# Ulam Stability of a Quartic Functional Equation

Authors: Abasalt Bodaghi; Idham Arif Alias; Mohammad Hosein Ghahramani
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

The oldest quartic functional equation was introduced by J. M. Rassias in (1999), and then was employed by other authors. The functional equationf(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f(x)-6f(y) is called a quartic functional equation, all of its solution is said to be a quartic function. In the current paper, the Hyers-Ulam stability and the superstability for quartic functional equations are established by using the fixed-point alternative theorem.


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## Body

## 1. Introduction

We say a functional equationℱ is stable if any function f satisfying the equation ℱ approximately is near to true solution of ℱ. Moreover, a functional equation ℱ is superstable if any function f satisfying the equation ℱ approximately is a true solution of ℱ (see [1] for another notion of the superstability which may be called superstability modulo the bounded functions).The stability problem for functional equations originated from a question by Ulam [2] in 1940, concerning the stability of group homomorphisms: let (G1,·) be a group, and let (G2,*) be a metric group with the metric d(·,·). Given ϵ>0, does there exist δ>0 such that, if a mapping h:G1→G2 satisfies the inequality d(h(s·t),h(s)*h(t))<δ for all s,t∈G1, then there exists a homomorphism H:G1→G2 with d(h(s),H(s))<ϵ for all s∈G1? In other words, under what condition a functional equation is stable? In the following year, Hyers [3] gave a partial affirmative answer to the question of Ulam for Banach spaces. In 1978, the generalized Hyers’ theorem was independently rediscovered by Th. M. Rassias [4] by obtaining a unique linear mapping under certain continuity assumption.The functional equations(1.1)f(x+y)+f(x-y)=2f(x)+2f(y),f(2x+y)+f(2x-y)=2f(x+y)+2f(x-y)+12f(x)
are called quadratic and cubic functional equations, respectively. During the last decades, several stability problems for functional equations especially the quadratic and cubic and their generalized have been extensively investigated by many mathematicians (for instances, [5–9]).In [10], Lee et al. considered the following quartic functional equation:
(1.2)f(2x+y)+f(2x-y)=4f(x+y)+4f(x-y)+24f(x)-6f(y).
It is easy to check that for every a∈ℝ, the function f(x)=ax4 is a solution of the above functional equation. They solved (1.2) and in fact showed that a function f:𝒳→𝒴 whenever 𝒳 and 𝒴 are real vector spaces is quadratic if and only if there exists a symmetric biquadratic function F:𝒳×𝒳→𝒴 such that f(x)=F(x,x) for all x∈𝒳. They also proved the stability of (1.2). Zhou Xu et al. in [11] used the fixed-point alternative (Theorem 2.1 of the current paper) to establish Hyers-Ulam-Rassias stability of the general mixed additive-cubic functional equation, where functions map a linear space into a complete quasifuzzy p-normed space. The generalized Hyers-Ulam stability of a general mixed AQCQ-functional in multi-Banach spaces is also proved by using the mentioned theorem in [12].Recently, Bodaghi et al. in [13, 14] investigated the stability and the superstability of quadratic and cubic functional equations by a fixed-point method and applied this method to prove the stability of (quadratic, cubic) multipliers on Banach algebras.In this paper we prove the generalized Hyers-Ulam stability and the superstability for quartic functional equation (1.2) by using the alternative fixed point (Theorem 2.1) under certain conditions.


## 2.  Main Results

Throughout this paper, assume that𝒳 is a normed vector space and 𝒴 is a Banach space. For a given mapping f:𝒳→𝒴, we consider(2.1)Df(x,y):=f(2x+y)+f(2x-y)-4f(x+y)-4f(x-y)-24f(x)+6f(y),for allx,y∈𝒳.To achieve our aim, we need the following known fixed-point theorem which has been proved in [15].Theorem 2.1.
Suppose that(Δ,d) is a complete generalized metric space, and let  𝒥:Δ→Δ be a strictly contractive mapping with Lipschitz constant  L<1, Then for each element  g∈Δ, either  d(𝒥ng,𝒥n+1g)=∞ for all  n≥0, or there exists a natural number  n0 such that (i)
d(𝒥ng,𝒥n+1g)<∞, for all n≥n0,(ii)
the sequence{𝒥ng} is convergent to a fixed-point g* of 𝒥,(iii)
g* is the unique fixed point of 𝒥 in the set
(2.2)Ω={g∈Δ:d(Jn0g,g)<∞};(iv)
d(g,g*)≤(1/(1-L))d(g,𝒥g), for all g∈Ω.Theorem 2.2.
Assume thatϕ:𝒳×𝒳→[0,∞) is a function satisfying
(2.3)‖Df(x,y)‖≤ϕ(x,y),
for all x,y∈𝒳. Let a mapping f:𝒳→𝒴 satisfy f(0)=0. If there exists K∈(0,1) such that
(2.4)ϕ(x,y)≤24Kϕ(x2,y2),
for all x,y∈𝒳, then there exists a unique quartic mapping Q:𝒳→𝒴 such that
(2.5)‖f(x)-Q(x)‖≤132(1-K)ϕ(x,0),
for all x∈𝒳.Proof.
By recurrence method, we can conclude from (2.4) that ϕ(2nx,2ny)/24n≤Knϕ(x,y) for all x,y∈𝒳. Passing to the limit, we get
(2.6)lim⁡n→∞ϕ(2nx,2ny)24n=0,
for all x,y∈𝒳. Here, we intend to build the conditions of Theorem 2.1 and so consider the set Δ:={h:𝒳→𝒴∣h(0)=0} and the mapping d defined on Δ×Δ by
(2.7)d(g,h):=inf⁡{C∈(0,∞):‖g(x)-h(x)‖≤Cϕ(x,0)∀x∈X}
if there exists such constant C, and d(g,h)=∞ otherwise. It is easy to see that d(h,h)=0 and d(g,h)=d(h,g), for all g,h∈Δ. For each g,h,p∈Δ, we have
(2.8)inf⁡{C∈(0,∞):‖g(x)-h(x)‖≤Cϕ(x,0)∀x∈X}≤inf⁡{C∈(0,∞):‖g(x)-p(x)‖≤Cϕ(x,0)∀x∈X}+inf⁡{C∈(0,∞):‖p(x)-h(x)‖≤Cϕ(x,0)∀x∈X}.
Hence,d(g,h)≤d(g,p)+d(p,h). Now if d(g,h)=0, then for every fixed x0∈𝒳, we have ∥g(x0)-h(x0)∥≤Cϕ(x0,0), for all C>0. This implies g=h. Let {hn} be a d-Cauchy sequence in Δ, then d(hm,hn)→0, and thus ∥hm(x)-hn(x)∥→0, for all x∈𝒳. Since 𝒴 is complete, then there exists h∈Δ such that hn→dh in Δ. Therefore, d is a generalized metric on Δ, and the metric space (Δ,d) is complete. Now, we define the mapping 𝒥:Δ→Δ by(2.9)Jg(x)=124g(2x),(x∈X).
Fix aC∈(0,∞) and take g,h∈Δ such that d(g,h)<C. The definitions of d and 𝒥 show that
(2.10)‖124g(2x)-124h(2x)‖≤124Cϕ(2x,0),
for allx∈𝒳. By using (2.4), we have
(2.11)‖124g(2x)-124h(2x)‖≤CKϕ(x,0),
for all x∈𝒳. It follows from the above inequality that d(𝒥g,𝒥h)≤Kd(g,h), for all g,h∈Δ. Hence, 𝒥 is a strictly contractive mapping on Δ with a Lipschitz constant K. Putting y=0 in (2.3) and dividing both sides of the resulting inequality by 32, we have
(2.12)‖f(x)-116f(2x)‖≤132ϕ(x,0),
for all x∈X. Thus, d(f,𝒥f)≤1/32<∞. Note that by Theorem 2.1, d(𝒥ng,𝒥n+1g)<∞, for all n≥0. Thus, we get n0=0 in this theorem, so (iii) and (iv) of Theorem 2.1 are true on the whole Δ. However, the sequence {𝒥nf} converges to a unique fixed-point Q:𝒳→𝒴 in the set {g∈Δ;d(f,g)<∞}, that is,
(2.13)Q(x)=lim⁡n→∞f(2nx)24n,
for all x∈X. By the part (iv) of Theorem 2.1, we have
(2.14)d(f,Q)≤d(f,Jf)1-K≤132(1-K).
From (2.14), we observe that the inequality (2.5) holds for all x∈𝒳. Substituting x,y by 2nx,2ny in (2.3), respectively, and applying (2.6) and (2.13), we have(2.15)‖DQ(x,y)‖=lim⁡n→∞124n‖Df(2nx,2ny)‖≤lim⁡n→∞124nϕ(2nx,2ny)=0,
for all x∈𝒳. Therefore, Q is a quartic mapping which is unique by part (iii) of Theorem 2.1.Corollary 2.3.
Letp,θ be nonnegative real numbers such that  p<4, and let  f:𝒳→𝒴 be a mapping (with  f(0)=0 when  p=0) satisfying
(2.16)‖Df(x,y)‖≤θ(‖x‖p+‖y‖p),
for all x,y∈𝒳, then there exists a unique quartic mapping Q:𝒳→𝒴 such that
(2.17)‖f(x)-Q(x)‖≤θ32-2p+1‖x‖p,
for all x∈𝒳.Proof.
The result follows from Theorem2.2 by using ϕ(x,y)=θ(∥x∥p+∥y∥p).Now, we establish the superstability of quartic mapping on Banach spaces under some conditions.Corollary 2.4.
Letp,q,θ be nonnegative real numbers such that  p+q∈(0,4). Suppose that a mapping  f:𝒳→𝒴 satisfies
(2.18)‖Df(x,y)‖≤θ‖x‖p‖y‖q,
for all x,y∈𝒳, then f is a quartic mapping on 𝒳.Proof.
Lettingϕ(x,y)=θ∥x∥p∥y∥q in Theorem 2.2, we have
(2.19)lim⁡n→∞ϕ(2nx,2ny)24n=0,
which shows (2.6) holds for ϕ. Putting x=y=0 in (2.18), we get f(0)=0. Furthermore, if we put y=0 in (2.18), then we have f(2x)=24f(x), for all x∈𝒳. It is easy to see that by induction, we have f(2nx)=24nf(x), and so f(x)=f(2nx)/24n, for all x∈𝒳 and n∈ℕ. Now, it follows from Theorem 2.2 that f is a quartic mapping.Letθ and p be positive real numbers. Suppose that a mapping f:𝒳→𝒴 satisfies
(2.20)‖Df(x,y)‖≤θ‖y‖p,
for all x,y∈𝒳, then by considering ϕ(x,y)=θ∥y∥p in Theorem 2.2, the mapping f is again a quartic mapping on 𝒳.The following result is proved in [16, Theorem 1].Theorem 2.5.
Let𝒳 be a linear space, and let  𝒴 be a Banach space. Let  f:𝒳→𝒴 be a mapping for which there exists a function  φ:𝒳×𝒳→[0,∞) such that
(2.21)φ̃(x,y):=∑k=0∞2-4kφ(2kx,2ky)<∞,‖Df(x,y)‖≤δ+φ(x,y)
for all x,y∈𝒳, where δ≥0, then there exists a unique quartic mapping Q:𝒳→𝒴 such that
(2.22)‖f(x)-Q(x)+15f(0)‖≤130δ+132φ̃(x,0)
for all x∈𝒳.One should note that in the above theorem,f(0) is not necessarily zero, but in the following result, we assume that f(0)=0 and also consider the case δ=0. By these hypotheses and by applying Theorem 2.1, we obtain the specific result which is a way to prove the superstability of a quartic functional equation.Theorem 2.6.
Letf:𝒳→𝒴 be a mapping with  f(0)=0, and let  ψ:𝒳×𝒳→[0,∞) be a function satisfying
(2.23)lim⁡n→∞24nψ(x2n,y2n)=0,(2.24)‖Df(x,y)‖≤ψ(x,y),
for all x,y∈𝒳. If there exists L∈(0,1) such that
(2.25)ψ(x,0)≤2-4Lψ(2x,0),
for all x∈𝒳, then there exists a unique quartic mapping Q:𝒳→𝒴 such that
(2.26)‖f(x)-Q(x)‖≤L32(1-L)ψ(x,0),
for all x∈𝒳.Proof.
We take the setΩ:={g:𝒳→𝒴∣g(0)=0} and consider the generalized metric on Ω,
(2.27)d(g1,g2):=inf⁡{C∈(0,∞):‖g1(x)-g2(x)‖≤Cψ(x,0)∀x∈X},
if there exists such a constant C, and d(g1,g2)=∞ otherwise. It follows from the proof of Theorem 2.2 that the metric space (Ω,d) is complete (see the proof of Theorem 2.2). We will show that the mapping 𝒥:Ω→Ω defined by 𝒥g(x)=24g(x/2)(x∈𝒳) is strictly contractive. Fix a C∈(0,∞) and take g1,g2∈Ω such that d(g1,g2)<C, then we have
(2.28)‖24g1(x2)-24g2(x2)‖≤24Cψ(x2,0),
for all x∈𝒳. By using (2.25), we obtain
(2.29)‖24g1(x2)-24g2(x2)‖≤CLψ(x,0),
for all x∈𝒳. It follows from the last inequality that d(𝒥g1,𝒥g2)≤Ld(g1,g2), for all g1,g2∈Ω. Hence, 𝒥 is a strictly contractive mapping on Ω with a Lipschitz constant L. By putting y=0, replacing x by x/2 in (2.24) and using (2.25), and then dividing both sides of the resulting inequality by 2, we have
(2.30)‖24f(x2)-f(x)‖≤12ψ(x2,0)≤2-5Lψ(x,0),
for all x∈𝒳. Hence, d(f,𝒥f)≤2-5L<∞. By applying the fixed-point alternative Theorem 2.1, there exists a unique mapping Q:𝒳→𝒴 in the set Ω1={g∈Ω;d(f,g)<∞} such that
(2.31)Q(x)=lim⁡n→∞24nf(x2n),
for all x∈𝒳. Again Theorem 2.1 shows that
(2.32)d(f,Q)≤d(f,Jf)1-L≤2-5L1-L.
Hence, inequality (2.32) implies (2.26). Replacing x,y by 2nx,2ny in (2.24), respectively, and using (2.23) and (2.31), we conclude that(2.33)‖DQ(x,y)‖=lim⁡n→∞24n‖Df(x2n,y2n)‖≤lim⁡n→∞24nψ(x2n,y2n)=0,
for all x∈𝒳. Therefore, Q is a quartic mapping.Corollary 2.7.
Letp and  λ be nonnegative real numbers such that  p>4. Suppose that  f:𝒳→𝒴 is a mapping satisfying
(2.34)‖Df(x,y)‖≤λ(‖x‖p+‖y‖p),
for all x,y∈𝒳, then there exists a unique quartic mapping Q:𝒳→𝒴 such that
(2.35)‖f(x)-Q(x)‖≤λ2(2p-24)‖x‖p
for all x∈𝒳.Proof.
It is enough to letψ(x,y)=λ(∥x∥p+∥y∥p) in Theorem 2.6.Corollary 2.8.
Letp,q,λ be nonnegative real numbers such that  p+q∈(4,∞). Suppose that a mapping  f:𝒳→𝒴 satisfies
(2.36)‖Df(x,y)‖≤λ‖x‖p‖y‖q
for all x,y∈𝒳. Then f is a quartic mapping on 𝒳.Proof.
Puttingψ(x,y)=θ∥x∥p∥y∥q in Theorem 2.6, we have
(2.37)lim⁡n→∞ψ(2nx,2ny)24n=0,
and thus, (2.6) holds. If we put x=y=0 in (2.36), then we get f(0)=0. Again, letting y=0 in (2.36), we conclude that f(x)=24f(x/2), and thus, f(x)=24nf(x/2n), for all x∈𝒳 and n∈ℕ. Now, we can obtain the desired result by Theorem 2.6.
From Corollaries2.4 and 2.8 we deduce the following result.Corollary 2.9.
Letp,q,andλ be nonnegative real numbers such that  p+q>0 and  p+q≠4. Suppose that a mapping  f:𝒳→𝒴 satisfies (2.36), for all  x,y∈𝒳 then  f is a quartic mapping on  𝒳.



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