# Quartic Functional Equation: Ulam-Type Stability inβ,p-Banach Space and Non-Archimedean β-Normed Space

Authors: Ravinder Kumar Sharma; Sumit Chandok
Journal: Journal of Mathematics
 (2022)

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## Abstract

In this study, we use the alternative fixed-point approach and the direct method to examine the generalized Hyers–Ulam stability of the quartic functional equationgu+mv+gu−mv=2m−17m−9gu+2m2−1m2gv−m−12g2u+m2gu+v+gu−v, with a fixed positive integer m/ge2 in the context of β,p-Banach space. In non-Archimedean β-normed space, we also verify Hyers–Ulam stability for the quartic functional equation stated. Many of the findings in the literature are improved and generalized by our findings.


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## Body

## 1. Introduction

In the theory of Ulam’s stability, one can find the efficient tool to evaluate the errors that are to study the existence of an exact solution of the perturbed functional equation, which is not far from the given function. Ulam gave the core reason for studying the stability of functional equations in a discussion at the University of Wisconsin in 1940 [1]. Hyers [2] presented a partial affirmative response to the question of Ulam in Banach space. Following that, Aoki [3] generalized Hyers theorem for additive mapping, and Rassias [4] generalized Hyers theorem for linear mapping by taking into account an unbounded Cauchy difference. In 1994, Găvruta [5] generalized Rassias theorem and discussed the stability of linear functional equations. Several mathematicians have obtained numerous results about the stability of functional equations over the last four decades. Hundreds of papers and numerous books have been produced on this vital topic over the previous five decades.The theory of functional equations is a vast area of nonlinear analysis, which is rather hard to explore. Geometry, economics, game theory, measure theory, dynamics, and a variety of other subjects all use functional equations. In the subject of analysis, the study of solutions and stability results of functional equations is a popular topic. In nonlinear analysis, notably in fixed-point theory (FPT), the stability results of functional equations are used. The stability results are utilised to investigate additive mappings’ asymptotic properties.The stability of several types of functional equations, such as additive and quadratic, established by Cauchy and D’ Alembert, was extensively addressed in many research articles in the mid-twentieth century. Many academics later looked at the stability of cubic-, quartic-, and mixed-type functional equations, as well as a variety of other properties (see [6–10]).The equation,(1)gu−v+gu+v=2gv+2gu,is called a quadratic functional equation. It is obvious that the quadratic function gu=Cu2 is a solution of the quadratic functional equation (1), where C is an arbitrary real constant. Skof [11] was the first author who intially worked on the generalized Hyers–Ulam (HU) stability of the quadratic functional equation. Cholewa [12] found that the result of Skof [11] is still valid if the domain normed linear space is replaced by an abelian group. Czerwik [13] generalized Skof’s result.In computational geometry and related subjects such as computer graphics, computer-aided design, computer-aided manufacturing, and optics, quartic equations are frequently encountered. A quartic equation is the characteristic equation of a fourth-order linear difference equation or differential equation. There are numerous types of quartic equations. In 1999, Rassias [14] investigates stability properties of the following quartic functional equation:(2)gu−2v+gu+2v+6gu=4gu−v+gu+v+24gv.Sinceg2u=16gu, we obtain(3)gu−2v+gu+2v=10gu+24gv−g2u+4gu−v+gu+v.Without imposing any regularity criteria on the unknown function, Chung and Sahoo [15] determine the general solution of (3). Furthermore, they showed that the function g:ℝ⟶ℝ is a solution of (3) if and only if gu=qu,u,u,u, where the function q:ℝ4⟶ℝ is symmetric and additive in each variable. The premise that every solution of (3) is even can be expressed as follows:(4)g2u−v+g2u+v=4gu−v+gu+v+24gu−6gv.The quartic functiongu=Cu4 is easily recognised as a solution of the quartic functional (4), where C is an arbitrary real constant.In 2005, Lee et al. [16] found the general solution of (4) and demonstrated the functional equation’s HU stability. Park [17] investigates the stability problem of functional (4) in orthogonality normed space. In 2009, Lee et al. [18] considered the following quartic functional equation, which is a generalization of functional (4):(5)gmu+v+gmu−v=2m2m2−1gu−2m2−1gv+m2gu−v+gu+v,for fixed integer m, where m≠±1,0. They found the general solution of functional (5) and showed its HU stability. There are several authors (see [14, 19–22]) who investigated the stability of quartic functional equations.The quasi-β-normed space is one of the interesting generalization of the normed space (see [22–26]). The variation between a norm and quasi-β-norm is that the modulus of concavity of a quasi-norm is greater than equal to 1, while that of a norm is equal to 1. In general, the quasi-β-norm is not continuous, whereas a norm is always continuous. However, every β,p-norm is continuous quasi-β-norm. Since it is much easier to work with β,p-norm than quasi-β-norm, henceforth, we restrict our attention mainly to β,p-norm.In this study, we investigate the generalized HU stability of quartic functional equation of the form:(6)gu+mv+gu−mv=2m−17m−9gu+2m2−1m2gv−m−12g2u+m2gu−v+gu+v,for a fixed integerm/ge2 and u,v∈E1, where g is a mapping from a β,p-normed space E1 to a complete β,p-normed space E2 over the same field as E1 by using the directed method and alternative FPT. In non-Archimedean β-normed space, we also demonstrate the HU stability for the quartic functional (6). The quartic functional (6) is a generalization of several types of functional equations that depends on the values of m.


## 2. Preliminaries

Now, we will go over the basic definitions and results of quasiβ-normed space, as well as a number of other results that are required for the main findings.Definition 1.
Letβ be a real number with 0<β≤1, and E1 be a linear space over field K with K=ℝ or ℂ. A function ⋅:E1⟶0,∞ is called a quasi-β-norm if it satisfies the following conditions:N1u≥0 and u  =  0 if and only if u=0N2ru  =  rβ.u for every real number rN3v+u≤Kv+Kufor all v,u∈E1 and K≥1.
The pairE1,⋅,K is called quasi-β-normed space [26] if ⋅ is a quasi-β-norm on E1. The smallest possible K is called the modulus of concavity of ⋅.
A quasi-β-normed space E1,.,K is called β,p-normed space if(7)u+vp≤up+vp,
For some0<p<1 and for all u,v∈E1, . is called a β,p-norm on E1.For a sequence to be convergent and Cauchy in quasi-β-normed space, see [26].Example 1.
Foru=u1,u2∈ℝ2, we define(8)up,β=u1β,p+u2β,p1/p,u2≠0,2u1β,u2=0,where 0<p,β≤1. Then, ℝ2,up,β is a β,p-norm space (see [24]).Definition 2.
LetE1 be a nonempty set, K≥1, and δ:E1×E1⟶0,∞ be a function satisfying the following conditions, for all u,v,w∈E1:δ1δu,v=0 if and only if u=vδ3δu,v=δv,u, for all u,v∈E1δ4 There is a constant K≥1 such that δv,u≤Kδv,w+Kδw,u
Then,δ is called a generalized b-metric on E1 and the pair E1,δ,K is called a generalized b-metric space (GBM) [27].The following result of Diaz et al. [28] will be used in the sequel.Theorem 1.
LetE1,D be a complete generalized metric space and T:E1⟶E1 be a contraction mapping satisfying the following.C1 There exists a constant M, with 0<M<1, such that whenever Du,v<∞, one has(9)DTu,Tv≤MDu,v.Then, for eachu∈E1, we have either of the following conditions hold.EitherDTnu,Tn+1u=∞, for all n∈ℕ, or the following assertions hold:(i)
limn⟶∞Tnu=u∗, where u∗ is a fixed point of T(ii)
Du,u∗≤1/1−MDu,TuThe following proposition of Paluszyński et al. [29] will be used in this sequel.Proposition 1.
LetE1,δ,K be a b-metric space, p=log2K2, and(10)Du,v=inf∑i=1nδpui,ui+1:u1=u,u2,…,un,un+1=v∈E1,n≥1,for all u,v∈E1. Then, D is a metric on E1 satisfying(11)14δpu,v≤Du,v≤δpu,v,for all u,v∈E1. In particular, if δ is a metric, then p=1 and D=δ.The following result of Bodaghi [30] will be used in Section 3.Theorem 2.
LetE1 and E2 be real vector spaces. Then, a mapping g:E1⟶E2 satisfies the functional equation (3) if and only if it satisfies the functional equation (6), where m≥3. Therefore, every solution of (6) is also a quartic function.


## 3. Alternative Fixed-Point Method

In this section, we will investigate the stability of a quartic functional (6) on β,p-normed spaces by applying Theorem 1 with Remark 1.Remark 1.
LetE1 and E2 be two vector spaces over the same field and g:E1⟶E2 satisfies functional equation (6). Then ,for all u,v∈E1 and l,k≥0, we haveCase 1: ifu=v=0 in (6), then g0=0.Case 2: ifv=0 in (6), then g2u=24gu.Case 3: if we replaceu with ku and v with lu in (6), then gku+lu=k+l4gu. If k+l=m, so we have gmu=m4gu.Case 4: ifu=0 in (6), then g−1.u=−14gu.Letm be an integer with m≥2; we use the abbreviation for the given mapping g:E1⟶E2 as follows:(12)Bmgu,v≔gu+mv+gu−mv−2m−17m−9gu−2m2−1m2gv+m−12g2u−m2gu+v+gu−v.Theorem 3.
LetE1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p- normed space over the same field with E1, Θ:E12⟶0.∞ be a mapping, and g:E1⟶E2 be a mapping with g0=0. Assume that the following assumptions are satisfies, for all u,v∈E1:(13)aΘmu,mv≤Mm4βΘu,v,where0≤M<1,0<β≤1,(14)bBmgu,vE2≤Θu,v.
Then, there exists a unique mappingG:E1⟶E2 satisfying functional (6) and(15)gu−GuE2≤41−M1/p124β|m−1|2βΘu,0,for all u∈E1andp=log2KE22.Proof.
Letσ=f:E1⟶E2. Defined a function δ:σ×σ⟶0,∞,(16)δf,J=infC≥0|fu−JuE2≤CΘu,0,∀u∈E1,for all f,J∈σ, where inf=∞. Firstly, we will prove that σ,δ is a GBM. Let f,J,a∈σ. It is easy to see that δf,J=δJ,f. Now, if f=J, then δf,J=0. Note that fu−JuE2≤δf,JΘu,0, for all u∈E1. If δf,J=0, then fu−JuE2=0, that is, fu=Ju, for all u∈E1. Then, f=J. Now, we assert the last property of a GBM. For each u∈E1, we have fu−auE2≤δf,aΘu,0, au−JuE2≤δa,JΘu,0, and fu−JuE2≤δf,JΘu,0.
It follows that, for allu∈E1,(17)fu−JuE2≤KE2‖fu−au‖E2+‖au−Ju‖E2≤KE2δf,a+KE2δa,JΘu,0,
Then, we can write(18)fu−Ju≤KE2δf,a+KE2δa,JΘu,0.
So, we have(19)δf,J≤KE2δf,a+KE2δa,J.
Therefore,σ,δ,KE2 is a GBM with the coefficient KE2 on σ.
Next, we will show that GBMσ,δ,KE2 is complete. Assume that gn is a Cauchy sequence in σ,δ,KE2. Then, we have limn,m⟶∞δgn,gm=0. Note that, for all u∈E1, we have(20)gnu−gmuE2≤δgn,gmΘu,0.
Then,limn,m⟶∞gnu−gmuE2=0. It implies that gnu is a Cauchy sequence in E2,⋅E2,KE2. Since E2,⋅E2,KE2 is a β,p-Banach, there exists limn⟶∞gnu=v in E2,⋅E2,KE2. Put gu=v; we have the mapping f:E1⟶E2. We will show that limn⟶∞gn=f in σ,δ,KE2. Indeed, for each ε>0, there exists n0 such that δgn,gm<ε for all m,n≥n0. So, from (20), for all u∈E1 and n≥n0,(21)gnu−gmuE2≤εΘu,0.
Takingm⟶∞ in (21), we get, for all u∈E1 and n≥n0,(22)gnu−fuE2≤εΘu,0.
This implies thatδgn,f≤ε for all n≥n0. So, limn⟶∞gn=f in σ,δ,KE2. Then, σ,δ,KE2 is complete.
Next, lettingu=0 in (14) and using g0=0, we have(23)m−12g2u−m−1216guE2≤Θu,0g2u16−guE2≤Θu,024β|m−1|2β,for all u∈E1. It yields that(24)g2u24−guE2≤Θu,024β|m−1|2β,for allu∈E1.
Define a mapT:σ⟶σ by Tfu=f2u/24, for all f∈σ and for all u∈E1.
Now, we have to show thatδTf,TJ≤Mδf,J. By (13) and f,J∈σ, we have(25)Tfu−TJuE2=f2u24−J2u24E2=124βf2u−J2uE2≤δf,J24βΘ2u,0≤δf,J24βM24βΘu,0=Mδf,JΘu,0.
So, we obtain(26)δTf,TJ≤Mδf,J,for all f,J∈σ. Since 0≤M<1, using Theorem 1, for the mapping T on the complete generalized metric space E1,D, we have either DTnf,Tn+1f=∞, for all n∈ℕ, or the following assertions hold:(1)
limn⟶∞Tnf=G, where G is a fixed point of T(2)
Df,G≤1/1−MDf,Tf
By (11), we have, for all n∈ℕ,(27)δf,G≤41/pD1/pf,G≤41−M1/pD1/pf,Tf≤41−M1/pδf,Tf.
From (24), we have, for all u∈E1, Tgu−guE2≤Θu,0/24βm−12β. So, δTg,g≤1/24βm−12β<∞. This shows that if we choose f=g, then(1)
limn⟶∞Tng=G(2)
δg,G≤4/1−M1/pδg,Tg
So, we find that(28)δg,G≤41−M1/pδg,Tg≤41−M1/p124β|m−1|2β.
Then, for allu∈E1, gu−GuE2≤4/1−M1/p1/24βm−12βΘu,0. That is, (15) holds.
Next, we will show thatq is quartic by using the continuity. For each u∈E1, note that Tgu=g2u/24. So,(29)T2gu=Tg2u24=g22u24.2,…,Tngu=g2nu24n.
So (13) and (14), we have, for all u,v∈E1,(30)Gu+mv+Gu−mv−2m−17m−9Gu−2m2−1m2Gv+m−12G2u−m2Gu+v+Gu−vE2p=limn⟶∞Tngu+mv+limn⟶∞Tngu−mv−2m−17m−9limn⟶∞Tngu−2m2−1m2limn⟶∞Tngv+m−12limn⟶∞Tng2u−m2limn⟶∞Tngu+v−m2limn⟶∞Tngu−vE2p⋅limn⟶∞Tngu+mv+Tngu−mv−2m−17m−9Tngu−2m2−1m2Tngv+m−12Tng2u−m2Tngu+v−m2Tngu−vE2p⋅limn⟶∞g2nu+mv24n+g2nu−mv24n−2m−17m−9g2nu24n−2m2−1m2g2nv24n+m−12g2n2u24n−m2g2nu+v24n−m2g2nu−v24nE2p=limn⟶∞124npβg2nu+2nmv+g2nu−2nmv−2m−17m−9g2nu−2m2−1m2g2nv+m−12g2n2u−m2g2nu+2nv−m2g2nu−2nvE2p≤limn⟶∞124npβΘp2nu,2nv≤limn⟶∞124npβMn24npβΘpu,u=limn⟶∞MnΘpu,u=0.
It implies that, for allu,v∈E1,(31)Gu+mv+Gu−mv−2m−17m−9Gu−2m2−1m2Gv+m−12G2u−m2Gu+v+Gu−v=0.
So,G satisfies functional (6). By Remark 1, G is a quartic mapping.
In the last, we show the uniqueness ofG. Assume that J:E1⟶E2 is also a quartic mapping satisfying (15). We need to prove that J=G. It follows from Remark 1 that G2u=24Gu and J2u=24Ju. By using (13) and (15), for each n∈ℕ, we obtain(32)Gu−JuE2p=G2nu24n−J2nu24nE2p=124npβG2nu−J2nuE2p=124npβG2nu−g2nu+g2nu−J2nuE2p≤124npβG2nu−g2nuE2p+g2nu−J2nxE2p≤124npβG2nu−g2nuE2p+g2nu−J2nuE2p≤124npβ41−M124pβ|m−1|2pβΘp2nu,0+41−M124pβ|m−1|2pβΘp2nu,0≤124npβ+4pβ|m−1|2pβ81−MΘp2nu,0≤124npβ+4pβ|m−1|2pβ81−MMnp24npβΘpu,0=124pβ|m−1|2pβ8Mnp1−MΘpu,0.
Since0≤M<1 and p=log2KE22, letting n⟶∞, we get Gu−JuE2p=0 for all u∈E1. This proves J=G.


## 4. Direct Method

In this section, we show the stability of the quartic functional (6) by using the direct method.Theorem 4.
LetE1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space over the same field with E1, and g:E1⟶E2 be a mapping for which there exists a function Θ:E1×E1⟶0,∞ such that(33)Θ′u,v≔∑k=0∞124kpβΘp2ku,2kv<∞,(34)Bmgu,vE2≤Θu,v,for all u,v∈E2.Then, there exists a unique quartic mappingG:E1⟶E2 such that(35)gu−GuE2≤124βΘ′u,01/pm−12β,for all u∈E1.Proof.
Puttingv=0 in (34), we have(36)m−12g2u−m−1216guE2≤Θu,0,for all u∈E1. Thus,(37)g2u16−guE2p≤Θpu,024pβ|m−1|2pβ,for all u∈E1. Replacing u by 2u in (37) and continuing this method, we obtain(38)g2nu24n−guE2p≤124pβ|m−1|2pβ∑k=0n−1Θp2ku,024kpβ.
On the contrary, we can use induction to obtain(39)g2nu24n−g2lu24lE2p≤124pβ|m−1|2pβ∑k=0n−1Θp2ku,024kpβ,for all u∈E1, and n>l≥0. Thus, the sequence g2nu/24n is a Cauchy by (33) and (39). Since E2 is complete, so we assume that there exists a mapping G so that(40)limn⟶∞g2nu24n=Gu.
We can see that inequality (35) holds if we take the limit as n⟶∞ in (38) and use (40). Now, we replace u,v by 2nu,2nv, respectively, in (34); then,(41)124npβBmg2nu,2nvE2p≤Θp2nu,2nv24npβ.
Letting the limit asn⟶∞, we obtain BmGu,v=0, all u,v∈E1. Therefore, by Theorem 2, it indicates that G:E1⟶E2 is a quartic mapping. Now, let G′:E1⟶E2 be another quartic mapping which satisfies (35). Then, we have(42)Gu−G′uE2p=124npβG2nu−G′2nuE2p≤124npβG2nu−g2nuE2p+g2nu−G′2nuE2p≤124npβΘ′2nu,024|m−1|2pβ=124|m−1|2pβ∑k=0∞124n+kpβΘp2n+ku,0=124|m−1|2pβ∑n=0∞124npβΘp2nu,0,for all u∈E1. Taking n⟶∞ in the preceding inequality, we can instantly see that G is unique. The proof is now complete.We have the following result which is analogous to Theorem4 for the quartic functional (6).Theorem 5.
LetE1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space over the same field with E1, and g:E1⟶E2 be a mapping for in which there exists a function φ:E1×E1⟶0,∞ such that(43)Θ′u,v≔∑k=0∞24kpβΘpu2k,v2k<∞,(44)Bmgu,vE2≤Θu,v,for all u,v∈E1. Then, there exists a unique quartic mapping G:E1⟶E2 such that(45)gu−GuE2≤Θ′u,0m−12β1/p,for all u∈E1.Proof.
It follows from (43) that Θ0,0=0. Thus, from (44), we have g0=0. Putting E2=0 in (44), we obtain(46)m−12g2u−m−1216guE2≤Θu,0,for all u∈E1. In the previous inequality, if we replace u by u/2 and divide both sides by m−12β, we obtain(47)gu−16gu2E2p≤Θpu/2,0m−12pβ,for all u∈E1. Using triangular inequality and moving forward in this manner,(48)gu−24ngu2nE2p≤1m−12pβ∑k=1n24kpβΘpu2k,0,for all u∈E1. If we show that the sequence 24ngu/2n is Cauchy, then the completeness of E2 will imply that it is convergent. For this, if we substitute u in (48) with u/2l and then multiply both sides by 24l, then we obtain(49)24l+ngu2n−24lgu2lE2p≤1m−12pβ∑k=1n24pk+lβΘpu2k+l,0=1m−12pβ∑k=l+1l+n24pkβΘpu2k,0,for all u∈E1, and n>l>0. Thus, the above sequence is convergent to the mapping G, that is,(50)Gu=limn⟶∞24ngu2n.
We now get the result, which is similar to the proof of Theorem4.


## 5. Some Consequences

In this section, we give some consequences of the results proved in the previous sections.Corollary 1.
LetE1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space space over the same field with E1, and g:E1⟶E2 be a mapping with g0=0. Assume that there is a positive real number λ and a real number r1<4 such that(51)Bmgu,vE2≤λur1+vr1,for all u,v∈E1/0.Then, there exists a unique mappingG:E1⟶E2 satisfying (6) and(52)gu−GuE2≤11−M1/p124β|m−1|2βur1,for all u∈E1,p=log2KE22.Proof.
Define a mappingΘ:E12⟶0,∞ by(53)φu,v≔0,ifu=0orv=0,λur1+vr1,otherwise.
Next, we will show that(54)Θmu,mv≤Mm4βΘu,v,for all u,v∈E1, where M≔mr1β−4β∈0,1. Let u,v∈E1. If u=0 or v=0, then(55)Θmu,mv=0≤Mm4βΘu,v.
Ifu≠0 and v≠0, then we have(56)Θmu,mv=λmur1+mvr1=mr1βλur1+vr1=Mm4βΘu,v.
Now, all the conditions of Theorem3 hold. Therefore, we obtain the result.Corollary 2.
LetE1,⋅E1,KE2 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space over the same field with E1, and g:E1⟶E2 be a mapping with g0=0. Assume that there is a positive real number λ and a real number r1<4 such that(57)Bmgu,vE2≤λur1vr1,for all u,v∈E1/0.
Then, there exists a unique mappingG:E1⟶E2 satisfying (6) and(58)gu−GuE2≤11−M1/p124β|m−1|2βur1,for all u∈E1,p=log2KE22.Proof.
Define a mappingΘ:E12⟶0,∞ by(59)Θu,v≔0,ifu=0orv=0,λur1vr1,otherwise.
Next, we will show that(60)Θmu,mv≤Mm4βΘu,v,for all u,v∈E1, where M≔m2r1β−4β∈0,1. Let u,v∈E1. If u=0 or v=0, then(61)Θmu,mv=0≤Mm4βΘu,v.
Ifu≠0 and v≠0, then we have(62)Θmu,mv=λmur1mvr1=m2r1βλur1vr1=Mm4βΘu,v.
All of the conditions in Theorem3 are now satisfied. Hence, we obtain the result.Corollary 3.
LetE1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space over the same field with E1, and g:E1⟶E2 be a mapping with g0=0. Assume that there is a positive real number λ and a negative real number r1 with mr1β<mβ such that(63)Bmgu,vE2≤λur1vr1+ur1+vr1,for all u,v∈E1/0.Then, there exists a unique mappingG:E1⟶E2 satisfying (6) and(64)gu−GuE2≤11−M1/p124β|m−1|2βur1,for all u∈E1,p=log2KE22.Proof.
Define a mappingΘ:E12⟶0,∞ by(65)Θu,v≔0,ifu=0orv=0,λur1vr1+ur1+vr1,otherwise.
Next, we will show that(66)Θmu,mv≤Mm4βΘu,v,for all u,v∈E1, where M:=mr1β−4β∈0,1. Let u,v∈E1. If u=0 or v=0, then(67)Θmu,mv=0≤Mm4βΘu,v.
Ifu≠0 and v≠0, then we have(68)Θmu,mv=λmur1mvr1+mur1+mvr1=mr1βλmr1βur1vr1+ur1+vr1≤mr1βλur1vr1+ur1+vr1=Lm4βΘu,v.
Now, all the conditions of Theorem3 hold. Therefore, we obtain the result.Corollary 4.
Letλ,γ,r1, and s1 be nonnegative real numbers such that s1>0 and r1,s1<4. Let E1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space over the same field with E1, and g:E1⟶E2 be a mapping fulfilling(69)Bmgu,vE2≤λur1+γvs1,for all u,v∈E1. Then, there exists a unique quartic mapping G:E1⟶E2 such that(70)gu−GuE2≤1m−12βλur124pβ−2r1pβ1/p,for all u∈E1 and all u∈E1/0 if r1<0.Proof.
If we putu=v=0 in (69), we have g0=0. Using Θu,v=λur1+γvs1 in Theorem 4, we obtain the desired result.Corollary 5.
Letλ,γ,r1, and s1 be nonnegative real numbers such that r1,s1>4. Let E1,⋅E1,KE1 be a β,p-normed space, E2,⋅E2,KE2 be a complete β,p-normed space over the same field with E1, and g:E1⟶E2 be a mapping fulfilling(71)Bmgu,vE2≤λur1+γvs1,for all u,v∈E1. Then, there exists a unique quartic mapping G:E1⟶E2 such that(72)gu−GuE2≤2βr1λur1m−12β24pβ−2r1pβ1/p,for all u∈E1.Proof.
If we putu=v=0 in (71), we have g0=0. Using Θu,v=λur1+γvs1 in Theorem 5, we obtain the desired result.Remark 2.
Ifβ=1=p in Theorems 4 and 5 and Corollaries 4 and 5, we get the corresponding results of [30].


## 6. Non-Archimedeanβ-Normed Space

In this section, we demonstrate the stability of functional (6) in the context of non-Archimedean β-normed space.In 1897, Hensel [31] presented a normed space without the Archimedean property. Later on, it was discovered that non-Archimedean spaces offer a wide range of useful applications (see [32–34]).Definition 3.
A valuation mapping on a fieldF is a function |.|:F⟶R such that, for any u,v∈K, we haveA1u≥0 and the equality holds if and only if u=0.A2uv=uv.A3u+v≤u+v.A field having a valuation mapping is referred to as a valued field. The valuation⋅ is said to be non-Archimedean if condition (A3) in the definition of a valuation mapping is substituted byA3′u+v≤maxu,v
ConditionA3′ is called the strict triangle inequality. By A2, we have 1=−1=1. Thus, by induction, it follows from A3′ that n≤1, for each n. We always assume in addition that . is nontrivial, that is, there is a w0∈F such that w0≠0,1. The p−adic numbers are the most notable examples of non-Archimedean spaces.Example 2.
Assume thatp is a prime number. Define the p-adic absolute value |a|p=p−r, for any nonzero rational number a=prs/t such that s and t are coprime to the prime number p. Then, on ℚ, ⋅ is a non-Archimedean norm. The p-adic number field is the completion of ℚ with respect to ⋅ and is denoted by ℚp.Definition 4.
Letβ be a real number with 0<β≤1 and E1 be a linear space over field F with F=ℝ or ℂ. A function ⋅:E1⟶0,∞ is called a non-Archimedean β-norm if it satisfies the following conditions:N1u≥0 and ∥u∥  =  0 if and only if u=0N2∥cu∥  =  ∣c∣β.∥u∥ for every real number cN3 The strong triangle inequality (ultrametric):u+v≤max∥u∥,∥v∥
For allu,v∈E1, the pair E1,∥⋅∥ is called a non-Archimedean β-normed space if ∥⋅∥ is a non-Archimedean β-norm on E1.A sequenceun in a non-Archimedean β-normed space E1 is a Cauchy sequence if and only if un+1−un converge to zero.Theorem 6.
LetE1 be a non-Archimedean β-normed space and E2 be a complete non-Archimedean β-normed space, and let Θ:E1×E1⟶0,∞ such that(73)limk⟶∞124kβΘ2ku,2kv=0.
Suppose thatg:E1⟶E2 is a mapping satisfying equality(74)Bmgu,vE2≤Θu,v,for all u,v∈E1. Then, there exists a unique quartic mapping G:E1⟶E2 such that(75)gu−GuE2≤Θ′u,0|24m−12|β,for all u∈E1, where Θ′u,0=supΘ2ju,0/|24|jβ:j∈ℕ∪0.Proof.
Puttingv=0 in (74), we obtain(76)m−12g2u−m−1216guE2≤Θu,0,for all u∈E1. Thus, we have(77)g2u−16guE2≤1m−12βΘu,0,for all u∈E1. Replacing u by 2ru in (77) and then dividing both sides by 24r+1β, we have(78)124r+1g2r+1u−124rg2ruE2≤1|m−1|2β|24|r+1βΘ2ru,0,for all u∈E1 and all nonnegative integers r. Thus, the sequence g2ru/24r is Cauchy by (73) and (78). Because of the completeness of E2 as a non-Archimedean β-normed space, there exists a mapping G so that(79)limr⟶∞g2ru24r=Gu.
For eachu∈E1 and nonnegative integers r, we have(80)gu−g2ru24rY=∑j=0r−1g2ju24j−g2j+1u24j+1E2≤max∑j=0r−1g2ju24j−g2j+1u24j+1Y:0≤j<r≤1|24m−12|βmaxΘ2ju,0|24|jβ:0≤j<r.
Takingr⟶∞ in (80) and using (79), we can see that inequality (75) holds when m≥2. It follows from (73), (74), and (79) that, for all u,v∈E1,(81)BmGu,vE2=limr⟶∞1|2|4rβBmg2ru,2rvE2≤limr⟶∞1|2|4rβΘ2ru,2rv=0.
Hence, the mappingG satisfies (6). Now, let G′:E1⟶E2 be another quartic map satisfying (75). Then, we have(82)Gu−G′uE2=limk⟶∞124kG2ku−G′2kuE2≤limk⟶∞124kmaxG2ku−g2kuE2,g2ku−G′2kuE2≤1|24m−12|βlimk⟶∞limr⟶∞maxΘ2ju,0|24|jβ:k≤j<r+k=1|24m−12|βlimk⟶∞supΘ2ju,0|24|jβ:k≤j<∞=0,for all u,v∈E1. This shows the uniqueness of G.We have the following results which is analogous to Theorem6 for (6).Theorem 7.
LetE1 be a non-Archimedean β-normed space and E2 be a complete non-Archimedean β-normed space, and let Θ:E1×E1⟶0,∞ such that(83)limk⟶∞24kβΘu2k,v2k=0.
Suppose thatg:E1⟶E2 is a mapping satisfying equality(84)Bmgu,vE2≤Θu,v,for all u,v∈E1. Then, there exists a unique quartic mapping R:E1⟶E2 such that(85)gu−RuE2≤Θ′u,0|24m−12|β,for all u∈E1, where Θ′u,0=sup|24|jβΘu/2j,0:j∈ℕ∪0.Proof.
In the same way as Theorem6, we have(86)g2u−16guE2≤1|m−1|2βΘu,0,for all u∈E1. Replacing u by u/2r+1 in (86) and then multiplying both sides by |2|4rβ, we obtain(87)24rgu2n−24r+1gu2r+1E2≤24rβ|m−1|2βΘu2r+1,0,for all u∈E1 and all nonnegative integers r. Thus, the sequence 2rgu/2r is Cauchy by (83) and (87). Because of the completeness of E2 as a non-Archimedean β-normed space, there exists a map R so that(88)limr⟶∞24rgu2r=Ru.
For eachu∈E1 and nonnegative integers r, we have(89)gu−24rgu2rE2≤1m−12βmax|2|4jβΘu2j+1,0:0≤j<r,for all u∈E1 and nonnegative integers r. Since the right-hand side of inequality (89) tends to 0 as r⟶∞, by using (88), we deduce inequality (85). We can now complete the rest of the proof in the similar manner as in Theorem 6.


### 6.1. Consequences

In this section, we give some consequences of Theorems6 and 7.Corollary 6.
Letλ>0, E1 be a non-Archimedean β-normed space, and E2 be a complete non-Archimedean β-normed space, and let ϖ:0,∞⟶0,∞ be a function satisfying ϖ|e|z≤ϖeϖz, for all e,z∈0,∞, for which ϖ2≤16. Assume that g:E1⟶E2 is a mapping satisfying the inequality(90)Bmgu,vE2≤λϖu+ϖv,for all u,v∈E1. Then, there exists a unique quartic mappings G:E1⟶E2 such that(91)gu−GuE2≤λϖu24m−12β,for all u∈E1.Proof.
Definingφ:E1×E1⟶0,∞ by φu,v=Cϖu+ϖdv, we have(92)limn⟶∞1|2|4nβφ2nu,2nv≤limn⟶∞ϖ224nβφu,v=0,for all u,v∈E1. We also have(93)φ′u,0=supφ2ju,0|2|4β:0≤j<∞=φu,0=Cϖu,for all u∈E1. Now, Theorem 6 implies the desired result.The proof of the following consequence is a direct consequence of Theorem7 and similar to the proof of Corollary 6.Corollary 7.
Letλ>0, E1 be a non-Archimedean β-normed space, and E2 be a complete non-Archimedean β-normed space, and let ϖ:0,∞⟶0,∞ be a function satisfying ϖ|e|z≤ϖeϖz for all e,z∈0,∞ for which ϖ|2|−1≤|16|−1. Assume that g:E1⟶E2 is a mapping satisfying the inequality(94)Bmgu,vE2≤λϖu+ϖv,for all u,v∈E1. Then, there exists a unique quartic mappings R:E1⟶E2 such that(95)gu−RuE2≤λϖu/2m−12β,for all u∈E1.Remark 3.
Ifβ=1 in Theorems 6 and 7 and Corollaries 6 and 7, we get the corresponding results of Bodagi ([30] Theorems 12, 14 and Corollaries 13, 15).


## 6.1. Consequences

In this section, we give some consequences of Theorems6 and 7.Corollary 6.
Letλ>0, E1 be a non-Archimedean β-normed space, and E2 be a complete non-Archimedean β-normed space, and let ϖ:0,∞⟶0,∞ be a function satisfying ϖ|e|z≤ϖeϖz, for all e,z∈0,∞, for which ϖ2≤16. Assume that g:E1⟶E2 is a mapping satisfying the inequality(90)Bmgu,vE2≤λϖu+ϖv,for all u,v∈E1. Then, there exists a unique quartic mappings G:E1⟶E2 such that(91)gu−GuE2≤λϖu24m−12β,for all u∈E1.Proof.
Definingφ:E1×E1⟶0,∞ by φu,v=Cϖu+ϖdv, we have(92)limn⟶∞1|2|4nβφ2nu,2nv≤limn⟶∞ϖ224nβφu,v=0,for all u,v∈E1. We also have(93)φ′u,0=supφ2ju,0|2|4β:0≤j<∞=φu,0=Cϖu,for all u∈E1. Now, Theorem 6 implies the desired result.The proof of the following consequence is a direct consequence of Theorem7 and similar to the proof of Corollary 6.Corollary 7.
Letλ>0, E1 be a non-Archimedean β-normed space, and E2 be a complete non-Archimedean β-normed space, and let ϖ:0,∞⟶0,∞ be a function satisfying ϖ|e|z≤ϖeϖz for all e,z∈0,∞ for which ϖ|2|−1≤|16|−1. Assume that g:E1⟶E2 is a mapping satisfying the inequality(94)Bmgu,vE2≤λϖu+ϖv,for all u,v∈E1. Then, there exists a unique quartic mappings R:E1⟶E2 such that(95)gu−RuE2≤λϖu/2m−12β,for all u∈E1.Remark 3.
Ifβ=1 in Theorems 6 and 7 and Corollaries 6 and 7, we get the corresponding results of Bodagi ([30] Theorems 12, 14 and Corollaries 13, 15).


## 7. Conclusion

Inβ,p Banach space and non-Archimedean β-normed space, we investigate the Hyers–Ulam–Rassias stability for the quartic functional equation using fixed-point approach and direct method. Also, we obtain some interesting consequences.



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