Lee, An and Park introduced the quadratic
functional equation f(2x+y)+f(2x−y)=8f(x)+2f(y) and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic
functional equation in Banach spaces.
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers’ Theorem was generalized by Aoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.
Theorem 1.1 (Th. M. Rassias).
Let f:E→E′ be a mapping from a normed vector space E into a Banach space E′ subject to the inequality
∥f(x+y)-f(x)-f(y)∥≤ϵ(∥x∥p+∥y∥p)
for all x,y∈E, where ϵ and p are constants with ϵ>0 and p<1. Then the limit
L(x)=limn→∞f(2nx)2n
exists for all x∈E, and L:E→E′ is the unique additive mapping which satisfies
∥f(x)-L(x)∥≤2ϵ2-2p∥x∥p
for all x∈E. Also, if for each x∈E the function f(tx) is continuous in t∈ℝ, then L is ℝ-linear.
The above inequality (1.1) has provided a lot of influence in the development of what is now known as a generalized Hyers-Ulam stability of functional equations. Beginning around the year 1980, the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Găvruta [5] generalized the Rassias’ result.
Theorem 1.2 (see [6–8]).
Let X be a real normed linear space and Y a real complete normed linear space. Assume that f:X→Y is an approximately additive mapping for which there exist constants θ≥0 and p∈ℝ-{1} such that f satisfies inequality
∥f(x+y)-f(x)-f(y)∥≤θ·∥x∥p/2·∥y∥p/2
for all x,y∈X. Then there exists a unique additive mapping L:X→Y satisfying
∥f(x)-L(x)∥≤θ|2p-2|∥x∥p
for all x∈X. If, in addition, f:X→Y is a mapping such that the transformation t→f(tx) is continuous in t∈ℝ for each fixed x∈X, then L is an ℝ-linear mapping.
The functional equation
f(x+y)+f(x-y)=2f(x)+2f(y)
is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings f:X→Y, where X is a normed space and Y is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain X is replaced by an Abelian group. Czerwik [11] proved the generalized Hyers-Ulam stability of the quadratic functional equation. Several functional equations have been investigated in [12–25].
Let X be a set. A function d:X×X→[0,∞] is called a generalized metric on X if d satisfies
d(x,y)=0 if and only if x=y;
d(x,y)=d(y,x) for all x,y∈X;
d(x,z)≤d(x,y)+d(y,z) for all x,y,z∈X.
We recall a fundamental result in fixed point theory.
Theorem 1.3 (see [26–28]).
Let (X,d) be a complete generalized metric space and let J:X→X be a strictly contractive mapping with Lipschitz constant L<1. Then for each given element x∈X, either
d(Jnx,Jn+1x)=∞
for all nonnegative integers n or there exists a positive integer n0 such that
d(Jnx,Jn+1x)<∞, for all n≥n0;
the sequence {Jnx} converges to a fixed point y* of J;
y* is the unique fixed point of J in the set Y={y∈X∣d(Jn0x,y)<∞};
d(y,y*)≤(1/(1-L))d(y,Jy) for all y∈Y.
Lee et al. [29] proved that a mapping f:X→Y satisfies
f(2x+y)+f(2x-y)=8f(x)+2f(y)
for all x,y∈X if and only if the mapping f:X→Y satisfies
f(x+y)+f(x-y)=2f(x)+2f(y)
for all x,y∈X.
Using the fixed point method, Park [14] proved the generalized Hyers-Ulam stability of the quadratic functional equation
f(2x+y)=4f(x)+f(y)+f(x+y)-f(x-y)
in Banach spaces.
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.8) in Banach spaces.
Throughout this paper, assume that X is a normed vector space with norm ||·|| and that Y is a Banach space with norm ∥·∥.
2. Fixed Points and Generalized Hyers-Ulam Stability of a Quadratic Functional Equation
For a given mapping f:X→Y, we define
Cf(x,y):=f(2x+y)+f(2x-y)-8f(x)-2f(y)
for all x,y∈X.
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation Cf(x,y)=0.
Theorem 2.1.
Let f:X→Y be a mapping for which there exists a function φ:X2→[0,∞) with f(0)=0 such that
∥Df(x,y)∥≤φ(x,y)
for all x,y∈X. If there exists an L<1 such that φ(x,y)≤4Lφ(x/2,y/2) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤18-8Lφ(x,0)
for all x∈X.
Proof.
Consider the set
S:={g:X→Y},
and introduce the generalized metric on S:
d(g,h)=inf{K∈ℝ+:∥g(x)-h(x)∥≤Kφ(x,0),∀x∈X}.
It is easy to show that (S,d) is complete.
Now we consider the linear mapping J:S→S such that
Jg(x):=14g(2x)
for all x∈X.
By [30, Theorem 3.1],
d(Jg,Jh)≤Ld(g,h)
for all g,h∈S.
Letting y=0 in (2.2), we get
∥2f(2x)-8f(x)∥≤φ(x,0)
for all x∈X. So
∥f(x)-14f(2x)∥≤18φ(x,0)
for all x∈X. Hence d(f,Jf)≤1/8.
By Theorem 1.3, there exists a mapping Q:X→Y such that
(1) Q is a fixed point of J, that is,
Q(2x)=4Q(x)
for all x∈X. The mapping Q is a unique fixed point of J in the set
M={g∈S:d(f,g)<∞}.
This implies that Q is a unique mapping satisfying (2.10) such that there exists K∈(0,∞) satisfying
∥f(x)-Q(x)∥≤Kφ(x,0)
for all x∈X.
(2) d(Jnf,Q)→0 as n→∞. This implies the equality
limn→∞f(2nx)4n=Q(x)
for all x∈X.
(3) d(f,Q)≤(1/(1-L))d(f,Jf), which implies the inequality
d(f,Q)≤18-8L.
This implies that the inequality (2.3) holds.
It follows from (2.2) and (2.13) that
∥CQ(x,y)∥=limn→∞14n∥Cf(2nx,2ny)∥≤limn→∞14nφ(2nx,2ny)≤limn→∞Lnφ(x,y)=0
for all x,y∈X. So CQ(x,y)=0 for all x,y∈X.
By [29, Proposition 2.1], the mapping Q:X→Y is quadratic, as desired.
Corollary 2.2.
Let 0<p<2 and θ be positive real numbers, and let f:X→Y be a mapping such that
∥Cf(x,y)∥≤θ(∥x∥p+∥y∥p)
for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤θ8-2p+1∥x∥p
for all x∈X.
Proof.
The proof follows from Theorem 2.1 by taking
φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=2p-2, and we get the desired result.
Theorem 2.3.
Let f:X→Y be a mapping for which there exists a function φ:X2→[0,∞) satisfying (2.2) and f(0)=0. If there exists an L<1 such that φ(x,y)≤(L/4)φ(2x,2y) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤L8-8Lφ(x,0)
for all x∈X.
Proof.
We consider the linear mapping J:S→S such that
Jg(x):=4g(x2)
for all x∈X.
It follows from (2.8) that
∥f(x)-4f(x2)∥≤12φ(x2,0)≤L8φ(x,0)
for all x∈X. Hence d(f,Jf)≤L/8.
By Theorem 1.3, there exists a mapping Q:X→Y such that
(1) Q is a fixed point of J, that is,
Q(2x)=4Q(x)
for all x∈X. The mapping Q is a unique fixed point of J in the set
M={g∈S:d(f,g)<∞}.
This implies that Q is a unique mapping satisfying (2.22) such that there exists K∈(0,∞) satisfying
∥f(x)-Q(x)∥≤Kφ(x,0)
for all x∈X.
(2) d(Jnf,Q)→0 as n→∞. This implies the equality
limn→∞4nf(x2n)=Q(x)
for all x∈X.
(3) d(f,Q)≤(1/(1-L))d(f,Jf), which implies the inequality
d(f,Q)≤L8-8L,
which implies that the inequality (2.19) holds.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.4.
Let p>2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying (2.16). Then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤θ2p+1-8∥x∥p
for all x∈X.
Proof.
The proof follows from Theorem 2.3 by taking
φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=22-p and, we get the desired result.
Theorem 2.5.
Let f:X→Y be a mapping for which there exists a function φ:X2→[0,∞) satisfying (2.2). If there exists an L<1 such that φ(x,y)≤9Lφ(x/3,y/3) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤19-9Lφ(x,x)
for all x∈X.
Proof.
Consider the set
S:={g:X→Y},
and introduce the generalized metric on S:
d(g,h)=inf{K∈ℝ+:∥g(x)-h(x)∥≤Kφ(x,x),∀x∈X}.
It is easy to show that (S,d) is complete.
Now we consider the linear mapping J:S→S such that
Jg(x):=19g(3x)
for all x∈X.
By [30, Theorem 3.1],
d(Jg,Jh)≤Ld(g,h)
for all g,h∈S.
Letting y=x in (2.2), we get
∥f(3x)-9f(x)∥≤φ(x,x)
for all x∈X. So
∥f(x)-19f(3x)∥≤19φ(x,x)
for all x∈X. Hence d(f,Jf)≤1/9.
By Theorem 1.3, there exists a mapping Q:X→Y such that
(1) Q is a fixed point of J, that is,
Q(3x)=9Q(x)
for all x∈X. The mapping Q is a unique fixed point of J in the set
M={g∈S:d(f,g)<∞}.
This implies that Q is a unique mapping satisfying (2.36) such that there exists K∈(0,∞) satisfying
∥f(x)-Q(x)∥≤Kφ(x,x)
for all x∈X.
(2) d(Jnf,Q)→0 as n→∞. This implies the equality
limn→∞f(3nx)9n=Q(x)
for all x∈X.
(3) d(f,Q)≤(1/(1-L))d(f,Jf), which implies the inequality
d(f,Q)≤19-9L.
This implies that the inequality (2.29) holds.
It follows from (2.2) and (2.39) that
∥CQ(x,y)∥=limn→∞19n∥Cf(3nx,3ny)∥≤limn→∞19nφ(3nx,3ny)≤limn→∞Lnφ(x,y)=0
for all x,y∈X. So CQ(x,y)=0 for all x,y∈X.
By [29, Proposition 2.1], the mapping Q:X→Y is quadratic, as desired.
Corollary 2.6.
Let 0<p<2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying (2.16). Then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤2θ9-3p∥x∥p
for all x∈X.
Proof.
The proof follows from Theorem 2.5 by taking
φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=3p-2 and, we get the desired result.
Corollary 2.7.
Let 0<p<1 and θ be positive real numbers, and let f:X→Y be a mapping such that
∥Df(x,y)∥≤θ·∥x∥p·∥y∥p
for all x,y∈X. Then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤θ9-9p∥x∥2p
for all x∈X.
Proof.
The proof follows from Theorem 2.5 by taking
φ(x,y):=θ·∥x∥p·∥y∥p
for all x,y∈X. Then L=9p-1 and, we get the desired result.
Theorem 2.8.
Let f:X→Y be a mapping for which there exists a function φ:X2→[0,∞) satisfying (2.2). If there exists an L<1 such that φ(x,y)≤(L/9)φ(3x,3y) for all x,y∈X, then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤L9-9Lφ(x,x)
for all x∈X.
Proof.
We consider the linear mapping J:S→S such that
Jg(x):=9g(x3)
for all x∈X.
The rest of the proof is similar to the proof of Theorem 2.1.
Corollary 2.9.
Let p>2 and θ be positive real numbers, and let f:X→Y be a mapping satisfying (2.16). Then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤2θ3p-9∥x∥p
for all x∈X.
Proof.
The proof follows from Theorem 2.8 by taking
φ(x,y):=θ(∥x∥p+∥y∥p)
for all x,y∈X. Then L=32-p, and we get the desired result.
Corollary 2.10.
Let p>1 and θ be positive real numbers, and let f:X→Y be a mapping satisfying (2.44). Then there exists a unique quadratic mapping Q:X→Y satisfying (1.8) and
∥f(x)-Q(x)∥≤θ9p-9∥x∥2p
for all x∈X.
Proof.
The proof follows from Theorem 2.8 by taking
φ(x,y):=θ·∥x∥p·∥y∥p
for all x,y∈X. Then L=91-p, and we get the desired result.
Acknowledgment
The first author was supported by Hanyang University in 2009.
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