The Stability of a Quadratic Functional Equation with the Fixed Point Alternative

Abstract

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) (1.1) for all 𝑥,𝑦∈𝐸, where 𝜖 and 𝑝 are constants with 𝜖>0 and 𝑝<1. Then the limit 𝐿(𝑥)=lim_𝑛→∞𝑓(2^𝑛𝑥)2^𝑛 (1.2) exists for all 𝑥∈𝐸, and 𝐿∶𝐸→𝐸^′ is the unique additive mapping which satisfies ‖𝑓(𝑥)−𝐿(𝑥)‖≤2𝜖2−2^𝑝‖𝑥‖^𝑝 (1.3) 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 (1.4) for all 𝑥,𝑦∈𝑋. Then there exists a unique additive mapping 𝐿∶𝑋→𝑌 satisfying 𝜃‖𝑓(𝑥)−𝐿(𝑥)‖≤|2^𝑝−2|‖𝑥‖^𝑝 (1.5) 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) (1.6) 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

(1)𝑑(𝑥,𝑦)=0 if and only if 𝑥=𝑦;(2)𝑑(𝑥,𝑦)=𝑑(𝑦,𝑥) for all 𝑥,𝑦∈𝑋;(3)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(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)=∞ (1.7) for all nonnegative integers 𝑛 or there exists a positive integer 𝑛₀ such that(1)𝑑(𝐽^𝑛𝑥,𝐽^𝑛+1𝑥)<∞, for all 𝑛≥𝑛₀;(2)the sequence {𝐽^𝑛𝑥} converges to a fixed point 𝑦^∗ of 𝐽;(3)𝑦^∗ is the unique fixed point of 𝐽 in the set 𝑌={𝑦∈𝑋∣𝑑(𝐽^𝑛0𝑥,𝑦)<∞};(4)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) (1.8) 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) (1.9) 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) (1.10) 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) (2.1) 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) (2.2) 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 1‖𝑓(𝑥)−𝑄(𝑥)‖≤8−8𝐿𝜑(𝑥,0) (2.3) for all 𝑥∈𝑋.

Proof. Consider the set 𝑆∶={𝑔∶𝑋→𝑌}, (2.4) and introduce the generalized metric on 𝑆: 𝑑(𝑔,ℎ)=inf{𝐾∈ℝ₊∶‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐾𝜑(𝑥,0),∀𝑥∈𝑋}. (2.5) It is easy to show that (𝑆,𝑑) is complete.
Now we consider the linear mapping 𝐽∶𝑆→𝑆 such that 1𝐽𝑔(𝑥)∶=4𝑔(2𝑥) (2.6) for all 𝑥∈𝑋.
By [30, Theorem 3.1], 𝑑(𝐽𝑔,𝐽ℎ)≤𝐿𝑑(𝑔,ℎ) (2.7) for all 𝑔,ℎ∈𝑆.
Letting 𝑦=0 in (2.2), we get ‖2𝑓(2𝑥)−8𝑓(𝑥)‖≤𝜑(𝑥,0) (2.8) for all 𝑥∈𝑋. So 1‖𝑓(𝑥)−41𝑓(2𝑥)‖≤8𝜑(𝑥,0) (2.9) 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) (2.10) for all 𝑥∈𝑋. The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}. (2.11) 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) (2.12) for all 𝑥∈𝑋.
(2) 𝑑(𝐽^𝑛𝑓,𝑄)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞𝑓(2^𝑛𝑥)4^𝑛=𝑄(𝑥) (2.13) for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝑄)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), which implies the inequality 1𝑑(𝑓,𝑄)≤.8−8𝐿 (2.14) This implies that the inequality (2.3) holds.
It follows from (2.2) and (2.13) that ‖𝐶𝑄(𝑥,𝑦)‖=lim_𝑛→∞14^𝑛‖𝐶𝑓(2^𝑛𝑥,2^𝑛𝑦)‖≤lim_𝑛→∞14^𝑛𝜑(2^𝑛𝑥,2^𝑛𝑦)≤lim_𝑛→∞𝐿^𝑛𝜑(𝑥,𝑦)=0 (2.15) 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) (2.16) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (1.8) and 𝜃‖𝑓(𝑥)−𝑄(𝑥)‖≤8−2^𝑝+1‖𝑥‖^𝑝 (2.17) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.1 by taking 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) (2.18) for all 𝑥,𝑦∈𝑋. Then 𝐿=2^𝑝−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 𝐿‖𝑓(𝑥)−𝑄(𝑥)‖≤8−8𝐿𝜑(𝑥,0) (2.19) for all 𝑥∈𝑋.

Proof. We consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝑥𝐽𝑔(𝑥)∶=4𝑔(2) (2.20) for all 𝑥∈𝑋.
It follows from (2.8) that 𝑥‖𝑓(𝑥)−4𝑓(21)‖≤2𝑥𝜑(2𝐿,0)≤8𝜑(𝑥,0) (2.21) 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) (2.22) for all 𝑥∈𝑋. The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}. (2.23) 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) (2.24) for all 𝑥∈𝑋.
(2) 𝑑(𝐽^𝑛𝑓,𝑄)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞4^𝑛𝑥𝑓(2^𝑛)=𝑄(𝑥) (2.25) for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝑄)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), which implies the inequality 𝐿𝑑(𝑓,𝑄)≤,8−8𝐿 (2.26) 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 𝜃‖𝑓(𝑥)−𝑄(𝑥)‖≤2^𝑝+1−8‖𝑥‖^𝑝 (2.27) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.3 by taking 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) (2.28) for all 𝑥,𝑦∈𝑋. Then 𝐿=2^2−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 1‖𝑓(𝑥)−𝑄(𝑥)‖≤9−9𝐿𝜑(𝑥,𝑥) (2.29) for all 𝑥∈𝑋.

Proof. Consider the set 𝑆∶={𝑔∶𝑋→𝑌}, (2.30) and introduce the generalized metric on 𝑆: 𝑑(𝑔,ℎ)=inf{𝐾∈ℝ₊∶‖𝑔(𝑥)−ℎ(𝑥)‖≤𝐾𝜑(𝑥,𝑥),∀𝑥∈𝑋}. (2.31) It is easy to show that (𝑆,𝑑) is complete.
Now we consider the linear mapping 𝐽∶𝑆→𝑆 such that 1𝐽𝑔(𝑥)∶=9𝑔(3𝑥) (2.32) for all 𝑥∈𝑋.
By [30, Theorem 3.1], 𝑑(𝐽𝑔,𝐽ℎ)≤𝐿𝑑(𝑔,ℎ) (2.33) for all 𝑔,ℎ∈𝑆.
Letting 𝑦=𝑥 in (2.2), we get ‖𝑓(3𝑥)−9𝑓(𝑥)‖≤𝜑(𝑥,𝑥) (2.34) for all 𝑥∈𝑋. So 1‖𝑓(𝑥)−91𝑓(3𝑥)‖≤9𝜑(𝑥,𝑥) (2.35) 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) (2.36) for all 𝑥∈𝑋. The mapping 𝑄 is a unique fixed point of 𝐽 in the set 𝑀={𝑔∈𝑆∶𝑑(𝑓,𝑔)<∞}. (2.37) 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) (2.38) for all 𝑥∈𝑋.
(2) 𝑑(𝐽^𝑛𝑓,𝑄)→0 as 𝑛→∞. This implies the equality lim_𝑛→∞𝑓(3^𝑛𝑥)9^𝑛=𝑄(𝑥) (2.39) for all 𝑥∈𝑋.
(3) 𝑑(𝑓,𝑄)≤(1/(1−𝐿))𝑑(𝑓,𝐽𝑓), which implies the inequality 1𝑑(𝑓,𝑄)≤.9−9𝐿 (2.40) This implies that the inequality (2.29) holds.
It follows from (2.2) and (2.39) that ‖𝐶𝑄(𝑥,𝑦)‖=lim_𝑛→∞19^𝑛‖𝐶𝑓(3^𝑛𝑥,3^𝑛𝑦)‖≤lim_𝑛→∞19^𝑛𝜑(3^𝑛𝑥,3^𝑛𝑦)≤lim_𝑛→∞𝐿^𝑛𝜑(𝑥,𝑦)=0 (2.41) 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)‖≤2𝜃9−3^𝑝‖𝑥‖^𝑝 (2.42) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.5 by taking 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) (2.43) for all 𝑥,𝑦∈𝑋. Then 𝐿=3^𝑝−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 (2.44) for all 𝑥,𝑦∈𝑋. Then there exists a unique quadratic mapping 𝑄∶𝑋→𝑌 satisfying (1.8) and 𝜃‖𝑓(𝑥)−𝑄(𝑥)‖≤9−9^𝑝‖𝑥‖^2𝑝 (2.45) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.5 by taking 𝜑(𝑥,𝑦)∶=𝜃⋅‖𝑥‖^𝑝⋅‖𝑦‖^𝑝 (2.46) for all 𝑥,𝑦∈𝑋. Then 𝐿=9^𝑝−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 𝐿‖𝑓(𝑥)−𝑄(𝑥)‖≤9−9𝐿𝜑(𝑥,𝑥) (2.47) for all 𝑥∈𝑋.

Proof. We consider the linear mapping 𝐽∶𝑆→𝑆 such that 𝑥𝐽𝑔(𝑥)∶=9𝑔(3) (2.48) 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)‖≤2𝜃3^𝑝−9‖𝑥‖^𝑝 (2.49) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.8 by taking 𝜑(𝑥,𝑦)∶=𝜃(‖𝑥‖^𝑝+‖𝑦‖^𝑝) (2.50) for all 𝑥,𝑦∈𝑋. Then 𝐿=3^2−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 𝜃‖𝑓(𝑥)−𝑄(𝑥)‖≤9^𝑝−9‖𝑥‖^2𝑝 (2.51) for all 𝑥∈𝑋.

Proof. The proof follows from Theorem 2.8 by taking 𝜑(𝑥,𝑦)∶=𝜃⋅‖𝑥‖^𝑝⋅‖𝑦‖^𝑝 (2.52) for all 𝑥,𝑦∈𝑋. Then 𝐿=9^1−𝑝, and we get the desired result.

Acknowledgment

The first author was supported by Hanyang University in 2009.