking the limit l⟶∞. This implies
(54)liml⟶∞4nξ2−nϑ=Qϑ,forallϑ∈V.(3)
dξ,Q≤1/1−Ldξ,Jξ, which implies
(55)ξϑ−Qϑ≤L221−Lρϑ,ϑ,ϑ,forallϑ∈V.
From (39) and (51),
(56)∥ΔQϑ1,ϑ2,ϑ3∥=limj⟶∞22j∥Δξ2−jϑ1,2−jϑ2,2−jϑ3∥≤limj⟶∞22jρ2−jϑ1,2−jϑ2,2−jϑ3=0,for all ϑ1,ϑ2,ϑ3∈V. Thus,
(57)ΔQϑ1,ϑ2,ϑ3=0,for all ϑ1,ϑ2,ϑ3∈V. By using Lemma 4, the function Q is quadratic. Hence, the proof of the theorem is now completed.Theorem 10.
Letρ:V3⟶0,∞ be a mapping such that there exists L<1 with
(58)ρϑ1,ϑ2,ϑ3≤L22ρ2−1ϑ1,2−1ϑ2,2−1ϑ3,ϑ1,ϑ2,ϑ3∈V.
If a mappingξ:V⟶W such that ξ0=0 and (19) for all ϑ1,ϑ2,ϑ3∈V, then there exists a unique quadratic mapping Q:V⟶W satisfying
(59)ξϑ−Qϑ≤1221−Lρϑ,ϑ,ϑ,for all ϑ∈V.Proof.
Settingϑ1=ϑ2=ϑ3=ϑ in (19), we have
(60)ξ2ϑ−4ξϑ≤ρϑ,ϑ,ϑ,for all ϑ∈V. From the inequality (60), we get
(61)ξϑ−122ξϑ≤122ρϑ,ϑ,ϑ,ϑ∈V.
The generalized metric spaceS,d is defined in the proof of Theorem 9. Consider a mapping J:S⟶S defined by
(62)Jpϑ≔122p2ϑ,ϑ∈V.
From inequality (61),
(63)dξ,Jξ≤122.
Hence,(64)ξϑ−Qϑ≤1221−Lρϑ,ϑ,ϑ,ϑ∈V.
The remaining proof is the same as in the proof of Theorem9.Corollary 11.
Letξ:V⟶W be a mapping such that ξ0=0 and
(65)Δξϑ1,ϑ2,ϑ3≤θ∑i=13ϑir,for all ϑ1,ϑ2,ϑ3∈V, where r and θ are in ℝ+ with r<2; then there exists a unique quadratic mapping Q:V⟶W satisfying
(66)ξϑ−Qϑ≤2θϑr2r−22,for all ϑ∈V.Corollary 12.
Letξ:V⟶W be a mapping such that ξ0=0 and
(67)Δξϑ1,ϑ2,ϑ3≤θ∑i=13ϑir,for all ϑ1,ϑ2,ϑ3∈V, where r and θ are in ℝ+ with r>2; then there exists a unique quadratic mapping Q:V⟶W satisfying
(68)ξϑ−Qϑ≤2θ22r−2ϑr,for all ϑ∈V.


## 6. Stability of (6) in Non-Archimedean Random Normed Space

Definition 13 [24].
A random normed space is tripleV,μ,T, where V is a vector space, T is a continuous t−norm, and a mapping μ:V⟶D+ satisfies
(RN1)μϑt=ε0t,∀t>0 if and only if ϑ=0.
(RN2)μλϑt=μϑt/λ for all ϑ∈V, λ≠0.
(RN3)μϑ1+ϑ2t1+t2≥Tμϑ1t1,μϑ2t2 for all ϑ1,ϑ2∈V and t1,t2≥0.Definition 14 [25].
A random normed spaceV,μ,T is said to be non-Archimedean random normed space if it satisfies
(NAR1)μϑt=ε0t for all t>0 if and only if ϑ=0.
(NAR2)μλϑt=μϑ1/λ for all ϑ∈V,t>0,λ≠0.
(NAR3)μϑ1+ϑ2maxt1,t2≥Tμϑ1t1,μϑ2t2 for all ϑ1,ϑ2∈V and t1,t2≥0.
It is clear that if (NAR3) holds, then so(69)RN3μϑ1+ϑ2t+s≥Tμϑ1t,μϑ2s.Example 1 [25].
Let a non-Archimedean normed spaceV,· and we define
(70)μϑt=tt+∥ϑ∥,for all ϑ∈V and all t>0. Then, the triple V,μ,TM is a non-Archimedean random normed space.Definition 15 [25].
LetV,μ,T be a non-Archimedean random normed space and a sequence ϑn in V. Then, the sequence ϑn is called as convergent if there exist ϑ∈V such that
(71)limn⟶∞μn−ϑt=1,for all t>0. In particular, ϑ is called the limit of the sequence ϑn.Here, letV be a vector space over a non-Archimedean field K and W,μ,T be a non-Archimedean random Banach space over K. And consider that 2≠0 in K.Next, we define a random approximately quadratic function. Let a distribution mappingψ:V×V⟶0,∞ satisfies ψϑ1,ϑ2,ϑ3,· which is symmetric and nondecreasing and
(72)ψλϑ,λϑ,λϑ,t≥ψϑ,ϑ,ϑ,tλ,for all ϑ∈V and all λ≠0.Definition 16.
A functionξ:V⟶W is called as a ψ-approximately quadratic if
(73)μ2ξϑ1+ϑ2/2+2ξϑ3−ξϑ1+ϑ2/2+ϑ3−λ−2ξλϑ1+ϑ2/2−ϑ3≥ψϑ1,ϑ2,ϑ3,t,for all ϑ1,ϑ2,ϑ3∈V and t>0.Theorem 17.
Let a functionξ:V⟶W be a ψ-approximately quadratic mapping. If for some real number α>0, and some integer k, k>1 with α>2k,
(74)ψ2−kϑ1,2−kϑ2,2−kϑ3,t≥ψϑ1,ϑ2,ϑ3,αt,for all ϑ1,ϑ2,ϑ3∈V and t>0, and
(75)limn⟶∞Tj=n∞Mϑ,αjt2kj=1,for all ϑ∈V and every t>0; then there exists a unique quadratic mapping Q:V⟶W such that
(76)μξϑ−Qϑt≥Ti=1∞Mϑ,αi+1t2ki,where
(77)Mϑ,t≔Tψϑ,ϑ,ϑ,tψ2ϑ,2ϑ,2ϑ,t,⋯,ψ2k−1ϑ,2k−1ϑ,2k−1ϑ,t,for all ϑ∈V and all t>0.Proof.
First, we demonstrate by induction onj that for all ϑ∈V, t>0 and j>0,
(78)μξ2jϑ−22jξϑt≥Mjϑ,t≔Tψϑ,ϑ,ϑ,t,⋯,ψ2j−1ϑ,2j−1ϑ,2j−1ϑ,t.
Settingϑ1=ϑ2=ϑ3=ϑ in (73), we obtain
(79)μξ2ϑ−22ξϑt≥ψϑ,ϑ,ϑ,t,for all ϑ∈V and all t>0. This proves that (78) for j=1. Suppose that (78) holds for some j>0. Replacing ϑ by 2jϑ in (73), we get
(80)μξ2j+1ϑ−22ξ2jϑt≥ψ2jϑ,2jϑ,2jϑ,t,for all ϑ∈V and all t>0. Since 22≤1,
(81)μξ2j+1ϑ−22j+1ξϑt≥Tμξ2j+1ϑ−22ξ2jϑt,μ22ξ2jϑ−22j+1ξϑt=Tμξ2j+1ϑ−22ξ2jϑt,μξ2jϑ−22jξϑt22Tμξ2j+1ϑ−22ξ2jϑt,μξ2jϑ−22jξϑt≥Tψ2jϑ,2jϑ,2jϑ,t,Mjϑ,t=Mj+1ϑ,t,for all ϑ∈V. Thus, condition (78) holds for all j>0. In particular,
(82)μξ2kϑ−22kξϑt≥Mϑ,t,for all ϑ∈V and all t>0. Replacing ϑ by 2−k+knϑ in (82) and using the inequality (74), we have
(83)μξϑ/2kn−22kξϑ/2kn+k≥Mϑ2kn+k,t≥Mϑ,αn+1t;n=0,1,2,⋯,for all ϑ∈V and all t>0. Then,
(84)μ22knξϑ/2kn−22kn+1ξϑ/2kn+1t≥Mϑ,αn+122knt;n=0,1,2,⋯,for all ϑ∈V and all t>0. Hence,
(85)μ2knξϑ/2kn−22kn+pξϑ/2kn+pt≥Tj=nn+pμ2kjξϑ/2kj−22kj+pξϑ/2kj+pt≥Tj=nn+pMϑ,αj+122kjt≥Tj=nn+pMϑ,αj+12kjt.
Sincelimn⟶∞Tj=n∞Mϑ,αj+1/2kjt=1 for all ϑ∈V and all t>0, 22knξϑ/2knn∈N is a Cauchy sequence in W,μ,T. Hence, we can define a mapping Q:V⟶W such that
(86)limn⟶∞μ22knξϑ/2kn−Qϑt=1,for all ϑ∈V and all t>0. Now, for all n≥1,
(87)μξϑ−22knξϑ/2knt=μ∑i=0n−122kiξϑ/2ki−22ki+1ξϑ/2ki+1t≥Ti=0n−1μ22kiξϑ/2ki−22ki+1ξϑ/2ki+1t≥Ti=0n−1Mϑ,αi+1t22ki,for all ϑ∈V and t>0. Thus,
(88)μξϑ−Qϑt≥Tμξϑ−22knξϑ/2kn,μ22knξϑ/2kn−Qϑt≥TTi=0n−1Mϑ,αi+1t22ki,μ22knξϑ/2kn−Qϑt.
By taking the limitn⟶∞, we have
(89)μξϑ−Qϑt≥Ti=1∞Mϑ,αi+1t2ki.
This shows that (76) holds. Since T is continuous, by a well-known result in probabilistic metric space (see, e.g., [[26], Chapter 12]), that
(90)limn⟶∞μ2knΔξ2−knϑ1,2−knϑ2,2−knϑ3t=μΔQϑ1,ϑ2,ϑ3t,for all t>0.
On the other hand, replacingϑ1,ϑ2,ϑ3 by 2−knϑ1,2−knϑ2,2−knϑ3, respectively, in (73) and using (NAR2) and (74), we get
(91)μ2knΔξ2−knϑ1,2−knϑ2,2−knϑ3t≥ψ2−knϑ1,2−knϑ2,2−knϑ3,t2kn≥ψϑ1,ϑ2,ϑ3,αnt2kn.
Sincelimn⟶∞ψϑ1,ϑ2,ϑ3,αnt/2kn=1, we can conclude that the function Q is quadratic. Consider another quadratic mapping Q′:V⟶W such that μQ′ϑ−ξϑt≥Mϑ,t for all ϑ∈V and all t>0; then for all n∈N and ϑ∈V and all t>0,
(92)μQϑ−Q′ϑt≥TμQϑ−24knξϑ/2knt,μ22knξϑ/2kn−Q′ϑt,t.
From condition (86), we arrive at the conclusion that Q=Q′.Corollary 18.
Let a functionξ:V⟶W be a ψ-approximately quadratic. If for some real number α>0 and some integer k, k>1, with 2k<α,
(93)ψ2−kϑ1,2−kϑ2,2−kϑ3,t≥ψϑ1,ϑ2,ϑ3,αt,for all ϑ1,ϑ2,ϑ3∈V and t>0, then there exists a unique quadratic mapping Q:V⟶W satisfying
(94)μξϑ−Qϑt≥Ti=1∞Mϑ,αi+1t2ki,where
(95)Mϑ,t≔Tψϑ,ϑ,ϑ,t,ψ2ϑ,2ϑ,2ϑ,t,⋯,ψ2k−1ϑ,2k−1ϑ,2k−1ϑ,t,for all ϑ∈V and all t>0.Proof.
Since(96)limn⟶∞Mϑ,αjt2kj=1,for all ϑ∈V and all t>0 and T is of Hadzic type, from Proposition 2.1 in [25], it follows that
(97)limn⟶∞Tj=n∞Mϑ,αjt2kj,for all ϑ∈V and t>0. Now, we can obtain our needed result by using Theorem 17Example 2.
Let a non-Archimedean random normed spaceV,μ,TM, in which
(98)μϑt=tt+∥ϑ∥,for all ϑ∈V and every t>0, and let W,μ,TM be a complete non-Archimedean random normed space (see Example 1). Now, we can define
(99)ψϑ1,ϑ2,ϑ3,t=t1+t.
It is obvious that (74) holds for α=1. Furthermore,
(100)Mϑ,t=t1+t.We obtain
(101)limn⟶∞TM,j=n∞Mϑ,αjt2kj=limn⟶∞limm⟶∞TM,j=nmMϑ,t2kj=limn⟶∞limm⟶∞tt+2kn=1,for all ϑ∈V and all t>0.


## 7. Conclusion

In this paper, we introducedλ-quadratic functional equation and obtained its general solution. In Section 4 and Section 5, we investigated Ulam-Hyers stability of equation (6) by using direct method and fixed point method in non-Archimedean Banach space, and also in Section 6, we investigated the Ulam-Hyers stability results in non-Archimedean random normed space. The direct method requires us to find the Cauchy sequence and prove that every Cauchy sequence is convergent, as well as prove the uniqueness of the function; this method was introduced by Hyers [2], and the fixed point method requires us to use the Banach contraction principle and Lipschitz constant L to obtain the stability results of the functional equation; this method was introduced by Radu [27]. The fixed point method gives more accurate stability results when compared with the direct method. Finally, these stability results generalized the findings of [11].



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