# Stability of an Additive-Cubic-Quartic Functional Equation in Multi-Banach Spaces

Authors: Zhihua Wang; Xiaopei Li; Themistocles M. Rassias
Journal: Abstract and Applied Analysis
 (2011)

---

## Abstract

We prove the Hyers-Ulam stability of the additive-cubic-quartic functional equation in multi-Banach spaces by using the fixed
point alternative method. The first results on the stability in the multi-Banach spaces were presented in (Dales and Moslehian 2007).


---

## Body

## 1. Introduction

Stability is investigated when one is asking whether a small error of parameters in one problem causes a large deviation of its solution. Given an approximate homomorphism, is it possible to approximate it by a true homomorphism? In other words, we are looking for situations when the homomorphisms are stable, that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it with small error as much as possible. This problem was posed by Ulam in 1940 (cf. [1]) and is called the stability of functional equations. For Banach spaces, the problem was solved by Hyers [2] in the case of approximately additive mappings. Later, Hyers' result was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by allowing the Cauchy difference to be unbounded. During the last decade, stability of functional equations was studied by several mathematicians for mappings in various spaces including random normed spaces and fuzzy Banach spaces (cf. [5, 6]). For various other results on the stability of functional equations, one is referred to [7–26].Most of the proofs of stability theorems in the Hyers-Ulam context have applied Hyers’ direct method. The exact solution of the functional equation is explicitly constructed as the limit of a sequence, which is originating from the given approximate solution. In 2003, Radu [27] proposed the fixed point alternative method for obtaining the existence of exact solutions and error estimations and noticed that a fixed point alternative method is essential for the solution of Ulam problem for approximate homomorphisms. Subsequently, some authors [28, 29] applied the fixed alternative method to investigate the stability problems of several functional equations.The notion of multi-normed space was introduced by Dales and Polyakov [30] (or see [31, 32]). This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples were given in [30, 31]. In 2007, stability of mappings on multi-normed spaces was first given in [31], and asymptotic aspect of the quadratic functional equation in multi-normed spaces was investigated in [33].In this paper, we consider the following functional equation derived from additive, cubic, and quartic mappings:(1.1)11[f(x+2y)+f(x-2y)]=44[f(x+y)+f(x-y)]+12f(3y)-48f(2y)+60f(y)-66f(x).
It is easy to see that the function f(x)=ax+bx3+cx4 satisfies the functional equation (1.1). Eshaghi Gordji et al. [34] established the general solution and proved the generalized Hyers-Ulam stability for (1.1). The main purpose of this paper is to establish the Hyers-Ulam stability of (1.1) in multi-Banach spaces by using the fixed point alternative method.


## 2. Preliminaries

In this section, some useful results are pointed out. We begin with the alternative of a fixed point of Diaz and Margolis, which we will refer to as follows.Lemma 2.1 (cf. [27, 35]).
Let(X,d) be a complete generalized metric space and J:X→X be a strictly contractive mapping, that is,
(2.1)d(Jx,Jy)≤Ld(x,y)∀x,y∈X,
for some L≤1. Then,  for each fixed element x∈X, either
(2.2)d(Jnx,Jn+1x)=+∞∀n≥0,or(2.3)d(Jnx,Jn+1x)<+∞∀n≥n0,
for some natural number n0. Moreover, if the second alternative holds, then(i)
the sequence{Jnx} is convergent to a fixed point y* of J;(ii)
y* is the unique fixed point of J in the set Y:={y∈X∣d(Jn0x,y)<+∞} and d(y,y*)≤1/(1-L)d(y,Jy),forallx,y∈Y.Following [30, 31], we recall some basic facts concerning multi-normed spaces and some preliminary results.Let(ℰ,∥·∥) be a complex normed space, and let k∈ℕ. We denote by ℰk the linear space ℰ⊕⋯⊕ℰ consisting of k-tuples (x1,…,xk), where x1,…,xk∈ℰ. The linear operations ℰk are defined coordinatewise. The zero element of either ℰ or ℰk is denoted by 0. We denote by ℕk the set {1,2,…,k} and by 𝔖k the group of permutations on k symbols.Definition 2.2 (cf. [30, 31]).
A multi-norm on{ℰk:k∈ℕ} is a sequence (∥·∥k)=(∥·∥k:k∈ℕ) such that ∥·∥k is a norm on ℰk for each k∈ℕ, ∥x∥1=∥x∥ for each x∈ℰ, and the following axioms are satisfied for each k∈ℕ with k≥2:
(N1)
∥(xσ(1),…,xσ(k))∥k=∥(x1,…,xk)∥k, for σ∈𝔖k, x1,…,xk∈ℰ;(N2)
∥(α1x1,…,αkxk)∥k≤(maxi∈ℕk|αi|)∥(x1,…,xk)∥k, for α1,…,αk∈ℂ,x1,…,xk∈ℰ;(N3)
∥(x1,…,xk-1,0)∥k=∥(x1,…,xk-1)∥k-1, for x1,…,xk-1∈ℰ;(N4)
∥(x1,…,xk-1,xk-1)∥k=∥(x1,…,xk-1)∥k-1, for x1,…,xk-1∈ℰ.In this case, we say that ((ℰk,∥·∥k):k∈ℕ) is a multi-normed space.Suppose that((ℰk,∥·∥k):k∈ℕ) is a multi-normed space, and take k∈ℕ. We need the following two properties of multi-norms. They can be found in [30](a)
∥(x,…,x)∥k=∥x∥, for x∈ℰ,(b)
maxi∈ℕk∥xi∥≤∥(x1,…,xk)∥k≤∑i=1k∥xi∥≤kmaxi∈ℕk∥xi∥, for x1,…,xk∈ℰ.It follows from(b) that if (ℰ,∥·∥) is a Banach space, then (ℰk,∥·∥k) is a Banach space for each k∈ℕ; in this case, ((ℰk,∥·∥k):k∈ℕ) is a multi-Banach space.Lemma 2.3 (cf. [30, 31]).
Suppose thatk∈ℕ and (x1,…,xk)∈ℰk. For each j∈{1,…,k}, let (xnj)n=1,2,… be a sequence in ℰ such that limn→∞xnj=xj. Then
(2.4)lim⁡n→∞(xn1-y1,…,xnk-yk)=(x1-y1,…,xk-yk)
holds for all (y1,…,yk)∈ℰk.Definition 2.4 (cf. [30, 31]).
Let((ℰk,∥·∥k):k∈ℕ) be a multi-normed space. A sequence (xn) in ℰ is a multi-null sequence if for each ɛ>0, there exists n0∈ℕ such that
(2.5)sup⁡k∈N‖(xn,…,xn+k-1)‖k≤ɛ(n≥n0).Letx∈ℰ, we say that the sequence (xn) is multi-convergent to x in ℰ and write limn→∞xn=x if (xn-x) is a multi-null sequence.


## 3. Main Results

Throughout this section, letɛ≥0, ℰ be a linear space, and let ((ℱn,∥·∥n):n∈ℕ) be a multi-Banach space. For convenience, we use the following abbreviation for a given mapping f:ℰ→ℱ:
(3.1)Df(x,y)=11[f(x+2y)+f(x-2y)]-44[f(x+y)+f(x-y)]-12f(3y)+48f(2y)-60f(y)+66f(x).
Before proceeding to the proof of the main results in this section, we shall need the following two lemmas.Lemma 3.1 (cf. [34]).
If an even mappingf:X→Y satisfies (1.1), then f is quartic.Lemma 3.2 (cf. [34]).
If an odd mappingf:X→Y satisfies (1.1), then f is cubic-additive.Theorem 3.3.
Suppose that an even mappingf:ℰ→ℱ satisfies f(0)=0 and
(3.2)sup⁡k∈N‖(Df(x1,y1),…,Df(xk,yk))‖k≤ɛ
for all x1,…,xk,y1,…,yk∈ℰ. Then there exists a unique quartic mapping Q:ℰ→ℱ satisfying (1.1) and
(3.3)sup⁡k∈N‖(f(x1)-Q(x1),…,f(xk)-Q(xk))‖k≤13330ɛ
for all x1,…,xk∈ℰ.Proof.
Lettingx1=⋯=xk=0 in (3.2), we get
(3.4)sup⁡k∈N‖(-12f(3y1)+70f(2y1)-148f(y1),…,-12f(3yk)+70f(2yk)-148f(yk))‖k≤ɛ
for all y1,…,yk∈ℰ. Replacing x1,…,xk with y1,…,yk in (3.2), we obtain
(3.5)sup⁡k∈N‖(-f(3y1)+4f(2y1)+17f(y1),…,-f(3yk)+4f(2yk)+17f(yk))‖k≤ɛ
for all y1,…,yk∈ℰ.
It follows from (3.4) and (3.5) that
(3.6)sup⁡k∈N‖(f(2x1)-16f(x1),…,f(2xk)-16f(xk))‖k≤1322ɛ
for all x1,…,xk∈ℰ.
LetE:={gg:ℰ→ℱ,g(0)=0}, and introduce the generalized metric d defined on E by
(3.7)d(g,h)=inf⁡⁡{c∈(0,∞)∣sup⁡k∈N‖(g(x1)-h(x1),…,g(xk)-h(xk))‖k≤c,∀x1,…,xk∈E}.
Then, it is easy to show that d is a complete generalized metric on E (see the proof in [36] or [5]). We now define a function J1:E→E by
(3.8)J1g(x)=116g(2x),∀x∈E.
We assert that J1 is a strictly contractive mapping. Given g,h∈E, let c∈(0,∞) be an arbitrary constant with d(g,h)≤c. From the definition of d, it follows that
(3.9)sup⁡k∈N‖(g(x1)-h(x1),…,g(xk)-h(xk))‖k≤c
for all x1,…,xk∈ℰ. Therefore,
(3.10)sup⁡k∈N‖(J1g(x1)-J1h(x1),…,J1g(xk)-J1h(xk))‖k=sup⁡k∈N‖(116g(2x1)-116h(2x1),…,116g(2xk)-116h(2xk))‖k≤116c
for all x1,…,xk∈ℰ. Hence, it holds that d(J1g,J1h)≤(1/16)c, that is, d(J1g,J1h)≤(1/16)d(g,h) for all g,h∈E.
By using (3.6), we have d(J1f,f)≤(13/352)ɛ. According to Lemma 2.1, we deduce the existence of a fixed point of J1, that is, the existence of a mapping Q:X→Y such that Q(2x)=16Q(x) for all x∈ℰ. Moreover, we have d(J1nf,Q)→0, which implies that
(3.11)Q(x)=lim⁡n→∞(J1nf)(x)=lim⁡n→∞f(2nx)16n,∀x∈E.
Also, d(f,Q)≤1/(1-L)d(J1f,f) implies the inequality
(3.12)d(f,Q)≤11-(1/16)d(J1f,f)≤13330ɛ.
Setx1=⋯=xk=2nx,y1=⋯=yk=2ny in (3.2), and divide both sides by 16n. Then, using property (a), we obtain
(3.13)‖DQ(x,y)‖=lim⁡n→∞116n‖Df(2nx,2ny)‖≤lim⁡n→∞ɛ16n=0
for all x,y∈ℰ. Hence, by Lemma 3.1, Q is quartic.
The uniqueness ofQ follows from the fact that Q is the unique fixed point of J1 with the property that there exists l∈(0,∞) such that
(3.14)sup⁡k∈N‖(f(x1)-Q(x1),…,f(xk)-Q(xk))‖k≤l
for all x1,…,xk∈ℰ. This completes the proof of the theorem.Theorem 3.4.
Suppose that an odd mappingf:ℰ→ℱ satisfies
(3.15)sup⁡k∈N‖(Df(x1,y1),…,Df(xk,yk))‖k≤ɛ
for all x1,…,xk,y1,…,yk∈ℰ. Then there exists a unique additive mapping A:ℰ→ℱ and a unique cubic mapping C:ℰ→ℱ satisfying (1.1) and
(3.16)sup⁡k∈N‖(f(2x1)-8f(x1)-A(x1),…,f(2xk)-8f(xk)-A(xk))‖k≤1733ɛ,sup⁡k∈N‖(f(2x1)-2f(x1)-C(x1),…,f(2xk)-2f(xk)-C(xk))‖k≤17231ɛ
for all x1,…,xk∈ℰ.Proof.
Putx1=⋯=xk=0 in (3.15). Then, by the oddness of f, we have
(3.17)sup⁡k∈N‖(12f(3y1)-48f(2y1)+60f(y1),…12f(3yk)-48f(2yk)+60f(yk))‖k≤ɛ
for all y1,…,yk∈ℰ. Replacing x1,…,xk with 2y1,…,2yk in (3.15), we obtain
(3.18)sup⁡k∈N⁡‖(11f(4y1)-56f(3y1)+114f(2y1)-104f(y1),…,11f(4yk)-56f(3yk)+114f(2yk)-104f(yk))‖k≤ɛ
for all y1,…,yk∈ℰ. By (3.17) and (3.18), we have
(3.19)sup⁡k∈N‖(f(4x1)-10f(2x1)+16f(x1),…,f(4xk)-10f(2xk)+16f(xk))‖k≤1733ɛ
for all x1,…,xk∈ℰ. Putting α(x):=f(2x)-8f(x) for all x∈ℰ, we get
(3.20)sup⁡k∈N‖(α(2x1)-2α(x1),…,α(2xk)-2α(xk))‖k≤1733ɛ
for all x1,…,xk∈ℰ.
Let the same definitions forE and d as in the proof of Theorem 3.3 such that (E,d) becomes a complete generalized metric space. We now define a function J2:E→E by
(3.21)J2g(x)=12g(2x),∀x∈E.
Applying a similar technique as in the proof of Theorem 3.3, we obtain d(J2g,J2h)≤(1/2)c, that is, d(J2g,J2h)≤(1/2)d(g,h) for all g,h∈E.
By (3.20), we have d(J2α,α)≤(17/66)ɛ. According to Lemma 2.1, we deduce the existence of a fixed point of J2, that is, the existence of a mapping A:X→Y such that A(2x)=2A(x) for all x∈ℰ. Moreover, we have d(J2nα,A)→0, which implies that
(3.22)A(x)=lim⁡n→∞(J2nα)(x)=lim⁡n→∞α(2nx)2n,∀x∈E.
Also, d(α,A)≤1/(1-L)d(J2α,α) implies the inequality
(3.23)d(α,A)≤11-(1/2)d(J2α,α)≤1733ɛ.
Hence, it follows that(3.24)‖DA(x,y)‖=lim⁡n→∞12n‖Dα(2nx,2ny)‖=lim⁡n→∞12n‖Df(2n+1x,2n+1y)-8Df(2nx,2ny)‖≤lim⁡n→∞ɛ2n=0
for all x,y∈ℰ. This means that A satisfies (1.1). Then, by Lemma 3.2, x↦A(2x)-8A(x) is additive. Thus, by A(2x)=2A(x), we conclude that A is additive.
Puttingβ(x):=f(2x)-2f(x) for all x∈ℰ, we get
(3.25)sup⁡k∈N‖(β(2x1)-8β(x1),…,α(2xk)-8β(xk))‖k≤1733ɛ
for all x1,…,xk∈ℰ. We now define a function J3:E→E by
(3.26)J3g(x)=18g(2x),∀x∈E.
Applying a similar technique as in the proof of Theorem 3.3, we obtain d(J3g,J3h)≤(1/8)c, that is, d(J3g,J3h)≤(1/8)d(g,h) for all g,h∈E.
By (3.25), we have d(J3β,β)≤(17/264)ɛ. According to Lemma 2.1, we deduce the existence of a fixed point of J3, that is, the existence of a mapping C:X→Y such that C(2x)=8C(x) for all x∈ℰ. Moreover, we have d(J3nβ,C)→0, which implies that
(3.27)C(x)=lim⁡n→∞(J3nβ)(x)=lim⁡n→∞β(2nx)8n,∀x∈E.
Also, d(β,C)≤1/(1-L)d(J3β,β) implies the inequality
(3.28)d(β,C)≤11-(1/8)d(J3β,β)≤17231ɛ.
Then we have(3.29)‖DC(x,y)‖=lim⁡n→∞18n‖Dβ(2nx,2ny)‖=lim⁡n→∞18n‖Df(2n+1x,2n+1y)-2Df(2nx,2ny)‖≤lim⁡n→∞ɛ8n=0
for all x,y∈ℰ. Hence, the mapping C satisfies (1.1). Therefore, by Lemma 3.2, x↦C(2x)-2C(x) is cubic. Thus, C(2x)=8C(x) implies that the mapping C is cubic.
The uniqueness ofA and C can be proved in the same reasoning as in the proof of Theorem 3.3. This completes the proof of the theorem.Theorem 3.5.
Suppose that an odd mappingf:ℰ→ℱ satisfies
(3.30)sup⁡k∈N‖(Df(x1,y1),…,Df(xk,yk))‖k≤ɛ
for all x1,…,xk,y1,…,yk∈ℰ. Then there exists a unique additive mapping A:ℰ→ℱ and a unique cubic mapping C:ℰ→ℱ such that
(3.31)sup⁡k∈N‖(f(x1)-A(x1)-C(x1),…,f(xk)-A(xk)-C(xk))‖k≤68693ɛ
for all x1,…,xk∈ℰ.Proof.
By Theorem3.4, there exist an additive mapping A0:ℰ→ℱ and a cubic mapping C0:ℰ→ℱ such that
(3.32)sup⁡k∈N‖(f(2x1)-8f(x1)-A0(x1),…,f(2xk)-8f(xk)-A0(xk))‖k≤1733ɛ,sup⁡k∈N‖(f(2x1)-2f(x1)-C0(x1),…,f(2xk)-2f(xk)-C0(xk))‖k≤17231ɛ
for all x1,…,xk∈ℰ. Combining the above two equations in (3.32) yields that
(3.33)sup⁡k∈N‖(6f(x1)+A0(x1)-C0(x1),…,f(6xk)+A0(xk)-C0(xk))‖k≤136231ɛ
for all x1,…,xk∈ℰ. So we obtain (3.31) by letting A=-A0/6 and C=C0/6.
To prove the uniqueness ofA and C, let A1,C1:ℰ→ℱ be other additive and cubic mappings satisfying (3.31). Let A′=A-A1 and C′=C-C1. Then, using property (a), we obtain
(3.34)‖A′(x)+C′(x)‖≤‖f(x)-A(x)-C(x)‖+‖f(x)-A1(x)-C1(x)‖≤136693ɛ
for all x∈ℰ, then (3.34) implies that
(3.35)lim⁡n→∞18n‖A′(2nx)+C′(2nx)‖=0
for all x∈ℰ. Therefore, C′(x)=0 for all x∈ℰ. By (3.34), we have A′(x)=0 for all x∈ℰ. This completes the proof of the theorem.Theorem 3.6.
Suppose that a mappingf:ℰ→ℱ satisfies f(0)=0 and
(3.36)sup⁡k∈N‖(Df(x1,y1),…,Df(xk,yk))‖k≤ɛ
for all x1,…,xk,y1,…,yk∈ℰ. Then there exists a unique additive mapping A:ℰ→ℱ, a unique cubic mapping C:ℰ→ℱ, and a unique quartic mapping Q:ℰ→ℱ such that
(3.37)sup⁡k∈N‖(f(x1)-A(x1)-C(x1-Q(x1),…,f(xk)-A(xk)-C(xk)-Q(xk))‖k≤9536930ɛ
for all x1,…,xk∈ℰ.Proof.
Letfo(x)=(1/2)(f(x)-f(-x)) for all x∈ℰ, then fo(0)=0,fo(-x)=-fo(x) and
(3.38)sup⁡k∈N‖(Dfo(x1,y1),…,Dfo(xk,yk))‖k≤ɛ
for all x1,…,xk,y1,…,yk∈ℰ. From Theorem 3.5, it follows that there exists a unique additive mapping A:ℰ→ℱ and a unique cubic mapping C:ℰ→ℱ satisfying (3.31).
Letfe(x)=(1/2)(f(x)+f(-x)) for all x∈ℰ, then fe(0)=0,fe(-x)=fe(x) and
(3.39)sup⁡k∈N‖(Dfe(x1,y1),…,Dfe(xk,yk))‖k≤ɛ
for all x1,…,xk,y1,…,yk∈ℰ. By Theorem 3.3, there exists a unique quartic mapping Q:ℰ→ℱ satisfying (3.3). Now it is obvious that (3.37) holds for all x1,…,xk∈ℰ. This completes the proof of the theorem.



---

*Source: 536520-2011-11-17.xml*