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Abstract and Applied Analysis
VolumeΒ 2008, Article IDΒ 628178, 8 pages
http://dx.doi.org/10.1155/2008/628178
Research Article

On the Stability of Quadratic Functional Equations

1Department of Mathematics, Daejin University, Kyeonggi 487-711, South Korea
2Department of Mathematics Education, Pusan National University, Pusan 609-735, South Korea
3Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 5 October 2007; Revised 27 November 2007; Accepted 4 January 2008

Academic Editor: PaulΒ Eloe

Copyright Β© 2008 Jung Rye Lee et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let 𝑋,π‘Œ be vector spaces and π‘˜ a fixed positive integer. It is shown that a mapping 𝑓(π‘˜π‘₯+𝑦)+𝑓(π‘˜π‘₯βˆ’π‘¦)=2π‘˜2𝑓(π‘₯)+2𝑓(𝑦) for all π‘₯,π‘¦βˆˆπ‘‹ if and only if the mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦) for all π‘₯,π‘¦βˆˆπ‘‹. Furthermore, the Hyers-Ulam-Rassias stability of the above functional equation in Banach spaces is proven.

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 answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [3] for additive mapping and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we now call Hyers-Ulam-Rassias stability of functional equations. Th. M. Rassias [5] during the 27th International Symposium on Functional Equations asked the question whether such a theorem can also be proved for 𝑝β‰₯1. Gajda [6], following the same approach as in [4], gave an affirmative solution to this question for 𝑝>1. It was shown by Gajda [6] as well as by Rassias and Ε emrl [7] that one cannot prove a Th.M. Rassias' type theorem when 𝑝=1. J. M. Rassias [8], following the spirit of the innovative approach of Th. M. Rassias [4] for the unbounded Cauchy difference, proved a similar stability theorem in which he replaced the factor β€–π‘₯‖𝑝+‖𝑦‖𝑝 by β€–π‘₯β€–π‘β‹…β€–π‘¦β€–π‘ž for 𝑝,π‘žβˆˆβ„ with 𝑝+π‘žβ‰ 1.

The functional equation𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦)(1.1)is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be a quadratic function. A Hyers-Ulam-Rassias stability problem for the quadratic functional equation was proved by Skof [9] for mappings π‘“βˆΆπ‘‹β†’π‘Œ, where 𝑋 is a normed space and π‘Œ 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. In [11], Czerwik proved the Hyers-Ulam-Rassias stability of the quadratic functional equation. Several functional equations have been investigated in [12–17].

Throughout this paper, assume that π‘˜ is a fixed positive integer.

In this paper, we solve the functional equation𝑓(π‘˜π‘₯+𝑦)+𝑓(π‘˜π‘₯βˆ’π‘¦)=2π‘˜2𝑓(π‘₯)+2𝑓(𝑦)(1.2)and prove the Hyers-Ulam-Rassias stability of the functional equation (1.2) in Banach spaces.

2. Hyers-Ulam-Rassias Stability of the Quadratic Functional Equation

Proposition 2.1. Let 𝑋 and π‘Œ be vector spaces. A mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(π‘˜π‘₯+𝑦)+𝑓(π‘˜π‘₯βˆ’π‘¦)=2π‘˜2𝑓(π‘₯)+2𝑓(𝑦)(2.1)for all π‘₯,π‘¦βˆˆπ‘‹ if and only if the mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦)(2.2)for all π‘₯,π‘¦βˆˆπ‘‹.

Proof. Assume that π‘“βˆΆπ‘‹β†’π‘Œ satisfies (2.1).
Letting π‘₯=𝑦=0 in (2.1), we get 𝑓(0)=0.
Letting 𝑦=0 in (2.1), we get 𝑓(π‘˜π‘₯)=π‘˜2𝑓(π‘₯) for all π‘₯βˆˆπ‘‹.
Letting π‘₯=0 in (2.1), we get 𝑓(βˆ’π‘¦)=𝑓(𝑦) for all π‘¦βˆˆπ‘‹.
It follows from (2.1) that𝑓(π‘˜π‘₯+𝑦)+𝑓(π‘˜π‘₯βˆ’π‘¦)=2π‘˜2𝑓(π‘₯)+2𝑓(𝑦)=2𝑓(π‘˜π‘₯)+2𝑓(𝑦)(2.3)for all π‘₯,π‘¦βˆˆπ‘‹. So the mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦)(2.4)for all π‘₯,π‘¦βˆˆπ‘‹.
Assume that π‘“βˆΆπ‘‹β†’π‘Œ satisfies 𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦) for all π‘₯,π‘¦βˆˆπ‘‹.
We prove (2.1) for π‘˜=𝑗 by induction on 𝑗.
For the case 𝑗=1, (2.1) holds by the assumption.
For the case 𝑗=2, since𝑓(2π‘₯+𝑦)+𝑓(2π‘₯βˆ’π‘¦)=𝑓(π‘₯+𝑦+π‘₯)+𝑓(π‘₯βˆ’π‘¦+π‘₯)=2𝑓(π‘₯+𝑦)+2𝑓(π‘₯)βˆ’π‘“(𝑦)+2𝑓(π‘₯βˆ’π‘¦)+2𝑓(π‘₯)βˆ’π‘“(βˆ’π‘¦)=2𝑓(π‘₯+𝑦)+2𝑓(π‘₯βˆ’π‘¦)+4𝑓(π‘₯)βˆ’2𝑓(𝑦)=4𝑓(π‘₯)+4𝑓(𝑦)+4𝑓(π‘₯)βˆ’2𝑓(𝑦)=8𝑓(π‘₯)+2𝑓(𝑦)(2.5)for all π‘₯,π‘¦βˆˆπ‘‹, then (2.1) holds.
Assume that (2.1) holds for 𝑗=π‘›βˆ’2 and 𝑗=π‘›βˆ’1 (2<π‘›β‰€π‘˜). By the assumption,𝑓(𝑛π‘₯+𝑦)+𝑓(𝑛π‘₯βˆ’π‘¦)=𝑓(π‘›βˆ’1)π‘₯+𝑦+π‘₯+𝑓(π‘›βˆ’1)π‘₯βˆ’π‘¦+π‘₯=2𝑓(π‘›βˆ’1)π‘₯+𝑦+2𝑓(π‘₯)βˆ’π‘“(π‘›βˆ’2)π‘₯+𝑦+2𝑓(π‘›βˆ’1)π‘₯βˆ’π‘¦+2𝑓(π‘₯)βˆ’π‘“(π‘›βˆ’2)π‘₯βˆ’π‘¦=4(π‘›βˆ’1)2𝑓(π‘₯)+4𝑓(𝑦)+4𝑓(π‘₯)βˆ’2(π‘›βˆ’2)2𝑓(π‘₯)βˆ’2𝑓(𝑦)=2𝑛2𝑓(π‘₯)+2𝑓(𝑦)(2.6)for all π‘₯,π‘¦βˆˆπ‘‹, (2.1) holds for 𝑗=𝑛. Hence the mapping π‘“βˆΆπ‘‹β†’π‘Œ satisfies (2.1) for 𝑗=π‘˜.

From now on, assume that 𝑋 is a normed vector space with norm β€–β‹…β€– and that π‘Œ is a Banach space with norm β€–β‹…β€–.

For a given mapping π‘“βˆΆπ‘‹β†’π‘Œ, we define𝐷𝑓(π‘₯,𝑦)∢=𝑓(π‘˜π‘₯+𝑦)+𝑓(π‘˜π‘₯βˆ’π‘¦)βˆ’2π‘˜2𝑓(π‘₯)βˆ’2𝑓(𝑦)(2.7)for all π‘₯,π‘¦βˆˆπ‘‹.

Now we prove the Hyers-Ulam-Rassias stability of the quadratic functional equation 𝐷𝑓(π‘₯,𝑦)=0.

Theorem 2.2. Let π‘“βˆΆπ‘‹β†’π‘Œ be a mapping with 𝑓(0)=0 for which there exists a function πœ‘βˆΆπ‘‹2β†’[0,∞) such that ξ‚πœ‘(π‘₯,𝑦)∢=βˆžξ“π‘—=01π‘˜2π‘—πœ‘ξ‚€π‘˜π‘—π‘₯,π‘˜π‘—π‘¦ξ‚β€–β€–β€–β€–<∞,(2.8)𝐷𝑓(π‘₯,𝑦)β‰€πœ‘(π‘₯,𝑦)(2.9)π‘₯,π‘¦βˆˆπ‘‹for all π‘„βˆΆπ‘‹β†’π‘Œ. Then there exists a unique quadratic mapping ‖‖‖‖≀1𝑓(π‘₯)βˆ’π‘„(π‘₯)2π‘˜2ξ‚πœ‘(π‘₯,0)(2.10) such that π‘₯βˆˆπ‘‹for all 𝑦=0.

Proof. Letting β€–β€–2𝑓(π‘˜π‘₯)βˆ’2π‘˜2‖‖𝑓(π‘₯)β‰€πœ‘(π‘₯,0)(2.11) in (2.9), we getπ‘₯βˆˆπ‘‹for all 1‖𝑓(π‘₯)βˆ’π‘˜21𝑓(π‘˜π‘₯)‖≀2π‘˜2πœ‘(π‘₯,0)(2.12). Soπ‘₯βˆˆπ‘‹for all β€–1π‘˜2π‘™π‘“ξ‚€π‘˜π‘™π‘₯ξ‚βˆ’1π‘˜2π‘šπ‘“ξ‚€π‘˜π‘šπ‘₯ξ‚β€–β‰€π‘šβˆ’1𝑗=𝑙12π‘˜2𝑗+2πœ‘ξ‚€π‘˜π‘—ξ‚π‘₯,0(2.13). Henceπ‘šfor all nonnegative integers 𝑙 and π‘š>𝑙 with π‘₯βˆˆπ‘‹ and all {(1/π‘˜2𝑛)𝑓(π‘˜π‘›π‘₯)}. It follows from (2.13) that the sequence π‘₯βˆˆπ‘‹ is Cauchy for all π‘Œ. Since {(1/π‘˜2𝑛)𝑓(π‘˜π‘›π‘₯)} is complete, the sequence π‘„βˆΆπ‘‹β†’π‘Œ converges. So one can define the mapping 𝑄(π‘₯)∢=limπ‘›β†’βˆž1π‘˜2π‘›π‘“ξ‚€π‘˜π‘›π‘₯(2.14) byπ‘₯βˆˆπ‘‹for all ‖‖‖‖𝐷𝑄(π‘₯,𝑦)=limπ‘›β†’βˆž1π‘˜2π‘›β€–β€–ξ‚€π‘˜π·π‘“π‘›π‘₯,π‘˜π‘›π‘¦ξ‚β€–β€–β‰€limπ‘›β†’βˆž1π‘˜2π‘›πœ‘ξ‚€π‘˜π‘›π‘₯,π‘˜π‘›π‘¦ξ‚=0(2.15).
By (2.8),π‘₯,π‘¦βˆˆπ‘‹for all 𝐷𝑄(π‘₯,𝑦)=0. So π‘„βˆΆπ‘‹β†’π‘Œ. By Proposition 2.1, the mapping 𝑙=0 is quadratic. Moreover, letting π‘šβ†’βˆž and passing the limit π‘‡βˆΆπ‘‹β†’π‘Œ in (2.13), we get (2.10).
Now, let β€–β€–β€–β€–=1𝑄(π‘₯)βˆ’π‘‡(π‘₯)π‘˜2π‘›β€–β€–π‘„ξ‚€π‘˜π‘›π‘₯ξ‚ξ‚€π‘˜βˆ’π‘‡π‘›π‘₯‖‖≀1π‘˜2π‘›ξ‚€β€–β€–π‘„ξ‚€π‘˜π‘›π‘₯ξ‚ξ‚€π‘˜βˆ’π‘“π‘›π‘₯‖‖+β€–β€–π‘‡ξ‚€π‘˜π‘›π‘₯ξ‚ξ‚€π‘˜βˆ’π‘“π‘›π‘₯‖‖≀1π‘˜2𝑛+2ξ‚€π‘˜ξ‚πœ‘π‘›ξ‚,π‘₯,0(2.16) be another quadratic mapping satisfying (2.1) and (2.10). Then we haveπ‘›β†’βˆžwhich tends to zero as π‘₯βˆˆπ‘‹ for all 𝑄(π‘₯)=𝑇(π‘₯). So we can conclude that π‘₯βˆˆπ‘‹ for all 𝑄. This proves the uniqueness of π‘„βˆΆπ‘‹β†’π‘Œ. So there exists a unique quadratic mapping 𝑝<2 satisfying (2.10).

Corollary 2.3. Let πœƒ and π‘“βˆΆπ‘‹β†’π‘Œ be positive real numbers, and let ‖‖‖‖𝐷𝑓(π‘₯,𝑦)β‰€πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(2.17) be a mapping such that π‘₯,π‘¦βˆˆπ‘‹for all π‘„βˆΆπ‘‹β†’π‘Œ. Then there exists a unique quadratic mapping β€–β€–β€–β€–β‰€πœƒπ‘“(π‘₯)βˆ’π‘„(π‘₯)8βˆ’2𝑝+1β€–π‘₯‖𝑝(2.18) such that π‘₯βˆˆπ‘‹for all ξ‚€πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(2.19).

Proof. The proof follows from Theorem 2.2 by takingπ‘₯,π‘¦βˆˆπ΄for all π‘“βˆΆπ‘‹β†’π‘Œ.

Theorem 2.4. Let 𝑓(0)=0 be a mapping with πœ‘βˆΆπ‘‹2β†’[0,∞) for which there exists a function ξ‚πœ‘(π‘₯,𝑦)∢=βˆžξ“π‘—=0π‘˜2𝑗π‘₯πœ‘(π‘˜π‘—,π‘¦π‘˜π‘—)<∞(2.20) satisfying (2.9) such that π‘₯,π‘¦βˆˆπ‘‹for all π‘„βˆΆπ‘‹β†’π‘Œ. Then there exists a unique quadratic mapping ‖‖‖‖≀1𝑓(π‘₯)βˆ’π‘„(π‘₯)2π‘₯ξ‚πœ‘(π‘˜,0)(2.21) such that π‘₯βˆˆπ‘‹for all ‖𝑓(π‘₯)βˆ’π‘˜2π‘₯𝑓(π‘˜1)‖≀2π‘₯πœ‘(π‘˜,0)(2.22).

Proof. It follows from (2.11) thatπ‘₯βˆˆπ‘‹for all β€–π‘˜2𝑙π‘₯𝑓(π‘˜π‘™)βˆ’π‘˜2π‘šπ‘₯𝑓(π‘˜π‘š)β€–β‰€π‘šβˆ’1𝑗=π‘™π‘˜2𝑗2π‘₯πœ‘(π‘˜π‘—+1,0)(2.23). Henceπ‘šfor all nonnegative integers 𝑙 and π‘š>𝑙 with π‘₯βˆˆπ‘‹ and all {π‘˜2𝑛𝑓(π‘₯/π‘˜π‘›)}. It follows from (2.23) that the sequence π‘₯βˆˆπ‘‹ is Cauchy for all π‘Œ. Since {π‘˜2𝑛𝑓(π‘₯/π‘˜π‘›)} is complete, the sequence π‘„βˆΆπ‘‹β†’π‘Œ converges. So one can define the mapping 𝑄(π‘₯)∢=limπ‘›β†’βˆžπ‘˜2𝑛π‘₯𝑓(π‘˜π‘›)(2.24) byπ‘₯βˆˆπ‘‹for all ‖‖‖‖𝐷𝑄(π‘₯,𝑦)=limπ‘›β†’βˆžπ‘˜2𝑛π‘₯‖𝐷𝑓(π‘˜π‘›,π‘¦π‘˜π‘›)‖≀limπ‘›β†’βˆžπ‘˜2𝑛π‘₯πœ‘(π‘˜π‘›,π‘¦π‘˜π‘›)=0(2.25).
By (2.20),π‘₯,π‘¦βˆˆπ‘‹for all 𝐷𝑄(π‘₯,𝑦)=0. So π‘„βˆΆπ‘‹β†’π‘Œ. By Proposition 2.1, the mapping 𝑙=0 is quadratic. Moreover, letting π‘šβ†’βˆž and passing the limit 𝑝>2 in (2.23), we get (2.21).
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.5. Let πœƒ and π‘“βˆΆπ‘‹β†’π‘Œ be positive real numbers, and let π‘„βˆΆπ‘‹β†’π‘Œ be a mapping satisfying (2.17). Then there exists a unique quadratic mapping β€–β€–β€–β€–β‰€πœƒπ‘“(π‘₯)βˆ’π‘„(π‘₯)2𝑝+1βˆ’8β€–π‘₯‖𝑝(2.26) such that π‘₯βˆˆπ‘‹for all ξ‚€πœ‘(π‘₯,𝑦)∢=πœƒβ€–π‘₯‖𝑝+‖𝑦‖𝑝(2.27).

Proof. The proof follows from Theorem 2.4 by takingπ‘₯,π‘¦βˆˆπ΄for all π‘˜=2.

From now on, assume that π‘“βˆΆπ‘‹β†’π‘Œ.

Theorem 2.6. Let 𝑓(0)=0 be a mapping with πœ‘βˆΆπ‘‹2β†’[0,∞) for which there exists a function ξ‚πœ‘(π‘₯,𝑦)∢=βˆžξ“π‘—=019π‘—πœ‘ξ‚€3𝑗π‘₯,3𝑗𝑦<∞(2.28) satisfying (2.9) such that π‘₯,π‘¦βˆˆπ‘‹for all π‘„βˆΆπ‘‹β†’π‘Œ. Then there exists a unique quadratic mapping ‖‖‖‖≀1𝑓(π‘₯)βˆ’π‘„(π‘₯)9ξ‚πœ‘(π‘₯,π‘₯)(2.29) such that π‘₯βˆˆπ‘‹for all 𝑦=π‘₯.

Proof. Letting ‖‖‖‖𝑓(3π‘₯)βˆ’9𝑓(π‘₯)β‰€πœ‘(π‘₯,π‘₯)(2.30) in (2.9), we getπ‘₯βˆˆπ‘‹for all 1‖𝑓(π‘₯)βˆ’91𝑓(3π‘₯)‖≀9πœ‘(π‘₯,π‘₯)(2.31). Soπ‘₯βˆˆπ‘‹for all β€–19𝑙𝑓(3𝑙1π‘₯)βˆ’9π‘šπ‘“ξ‚€3π‘šπ‘₯ξ‚β€–β‰€π‘šβˆ’1𝑗=𝑙19𝑗+1πœ‘ξ‚€3𝑗π‘₯,3𝑗π‘₯(2.32). Henceπ‘šfor all nonnegative integers 𝑙 and π‘š>𝑙 with π‘₯βˆˆπ‘‹ and all {(1/9𝑛)𝑓(3𝑛π‘₯)}. It follows from (2.32) that the sequence π‘₯βˆˆπ‘‹ is Cauchy for all π‘Œ. Since {(1/9𝑛)𝑓(3𝑛π‘₯)} is complete, the sequence π‘„βˆΆπ‘‹β†’π‘Œ converges. So one can define the mapping 𝑄(π‘₯)∢=limπ‘›β†’βˆž19𝑛𝑓3𝑛π‘₯(2.33) byπ‘₯βˆˆπ‘‹for all ‖‖‖‖𝐷𝑄(π‘₯,𝑦)=limπ‘›β†’βˆž19𝑛‖‖3𝐷𝑓𝑛π‘₯,3𝑛𝑦‖‖≀limπ‘›β†’βˆž19π‘›πœ‘ξ‚€3𝑛π‘₯,3𝑛𝑦=0(2.34).
By (2.28),π‘₯,π‘¦βˆˆπ‘‹for all 𝐷𝑄(π‘₯,𝑦)=0. So π‘„βˆΆπ‘‹β†’π‘Œ. By Proposition 2.1, the mapping 𝑙=0 is quadratic. Moreover, letting π‘šβ†’βˆž and passing the limit 𝑝<1 in (2.32), we get (2.29).
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.7. Let πœƒ and π‘“βˆΆπ‘‹β†’π‘Œ be positive real numbers, and let ‖‖‖‖𝐷𝑓(π‘₯,𝑦)β‰€πœƒβ‹…β€–π‘₯‖𝑝⋅‖𝑦‖𝑝(2.35) be a mapping such that π‘₯,π‘¦βˆˆπ‘‹for all π‘„βˆΆπ‘‹β†’π‘Œ. Then there exists a unique quadratic mapping β€–β€–β€–β€–β‰€πœƒπ‘“(π‘₯)βˆ’π‘„(π‘₯)9βˆ’9𝑝‖π‘₯β€–2𝑝(2.36) such that π‘₯βˆˆπ‘‹for all πœ‘(π‘₯,𝑦)∢=πœƒβ‹…β€–π‘₯‖𝑝⋅‖𝑦‖𝑝(2.37).

Proof. The proof follows from Theorem 2.6 by takingπ‘₯,π‘¦βˆˆπ΄for all π‘“βˆΆπ‘‹β†’π‘Œ.

Theorem 2.8. Let 𝑓(0)=0 be a mapping with πœ‘βˆΆπ‘‹2β†’[0,∞) for which there exists a function ξ‚πœ‘(π‘₯,𝑦)∢=βˆžξ“π‘—=09𝑗π‘₯πœ‘(3𝑗,𝑦3𝑗)<∞(2.38) satisfying (2.9) such that π‘₯,π‘¦βˆˆπ‘‹for all π‘„βˆΆπ‘‹β†’π‘Œ. Then there exists a unique quadratic mapping β€–β€–β€–β€–π‘₯𝑓(π‘₯)βˆ’π‘„(π‘₯)β‰€ξ‚πœ‘(3,π‘₯3)(2.39) such that π‘₯βˆˆπ‘‹for all π‘₯‖𝑓(π‘₯)βˆ’9𝑓(3π‘₯)β€–β‰€πœ‘(3,π‘₯3)(2.40).

Proof. It follows from (2.30) thatπ‘₯βˆˆπ‘‹for all β€–9𝑙π‘₯𝑓(3𝑙)βˆ’9π‘šπ‘₯𝑓(3π‘š)β€–β‰€π‘šβˆ’1𝑗=𝑙9𝑗π‘₯πœ‘(3𝑗+1,π‘₯3𝑗+1)(2.41). Henceπ‘šfor all nonnegative integers 𝑙 and π‘š>𝑙 with π‘₯βˆˆπ‘‹ and all {9𝑛𝑓(π‘₯/3𝑛)}. It follows from (2.41) that the sequence π‘₯βˆˆπ‘‹ is Cauchy for all π‘Œ. Since {9𝑛𝑓(π‘₯/3𝑛)} is complete, the sequence π‘„βˆΆπ‘‹β†’π‘Œ converges. So one can define the mapping 𝑄(π‘₯)∢=limπ‘›β†’βˆž9𝑛π‘₯𝑓(3𝑛)(2.42) byπ‘₯βˆˆπ‘‹for all ‖‖‖‖𝐷𝑄(π‘₯,𝑦)=limπ‘›β†’βˆž19𝑛‖‖3𝐷𝑓𝑛π‘₯,3𝑛𝑦‖‖≀limπ‘›β†’βˆž19π‘›πœ‘ξ‚€3𝑛π‘₯,3𝑛𝑦=0(2.43).
By (2.38),π‘₯,π‘¦βˆˆπ‘‹for all 𝐷𝑄(π‘₯,𝑦)=0. So π‘„βˆΆπ‘‹β†’π‘Œ. By Proposition 2.1, the mapping 𝑙=0 is quadratic. Moreover, letting π‘šβ†’βˆž and passing the limit 𝑝>1 in (2.41), we get (2.39).
The rest of the proof is similar to the proof of Theorem 2.2.

Corollary 2.9. Let πœƒ and π‘“βˆΆπ‘‹β†’π‘Œ be positive real numbers, and let π‘„βˆΆπ‘‹β†’π‘Œ be a mapping satisfying (2.35). Then there exists a unique quadratic mapping β€–β€–β€–β€–β‰€πœƒπ‘“(π‘₯)βˆ’π‘„(π‘₯)9π‘βˆ’9β€–π‘₯β€–2𝑝(2.44) such that π‘₯βˆˆπ‘‹for all πœ‘(π‘₯,𝑦)∢=πœƒβ‹…β€–π‘₯‖𝑝⋅‖𝑦‖𝑝(2.45).

Proof. The proof follows from Theorem 2.8 by takingπ‘₯,π‘¦βˆˆπ΄for all .

Acknowledgments

Jung Rye Lee was supported by Daejin University grants in 2007. The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.

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