Table of Contents
ISRN Mathematical Analysis
VolumeΒ 2011, Article IDΒ 503164, 8 pages
http://dx.doi.org/10.5402/2011/503164
Research Article

Fuzzy Stability of Quadratic Functional Equations in General Cases

Department of Mathematics, Damghan University, Damghan, Iran

Received 21 February 2011; Accepted 17 April 2011

Academic Editor: W.Β Sun

Copyright Β© 2011 Ehsan Movahednia. 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

The aim of this paper is to investigate fuzzy Hyers-Ulam-Rassias stability of the general case of quadratic functional equation 𝑓(π‘Žπ‘₯+𝑏𝑦)+𝑓(π‘Žπ‘₯βˆ’π‘π‘¦)=(π‘Ž/2)𝑓(π‘₯+𝑦)+(π‘Ž/2)𝑓(π‘₯βˆ’π‘¦)+(2π‘Ž2βˆ’π‘Ž)𝑓(π‘₯)+(2𝑏2βˆ’π‘Ž)𝑓(𝑦), where π‘Ž,𝑏β‰₯1 and fixed integers with π‘Žβ‰ 2𝑏2. These functional equations are equivalent. This has been proven by Ulam, 1964.

1. Introduction and Preliminaries

The stability problem of functional equations was raised by Ulam [1] in 1964. In fact he posed the question β€œAssume that a function satisfies a functional equation approximately according to some convention. Is it then possible to find near this function a function satisfying the equation accurately?” In 1941 Hyers gave a significant partial solution to this problem in his paper [2].

Hyers’ result was generalized by Aoki [3] for additive mappings. In 1978, Rassias and Song [4] generalized Hyers’ result, a fact which rekindled interest in the field. Such type of stability is now called the Ulam-Hyers-Rassias stability of functional equations.We refer the curious readers for further information on such problems to, for example, [5–7].

The functional equation 𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)=2𝑓(π‘₯)+2𝑓(𝑦)(1.1) is said to be a simple quadratic functional equation. The first person that investigated the stability of the simple quadratic equation was Skof [8]. He proved that, if 𝑓 is a mapping from a normed space 𝑋 into a Banach space π‘Œ satisfying ‖𝑓(π‘₯+𝑦)+𝑓(π‘₯βˆ’π‘¦)βˆ’2𝑓(π‘₯)βˆ’2𝑓(𝑦)β€–β‰€πœ–forsomeπœ–>0,(1.2) then there is a unique simple quadratic function π‘”βˆΆπ‘‹β†’π‘Œ such that πœ–β€–π‘“(π‘₯)βˆ’π‘”(π‘₯)‖≀2.(1.3) In 1984, Katsaras [9] defined a fuzzy norm on a linear space to construct a fuzzy vector topological structure on the space. Later, some mathematicians have defined fuzzy norms on a linear space from various points of view [10, 11]. In particular, in 2003, Bag and Samanta [12], following Cheng and Mordeson [13], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michalek type [14]. They also established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces.

Recently, considerable attention has been increasing to the problem of fuzzy stability of functional equations. Several fuzzy stability results concerning Cauchy, Jensen, simple quadratic, and cubic functional equations have been investigated [15–18].

Definition 1.1. Let 𝑋 be a real vector space. A function π‘βˆΆπ‘‹Γ—π‘…β†’[0,1] is called fuzzy normed on 𝑋 if for all π‘₯,π‘¦βˆˆπ‘‹ and all 𝑠,π‘‘βˆˆπ‘…(N1)𝑁(π‘₯,𝑑)=0 for 𝑑≀0,(N2)π‘₯=0 if and only if 𝑁(π‘₯,𝑑)=1 for all 𝑑>0,(N3)𝑁(𝑐π‘₯,𝑑)=𝑁(π‘₯,𝑑/|𝑐|) if 𝑐≠0,(N4)𝑁(π‘₯+𝑦,𝑑+𝑠)β‰₯min{𝑁(π‘₯,𝑑),𝑁(𝑦,𝑠)}, (N5)𝑁(π‘₯,.) is a nondecreasing function of 𝑅 and limπ‘‘β†’βˆžπ‘(π‘₯,𝑑)=1,(N6)for π‘₯β‰ 0,𝑁(π‘₯,.) is continuous on 𝑅,the pair (𝑋,𝑁) is called a fuzzy normed vector space.

Example 1.2. Let (𝑋,β€–β‹…β€–) be a normed linear space. One can easily verify that, for each π‘˜>0, π‘π‘˜βŽ§βŽͺ⎨βŽͺβŽ©π‘‘(π‘₯,𝑑)=𝑑+π‘˜β€–π‘₯β€–,if𝑑>0,0,if𝑑≀0,(1.4) defines a fuzzy norm on 𝑋.

Definition 1.3. Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {π‘₯𝑛} in 𝑋 is said to be convergent or converges if there exists an π‘₯βˆˆπ‘‹ such that limπ‘›β†’βˆžπ‘(π‘₯π‘›βˆ’π‘₯,𝑑)=1 for all 𝑑>0. In this case, π‘₯ is called the limit of the sequence {π‘₯𝑛}, and one denotes it by π‘βˆ’limπ‘›β†’βˆžπ‘ξ€·π‘₯π‘›ξ€Έβˆ’π‘₯,𝑑=π‘₯.(1.5)

Definition 1.4. Let (𝑋,𝑁) be a fuzzy normed vector space. A sequence {π‘₯𝑛} in 𝑋 is said to be Cauchy if for each πœ–>0 and each 𝛿>0 there exists an 𝑛0βˆˆπ‘ such that 𝑁π‘₯π‘šβˆ’π‘₯𝑛,𝛿>1βˆ’πœ–π‘š,𝑛β‰₯𝑛0ξ€Έ.(1.6)
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

2. Main Results

Let 𝑋 be a linear space and (𝑍,𝑁) a fuzzy normed space. Let (π‘Œ,π‘ξ…ž) be a fuzzy Banach space and π‘“βˆΆπ‘‹β†’π‘Œ a function satisfying π‘ξ…žξ‚€π‘Žπ‘“(π‘Žπ‘₯+𝑏𝑦)+𝑓(π‘Žπ‘₯βˆ’π‘π‘¦)βˆ’2π‘Žπ‘“(π‘₯+𝑦)βˆ’2𝑓(π‘₯βˆ’π‘¦)βˆ’2π‘Ž2ξ€Έβˆ’ξ€·βˆ’π‘Žπ‘“(π‘₯)2𝑏2ξ€Έξ‚βˆ’π‘Žπ‘“(𝑦),𝑑+𝑠β‰₯min{𝑁(πœ‘(π‘₯),π‘‘π‘ž),𝑁(πœ‘(𝑦),π‘ π‘ž)},(2.1) where π‘Ž,𝑏β‰₯1,π‘Žβ‰ 2𝑏2 and for all 𝑑,𝑠>0,π‘ž>1/2, such that πœ‘βˆΆπ‘‹β†’π‘ is a function, and πœ‘(π‘Žπ‘₯)=π›Όπœ‘(π‘₯),βˆ€π‘₯βˆˆπ‘‹,(2.2) for some number 𝛼 with 0<|𝛼|<π‘Ž. Then, there exists a unique quadratic functional equation π‘„βˆΆπ‘‹β†’π‘Œ such that π‘ξ…žξƒ©ξƒ©ξ€·π‘Ž(𝑄(π‘₯)βˆ’π‘“(π‘₯),𝑑)β‰₯π‘πœ‘(π‘₯),2βˆ’π›Όπ‘ξ€Έπ‘‘2ξƒͺπ‘žξƒͺ.(2.3)

Proof. Putting 𝑦=0 and 𝑑=𝑠 in (2.1), we have that π‘ξ…žξ€·2𝑓(π‘Žπ‘₯)βˆ’2π‘Ž2𝑓(π‘₯),2𝑑β‰₯𝑁(πœ‘(π‘₯),π‘‘π‘ž)(βˆ€π‘₯βˆˆπ‘‹,𝑑>0).(2.4) Therefore π‘ξ…žξ‚€1π‘Ž2𝑑𝑓(π‘Žπ‘₯)βˆ’π‘“(π‘₯),π‘Ž2β‰₯𝑁(πœ‘(π‘₯),π‘‘π‘ž)(βˆ€π‘₯βˆˆπ‘‹,𝑑>0),(2.5) and π‘ξ…žξ‚΅1π‘Ž2𝑑𝑓(π‘Žπ‘₯)βˆ’π‘“(π‘₯),π‘π‘Ž2ξ‚Άβ‰₯𝑁(πœ‘(π‘₯),𝑑).(2.6) Now, replacing π‘₯=π‘Žπ‘₯ in (2.6), π‘ξ…žξ‚΅1π‘Ž4π‘“ξ€·π‘Ž2π‘₯ξ€Έβˆ’1π‘Ž2𝑑𝑓(π‘Žπ‘₯),π‘π‘Ž4ξ‚Άβ‰₯𝑁(πœ‘(π‘Žπ‘₯),𝑑),(2.7) and then by the assumption that πœ‘(π‘Žπ‘₯)=π›Όπœ‘(π‘₯) and property (𝑁3) of Definition 1.1 we obtain that π‘ξ…žξ‚΅1π‘Ž4π‘“ξ€·π‘Ž2π‘₯ξ€Έβˆ’1π‘Ž2𝑓(π‘Žπ‘₯),(𝛼𝑑)π‘π‘Ž4ξ‚Άβ‰₯𝑁(πœ‘(π‘₯),𝑑).(2.8) By comparing (2.6) and (2.8) and using property (𝑁4) we obtain that π‘ξ…žξ‚΅1π‘Ž4π‘“ξ€·π‘Ž2π‘₯ξ€Έπ‘‘βˆ’π‘“(π‘₯),π‘π‘Ž2+(𝛼𝑑)π‘π‘Ž4ξ‚Άβ‰₯𝑁(πœ‘(π‘₯),𝑑)(βˆ€π‘₯βˆˆπ‘‹,𝑑>0).(2.9) Again, by replacing π‘₯=π‘Žπ‘₯, in (2.9), π‘ξ…žξ‚΅1π‘Ž6π‘“ξ€·π‘Ž3π‘₯ξ€Έβˆ’1π‘Ž2𝑑𝑓(π‘Žπ‘₯),π‘π‘Ž4+(𝛼𝑑)π‘π‘Ž6ξ‚Άβ‰₯𝑁(πœ‘(π‘Žπ‘₯),𝑑).(2.10) Thus π‘ξ…žξ‚΅1π‘Ž6π‘“ξ€·π‘Ž3π‘₯ξ€Έβˆ’1π‘Ž2𝑓(π‘₯),(𝛼𝑑)π‘π‘Ž4+𝛼2π‘π‘‘π‘π‘Ž6ξ‚Άβ‰₯𝑁(πœ‘(π‘₯),𝑑).(2.11) By comparing (2.6), and (2.11) we obtain that π‘ξ…žξ‚΅1π‘Ž6π‘“ξ€·π‘Ž3π‘₯ξ€Έπ‘‘βˆ’π‘“(π‘₯),π‘π‘Ž2+(𝛼𝑑)π‘π‘Ž4+𝛼2π‘π‘‘π‘π‘Ž6ξ‚Άβ‰₯𝑁(πœ‘(π‘₯),𝑑)(βˆ€π‘₯βˆˆπ‘‹,𝑑>0).(2.12) With following this process we obtain that π‘ξ…žξƒ©1π‘Ž2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘“(π‘₯),π‘›ξ“π‘˜=1𝛼(π‘˜βˆ’1)π‘π‘Žβˆ’2π‘˜π‘‘π‘ξƒͺβ‰₯𝑁(πœ‘(π‘₯),𝑑).(2.13) If π‘šβˆˆπ‘, 𝑛>π‘š>0, then π‘›βˆ’π‘šβˆˆπ‘. Replacing 𝑛 by π‘›βˆ’π‘š in (2.13) gives π‘ξ…žξƒ©π‘Ž2π‘šβˆ’2𝑛𝑓(π‘Žπ‘›βˆ’π‘šπ‘₯)βˆ’π‘“(π‘₯),π‘›βˆ’π‘šξ“π‘˜=1𝛼(π‘˜βˆ’1)π‘π‘Žβˆ’2π‘˜π‘‘π‘ξƒͺβ‰₯𝑁(πœ‘(π‘₯),𝑑)(βˆ€π‘₯βˆˆπ‘‹,𝑑>0,π‘›βˆˆπ‘).(2.14) By replacing π‘₯=π‘Žπ‘šπ‘₯ in (2.14) we obtain that π‘ξ…žξƒ©π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘Žβˆ’2π‘šπ‘“(π‘Žπ‘šπ‘₯),π‘Žβˆ’2π‘šπ‘›βˆ’π‘šξ“π‘˜=1𝛼(π‘˜βˆ’1)π‘π‘Žβˆ’2π‘˜π‘‘π‘ξƒͺβ‰₯𝑁(πœ‘(π‘Žπ‘šπ‘₯),𝑑).(2.15) Thus π‘ξ…žξƒ©π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘Žβˆ’2π‘šπ‘“(π‘Žπ‘šπ‘₯),π‘Žβˆ’2π‘šπ‘›βˆ’π‘šξ“π‘˜=1𝛼(π‘˜βˆ’1)π‘π‘Žβˆ’2π‘˜π›Όπ‘šπ‘π‘‘π‘ξƒͺβ‰₯𝑁(πœ‘(π‘₯),𝑑).(2.16) It follows that π‘ξ…žξƒ©π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘Žβˆ’2π‘šπ‘“(π‘Žπ‘šπ‘₯),π‘›ξ“π‘˜=π‘š+1𝛼(π‘˜βˆ’1)π‘π‘Žβˆ’2π‘˜π‘‘π‘ξƒͺβ‰₯𝑁(πœ‘(π‘₯),𝑑)(βˆ€π‘₯βˆˆπ‘‹,𝑑>0).(2.17) Let 𝑐>0, and let πœ– be given. Since limπ‘‘β†’βˆžπ‘(πœ‘(π‘₯),𝑑)=1, there is some 𝑑0>0 such that π‘ξ€·πœ‘(π‘₯),𝑑0ξ€Έβ‰₯1βˆ’πœ–.(2.18) Fix some 𝑑>𝑑0. The convergence of series βˆ‘βˆžπ‘˜=1(𝛼𝑝/π‘Ž2)π‘˜π›Όβˆ’π‘π‘‘π‘ guarantees that there exists some 𝑛0β‰₯0 such that, for each 𝑛>π‘š>𝑛0, the inequality βˆ‘π‘›π‘˜=π‘š+1(𝛼𝑝/π‘Ž2)π‘˜π›Όβˆ’π‘π‘‘π‘<𝑐 holds. It follows that π‘ξ…žξ€·π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘Žβˆ’2π‘šπ‘“(π‘Žπ‘šξ€Έπ‘₯),𝑐β‰₯π‘ξ…žξƒ©π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘Žβˆ’2π‘šπ‘“(π‘Žπ‘šπ‘₯),π‘›ξ“π‘˜=π‘š+1ξ‚΅π›Όπ‘π‘Ž2ξ‚Άπ‘˜π›Όβˆ’π‘π‘‘π‘0ξƒͺξ€·β‰₯π‘πœ‘(π‘₯),𝑑0ξ€Έβ‰₯1βˆ’πœ–.(2.19) Hence {𝑓(π‘Žπ‘›π‘₯)/π‘Ž2𝑛} is a Cauchy sequence in fuzzy Banach space (π‘Œ,π‘ξ…ž), and thus this sequence converges to some 𝑄(π‘₯)βˆˆπ‘Œ. It means that 𝑄(π‘₯)=π‘ξ…žβˆ’limπ‘›β†’βˆžπ‘“(π‘Žπ‘›π‘₯)π‘Ž2𝑛.(2.20) Furthermore by putting π‘š=0 in (2.17), π‘ξ…žξƒ©π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›π‘₯)βˆ’π‘“(π‘₯),π‘›ξ“π‘˜=1ξ‚΅π›Όπ‘π‘Ž2ξ‚Άπ‘˜π›Όβˆ’π‘π‘‘π‘ξƒͺ𝑁β‰₯𝑁(πœ‘(π‘₯),𝑑),(2.21)ξ…žξ€·π‘Žβˆ’2𝑛𝑓(π‘Žπ‘›ξ€ΈβŽ›βŽœβŽœβŽπ‘‘π‘₯)βˆ’π‘“(π‘₯),𝑑β‰₯π‘πœ‘(π‘₯),π‘žξ‚€βˆ‘π‘›π‘˜=1𝛼𝑝/π‘Ž2ξ€Έπ‘˜π›Όβˆ’π‘ξ‚π‘žβŽžβŽŸβŽŸβŽ (βˆ€π‘₯βˆˆπ‘‹,𝑑>0).(2.22) Next we will show that 𝑄 is quadratic. Let π‘₯,π‘¦βˆˆπ‘‹, and then we have that π‘ξ…žξ‚€π‘Žπ‘„(π‘Žπ‘₯+𝑏𝑦)+𝑄(π‘Žπ‘₯βˆ’π‘π‘¦)βˆ’2π‘Žπ‘„(π‘₯+𝑦)βˆ’2ξ€·(π‘₯βˆ’π‘¦)βˆ’2π‘Ž2ξ€Έξ€·βˆ’π‘Žπ‘„(π‘₯)βˆ’2𝑏2ξ€Έξ‚βˆ’π‘π‘„(𝑦),𝑑β‰₯π‘ξ…žξ‚€π‘Žπ‘„(π‘Žπ‘₯+𝑏𝑦)+𝑄(π‘Žπ‘₯βˆ’π‘π‘¦)βˆ’2π‘Žπ‘„(π‘₯+𝑦)βˆ’2βˆ’ξ€·(π‘₯βˆ’π‘¦)2π‘Ž2ξ€Έξ€·βˆ’π‘Žπ‘„(π‘₯)βˆ’2𝑏2ξ€Έξ€Έξ‚»π‘βˆ’π‘π‘„(𝑦),𝑑β‰₯minξ…žξ‚΅π‘“π‘„(π‘Žπ‘₯+𝑏𝑦)βˆ’(π‘Žπ‘›(π‘Žπ‘₯+𝑏𝑦))π‘Ž2𝑛,𝑑7ξ‚Ά,π‘ξ…žξ‚΅π‘„(π‘Žπ‘₯βˆ’π‘π‘¦)βˆ’π‘“(π‘Žπ‘›(π‘Žπ‘₯βˆ’π‘π‘¦))π‘Ž2𝑛,𝑑7ξ‚Ά,π‘ξ…žξ‚΅π‘Ž2𝑓(π‘Žπ‘›(π‘₯+𝑦))π‘Ž2π‘›βˆ’π‘Ž2𝑄𝑑(π‘₯+𝑦),7ξ‚Ά,π‘ξ…žξ‚΅π‘Ž2𝑓(π‘Žπ‘›(π‘₯βˆ’π‘¦))π‘Ž2π‘›βˆ’π‘Ž2𝑑𝑄(π‘₯βˆ’π‘¦),7ξ‚Ά,π‘ξ…žξ‚΅ξ€·2π‘Ž2ξ€Έβˆ’π‘Žπ‘“(π‘Žπ‘›(π‘₯))π‘Ž2π‘›βˆ’ξ€·2π‘Ž2ξ€Έπ‘‘βˆ’π‘Žπ‘„(π‘₯),7ξ‚Ά,π‘ξ…žξ‚΅ξ€·2𝑏2ξ€Έβˆ’π‘Žπ‘“(π‘Žπ‘›(𝑦))π‘Ž2π‘›βˆ’ξ€·2𝑏2ξ€Έπ‘‘βˆ’π‘Žπ‘„(𝑦),7ξ‚Ά,π‘ξ…žξ‚΅π‘“(π‘Žπ‘›(π‘Žπ‘₯+𝑏𝑦))π‘Ž2𝑛+𝑓(π‘Žπ‘›(π‘Žπ‘₯βˆ’π‘π‘¦))π‘Ž2π‘›βˆ’π‘Ž2𝑓(π‘Žπ‘›(π‘₯+𝑦))π‘Ž2π‘›βˆ’π‘Ž2𝑓(π‘Žπ‘›(π‘₯βˆ’π‘¦))π‘Ž2π‘›βˆ’ξ€·2π‘Ž2ξ€Έβˆ’π‘Žπ‘“(π‘Žπ‘›(π‘₯))π‘Ž2π‘›βˆ’ξ€·2𝑏2ξ€Έβˆ’π‘Žπ‘“(π‘Žπ‘›(𝑦))π‘Ž2𝑛,𝑑7.ξ‚Άξ‚Ό(2.23) The first six terms on the right-hand side of the above inequality tend to 1 as π‘›β†’βˆž, and the seventh term, by (2.1), is greater than or equal to ξ‚»π‘ξ‚΅ξ‚΅π‘Žminπœ‘(π‘₯),2π‘žπ›Όξ‚Άπ‘›ξ‚€π‘‘ξ‚14π‘žξ‚Άξ‚΅ξ‚΅π‘Ž,π‘πœ‘(𝑦),2π‘žπ›Όξ‚Άπ‘›ξ‚€π‘‘ξ‚14π‘žξ‚Άξ‚Ό,(2.24) which tends to 1 as π‘›β†’βˆž. Therefore π‘ξ…žξ‚€π‘Žπ‘„(π‘Žπ‘₯+𝑏𝑦)+𝑄(π‘Žπ‘₯βˆ’π‘π‘¦)βˆ’2π‘Žπ‘„(π‘₯+𝑦)βˆ’2βˆ’ξ€·π‘„(π‘₯βˆ’π‘¦)2π‘Ž2ξ€Έξ€·βˆ’π‘Žπ‘„(π‘₯)βˆ’2𝑏2ξ€Έξ‚βˆ’π‘Žπ‘„(𝑦),𝑑=1(2.25) for each π‘₯,π‘¦βˆˆπ‘‹ and 𝑑>0. So by property (𝑁2), we have that π‘„π‘Ž(π‘Žπ‘₯+𝑏𝑦)+𝑄(π‘Žπ‘₯βˆ’π‘π‘¦)βˆ’2π‘„π‘Ž(π‘₯+𝑦)βˆ’2𝑄(π‘₯βˆ’π‘¦)βˆ’2π‘Ž2ξ€Έπ‘„βˆ’ξ€·βˆ’π‘Ž(π‘₯)2𝑏2ξ€Έβˆ’π‘Žπ‘„(𝑦)=0,βˆ€π‘₯,π‘¦βˆˆπ‘‹.(2.26) Therefore 𝑄 is quadratic function. For every π‘₯βˆˆπ‘‹ and 𝑑,𝑠>0, by (2.22), for large enough 𝑛, we have that π‘ξ…žξ‚»π‘(𝑄(π‘₯)βˆ’π‘“(π‘₯),𝑑)β‰₯minξ…žξ‚΅π‘„(π‘₯)βˆ’π‘“(π‘Žπ‘›π‘₯)π‘Ž2𝑛,𝑑2ξ‚Ά,π‘ξ…žξ‚΅π‘“(π‘Žπ‘›π‘₯)π‘Ž2π‘›π‘‘βˆ’π‘“(π‘₯),2βŽ›βŽœβŽœβŽœβŽœβŽξ‚Άξ‚Όβ‰₯π‘πœ‘(π‘₯),(𝑑/2)π‘žξ‚΅π‘›βˆ‘π‘˜=1𝛼𝑝/π‘Ž2ξ€Έπ‘˜π›Όβˆ’π‘ξ‚Άπ‘žβŽžβŽŸβŽŸβŽŸβŽŸβŽ ξƒ©ξƒ©ξ€·π‘Ž=π‘πœ‘(π‘₯),2βˆ’π›Όπ‘ξ€Έπ‘‘2ξƒͺπ‘žξƒͺ.(2.27) Let π‘„ξ…ž be another quadratic function from 𝑋 to π‘Œ which satisfies (2.3). Since, for each π‘›βˆˆπ‘, 𝑄(π‘Žπ‘›π‘₯)=π‘Ž2𝑛𝑄(π‘₯),π‘„ξ…ž(π‘Žπ‘›π‘₯)=π‘Ž2π‘›π‘„ξ…ž(π‘₯),(2.28) We have that π‘ξ…žξ€·π‘„(π‘₯)βˆ’π‘„ξ…žξ€Έ(π‘₯),𝑑=π‘ξ…žξ€·π‘„(π‘Žπ‘›π‘₯)βˆ’π‘„ξ…ž(π‘Žπ‘›π‘₯),π‘Ž2𝑛𝑑𝑁β‰₯minξ…žξ‚΅π‘„(π‘Žπ‘›π‘₯)βˆ’π‘“(π‘Žπ‘›π‘Žπ‘₯),2𝑛𝑑2ξ‚Ά,π‘ξ…žξ‚΅π‘“(π‘Žπ‘›π‘₯)βˆ’π‘„ξ…ž(π‘Žπ‘›π‘Žπ‘₯),2𝑛𝑑2ξ‚΅ξ‚΅π‘Žξ‚Άξ‚Όβ‰₯π‘πœ‘(π‘₯),2βˆ’π›Όπ‘2ξ‚Άπ‘žπ‘Ž2π‘›π‘žπ›Όπ‘›π‘‘π‘ž2π‘žξ‚Ά(2.29) for each π‘›βˆˆπ‘. Due to π‘ž>1/2, limπ‘›β†’βˆžπ‘ξ‚΅ξ‚΅π‘Žπœ‘(π‘₯),2βˆ’π›Όπ‘2ξ‚Άπ‘žπ‘Ž2π‘›π‘žπ›Όπ‘›π‘‘π‘ž2π‘žξ‚Ά=1(2.30) for each π‘₯βˆˆπ‘‹ and 𝑑>0. Therefore 𝑄=π‘„ξ…ž.

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