Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 496361 6 pageshttpdxdoiorg1011552013496361

Research ArticleHyperstability of the Freacutechet Equation anda Characterization of Inner Product Spaces

Anna Bahyrycz1 Janusz Brzdwk1 Magdalena Piszczek1 and Justyna Sikorska2

1 Department of Mathematics Pedagogical University Podchorazych 2 30-084 Krakow Poland2 Institute of Mathematics University of Silesia Bankowa 14 40-007 Katowice Poland

Correspondence should be addressed to Anna Bahyrycz bahupkrakowpl

Received 30 May 2013 Accepted 4 October 2013

Academic Editor Bjorn Birnir

Copyright copy 2013 Anna Bahyrycz et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We prove some stability and hyperstability results for the well-known Frechet equation stemming from one of the characterizationsof the inner product spaces As the main tool we use a fixed point theorem for the function spaces We finish the paper with somenew inequalities characterizing the inner product spaces

1 Introduction

In the literature there are many characterizations of innerproduct spaces The first norm characterization of innerproduct space was given by Frechet [1] in 1935 He proved thata normed space (119883 sdot ) is an inner product space if and onlyif for all 119909 119910 119911 isin 119883

1003817100381710038171003817119909 + 119910 + 11991110038171003817100381710038172

+ 1199092+

100381710038171003817100381711991010038171003817100381710038172

+ 1199112

=1003817100381710038171003817119909 + 119910

10038171003817100381710038172

+ 119909 + 1199112+

1003817100381710038171003817119910 + 11991110038171003817100381710038172

(1)

In the same year Jordan and von Neumann [2] gave the cele-brated parallelogram law characterization of an inner productspace Since then numerous further conditions character-izing the inner product spaces among the normed spaceshave been shown More than 300 such conditions have beencollected in the book of Amir [3] Many geometrical charac-terizations are presented in the book by Alsina et al [4] forsome other see for example [5ndash8]

The results that we obtain are motivated by the notionof hyperstability of functional equations (see eg [9ndash14])which has been introduced in connection with the issue ofstability of functional equations (for more details see eg[15 16])

The main tool in the proof of the main theorem is afixed point result for function spaces from [17] (for related

outcomes see [18 19]) Similar method of the proof has beenalready applied in [11 20]

To present the fixed point theorem we introduce the fol-lowing necessary hypotheses (R

+stands for the set of non-

negative reals and119860119861 denotes the family of all functionsmap-ping a set 119861 = 0 into a set 119860 = 0)

(H1) 119878 is a nonempty set119864 is a Banach space and functions1198911 119891

119896 119878 rarr 119878 and 119871

1 119871

119896 119878 rarr R

+are

given

(H2) T 119864119878 rarr 119864119878 is an operator satisfying the inequality

1003817100381710038171003817T120585 (119909) minus T120583 (119909)1003817100381710038171003817

le

119896

sum119894=1

119871119894(119909)

1003817100381710038171003817120585 (119891119894(119909)) minus 120583 (119891

119894(119909))

1003817100381710038171003817 120585 120583 isin 119864119883 119909 isin 119878

(2)

(H3) Λ R119878+

rarr R119878+is defined by

Λ120575 (119909) =

119896

sum119894=1

119871119894(119909) 120575 (119891

119894(119909)) 120575 isin R

119878

+ 119909 isin 119878 (3)

Now we are in a position to present the above mentionedfixed point theorem for function spaces (see [17])

2 Journal of Function Spaces and Applications

Theorem 1 Let hypotheses (H1)ndash(H3) be valid and functions120576 119878 rarr R

+and 120593 119878 rarr 119864 fulfil the following two conditions1003817100381710038171003817T120593 (119909) minus 120593 (119909)

1003817100381710038171003817 le 120576 (119909) 119909 isin 119878

120576lowast(119909) =

infin

sum119899=0

Λ119899120576 (119909) lt infin 119909 isin 119878

(4)

Then there exists a unique fixed point 120595 ofT with1003817100381710038171003817120593 (119909) minus 120595 (119909)

1003817100381710038171003817 le 120576lowast(119909) 119909 isin 119878 (5)

Moreover

120595 (119909) = lim119899rarrinfin

T119899120593 (119909) 119909 isin 119878 (6)

We start our considerations from the functional equation

119891 (119909 + 119910 + 119911) + 119891 (119909) + 119891 (119910) + 119891 (119911)

= 119891 (119909 + 119910) + 119891 (119909 + 119911) + 119891 (119910 + 119911) (7)

that is patterned on (1) and therefore quite often named afterFrechet (see eg [21])

Note that (7) can be written in the form

Δ119909119910119911

119891 (0) = 0 (8)

where Δ denotes the Frechet difference operator defined (forfunctions mapping a commutative semigroup (119878 +) into agroup) by

Δ119910119891 (119909) = Δ

1

119910119891 (119909) = 119891 (119909 + 119910) minus 119891 (119909) 119909 119910 isin 119878

Δ119905119911

= Δ119905∘ Δ119911 Δ

2

119905= Δ119905119905 119905 119911 isin 119878

Δ119905119906119911

= Δ119905∘ Δ119906∘ Δ119911 Δ3

119905= Δ119905119905119905

119905 119906 119911 isin 119878

(9)

It is easy to check that

Δ119905119911

119891 (119909) = 119891 (119909 + 119905 + 119911) minus 119891 (119909 + 119905) minus 119891 (119909 + 119911)

+ 119891 (119909) 119909 119905 119911 isin 119878

Δ119905119911119906

119891 (119909) = 119891 (119909 + 119905 + 119911 + 119906) minus 119891 (119909 + 119905 + 119911)

minus 119891 (119909 + 119905 + 119906) minus 119891 (119909 + 119911 + 119906)

+ 119891 (119909 + 119905) + 119891 (119909 + 119911) + 119891 (119909 + 119906)

minus 119891 (119909) 119909 119905 119911 119906 isin 119878

(10)

Such operators were first considered by Frechet in [22 23](we refer to [24] for more information and further referencesconcerning this subject) so it is still another motivation for(7) to be called the Frechet equation

Let us yet observe (see [25]) that alternatively (7) can bewritten in the form

1198622119891 (119909 119910 119911) = 0 (11)

where 1198622119891(119909 119910 119911) = 119862119891(119909 119910 + 119911) minus 119862119891(119909 119910) minus 119862119891(119909 119911) and119862119891(119909 119910) = 119891(119909 +119910) minus119891(119909) minus119891(119910) that is 1198622119891 is the Cauchydifference of 119891 of the second order

We prove the subsequent theorem which corresponds to[26 Theorem 31] where the equation

Δ3

119911119891 (119909) = 0 (12)

has been investigated (the author has named it the super-stability result which is not a precise description becauseaccording to the terminology applied in [10ndash14] it should berather called the hyperstability result) For some analogousinvestigations see [27ndash29] Let us mention yet that stability of(7) has been already studied in [30ndash33] and our results com-plement the outcomes included there

It is easy to show that every solution 119891 of (7) mapping acommutative group (119866 +) into a real linear space 119883 must beof the form 119891 = 119886 + 119902 with some additive 119886 119866 rarr 119883 andquadratic 119902 119866 rarr 119883 (see eg [21]) Namely with 119909 = 119910 =

119911 = 0 from (7) we deduce that 119891(0) = 0 and next taking 119911 =

minus119910 in (7) we obtain that the even part of 119891 is quadratic whilethe odd part is a solution of the Jensen equation whence it isadditive

2 Main Results

The next theorem and corollary are the main results of thepaper (N and Z stand as usual for the sets of all positiveintegers and integers respectively moreover Z

0= Z 0)

Theorem 2 Let (119883 +) be a commutative group1198830= 1198830

119884 a Banach space and 119891 119883 rarr 119884 119888 Z0

rarr [0infin) and119871 11988330

rarr [0infin) satisfying the following three conditions

M = 119898 isin Z0 119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1) lt 1 = 0(13)

119871 (119896119909 119896119910 119896119911) le 119888 (119896) 119871 (119909 119910 119911) 119909 119910 119911 isin 1198830

119896 isin minus2119898119898 + 1 minus119898 2119898 + 1 119898 isin M(14)

1003817100381710038171003817119891 (119909 + 119910 + 119911) + 119891 (119909) + 119891 (119910) + 119891 (119911)

minus119891 (119909 + 119910) minus 119891 (119909 + 119911) minus 119891 (119910 + 119911)1003817100381710038171003817

le 119871 (119909 119910 119911) 119909 119910 119911 isin 1198830

(15)

Then there is a unique function 119865 119883 rarr 119884 satisfying (7) forall 119909 119910 119911 isin 119883 and such that

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le 120588119871(119909) 119909 isin 119883

0 (16)

where

120588119871(119909)

= inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(17)

Journal of Function Spaces and Applications 3

Proof Replacing 119909 by (2119898 + 1) 119909 and taking 119910 = 119911 = minus119898119909 in(15) we get1003817100381710038171003817119891 (119909) + 119891 ((2119898 + 1) 119909) + 2119891 (minus119898119909)

minus2119891 ((119898 + 1) 119909) minus 119891 (minus2119898119909)1003817100381710038171003817

le 119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 120576119898

(119909) 119909 isin 1198830 119898 isin Z

0

(18)

Further put

T119898120585 (119909) = 120585 (minus2119898119909) + 2120585 ((119898 + 1) 119909) minus 2120585 (minus119898119909)

minus 120585 ((2119898 + 1) 119909) 120585 isin 119884119883 119909 isin 119883 119898 isin Z

0

(19)

Then

T119899

119898119891 (0) = 0 119899 isin N 119898 isin Z

0 (20)

and inequality (18) takes the form1003817100381710038171003817T119898119891 (119909) minus 119891 (119909)

1003817100381710038171003817 le 120576119898

(119909) 119909 isin 1198830 119898 isin Z

0 (21)

Define an operator Λ119898

R1198830

+rarr R1198830

+for 119898 isin Z

0by

Λ119898120578 (119909) = 120578 (minus2119898119909) + 2120578 ((119898 + 1) 119909)

+ 2120578 (minus119898119909) + 120578 ((2119898 + 1) 119909)(22)

for 120578 isin R1198830

+and 119909 isin 119883

0 Then it is easily seen that for each

119898 isin Z0 the operatorΛ = Λ

119898has the formdescribed in (H3)

with 119896 = 4 119878 = 1198830 119864 = 119884 and

1198911(119909) = minus2119898119909 119891

2(119909) = (119898 + 1) 119909

1198913(119909) = minus119898119909 119891

4(119909) = (2119898 + 1) 119909

1198711(119909) = 119871

4(119909) = 1 119871

2(119909) = 119871

3(119909) = 2 119909 isin 119883

0

(23)

Moreover for every 120585 120583 isin 1198841198830 119909 isin 1198830 119898 isin Z

0

1003817100381710038171003817T119898120585 (119909) minus T119898120583 (119909)

1003817100381710038171003817

=1003817100381710038171003817120585 (minus2119898119909) + 2120585 ((119898 + 1) 119909) minus 2120585 (minus119898119909)

minus 120585 ((2119898 + 1) 119909) minus 120583 (minus2119898119909) minus 2120583 ((119898 + 1) 119909)

+ 2120583 (minus119898119909) + 120583 ((2119898 + 1) 119909)1003817100381710038171003817

le1003817100381710038171003817(120585 minus 120583) (minus2119898119909)

1003817100381710038171003817 + 21003817100381710038171003817(120585 minus 120583) ((119898 + 1) 119909)

1003817100381710038171003817

+ 21003817100381710038171003817(120585 minus 120583) (minus119898119909)

1003817100381710038171003817 +1003817100381710038171003817(120585 minus 120583) ((2119898 + 1) 119909)

1003817100381710038171003817

=

4

sum119894=1

119871119894(119909)

1003817100381710038171003817(120585 minus 120583) (119891119894(119909))

1003817100381710038171003817

(24)

where (120585 minus 120583) (119910) = 120585(119910) minus 120583(119910) for 119910 isin 1198830 It is easy to check

that in view of (14)

Λ119898120576119896(119909) le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1)) 120576119896(119909) 119896 119898 isin Z

0 119909 isin 119883

0

(25)

Therefore since the operator Λ119898is linear we have

120576lowast

119898(119909) =

infin

sum119899=0

Λ119898

119899120576119898

(119909)

le

infin

sum119899=0

(119888 (minus2119898) + 2119888 (119898 + 1)

+2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)

=120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119898 isin M 119909 isin 1198830

(26)

Thus byTheorem 1 (with 119878 = 1198830and119864 = 119884) for each119898 isin M

there exists a function 1198651015840119898

1198830

rarr 119884 with

1198651015840

119898(119909) = 119865

1015840

119898(minus2119898119909) + 2119865

1015840

119898((119898 + 1) 119909)

minus 21198651015840

119898(minus119898119909) minus 119865

1015840

119898((2119898 + 1) 119909) 119909 isin 119883

0

10038171003817100381710038171003817119891 (119909) minus 119865

1015840

119898(119909)

10038171003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(27)

Moreover

1198651015840

119898(119909) = lim

119899rarrinfinT119899

119898119891 (119909) 119909 isin 119883

0 119898 isin M (28)

Define 119865119898

119883 rarr 119884 by 119865119898(0) = 0 and 119865

119898(119909) = 1198651015840

119898(119909) for

119909 isin 1198830and 119898 isin M Then it is easily seen that by (20)

119865119898

(119909) = lim119899rarrinfin

T119899

119898119891 (119909) 119909 isin 119883 119898 isin M (29)

Next we show that

1003817100381710038171003817T119899

119898119891 (119909 + 119910 + 119911) + T

119899

119898119891 (119909) + T

119899

119898119891 (119910) + T

119899

119898119891 (119911)

minusT119899

119898119891 (119909 + 119910) minus T

119899

119898119891 (119909 + 119911) minus T

119899

119898119891 (119910 + 119911)

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119899

times 119871 (119909 119910 119911)

(30)

for every 119909 119910 119911 isin 1198830 119899 isin N

0= N cup 0 119898 isin M

4 Journal of Function Spaces and Applications

Fix 119898 isin M For 119899 = 0 the condition (30) is simply (15)So take 119903 isin N

0and suppose that (30) holds for 119899 = 119903 and

119909 119910 119911 isin 1198830 Then

10038171003817100381710038171003817T119903+1

119898119891 (119909 + 119910 + 119911) + T

119903+1

119898119891 (119909) + T

119903+1

119898119891 (119910) + T

119903+1

119898119891 (119911)

minusT119903+1

119898119891 (119909 + 119910) minus T

119903+1

119898119891 (119909 + 119911) minus T

119903+1

119898119891 (119910 + 119911)

10038171003817100381710038171003817

=1003817100381710038171003817T119903

119898119891 (minus2119898 (119909 + 119910 + 119911))

+ 2T119903

119898119891 ((119898 + 1) (119909 + 119910 + 119911))

minus 2T119903

119898119891 (minus119898 (119909 + 119910 + 119911))

minus T119903

119898119891 ((2119898 + 1) (119909 + 119910 + 119911))

+ T119903

119898119891 (minus2119898119909) + 2T

119903

119898119891 ((119898 + 1) 119909)

minus 2T119903

119898119891 (minus119898119909) minus T

119903

119898119891 ((2119898 + 1) 119909)

+ T119903

119898119891 (minus2119898119910) + 2T

119903

119898119891 ((119898 + 1) 119910)

minus 2T119903

119898119891 (minus119898119910) minus T

119903

119898119891 ((2119898 + 1) 119910)

+ T119903

119898119891 (minus2119898119911) + 2T

119903

119898119891 ((119898 + 1) 119911)

minus 2T119903

119898119891 (minus119898119911) minus T

119903

119898119891 ((2119898 + 1) 119911)

minus T119903

119898119891 (minus2119898 (119909 + 119910)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119910))

+ 2T119903

119898119891 (minus119898 (119909 + 119910)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119910))

minus T119903

119898119891 (minus2119898 (119909 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119911))

+ 2T119903

119898119891 (minus119898 (119909 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119911))

minus T119903

119898119891 (minus2119898 (119910 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119910 + 119911))

+ 2T119903

119898119891 (minus119898 (119910 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119910 + 119911))

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119903

times (119871 (minus2119898119909 minus2119898119910 minus2119898119911)

+ 2119871 ((119898 + 1) 119909 (119898 + 1) 119910 (119898 + 1) 119911)

+ 2119871 (minus119898119909 minus119898119910 minus119898119911)

+119871 ((2119898 + 1) 119909 (2119898 + 1) 119910 (2119898 + 1) 119911))

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1))119903+1

119871 (119909 119910 119911)

(31)

for every 119909 119910 119911 isin 1198830 which completes the proof of (30)

Letting 119899 rarr infin in (30) we obtain that

119865119898

(119909 + 119910 + 119911) + 119865119898

(119909) + 119865119898

(119910) + 119865119898

(119911)

= 119865119898

(119909 + 119910) + 119865119898

(119909 + 119911) + 119865119898

(119910 + 119911) 119909 119910 119911 isin 1198830

(32)

So we have proved that for each 119898 isin M there exists afunction 119865

119898 119883 rarr 119884 satisfying (7) for 119909 119910 119911 isin 119883

0and

such that1003817100381710038171003817119891 (119909) minus 119865

119898(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(33)

Now we show that119865119898

= 119865119896for all119898 119896 isin M So fix119898 119896 isin

M Note that 119865119896satisfies (32) with 119898 replaced by 119896 Hence

replacing 119909 by (2119898+ 1)119909 and taking 119910 = 119911 = minus119898119909 in (32) weobtain thatT

119898119865119895= 119865119895for 119895 = 119898 119896 and

1003817100381710038171003817119865119898 (119909) minus 119865119896(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

119909 isin 1198830

(34)

whence by the linearity of Λ and (25)1003817100381710038171003817119865119898 (119909) minus 119865

119896(119909)

1003817100381710038171003817

=1003817100381710038171003817T119899

119898119865119898

(119909) minus T119899

119898119865119896(119909)

1003817100381710038171003817

leΛ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+Λ119899

119898120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

le(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)(35)

for every 119909 isin 1198830and 119899 isin N Now letting 119899 rarr infin we get

119865119898

= 119865119896= 119865

Thus in view of (33) we have proved that1003817100381710038171003817119891 (119909) minus 119865 (119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830 119898 isin M

(36)

whence we derive (16)Since (in view of (32)) it is easy to notice that 119865 is a

solution to (7) (ie (7) holds for all 119909 119910 119911 isin 119883) it remains to

Journal of Function Spaces and Applications 5

prove the statement concerning the uniqueness of 119865 So let119866 119883 rarr 119884 be also a solution of (7) and 119891(119909) minus 119866(119909) le

120588119871(119909) for 119909 isin 119883

0 Then

119866 (119909) minus 119865 (119909) le 2120588119871(119909) 119909 isin 119883

0 (37)

Further T119898119866 = 119866 for each 119898 isin Z

0 Consequently with a

fixed 119898 isin M

119866 (119909) minus 119865 (119909)

=1003817100381710038171003817T119899

119898119866 (119909) minus T

119899

119898119865 (119909)

1003817100381710038171003817 le 2Λ119899

119898120588119871(119909)

le2Λ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)(38)

for 119909 isin 1198830and 119899 isin N Next analogously as (25) by induction

we get

Λ119899

119898120576119898

(119909) le (119888 (minus2119898) + 2119888 (119898 + 1)

+ 2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)(39)

for 119909 isin 1198830 119899 isin N This implies that 119866 = 119865

Theorem 2 yields at once the following hyperstabilityresult

Corollary 3 Let (119883 +) be a commutative group1198830= 1198830

let 119884 be a normed space 119891 119883 rarr 119884 119888 Z0

rarr [0infin)119871 11988330

rarr [0infin) and let the conditions (13) (14) and (15) bevalid Assume that

inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 0 119909 isin 1198830 (40)

Then 119891 satisfies (7) for all 119909 119910 119911 isin 119883

Proof Note that without loss of generality we may assumethat 119884 is complete because otherwise we can replace it by itscompletion Next in view of (40) 120588

119871(119909) = 0 for each 119909 isin 119883

0

where 120588119871is defined by (17) Hence fromTheorem 2 we easily

derive that 119891 is a solution to (7)

3 Final Remarks

Remark 4 Note that if in Theorem 2

lim inf119898rarrinfin

119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1) = 0

(41)

(this is the case when eg lim|119898|rarrinfin

119888(119898) = 0) then (13)holds and

120588119871(119909) = inf

119898isinM120576119898

(119909) 119909 isin 1198830 (42)

Further let 119883 be a normed space and

119871 (119909 119910 119911) = 120576 (119909119901+

10038171003817100381710038171199101003817100381710038171003817119901

+ 119911119901) 119909 119910 119911 isin 119883

0(43)

with some reals 120576 gt 0 and 119901 lt 0 Then the condition (14)is valid for instance with 119888(119896) = |119896|

119901 for 119896 isin Z0 Obviously

(40) holds and there exists 1198980isin N such that

|2119898|119901+ 2|119898 + 1|

119901+ 2|119898|

119901+ |2119898 + 1|

119901lt 1 119898 ge 119898

0

(44)

so we obtain (13) as well Consequently by Corollary 3 everyfunction 119891 119883 rarr 119884 fulfilling the inequality (15) satisfies(7) for all 119909 119910 119911 isin 119883 In this way we have obtained ahyperstability result that corresponds to the recent hypersta-bility outcomes in [11 20] and some classical stability resultsconcerning the Cauchy equation (see eg [9 page 3] [15page 15 16] and [16 page 2])

Below we provide two further simple and natural exam-ples of functions 119871 and 119888 satisfying the conditions (13) and(14) The first one clearly includes the case just described

(a) 119871(119909 119910 119911) = (12057211199091199041 +12057221199101199042 +12057231199111199043)119908 for 119909 119910 119911 isin

1198830with some 119908 120572

119894 119904119894

isin R such that 120572119894

gt 0 and119908119904119894lt 0 for 119894 = 1 2 3 and 119888(119898) equiv |119898|

minusV|119908| whereV = min |119904

1| |1199042| |1199043|

(b) 119871(119909 119910 119911) = 120572119909119904119910119905119911119906 for 119909 119910 119911 isin 119883

0with some

reals 120572 gt 0 and 119904 119905 119906 such that 119904 + 119905 + 119906 lt 0Clearly if two functions 119871 satisfy the condition (14)

then so do their sum and product with suitable functions 119888Therefore we can easily produce numerous examples of suchfunctions Of course there are some other such examples thatare a bit more artificial for instance 119871(119909 119910 119911) = 119861(119860(119909)

minus119899)

for 119909 isin 1198830 where 119899 isin N 119860 119883 rarr 119883 and 119861 R rarr R

are functions with 119860(119909119898) = 119898119860(119909) and 119861(119905119898) = 119898119861(119905) for119909 isin 119883 119905 isin R and 119898 isin Z

We end the paper with a simple example of applicationsof our main result

Corollary 5 Let119883 be a normed space and1198830= 1198830Write

119863(119909 119910 119911) =100381610038161003816100381610038161003817100381710038171003817119909 + 119910 + 119911

10038171003817100381710038172

+ 1199092+

100381710038171003817100381711991010038171003817100381710038172

+ 1199112

minus1003817100381710038171003817119909 + 119910

10038171003817100381710038172

minus 119909 + 1199112minus

1003817100381710038171003817119910 + 1199111003817100381710038171003817210038161003816100381610038161003816

(45)

for119909 119910 119911 isin 119883 Assume that one of the following two hypothesesis valid

(i) There exist 1199080 120572119894 119904119894

isin R such that 120572119894 1199080119904119894

gt 0 for119894 = 1 2 3 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

(12057211199091199041 + 1205722

100381710038171003817100381711991010038171003817100381710038171199042 + 12057231199111199043)1199080

lt infin (46)

(ii) There exist reals 119904 119905 119906 such that 119904 + 119905 + 119906 gt 0 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

1199091199041003817100381710038171003817119910

1003817100381710038171003817119905

119911119906

lt infin (47)

Then 119883 is an inner product space

Proof Write 119891(119909) = 1199092 for 119909 isin 119883 Then with 119871 and 119888

of the forms described in Remark 4 (with 119908 = minus1199080) from

Corollary 3 we easily derive that 119891 is a solution to (7) whichyields the statement

6 Journal of Function Spaces and Applications

References

[1] M Frechet ldquoSur la definition axiomatique drsquoune classe drsquoespacesvectoriels distancies applicables vectoriellement sur lrsquoespace deHilbertrdquoAnnals of Mathematics Second Series vol 36 no 3 pp705ndash718 1935

[2] P Jordan and J von Neumann ldquoOn inner products in linearmetric spacesrdquoAnnals ofMathematics Second Series vol 36 no3 pp 719ndash723 1935

[3] D Amir Characterizations of Inner Product Spaces OperatorTheory Advances and Applications vol 20 Birkhauser BaselSwitzerland 1986

[4] C Alsina J Sikorska and M S Tomas Norm Derivatives andCharacterizations of Inner Product Spaces World ScientificPublishing 2010

[5] S S Dragomir ldquoSome characterizations of inner product spacesand applicationsrdquo Universitatis Babes-Bolyai vol 34 no 1 pp50ndash55 1989

[6] M S Moslehian and J M Rassias ldquoA characterization of innerproduct spaces concerning an Euler-Lagrange identityrdquo Com-munications in Mathematical Analysis vol 8 no 2 pp 16ndash212010

[7] K Nikodem and Z Pales ldquoCharacterizations of inner productspaces by strongly convex functionsrdquo Banach Journal of Mathe-matical Analysis vol 5 no 1 pp 83ndash87 2011

[8] T M Rassias ldquoNew characterizations of inner product spacesrdquoBulletin des Sciences Mathematiques 2nd Serie vol 108 no 1pp 95ndash99 1984

[9] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[10] J Brzdęk ldquoRemarks on hyperstability of the Cauchy functionalequationrdquo Aequationes Mathematicae vol 86 no 3 pp 255ndash267 2013

[11] J Brzdęk ldquoHyperstability of the Cauchy equation on restricteddomainsrdquoActaMathematica Hungarica vol 141 no 1-2 pp 58ndash67 2013

[12] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[13] G Maksa ldquoThe stability of the entropy of degree alphardquo Journalof Mathematical Analysis and Applications vol 346 no 1 pp17ndash21 2008

[14] G Maksa and Z Pales ldquoHyperstability of a class of linearfunctional equationsrdquo Acta Mathematica vol 17 no 2 pp 107ndash112 2001

[15] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[16] S-M Jung Hvers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[17] J Brzdęk J Chudziak and Z Pales ldquoA fixed point approachto stability of functional equationsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 74 no 17 pp 6728ndash6732 2011

[18] J Brzdęk and K Cieplinski ldquoA fixed point approach to thestability of functional equations in non-Archimedean metricspacesrdquo Nonlinear Analysis Theory Methods amp Applications Avol 74 no 18 pp 6861ndash6867 2011

[19] L Cadariu L Gavruta and P Gavruta ldquoFixed points and gener-alized Hyers-Ulam stabilityrdquo Abstract and Applied Analysis vol2012 Article ID 712743 10 pages 2012

[20] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[21] Pl Kannappan Functional Equations and Inequalities withApplications Springer Monographs in Mathematics SpringerNew York NY USA 2009

[22] M Frechet ldquoUne definition fonctionnelles des polvnomesrdquoNouvelles Annales de Mathematique 4th serie vol 9 pp 145ndash162 1909

[23] M Frechet ldquoLes polvnomes abstraitsrdquo Journal de Mathema-tiques Pures et Appliquees 9th serie vol 8 pp 71ndash92 1929

[24] M Kuczma An Introduction to the Theory of Functional Equa-tions and Inequalities Cauchyrsquos Equation and Jensenrsquos InequalityBirkhauser Basel Switzerland 2nd edition 2009

[25] H Stetkaer Functional Equations on Groups World ScientificPublishing Singapore 2013

[26] Y H Lee ldquoOn the Hvers-Ulam-Rassias stability of the gen-eralized polynomial function of degree 2rdquo Journal of theChungcheong Mathematical Society vol 22 no 2 pp 201ndash2092009

[27] M Albert and J A Baker ldquoFunctions with bounded m-thdifferencesrdquoAnnales Polonici Mathematici vol 43 no 1 pp 93ndash103 1983

[28] D H Hyers ldquoTransformations with boundedm-th differencesrdquoPacific Journal of Mathematics vol 11 pp 591ndash602 1961

[29] Y-H Lee ldquoOn the stability of the monomial functional equa-tionrdquo Bulletin of the Korean Mathematical Society vol 45 no 2pp 397ndash403 2008

[30] M Chudziak On solutions and stability of functional equationsconnected to the Popoviciu inequality [PhD thesis] PedagogicalUniversity of Cracow Cracow Poland 2012 (Polish)

[31] W Fechner ldquoOn the Hyers-Ulam stability of functional equa-tions connected with additive and quadratic mappingsrdquo Journalof Mathematical Analysis and Applications vol 322 no 2 pp774ndash786 2006

[32] S-M Jung ldquoOn the Hyers-Ulam stability of the functionalequations that have the quadratic propertyrdquo Journal of Mathe-matical Analysis and Applications vol 222 no 1 pp 126ndash1371998

[33] J Sikorska ldquoOn a direct method for proving the Hyers-Ulam stability of functional equationsrdquo Journal of MathematicalAnalysis and Applications vol 372 no 1 pp 99ndash109 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

Theorem 1 Let hypotheses (H1)ndash(H3) be valid and functions120576 119878 rarr R

+and 120593 119878 rarr 119864 fulfil the following two conditions1003817100381710038171003817T120593 (119909) minus 120593 (119909)

1003817100381710038171003817 le 120576 (119909) 119909 isin 119878

120576lowast(119909) =

infin

sum119899=0

Λ119899120576 (119909) lt infin 119909 isin 119878

(4)

Then there exists a unique fixed point 120595 ofT with1003817100381710038171003817120593 (119909) minus 120595 (119909)

1003817100381710038171003817 le 120576lowast(119909) 119909 isin 119878 (5)

Moreover

120595 (119909) = lim119899rarrinfin

T119899120593 (119909) 119909 isin 119878 (6)

We start our considerations from the functional equation

119891 (119909 + 119910 + 119911) + 119891 (119909) + 119891 (119910) + 119891 (119911)

= 119891 (119909 + 119910) + 119891 (119909 + 119911) + 119891 (119910 + 119911) (7)

that is patterned on (1) and therefore quite often named afterFrechet (see eg [21])

Note that (7) can be written in the form

Δ119909119910119911

119891 (0) = 0 (8)

where Δ denotes the Frechet difference operator defined (forfunctions mapping a commutative semigroup (119878 +) into agroup) by

Δ119910119891 (119909) = Δ

1

119910119891 (119909) = 119891 (119909 + 119910) minus 119891 (119909) 119909 119910 isin 119878

Δ119905119911

= Δ119905∘ Δ119911 Δ

2

119905= Δ119905119905 119905 119911 isin 119878

Δ119905119906119911

= Δ119905∘ Δ119906∘ Δ119911 Δ3

119905= Δ119905119905119905

119905 119906 119911 isin 119878

(9)

It is easy to check that

Δ119905119911

119891 (119909) = 119891 (119909 + 119905 + 119911) minus 119891 (119909 + 119905) minus 119891 (119909 + 119911)

+ 119891 (119909) 119909 119905 119911 isin 119878

Δ119905119911119906

119891 (119909) = 119891 (119909 + 119905 + 119911 + 119906) minus 119891 (119909 + 119905 + 119911)

minus 119891 (119909 + 119905 + 119906) minus 119891 (119909 + 119911 + 119906)

+ 119891 (119909 + 119905) + 119891 (119909 + 119911) + 119891 (119909 + 119906)

minus 119891 (119909) 119909 119905 119911 119906 isin 119878

(10)

Such operators were first considered by Frechet in [22 23](we refer to [24] for more information and further referencesconcerning this subject) so it is still another motivation for(7) to be called the Frechet equation

Let us yet observe (see [25]) that alternatively (7) can bewritten in the form

1198622119891 (119909 119910 119911) = 0 (11)

where 1198622119891(119909 119910 119911) = 119862119891(119909 119910 + 119911) minus 119862119891(119909 119910) minus 119862119891(119909 119911) and119862119891(119909 119910) = 119891(119909 +119910) minus119891(119909) minus119891(119910) that is 1198622119891 is the Cauchydifference of 119891 of the second order

We prove the subsequent theorem which corresponds to[26 Theorem 31] where the equation

Δ3

119911119891 (119909) = 0 (12)

has been investigated (the author has named it the super-stability result which is not a precise description becauseaccording to the terminology applied in [10ndash14] it should berather called the hyperstability result) For some analogousinvestigations see [27ndash29] Let us mention yet that stability of(7) has been already studied in [30ndash33] and our results com-plement the outcomes included there

It is easy to show that every solution 119891 of (7) mapping acommutative group (119866 +) into a real linear space 119883 must beof the form 119891 = 119886 + 119902 with some additive 119886 119866 rarr 119883 andquadratic 119902 119866 rarr 119883 (see eg [21]) Namely with 119909 = 119910 =

119911 = 0 from (7) we deduce that 119891(0) = 0 and next taking 119911 =

minus119910 in (7) we obtain that the even part of 119891 is quadratic whilethe odd part is a solution of the Jensen equation whence it isadditive

2 Main Results

The next theorem and corollary are the main results of thepaper (N and Z stand as usual for the sets of all positiveintegers and integers respectively moreover Z

0= Z 0)

Theorem 2 Let (119883 +) be a commutative group1198830= 1198830

119884 a Banach space and 119891 119883 rarr 119884 119888 Z0

rarr [0infin) and119871 11988330

rarr [0infin) satisfying the following three conditions

M = 119898 isin Z0 119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1) lt 1 = 0(13)

119871 (119896119909 119896119910 119896119911) le 119888 (119896) 119871 (119909 119910 119911) 119909 119910 119911 isin 1198830

119896 isin minus2119898119898 + 1 minus119898 2119898 + 1 119898 isin M(14)

1003817100381710038171003817119891 (119909 + 119910 + 119911) + 119891 (119909) + 119891 (119910) + 119891 (119911)

minus119891 (119909 + 119910) minus 119891 (119909 + 119911) minus 119891 (119910 + 119911)1003817100381710038171003817

le 119871 (119909 119910 119911) 119909 119910 119911 isin 1198830

(15)

Then there is a unique function 119865 119883 rarr 119884 satisfying (7) forall 119909 119910 119911 isin 119883 and such that

1003817100381710038171003817119891 (119909) minus 119865 (119909)1003817100381710038171003817 le 120588119871(119909) 119909 isin 119883

0 (16)

where

120588119871(119909)

= inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(17)

Journal of Function Spaces and Applications 3

Proof Replacing 119909 by (2119898 + 1) 119909 and taking 119910 = 119911 = minus119898119909 in(15) we get1003817100381710038171003817119891 (119909) + 119891 ((2119898 + 1) 119909) + 2119891 (minus119898119909)

minus2119891 ((119898 + 1) 119909) minus 119891 (minus2119898119909)1003817100381710038171003817

le 119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 120576119898

(119909) 119909 isin 1198830 119898 isin Z

0

(18)

Further put

T119898120585 (119909) = 120585 (minus2119898119909) + 2120585 ((119898 + 1) 119909) minus 2120585 (minus119898119909)

minus 120585 ((2119898 + 1) 119909) 120585 isin 119884119883 119909 isin 119883 119898 isin Z

0

(19)

Then

T119899

119898119891 (0) = 0 119899 isin N 119898 isin Z

0 (20)

and inequality (18) takes the form1003817100381710038171003817T119898119891 (119909) minus 119891 (119909)

1003817100381710038171003817 le 120576119898

(119909) 119909 isin 1198830 119898 isin Z

0 (21)

Define an operator Λ119898

R1198830

+rarr R1198830

+for 119898 isin Z

0by

Λ119898120578 (119909) = 120578 (minus2119898119909) + 2120578 ((119898 + 1) 119909)

+ 2120578 (minus119898119909) + 120578 ((2119898 + 1) 119909)(22)

for 120578 isin R1198830

+and 119909 isin 119883

0 Then it is easily seen that for each

119898 isin Z0 the operatorΛ = Λ

119898has the formdescribed in (H3)

with 119896 = 4 119878 = 1198830 119864 = 119884 and

1198911(119909) = minus2119898119909 119891

2(119909) = (119898 + 1) 119909

1198913(119909) = minus119898119909 119891

4(119909) = (2119898 + 1) 119909

1198711(119909) = 119871

4(119909) = 1 119871

2(119909) = 119871

3(119909) = 2 119909 isin 119883

0

(23)

Moreover for every 120585 120583 isin 1198841198830 119909 isin 1198830 119898 isin Z

0

1003817100381710038171003817T119898120585 (119909) minus T119898120583 (119909)

1003817100381710038171003817

=1003817100381710038171003817120585 (minus2119898119909) + 2120585 ((119898 + 1) 119909) minus 2120585 (minus119898119909)

minus 120585 ((2119898 + 1) 119909) minus 120583 (minus2119898119909) minus 2120583 ((119898 + 1) 119909)

+ 2120583 (minus119898119909) + 120583 ((2119898 + 1) 119909)1003817100381710038171003817

le1003817100381710038171003817(120585 minus 120583) (minus2119898119909)

1003817100381710038171003817 + 21003817100381710038171003817(120585 minus 120583) ((119898 + 1) 119909)

1003817100381710038171003817

+ 21003817100381710038171003817(120585 minus 120583) (minus119898119909)

1003817100381710038171003817 +1003817100381710038171003817(120585 minus 120583) ((2119898 + 1) 119909)

1003817100381710038171003817

=

4

sum119894=1

119871119894(119909)

1003817100381710038171003817(120585 minus 120583) (119891119894(119909))

1003817100381710038171003817

(24)

where (120585 minus 120583) (119910) = 120585(119910) minus 120583(119910) for 119910 isin 1198830 It is easy to check

that in view of (14)

Λ119898120576119896(119909) le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1)) 120576119896(119909) 119896 119898 isin Z

0 119909 isin 119883

0

(25)

Therefore since the operator Λ119898is linear we have

120576lowast

119898(119909) =

infin

sum119899=0

Λ119898

119899120576119898

(119909)

le

infin

sum119899=0

(119888 (minus2119898) + 2119888 (119898 + 1)

+2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)

=120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119898 isin M 119909 isin 1198830

(26)

Thus byTheorem 1 (with 119878 = 1198830and119864 = 119884) for each119898 isin M

there exists a function 1198651015840119898

1198830

rarr 119884 with

1198651015840

119898(119909) = 119865

1015840

119898(minus2119898119909) + 2119865

1015840

119898((119898 + 1) 119909)

minus 21198651015840

119898(minus119898119909) minus 119865

1015840

119898((2119898 + 1) 119909) 119909 isin 119883

0

10038171003817100381710038171003817119891 (119909) minus 119865

1015840

119898(119909)

10038171003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(27)

Moreover

1198651015840

119898(119909) = lim

119899rarrinfinT119899

119898119891 (119909) 119909 isin 119883

0 119898 isin M (28)

Define 119865119898

119883 rarr 119884 by 119865119898(0) = 0 and 119865

119898(119909) = 1198651015840

119898(119909) for

119909 isin 1198830and 119898 isin M Then it is easily seen that by (20)

119865119898

(119909) = lim119899rarrinfin

T119899

119898119891 (119909) 119909 isin 119883 119898 isin M (29)

Next we show that

1003817100381710038171003817T119899

119898119891 (119909 + 119910 + 119911) + T

119899

119898119891 (119909) + T

119899

119898119891 (119910) + T

119899

119898119891 (119911)

minusT119899

119898119891 (119909 + 119910) minus T

119899

119898119891 (119909 + 119911) minus T

119899

119898119891 (119910 + 119911)

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119899

times 119871 (119909 119910 119911)

(30)

for every 119909 119910 119911 isin 1198830 119899 isin N

0= N cup 0 119898 isin M

4 Journal of Function Spaces and Applications

Fix 119898 isin M For 119899 = 0 the condition (30) is simply (15)So take 119903 isin N

0and suppose that (30) holds for 119899 = 119903 and

119909 119910 119911 isin 1198830 Then

10038171003817100381710038171003817T119903+1

119898119891 (119909 + 119910 + 119911) + T

119903+1

119898119891 (119909) + T

119903+1

119898119891 (119910) + T

119903+1

119898119891 (119911)

minusT119903+1

119898119891 (119909 + 119910) minus T

119903+1

119898119891 (119909 + 119911) minus T

119903+1

119898119891 (119910 + 119911)

10038171003817100381710038171003817

=1003817100381710038171003817T119903

119898119891 (minus2119898 (119909 + 119910 + 119911))

+ 2T119903

119898119891 ((119898 + 1) (119909 + 119910 + 119911))

minus 2T119903

119898119891 (minus119898 (119909 + 119910 + 119911))

minus T119903

119898119891 ((2119898 + 1) (119909 + 119910 + 119911))

+ T119903

119898119891 (minus2119898119909) + 2T

119903

119898119891 ((119898 + 1) 119909)

minus 2T119903

119898119891 (minus119898119909) minus T

119903

119898119891 ((2119898 + 1) 119909)

+ T119903

119898119891 (minus2119898119910) + 2T

119903

119898119891 ((119898 + 1) 119910)

minus 2T119903

119898119891 (minus119898119910) minus T

119903

119898119891 ((2119898 + 1) 119910)

+ T119903

119898119891 (minus2119898119911) + 2T

119903

119898119891 ((119898 + 1) 119911)

minus 2T119903

119898119891 (minus119898119911) minus T

119903

119898119891 ((2119898 + 1) 119911)

minus T119903

119898119891 (minus2119898 (119909 + 119910)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119910))

+ 2T119903

119898119891 (minus119898 (119909 + 119910)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119910))

minus T119903

119898119891 (minus2119898 (119909 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119911))

+ 2T119903

119898119891 (minus119898 (119909 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119911))

minus T119903

119898119891 (minus2119898 (119910 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119910 + 119911))

+ 2T119903

119898119891 (minus119898 (119910 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119910 + 119911))

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119903

times (119871 (minus2119898119909 minus2119898119910 minus2119898119911)

+ 2119871 ((119898 + 1) 119909 (119898 + 1) 119910 (119898 + 1) 119911)

+ 2119871 (minus119898119909 minus119898119910 minus119898119911)

+119871 ((2119898 + 1) 119909 (2119898 + 1) 119910 (2119898 + 1) 119911))

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1))119903+1

119871 (119909 119910 119911)

(31)

for every 119909 119910 119911 isin 1198830 which completes the proof of (30)

Letting 119899 rarr infin in (30) we obtain that

119865119898

(119909 + 119910 + 119911) + 119865119898

(119909) + 119865119898

(119910) + 119865119898

(119911)

= 119865119898

(119909 + 119910) + 119865119898

(119909 + 119911) + 119865119898

(119910 + 119911) 119909 119910 119911 isin 1198830

(32)

So we have proved that for each 119898 isin M there exists afunction 119865

119898 119883 rarr 119884 satisfying (7) for 119909 119910 119911 isin 119883

0and

such that1003817100381710038171003817119891 (119909) minus 119865

119898(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(33)

Now we show that119865119898

= 119865119896for all119898 119896 isin M So fix119898 119896 isin

M Note that 119865119896satisfies (32) with 119898 replaced by 119896 Hence

replacing 119909 by (2119898+ 1)119909 and taking 119910 = 119911 = minus119898119909 in (32) weobtain thatT

119898119865119895= 119865119895for 119895 = 119898 119896 and

1003817100381710038171003817119865119898 (119909) minus 119865119896(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

119909 isin 1198830

(34)

whence by the linearity of Λ and (25)1003817100381710038171003817119865119898 (119909) minus 119865

119896(119909)

1003817100381710038171003817

=1003817100381710038171003817T119899

119898119865119898

(119909) minus T119899

119898119865119896(119909)

1003817100381710038171003817

leΛ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+Λ119899

119898120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

le(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)(35)

for every 119909 isin 1198830and 119899 isin N Now letting 119899 rarr infin we get

119865119898

= 119865119896= 119865

Thus in view of (33) we have proved that1003817100381710038171003817119891 (119909) minus 119865 (119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830 119898 isin M

(36)

whence we derive (16)Since (in view of (32)) it is easy to notice that 119865 is a

solution to (7) (ie (7) holds for all 119909 119910 119911 isin 119883) it remains to

Journal of Function Spaces and Applications 5

prove the statement concerning the uniqueness of 119865 So let119866 119883 rarr 119884 be also a solution of (7) and 119891(119909) minus 119866(119909) le

120588119871(119909) for 119909 isin 119883

0 Then

119866 (119909) minus 119865 (119909) le 2120588119871(119909) 119909 isin 119883

0 (37)

Further T119898119866 = 119866 for each 119898 isin Z

0 Consequently with a

fixed 119898 isin M

119866 (119909) minus 119865 (119909)

=1003817100381710038171003817T119899

119898119866 (119909) minus T

119899

119898119865 (119909)

1003817100381710038171003817 le 2Λ119899

119898120588119871(119909)

le2Λ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)(38)

for 119909 isin 1198830and 119899 isin N Next analogously as (25) by induction

we get

Λ119899

119898120576119898

(119909) le (119888 (minus2119898) + 2119888 (119898 + 1)

+ 2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)(39)

for 119909 isin 1198830 119899 isin N This implies that 119866 = 119865

Theorem 2 yields at once the following hyperstabilityresult

Corollary 3 Let (119883 +) be a commutative group1198830= 1198830

let 119884 be a normed space 119891 119883 rarr 119884 119888 Z0

rarr [0infin)119871 11988330

rarr [0infin) and let the conditions (13) (14) and (15) bevalid Assume that

inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 0 119909 isin 1198830 (40)

Then 119891 satisfies (7) for all 119909 119910 119911 isin 119883

Proof Note that without loss of generality we may assumethat 119884 is complete because otherwise we can replace it by itscompletion Next in view of (40) 120588

119871(119909) = 0 for each 119909 isin 119883

0

where 120588119871is defined by (17) Hence fromTheorem 2 we easily

derive that 119891 is a solution to (7)

3 Final Remarks

Remark 4 Note that if in Theorem 2

lim inf119898rarrinfin

119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1) = 0

(41)

(this is the case when eg lim|119898|rarrinfin

119888(119898) = 0) then (13)holds and

120588119871(119909) = inf

119898isinM120576119898

(119909) 119909 isin 1198830 (42)

Further let 119883 be a normed space and

119871 (119909 119910 119911) = 120576 (119909119901+

10038171003817100381710038171199101003817100381710038171003817119901

+ 119911119901) 119909 119910 119911 isin 119883

0(43)

with some reals 120576 gt 0 and 119901 lt 0 Then the condition (14)is valid for instance with 119888(119896) = |119896|

119901 for 119896 isin Z0 Obviously

(40) holds and there exists 1198980isin N such that

|2119898|119901+ 2|119898 + 1|

119901+ 2|119898|

119901+ |2119898 + 1|

119901lt 1 119898 ge 119898

0

(44)

so we obtain (13) as well Consequently by Corollary 3 everyfunction 119891 119883 rarr 119884 fulfilling the inequality (15) satisfies(7) for all 119909 119910 119911 isin 119883 In this way we have obtained ahyperstability result that corresponds to the recent hypersta-bility outcomes in [11 20] and some classical stability resultsconcerning the Cauchy equation (see eg [9 page 3] [15page 15 16] and [16 page 2])

Below we provide two further simple and natural exam-ples of functions 119871 and 119888 satisfying the conditions (13) and(14) The first one clearly includes the case just described

(a) 119871(119909 119910 119911) = (12057211199091199041 +12057221199101199042 +12057231199111199043)119908 for 119909 119910 119911 isin

1198830with some 119908 120572

119894 119904119894

isin R such that 120572119894

gt 0 and119908119904119894lt 0 for 119894 = 1 2 3 and 119888(119898) equiv |119898|

minusV|119908| whereV = min |119904

1| |1199042| |1199043|

(b) 119871(119909 119910 119911) = 120572119909119904119910119905119911119906 for 119909 119910 119911 isin 119883

0with some

reals 120572 gt 0 and 119904 119905 119906 such that 119904 + 119905 + 119906 lt 0Clearly if two functions 119871 satisfy the condition (14)

then so do their sum and product with suitable functions 119888Therefore we can easily produce numerous examples of suchfunctions Of course there are some other such examples thatare a bit more artificial for instance 119871(119909 119910 119911) = 119861(119860(119909)

minus119899)

for 119909 isin 1198830 where 119899 isin N 119860 119883 rarr 119883 and 119861 R rarr R

are functions with 119860(119909119898) = 119898119860(119909) and 119861(119905119898) = 119898119861(119905) for119909 isin 119883 119905 isin R and 119898 isin Z

We end the paper with a simple example of applicationsof our main result

Corollary 5 Let119883 be a normed space and1198830= 1198830Write

119863(119909 119910 119911) =100381610038161003816100381610038161003817100381710038171003817119909 + 119910 + 119911

10038171003817100381710038172

+ 1199092+

100381710038171003817100381711991010038171003817100381710038172

+ 1199112

minus1003817100381710038171003817119909 + 119910

10038171003817100381710038172

minus 119909 + 1199112minus

1003817100381710038171003817119910 + 1199111003817100381710038171003817210038161003816100381610038161003816

(45)

for119909 119910 119911 isin 119883 Assume that one of the following two hypothesesis valid

(i) There exist 1199080 120572119894 119904119894

isin R such that 120572119894 1199080119904119894

gt 0 for119894 = 1 2 3 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

(12057211199091199041 + 1205722

100381710038171003817100381711991010038171003817100381710038171199042 + 12057231199111199043)1199080

lt infin (46)

(ii) There exist reals 119904 119905 119906 such that 119904 + 119905 + 119906 gt 0 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

1199091199041003817100381710038171003817119910

1003817100381710038171003817119905

119911119906

lt infin (47)

Then 119883 is an inner product space

Proof Write 119891(119909) = 1199092 for 119909 isin 119883 Then with 119871 and 119888

of the forms described in Remark 4 (with 119908 = minus1199080) from

Corollary 3 we easily derive that 119891 is a solution to (7) whichyields the statement

6 Journal of Function Spaces and Applications

References

[1] M Frechet ldquoSur la definition axiomatique drsquoune classe drsquoespacesvectoriels distancies applicables vectoriellement sur lrsquoespace deHilbertrdquoAnnals of Mathematics Second Series vol 36 no 3 pp705ndash718 1935

[2] P Jordan and J von Neumann ldquoOn inner products in linearmetric spacesrdquoAnnals ofMathematics Second Series vol 36 no3 pp 719ndash723 1935

[3] D Amir Characterizations of Inner Product Spaces OperatorTheory Advances and Applications vol 20 Birkhauser BaselSwitzerland 1986

[4] C Alsina J Sikorska and M S Tomas Norm Derivatives andCharacterizations of Inner Product Spaces World ScientificPublishing 2010

[5] S S Dragomir ldquoSome characterizations of inner product spacesand applicationsrdquo Universitatis Babes-Bolyai vol 34 no 1 pp50ndash55 1989

[6] M S Moslehian and J M Rassias ldquoA characterization of innerproduct spaces concerning an Euler-Lagrange identityrdquo Com-munications in Mathematical Analysis vol 8 no 2 pp 16ndash212010

[7] K Nikodem and Z Pales ldquoCharacterizations of inner productspaces by strongly convex functionsrdquo Banach Journal of Mathe-matical Analysis vol 5 no 1 pp 83ndash87 2011

[8] T M Rassias ldquoNew characterizations of inner product spacesrdquoBulletin des Sciences Mathematiques 2nd Serie vol 108 no 1pp 95ndash99 1984

[9] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[10] J Brzdęk ldquoRemarks on hyperstability of the Cauchy functionalequationrdquo Aequationes Mathematicae vol 86 no 3 pp 255ndash267 2013

[11] J Brzdęk ldquoHyperstability of the Cauchy equation on restricteddomainsrdquoActaMathematica Hungarica vol 141 no 1-2 pp 58ndash67 2013

[12] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[13] G Maksa ldquoThe stability of the entropy of degree alphardquo Journalof Mathematical Analysis and Applications vol 346 no 1 pp17ndash21 2008

[14] G Maksa and Z Pales ldquoHyperstability of a class of linearfunctional equationsrdquo Acta Mathematica vol 17 no 2 pp 107ndash112 2001

[15] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[16] S-M Jung Hvers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[17] J Brzdęk J Chudziak and Z Pales ldquoA fixed point approachto stability of functional equationsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 74 no 17 pp 6728ndash6732 2011

[18] J Brzdęk and K Cieplinski ldquoA fixed point approach to thestability of functional equations in non-Archimedean metricspacesrdquo Nonlinear Analysis Theory Methods amp Applications Avol 74 no 18 pp 6861ndash6867 2011

[19] L Cadariu L Gavruta and P Gavruta ldquoFixed points and gener-alized Hyers-Ulam stabilityrdquo Abstract and Applied Analysis vol2012 Article ID 712743 10 pages 2012

[20] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[21] Pl Kannappan Functional Equations and Inequalities withApplications Springer Monographs in Mathematics SpringerNew York NY USA 2009

[22] M Frechet ldquoUne definition fonctionnelles des polvnomesrdquoNouvelles Annales de Mathematique 4th serie vol 9 pp 145ndash162 1909

[23] M Frechet ldquoLes polvnomes abstraitsrdquo Journal de Mathema-tiques Pures et Appliquees 9th serie vol 8 pp 71ndash92 1929

[24] M Kuczma An Introduction to the Theory of Functional Equa-tions and Inequalities Cauchyrsquos Equation and Jensenrsquos InequalityBirkhauser Basel Switzerland 2nd edition 2009

[25] H Stetkaer Functional Equations on Groups World ScientificPublishing Singapore 2013

[26] Y H Lee ldquoOn the Hvers-Ulam-Rassias stability of the gen-eralized polynomial function of degree 2rdquo Journal of theChungcheong Mathematical Society vol 22 no 2 pp 201ndash2092009

[27] M Albert and J A Baker ldquoFunctions with bounded m-thdifferencesrdquoAnnales Polonici Mathematici vol 43 no 1 pp 93ndash103 1983

[28] D H Hyers ldquoTransformations with boundedm-th differencesrdquoPacific Journal of Mathematics vol 11 pp 591ndash602 1961

[29] Y-H Lee ldquoOn the stability of the monomial functional equa-tionrdquo Bulletin of the Korean Mathematical Society vol 45 no 2pp 397ndash403 2008

[30] M Chudziak On solutions and stability of functional equationsconnected to the Popoviciu inequality [PhD thesis] PedagogicalUniversity of Cracow Cracow Poland 2012 (Polish)

[31] W Fechner ldquoOn the Hyers-Ulam stability of functional equa-tions connected with additive and quadratic mappingsrdquo Journalof Mathematical Analysis and Applications vol 322 no 2 pp774ndash786 2006

[32] S-M Jung ldquoOn the Hyers-Ulam stability of the functionalequations that have the quadratic propertyrdquo Journal of Mathe-matical Analysis and Applications vol 222 no 1 pp 126ndash1371998

[33] J Sikorska ldquoOn a direct method for proving the Hyers-Ulam stability of functional equationsrdquo Journal of MathematicalAnalysis and Applications vol 372 no 1 pp 99ndash109 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

Proof Replacing 119909 by (2119898 + 1) 119909 and taking 119910 = 119911 = minus119898119909 in(15) we get1003817100381710038171003817119891 (119909) + 119891 ((2119898 + 1) 119909) + 2119891 (minus119898119909)

minus2119891 ((119898 + 1) 119909) minus 119891 (minus2119898119909)1003817100381710038171003817

le 119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 120576119898

(119909) 119909 isin 1198830 119898 isin Z

0

(18)

Further put

T119898120585 (119909) = 120585 (minus2119898119909) + 2120585 ((119898 + 1) 119909) minus 2120585 (minus119898119909)

minus 120585 ((2119898 + 1) 119909) 120585 isin 119884119883 119909 isin 119883 119898 isin Z

0

(19)

Then

T119899

119898119891 (0) = 0 119899 isin N 119898 isin Z

0 (20)

and inequality (18) takes the form1003817100381710038171003817T119898119891 (119909) minus 119891 (119909)

1003817100381710038171003817 le 120576119898

(119909) 119909 isin 1198830 119898 isin Z

0 (21)

Define an operator Λ119898

R1198830

+rarr R1198830

+for 119898 isin Z

0by

Λ119898120578 (119909) = 120578 (minus2119898119909) + 2120578 ((119898 + 1) 119909)

+ 2120578 (minus119898119909) + 120578 ((2119898 + 1) 119909)(22)

for 120578 isin R1198830

+and 119909 isin 119883

0 Then it is easily seen that for each

119898 isin Z0 the operatorΛ = Λ

119898has the formdescribed in (H3)

with 119896 = 4 119878 = 1198830 119864 = 119884 and

1198911(119909) = minus2119898119909 119891

2(119909) = (119898 + 1) 119909

1198913(119909) = minus119898119909 119891

4(119909) = (2119898 + 1) 119909

1198711(119909) = 119871

4(119909) = 1 119871

2(119909) = 119871

3(119909) = 2 119909 isin 119883

0

(23)

Moreover for every 120585 120583 isin 1198841198830 119909 isin 1198830 119898 isin Z

0

1003817100381710038171003817T119898120585 (119909) minus T119898120583 (119909)

1003817100381710038171003817

=1003817100381710038171003817120585 (minus2119898119909) + 2120585 ((119898 + 1) 119909) minus 2120585 (minus119898119909)

minus 120585 ((2119898 + 1) 119909) minus 120583 (minus2119898119909) minus 2120583 ((119898 + 1) 119909)

+ 2120583 (minus119898119909) + 120583 ((2119898 + 1) 119909)1003817100381710038171003817

le1003817100381710038171003817(120585 minus 120583) (minus2119898119909)

1003817100381710038171003817 + 21003817100381710038171003817(120585 minus 120583) ((119898 + 1) 119909)

1003817100381710038171003817

+ 21003817100381710038171003817(120585 minus 120583) (minus119898119909)

1003817100381710038171003817 +1003817100381710038171003817(120585 minus 120583) ((2119898 + 1) 119909)

1003817100381710038171003817

=

4

sum119894=1

119871119894(119909)

1003817100381710038171003817(120585 minus 120583) (119891119894(119909))

1003817100381710038171003817

(24)

where (120585 minus 120583) (119910) = 120585(119910) minus 120583(119910) for 119910 isin 1198830 It is easy to check

that in view of (14)

Λ119898120576119896(119909) le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1)) 120576119896(119909) 119896 119898 isin Z

0 119909 isin 119883

0

(25)

Therefore since the operator Λ119898is linear we have

120576lowast

119898(119909) =

infin

sum119899=0

Λ119898

119899120576119898

(119909)

le

infin

sum119899=0

(119888 (minus2119898) + 2119888 (119898 + 1)

+2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)

=120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119898 isin M 119909 isin 1198830

(26)

Thus byTheorem 1 (with 119878 = 1198830and119864 = 119884) for each119898 isin M

there exists a function 1198651015840119898

1198830

rarr 119884 with

1198651015840

119898(119909) = 119865

1015840

119898(minus2119898119909) + 2119865

1015840

119898((119898 + 1) 119909)

minus 21198651015840

119898(minus119898119909) minus 119865

1015840

119898((2119898 + 1) 119909) 119909 isin 119883

0

10038171003817100381710038171003817119891 (119909) minus 119865

1015840

119898(119909)

10038171003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(27)

Moreover

1198651015840

119898(119909) = lim

119899rarrinfinT119899

119898119891 (119909) 119909 isin 119883

0 119898 isin M (28)

Define 119865119898

119883 rarr 119884 by 119865119898(0) = 0 and 119865

119898(119909) = 1198651015840

119898(119909) for

119909 isin 1198830and 119898 isin M Then it is easily seen that by (20)

119865119898

(119909) = lim119899rarrinfin

T119899

119898119891 (119909) 119909 isin 119883 119898 isin M (29)

Next we show that

1003817100381710038171003817T119899

119898119891 (119909 + 119910 + 119911) + T

119899

119898119891 (119909) + T

119899

119898119891 (119910) + T

119899

119898119891 (119911)

minusT119899

119898119891 (119909 + 119910) minus T

119899

119898119891 (119909 + 119911) minus T

119899

119898119891 (119910 + 119911)

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119899

times 119871 (119909 119910 119911)

(30)

for every 119909 119910 119911 isin 1198830 119899 isin N

0= N cup 0 119898 isin M

4 Journal of Function Spaces and Applications

Fix 119898 isin M For 119899 = 0 the condition (30) is simply (15)So take 119903 isin N

0and suppose that (30) holds for 119899 = 119903 and

119909 119910 119911 isin 1198830 Then

10038171003817100381710038171003817T119903+1

119898119891 (119909 + 119910 + 119911) + T

119903+1

119898119891 (119909) + T

119903+1

119898119891 (119910) + T

119903+1

119898119891 (119911)

minusT119903+1

119898119891 (119909 + 119910) minus T

119903+1

119898119891 (119909 + 119911) minus T

119903+1

119898119891 (119910 + 119911)

10038171003817100381710038171003817

=1003817100381710038171003817T119903

119898119891 (minus2119898 (119909 + 119910 + 119911))

+ 2T119903

119898119891 ((119898 + 1) (119909 + 119910 + 119911))

minus 2T119903

119898119891 (minus119898 (119909 + 119910 + 119911))

minus T119903

119898119891 ((2119898 + 1) (119909 + 119910 + 119911))

+ T119903

119898119891 (minus2119898119909) + 2T

119903

119898119891 ((119898 + 1) 119909)

minus 2T119903

119898119891 (minus119898119909) minus T

119903

119898119891 ((2119898 + 1) 119909)

+ T119903

119898119891 (minus2119898119910) + 2T

119903

119898119891 ((119898 + 1) 119910)

minus 2T119903

119898119891 (minus119898119910) minus T

119903

119898119891 ((2119898 + 1) 119910)

+ T119903

119898119891 (minus2119898119911) + 2T

119903

119898119891 ((119898 + 1) 119911)

minus 2T119903

119898119891 (minus119898119911) minus T

119903

119898119891 ((2119898 + 1) 119911)

minus T119903

119898119891 (minus2119898 (119909 + 119910)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119910))

+ 2T119903

119898119891 (minus119898 (119909 + 119910)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119910))

minus T119903

119898119891 (minus2119898 (119909 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119911))

+ 2T119903

119898119891 (minus119898 (119909 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119911))

minus T119903

119898119891 (minus2119898 (119910 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119910 + 119911))

+ 2T119903

119898119891 (minus119898 (119910 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119910 + 119911))

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119903

times (119871 (minus2119898119909 minus2119898119910 minus2119898119911)

+ 2119871 ((119898 + 1) 119909 (119898 + 1) 119910 (119898 + 1) 119911)

+ 2119871 (minus119898119909 minus119898119910 minus119898119911)

+119871 ((2119898 + 1) 119909 (2119898 + 1) 119910 (2119898 + 1) 119911))

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1))119903+1

119871 (119909 119910 119911)

(31)

for every 119909 119910 119911 isin 1198830 which completes the proof of (30)

Letting 119899 rarr infin in (30) we obtain that

119865119898

(119909 + 119910 + 119911) + 119865119898

(119909) + 119865119898

(119910) + 119865119898

(119911)

= 119865119898

(119909 + 119910) + 119865119898

(119909 + 119911) + 119865119898

(119910 + 119911) 119909 119910 119911 isin 1198830

(32)

So we have proved that for each 119898 isin M there exists afunction 119865

119898 119883 rarr 119884 satisfying (7) for 119909 119910 119911 isin 119883

0and

such that1003817100381710038171003817119891 (119909) minus 119865

119898(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(33)

Now we show that119865119898

= 119865119896for all119898 119896 isin M So fix119898 119896 isin

M Note that 119865119896satisfies (32) with 119898 replaced by 119896 Hence

replacing 119909 by (2119898+ 1)119909 and taking 119910 = 119911 = minus119898119909 in (32) weobtain thatT

119898119865119895= 119865119895for 119895 = 119898 119896 and

1003817100381710038171003817119865119898 (119909) minus 119865119896(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

119909 isin 1198830

(34)

whence by the linearity of Λ and (25)1003817100381710038171003817119865119898 (119909) minus 119865

119896(119909)

1003817100381710038171003817

=1003817100381710038171003817T119899

119898119865119898

(119909) minus T119899

119898119865119896(119909)

1003817100381710038171003817

leΛ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+Λ119899

119898120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

le(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)(35)

for every 119909 isin 1198830and 119899 isin N Now letting 119899 rarr infin we get

119865119898

= 119865119896= 119865

Thus in view of (33) we have proved that1003817100381710038171003817119891 (119909) minus 119865 (119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830 119898 isin M

(36)

whence we derive (16)Since (in view of (32)) it is easy to notice that 119865 is a

solution to (7) (ie (7) holds for all 119909 119910 119911 isin 119883) it remains to

Journal of Function Spaces and Applications 5

prove the statement concerning the uniqueness of 119865 So let119866 119883 rarr 119884 be also a solution of (7) and 119891(119909) minus 119866(119909) le

120588119871(119909) for 119909 isin 119883

0 Then

119866 (119909) minus 119865 (119909) le 2120588119871(119909) 119909 isin 119883

0 (37)

Further T119898119866 = 119866 for each 119898 isin Z

0 Consequently with a

fixed 119898 isin M

119866 (119909) minus 119865 (119909)

=1003817100381710038171003817T119899

119898119866 (119909) minus T

119899

119898119865 (119909)

1003817100381710038171003817 le 2Λ119899

119898120588119871(119909)

le2Λ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)(38)

for 119909 isin 1198830and 119899 isin N Next analogously as (25) by induction

we get

Λ119899

119898120576119898

(119909) le (119888 (minus2119898) + 2119888 (119898 + 1)

+ 2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)(39)

for 119909 isin 1198830 119899 isin N This implies that 119866 = 119865

Theorem 2 yields at once the following hyperstabilityresult

Corollary 3 Let (119883 +) be a commutative group1198830= 1198830

let 119884 be a normed space 119891 119883 rarr 119884 119888 Z0

rarr [0infin)119871 11988330

rarr [0infin) and let the conditions (13) (14) and (15) bevalid Assume that

inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 0 119909 isin 1198830 (40)

Then 119891 satisfies (7) for all 119909 119910 119911 isin 119883

Proof Note that without loss of generality we may assumethat 119884 is complete because otherwise we can replace it by itscompletion Next in view of (40) 120588

119871(119909) = 0 for each 119909 isin 119883

0

where 120588119871is defined by (17) Hence fromTheorem 2 we easily

derive that 119891 is a solution to (7)

3 Final Remarks

Remark 4 Note that if in Theorem 2

lim inf119898rarrinfin

119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1) = 0

(41)

(this is the case when eg lim|119898|rarrinfin

119888(119898) = 0) then (13)holds and

120588119871(119909) = inf

119898isinM120576119898

(119909) 119909 isin 1198830 (42)

Further let 119883 be a normed space and

119871 (119909 119910 119911) = 120576 (119909119901+

10038171003817100381710038171199101003817100381710038171003817119901

+ 119911119901) 119909 119910 119911 isin 119883

0(43)

with some reals 120576 gt 0 and 119901 lt 0 Then the condition (14)is valid for instance with 119888(119896) = |119896|

119901 for 119896 isin Z0 Obviously

(40) holds and there exists 1198980isin N such that

|2119898|119901+ 2|119898 + 1|

119901+ 2|119898|

119901+ |2119898 + 1|

119901lt 1 119898 ge 119898

0

(44)

so we obtain (13) as well Consequently by Corollary 3 everyfunction 119891 119883 rarr 119884 fulfilling the inequality (15) satisfies(7) for all 119909 119910 119911 isin 119883 In this way we have obtained ahyperstability result that corresponds to the recent hypersta-bility outcomes in [11 20] and some classical stability resultsconcerning the Cauchy equation (see eg [9 page 3] [15page 15 16] and [16 page 2])

Below we provide two further simple and natural exam-ples of functions 119871 and 119888 satisfying the conditions (13) and(14) The first one clearly includes the case just described

(a) 119871(119909 119910 119911) = (12057211199091199041 +12057221199101199042 +12057231199111199043)119908 for 119909 119910 119911 isin

1198830with some 119908 120572

119894 119904119894

isin R such that 120572119894

gt 0 and119908119904119894lt 0 for 119894 = 1 2 3 and 119888(119898) equiv |119898|

minusV|119908| whereV = min |119904

1| |1199042| |1199043|

(b) 119871(119909 119910 119911) = 120572119909119904119910119905119911119906 for 119909 119910 119911 isin 119883

0with some

reals 120572 gt 0 and 119904 119905 119906 such that 119904 + 119905 + 119906 lt 0Clearly if two functions 119871 satisfy the condition (14)

then so do their sum and product with suitable functions 119888Therefore we can easily produce numerous examples of suchfunctions Of course there are some other such examples thatare a bit more artificial for instance 119871(119909 119910 119911) = 119861(119860(119909)

minus119899)

for 119909 isin 1198830 where 119899 isin N 119860 119883 rarr 119883 and 119861 R rarr R

are functions with 119860(119909119898) = 119898119860(119909) and 119861(119905119898) = 119898119861(119905) for119909 isin 119883 119905 isin R and 119898 isin Z

We end the paper with a simple example of applicationsof our main result

Corollary 5 Let119883 be a normed space and1198830= 1198830Write

119863(119909 119910 119911) =100381610038161003816100381610038161003817100381710038171003817119909 + 119910 + 119911

10038171003817100381710038172

+ 1199092+

100381710038171003817100381711991010038171003817100381710038172

+ 1199112

minus1003817100381710038171003817119909 + 119910

10038171003817100381710038172

minus 119909 + 1199112minus

1003817100381710038171003817119910 + 1199111003817100381710038171003817210038161003816100381610038161003816

(45)

for119909 119910 119911 isin 119883 Assume that one of the following two hypothesesis valid

(i) There exist 1199080 120572119894 119904119894

isin R such that 120572119894 1199080119904119894

gt 0 for119894 = 1 2 3 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

(12057211199091199041 + 1205722

100381710038171003817100381711991010038171003817100381710038171199042 + 12057231199111199043)1199080

lt infin (46)

(ii) There exist reals 119904 119905 119906 such that 119904 + 119905 + 119906 gt 0 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

1199091199041003817100381710038171003817119910

1003817100381710038171003817119905

119911119906

lt infin (47)

Then 119883 is an inner product space

Proof Write 119891(119909) = 1199092 for 119909 isin 119883 Then with 119871 and 119888

of the forms described in Remark 4 (with 119908 = minus1199080) from

Corollary 3 we easily derive that 119891 is a solution to (7) whichyields the statement

6 Journal of Function Spaces and Applications

References

[1] M Frechet ldquoSur la definition axiomatique drsquoune classe drsquoespacesvectoriels distancies applicables vectoriellement sur lrsquoespace deHilbertrdquoAnnals of Mathematics Second Series vol 36 no 3 pp705ndash718 1935

[2] P Jordan and J von Neumann ldquoOn inner products in linearmetric spacesrdquoAnnals ofMathematics Second Series vol 36 no3 pp 719ndash723 1935

[3] D Amir Characterizations of Inner Product Spaces OperatorTheory Advances and Applications vol 20 Birkhauser BaselSwitzerland 1986

[4] C Alsina J Sikorska and M S Tomas Norm Derivatives andCharacterizations of Inner Product Spaces World ScientificPublishing 2010

[5] S S Dragomir ldquoSome characterizations of inner product spacesand applicationsrdquo Universitatis Babes-Bolyai vol 34 no 1 pp50ndash55 1989

[6] M S Moslehian and J M Rassias ldquoA characterization of innerproduct spaces concerning an Euler-Lagrange identityrdquo Com-munications in Mathematical Analysis vol 8 no 2 pp 16ndash212010

[7] K Nikodem and Z Pales ldquoCharacterizations of inner productspaces by strongly convex functionsrdquo Banach Journal of Mathe-matical Analysis vol 5 no 1 pp 83ndash87 2011

[8] T M Rassias ldquoNew characterizations of inner product spacesrdquoBulletin des Sciences Mathematiques 2nd Serie vol 108 no 1pp 95ndash99 1984

[9] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[10] J Brzdęk ldquoRemarks on hyperstability of the Cauchy functionalequationrdquo Aequationes Mathematicae vol 86 no 3 pp 255ndash267 2013

[11] J Brzdęk ldquoHyperstability of the Cauchy equation on restricteddomainsrdquoActaMathematica Hungarica vol 141 no 1-2 pp 58ndash67 2013

[12] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[13] G Maksa ldquoThe stability of the entropy of degree alphardquo Journalof Mathematical Analysis and Applications vol 346 no 1 pp17ndash21 2008

[14] G Maksa and Z Pales ldquoHyperstability of a class of linearfunctional equationsrdquo Acta Mathematica vol 17 no 2 pp 107ndash112 2001

[15] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[16] S-M Jung Hvers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[17] J Brzdęk J Chudziak and Z Pales ldquoA fixed point approachto stability of functional equationsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 74 no 17 pp 6728ndash6732 2011

[18] J Brzdęk and K Cieplinski ldquoA fixed point approach to thestability of functional equations in non-Archimedean metricspacesrdquo Nonlinear Analysis Theory Methods amp Applications Avol 74 no 18 pp 6861ndash6867 2011

[19] L Cadariu L Gavruta and P Gavruta ldquoFixed points and gener-alized Hyers-Ulam stabilityrdquo Abstract and Applied Analysis vol2012 Article ID 712743 10 pages 2012

[20] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[21] Pl Kannappan Functional Equations and Inequalities withApplications Springer Monographs in Mathematics SpringerNew York NY USA 2009

[22] M Frechet ldquoUne definition fonctionnelles des polvnomesrdquoNouvelles Annales de Mathematique 4th serie vol 9 pp 145ndash162 1909

[23] M Frechet ldquoLes polvnomes abstraitsrdquo Journal de Mathema-tiques Pures et Appliquees 9th serie vol 8 pp 71ndash92 1929

[24] M Kuczma An Introduction to the Theory of Functional Equa-tions and Inequalities Cauchyrsquos Equation and Jensenrsquos InequalityBirkhauser Basel Switzerland 2nd edition 2009

[25] H Stetkaer Functional Equations on Groups World ScientificPublishing Singapore 2013

[26] Y H Lee ldquoOn the Hvers-Ulam-Rassias stability of the gen-eralized polynomial function of degree 2rdquo Journal of theChungcheong Mathematical Society vol 22 no 2 pp 201ndash2092009

[27] M Albert and J A Baker ldquoFunctions with bounded m-thdifferencesrdquoAnnales Polonici Mathematici vol 43 no 1 pp 93ndash103 1983

[28] D H Hyers ldquoTransformations with boundedm-th differencesrdquoPacific Journal of Mathematics vol 11 pp 591ndash602 1961

[29] Y-H Lee ldquoOn the stability of the monomial functional equa-tionrdquo Bulletin of the Korean Mathematical Society vol 45 no 2pp 397ndash403 2008

[30] M Chudziak On solutions and stability of functional equationsconnected to the Popoviciu inequality [PhD thesis] PedagogicalUniversity of Cracow Cracow Poland 2012 (Polish)

[31] W Fechner ldquoOn the Hyers-Ulam stability of functional equa-tions connected with additive and quadratic mappingsrdquo Journalof Mathematical Analysis and Applications vol 322 no 2 pp774ndash786 2006

[32] S-M Jung ldquoOn the Hyers-Ulam stability of the functionalequations that have the quadratic propertyrdquo Journal of Mathe-matical Analysis and Applications vol 222 no 1 pp 126ndash1371998

[33] J Sikorska ldquoOn a direct method for proving the Hyers-Ulam stability of functional equationsrdquo Journal of MathematicalAnalysis and Applications vol 372 no 1 pp 99ndash109 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

Fix 119898 isin M For 119899 = 0 the condition (30) is simply (15)So take 119903 isin N

0and suppose that (30) holds for 119899 = 119903 and

119909 119910 119911 isin 1198830 Then

10038171003817100381710038171003817T119903+1

119898119891 (119909 + 119910 + 119911) + T

119903+1

119898119891 (119909) + T

119903+1

119898119891 (119910) + T

119903+1

119898119891 (119911)

minusT119903+1

119898119891 (119909 + 119910) minus T

119903+1

119898119891 (119909 + 119911) minus T

119903+1

119898119891 (119910 + 119911)

10038171003817100381710038171003817

=1003817100381710038171003817T119903

119898119891 (minus2119898 (119909 + 119910 + 119911))

+ 2T119903

119898119891 ((119898 + 1) (119909 + 119910 + 119911))

minus 2T119903

119898119891 (minus119898 (119909 + 119910 + 119911))

minus T119903

119898119891 ((2119898 + 1) (119909 + 119910 + 119911))

+ T119903

119898119891 (minus2119898119909) + 2T

119903

119898119891 ((119898 + 1) 119909)

minus 2T119903

119898119891 (minus119898119909) minus T

119903

119898119891 ((2119898 + 1) 119909)

+ T119903

119898119891 (minus2119898119910) + 2T

119903

119898119891 ((119898 + 1) 119910)

minus 2T119903

119898119891 (minus119898119910) minus T

119903

119898119891 ((2119898 + 1) 119910)

+ T119903

119898119891 (minus2119898119911) + 2T

119903

119898119891 ((119898 + 1) 119911)

minus 2T119903

119898119891 (minus119898119911) minus T

119903

119898119891 ((2119898 + 1) 119911)

minus T119903

119898119891 (minus2119898 (119909 + 119910)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119910))

+ 2T119903

119898119891 (minus119898 (119909 + 119910)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119910))

minus T119903

119898119891 (minus2119898 (119909 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119909 + 119911))

+ 2T119903

119898119891 (minus119898 (119909 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119909 + 119911))

minus T119903

119898119891 (minus2119898 (119910 + 119911)) minus 2T

119903

119898119891 ((119898 + 1) (119910 + 119911))

+ 2T119903

119898119891 (minus119898 (119910 + 119911)) + T

119903

119898119891 ((2119898 + 1) (119910 + 119911))

1003817100381710038171003817

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))119903

times (119871 (minus2119898119909 minus2119898119910 minus2119898119911)

+ 2119871 ((119898 + 1) 119909 (119898 + 1) 119910 (119898 + 1) 119911)

+ 2119871 (minus119898119909 minus119898119910 minus119898119911)

+119871 ((2119898 + 1) 119909 (2119898 + 1) 119910 (2119898 + 1) 119911))

le (119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898)

+ 119888 (2119898 + 1))119903+1

119871 (119909 119910 119911)

(31)

for every 119909 119910 119911 isin 1198830 which completes the proof of (30)

Letting 119899 rarr infin in (30) we obtain that

119865119898

(119909 + 119910 + 119911) + 119865119898

(119909) + 119865119898

(119910) + 119865119898

(119911)

= 119865119898

(119909 + 119910) + 119865119898

(119909 + 119911) + 119865119898

(119910 + 119911) 119909 119910 119911 isin 1198830

(32)

So we have proved that for each 119898 isin M there exists afunction 119865

119898 119883 rarr 119884 satisfying (7) for 119909 119910 119911 isin 119883

0and

such that1003817100381710038171003817119891 (119909) minus 119865

119898(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830

(33)

Now we show that119865119898

= 119865119896for all119898 119896 isin M So fix119898 119896 isin

M Note that 119865119896satisfies (32) with 119898 replaced by 119896 Hence

replacing 119909 by (2119898+ 1)119909 and taking 119910 = 119911 = minus119898119909 in (32) weobtain thatT

119898119865119895= 119865119895for 119895 = 119898 119896 and

1003817100381710038171003817119865119898 (119909) minus 119865119896(119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

119909 isin 1198830

(34)

whence by the linearity of Λ and (25)1003817100381710038171003817119865119898 (119909) minus 119865

119896(119909)

1003817100381710038171003817

=1003817100381710038171003817T119899

119898119865119898

(119909) minus T119899

119898119865119896(119909)

1003817100381710038171003817

leΛ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+Λ119899

119898120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)

le(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

+(119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1))

119899120576119896(119909)

1 minus 119888 (minus2119896) minus 2119888 (119896 + 1) minus 2119888 (minus119896) minus 119888 (2119896 + 1)(35)

for every 119909 isin 1198830and 119899 isin N Now letting 119899 rarr infin we get

119865119898

= 119865119896= 119865

Thus in view of (33) we have proved that1003817100381710038171003817119891 (119909) minus 119865 (119909)

1003817100381710038171003817

le120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)

119909 isin 1198830 119898 isin M

(36)

whence we derive (16)Since (in view of (32)) it is easy to notice that 119865 is a

solution to (7) (ie (7) holds for all 119909 119910 119911 isin 119883) it remains to

Journal of Function Spaces and Applications 5

prove the statement concerning the uniqueness of 119865 So let119866 119883 rarr 119884 be also a solution of (7) and 119891(119909) minus 119866(119909) le

120588119871(119909) for 119909 isin 119883

0 Then

119866 (119909) minus 119865 (119909) le 2120588119871(119909) 119909 isin 119883

0 (37)

Further T119898119866 = 119866 for each 119898 isin Z

0 Consequently with a

fixed 119898 isin M

119866 (119909) minus 119865 (119909)

=1003817100381710038171003817T119899

119898119866 (119909) minus T

119899

119898119865 (119909)

1003817100381710038171003817 le 2Λ119899

119898120588119871(119909)

le2Λ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)(38)

for 119909 isin 1198830and 119899 isin N Next analogously as (25) by induction

we get

Λ119899

119898120576119898

(119909) le (119888 (minus2119898) + 2119888 (119898 + 1)

+ 2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)(39)

for 119909 isin 1198830 119899 isin N This implies that 119866 = 119865

Theorem 2 yields at once the following hyperstabilityresult

Corollary 3 Let (119883 +) be a commutative group1198830= 1198830

let 119884 be a normed space 119891 119883 rarr 119884 119888 Z0

rarr [0infin)119871 11988330

rarr [0infin) and let the conditions (13) (14) and (15) bevalid Assume that

inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 0 119909 isin 1198830 (40)

Then 119891 satisfies (7) for all 119909 119910 119911 isin 119883

Proof Note that without loss of generality we may assumethat 119884 is complete because otherwise we can replace it by itscompletion Next in view of (40) 120588

119871(119909) = 0 for each 119909 isin 119883

0

where 120588119871is defined by (17) Hence fromTheorem 2 we easily

derive that 119891 is a solution to (7)

3 Final Remarks

Remark 4 Note that if in Theorem 2

lim inf119898rarrinfin

119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1) = 0

(41)

(this is the case when eg lim|119898|rarrinfin

119888(119898) = 0) then (13)holds and

120588119871(119909) = inf

119898isinM120576119898

(119909) 119909 isin 1198830 (42)

Further let 119883 be a normed space and

119871 (119909 119910 119911) = 120576 (119909119901+

10038171003817100381710038171199101003817100381710038171003817119901

+ 119911119901) 119909 119910 119911 isin 119883

0(43)

with some reals 120576 gt 0 and 119901 lt 0 Then the condition (14)is valid for instance with 119888(119896) = |119896|

119901 for 119896 isin Z0 Obviously

(40) holds and there exists 1198980isin N such that

|2119898|119901+ 2|119898 + 1|

119901+ 2|119898|

119901+ |2119898 + 1|

119901lt 1 119898 ge 119898

0

(44)

so we obtain (13) as well Consequently by Corollary 3 everyfunction 119891 119883 rarr 119884 fulfilling the inequality (15) satisfies(7) for all 119909 119910 119911 isin 119883 In this way we have obtained ahyperstability result that corresponds to the recent hypersta-bility outcomes in [11 20] and some classical stability resultsconcerning the Cauchy equation (see eg [9 page 3] [15page 15 16] and [16 page 2])

Below we provide two further simple and natural exam-ples of functions 119871 and 119888 satisfying the conditions (13) and(14) The first one clearly includes the case just described

(a) 119871(119909 119910 119911) = (12057211199091199041 +12057221199101199042 +12057231199111199043)119908 for 119909 119910 119911 isin

1198830with some 119908 120572

119894 119904119894

isin R such that 120572119894

gt 0 and119908119904119894lt 0 for 119894 = 1 2 3 and 119888(119898) equiv |119898|

minusV|119908| whereV = min |119904

1| |1199042| |1199043|

(b) 119871(119909 119910 119911) = 120572119909119904119910119905119911119906 for 119909 119910 119911 isin 119883

0with some

reals 120572 gt 0 and 119904 119905 119906 such that 119904 + 119905 + 119906 lt 0Clearly if two functions 119871 satisfy the condition (14)

then so do their sum and product with suitable functions 119888Therefore we can easily produce numerous examples of suchfunctions Of course there are some other such examples thatare a bit more artificial for instance 119871(119909 119910 119911) = 119861(119860(119909)

minus119899)

for 119909 isin 1198830 where 119899 isin N 119860 119883 rarr 119883 and 119861 R rarr R

are functions with 119860(119909119898) = 119898119860(119909) and 119861(119905119898) = 119898119861(119905) for119909 isin 119883 119905 isin R and 119898 isin Z

We end the paper with a simple example of applicationsof our main result

Corollary 5 Let119883 be a normed space and1198830= 1198830Write

119863(119909 119910 119911) =100381610038161003816100381610038161003817100381710038171003817119909 + 119910 + 119911

10038171003817100381710038172

+ 1199092+

100381710038171003817100381711991010038171003817100381710038172

+ 1199112

minus1003817100381710038171003817119909 + 119910

10038171003817100381710038172

minus 119909 + 1199112minus

1003817100381710038171003817119910 + 1199111003817100381710038171003817210038161003816100381610038161003816

(45)

for119909 119910 119911 isin 119883 Assume that one of the following two hypothesesis valid

(i) There exist 1199080 120572119894 119904119894

isin R such that 120572119894 1199080119904119894

gt 0 for119894 = 1 2 3 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

(12057211199091199041 + 1205722

100381710038171003817100381711991010038171003817100381710038171199042 + 12057231199111199043)1199080

lt infin (46)

(ii) There exist reals 119904 119905 119906 such that 119904 + 119905 + 119906 gt 0 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

1199091199041003817100381710038171003817119910

1003817100381710038171003817119905

119911119906

lt infin (47)

Then 119883 is an inner product space

Proof Write 119891(119909) = 1199092 for 119909 isin 119883 Then with 119871 and 119888

of the forms described in Remark 4 (with 119908 = minus1199080) from

Corollary 3 we easily derive that 119891 is a solution to (7) whichyields the statement

6 Journal of Function Spaces and Applications

References

[1] M Frechet ldquoSur la definition axiomatique drsquoune classe drsquoespacesvectoriels distancies applicables vectoriellement sur lrsquoespace deHilbertrdquoAnnals of Mathematics Second Series vol 36 no 3 pp705ndash718 1935

[2] P Jordan and J von Neumann ldquoOn inner products in linearmetric spacesrdquoAnnals ofMathematics Second Series vol 36 no3 pp 719ndash723 1935

[3] D Amir Characterizations of Inner Product Spaces OperatorTheory Advances and Applications vol 20 Birkhauser BaselSwitzerland 1986

[4] C Alsina J Sikorska and M S Tomas Norm Derivatives andCharacterizations of Inner Product Spaces World ScientificPublishing 2010

[5] S S Dragomir ldquoSome characterizations of inner product spacesand applicationsrdquo Universitatis Babes-Bolyai vol 34 no 1 pp50ndash55 1989

[6] M S Moslehian and J M Rassias ldquoA characterization of innerproduct spaces concerning an Euler-Lagrange identityrdquo Com-munications in Mathematical Analysis vol 8 no 2 pp 16ndash212010

[7] K Nikodem and Z Pales ldquoCharacterizations of inner productspaces by strongly convex functionsrdquo Banach Journal of Mathe-matical Analysis vol 5 no 1 pp 83ndash87 2011

[8] T M Rassias ldquoNew characterizations of inner product spacesrdquoBulletin des Sciences Mathematiques 2nd Serie vol 108 no 1pp 95ndash99 1984

[9] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[10] J Brzdęk ldquoRemarks on hyperstability of the Cauchy functionalequationrdquo Aequationes Mathematicae vol 86 no 3 pp 255ndash267 2013

[11] J Brzdęk ldquoHyperstability of the Cauchy equation on restricteddomainsrdquoActaMathematica Hungarica vol 141 no 1-2 pp 58ndash67 2013

[12] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[13] G Maksa ldquoThe stability of the entropy of degree alphardquo Journalof Mathematical Analysis and Applications vol 346 no 1 pp17ndash21 2008

[14] G Maksa and Z Pales ldquoHyperstability of a class of linearfunctional equationsrdquo Acta Mathematica vol 17 no 2 pp 107ndash112 2001

[15] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[16] S-M Jung Hvers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[17] J Brzdęk J Chudziak and Z Pales ldquoA fixed point approachto stability of functional equationsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 74 no 17 pp 6728ndash6732 2011

[18] J Brzdęk and K Cieplinski ldquoA fixed point approach to thestability of functional equations in non-Archimedean metricspacesrdquo Nonlinear Analysis Theory Methods amp Applications Avol 74 no 18 pp 6861ndash6867 2011

[19] L Cadariu L Gavruta and P Gavruta ldquoFixed points and gener-alized Hyers-Ulam stabilityrdquo Abstract and Applied Analysis vol2012 Article ID 712743 10 pages 2012

[20] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[21] Pl Kannappan Functional Equations and Inequalities withApplications Springer Monographs in Mathematics SpringerNew York NY USA 2009

[22] M Frechet ldquoUne definition fonctionnelles des polvnomesrdquoNouvelles Annales de Mathematique 4th serie vol 9 pp 145ndash162 1909

[23] M Frechet ldquoLes polvnomes abstraitsrdquo Journal de Mathema-tiques Pures et Appliquees 9th serie vol 8 pp 71ndash92 1929

[24] M Kuczma An Introduction to the Theory of Functional Equa-tions and Inequalities Cauchyrsquos Equation and Jensenrsquos InequalityBirkhauser Basel Switzerland 2nd edition 2009

[25] H Stetkaer Functional Equations on Groups World ScientificPublishing Singapore 2013

[26] Y H Lee ldquoOn the Hvers-Ulam-Rassias stability of the gen-eralized polynomial function of degree 2rdquo Journal of theChungcheong Mathematical Society vol 22 no 2 pp 201ndash2092009

[27] M Albert and J A Baker ldquoFunctions with bounded m-thdifferencesrdquoAnnales Polonici Mathematici vol 43 no 1 pp 93ndash103 1983

[28] D H Hyers ldquoTransformations with boundedm-th differencesrdquoPacific Journal of Mathematics vol 11 pp 591ndash602 1961

[29] Y-H Lee ldquoOn the stability of the monomial functional equa-tionrdquo Bulletin of the Korean Mathematical Society vol 45 no 2pp 397ndash403 2008

[30] M Chudziak On solutions and stability of functional equationsconnected to the Popoviciu inequality [PhD thesis] PedagogicalUniversity of Cracow Cracow Poland 2012 (Polish)

[31] W Fechner ldquoOn the Hyers-Ulam stability of functional equa-tions connected with additive and quadratic mappingsrdquo Journalof Mathematical Analysis and Applications vol 322 no 2 pp774ndash786 2006

[32] S-M Jung ldquoOn the Hyers-Ulam stability of the functionalequations that have the quadratic propertyrdquo Journal of Mathe-matical Analysis and Applications vol 222 no 1 pp 126ndash1371998

[33] J Sikorska ldquoOn a direct method for proving the Hyers-Ulam stability of functional equationsrdquo Journal of MathematicalAnalysis and Applications vol 372 no 1 pp 99ndash109 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

prove the statement concerning the uniqueness of 119865 So let119866 119883 rarr 119884 be also a solution of (7) and 119891(119909) minus 119866(119909) le

120588119871(119909) for 119909 isin 119883

0 Then

119866 (119909) minus 119865 (119909) le 2120588119871(119909) 119909 isin 119883

0 (37)

Further T119898119866 = 119866 for each 119898 isin Z

0 Consequently with a

fixed 119898 isin M

119866 (119909) minus 119865 (119909)

=1003817100381710038171003817T119899

119898119866 (119909) minus T

119899

119898119865 (119909)

1003817100381710038171003817 le 2Λ119899

119898120588119871(119909)

le2Λ119899119898120576119898

(119909)

1 minus 119888 (minus2119898) minus 2119888 (119898 + 1) minus 2119888 (minus119898) minus 119888 (2119898 + 1)(38)

for 119909 isin 1198830and 119899 isin N Next analogously as (25) by induction

we get

Λ119899

119898120576119898

(119909) le (119888 (minus2119898) + 2119888 (119898 + 1)

+ 2119888 (minus119898) + 119888 (2119898 + 1))119899120576119898

(119909)(39)

for 119909 isin 1198830 119899 isin N This implies that 119866 = 119865

Theorem 2 yields at once the following hyperstabilityresult

Corollary 3 Let (119883 +) be a commutative group1198830= 1198830

let 119884 be a normed space 119891 119883 rarr 119884 119888 Z0

rarr [0infin)119871 11988330

rarr [0infin) and let the conditions (13) (14) and (15) bevalid Assume that

inf119898isinM

119871 ((2119898 + 1) 119909 minus119898119909 minus119898119909) = 0 119909 isin 1198830 (40)

Then 119891 satisfies (7) for all 119909 119910 119911 isin 119883

Proof Note that without loss of generality we may assumethat 119884 is complete because otherwise we can replace it by itscompletion Next in view of (40) 120588

119871(119909) = 0 for each 119909 isin 119883

0

where 120588119871is defined by (17) Hence fromTheorem 2 we easily

derive that 119891 is a solution to (7)

3 Final Remarks

Remark 4 Note that if in Theorem 2

lim inf119898rarrinfin

119888 (minus2119898) + 2119888 (119898 + 1) + 2119888 (minus119898) + 119888 (2119898 + 1) = 0

(41)

(this is the case when eg lim|119898|rarrinfin

119888(119898) = 0) then (13)holds and

120588119871(119909) = inf

119898isinM120576119898

(119909) 119909 isin 1198830 (42)

Further let 119883 be a normed space and

119871 (119909 119910 119911) = 120576 (119909119901+

10038171003817100381710038171199101003817100381710038171003817119901

+ 119911119901) 119909 119910 119911 isin 119883

0(43)

with some reals 120576 gt 0 and 119901 lt 0 Then the condition (14)is valid for instance with 119888(119896) = |119896|

119901 for 119896 isin Z0 Obviously

(40) holds and there exists 1198980isin N such that

|2119898|119901+ 2|119898 + 1|

119901+ 2|119898|

119901+ |2119898 + 1|

119901lt 1 119898 ge 119898

0

(44)

so we obtain (13) as well Consequently by Corollary 3 everyfunction 119891 119883 rarr 119884 fulfilling the inequality (15) satisfies(7) for all 119909 119910 119911 isin 119883 In this way we have obtained ahyperstability result that corresponds to the recent hypersta-bility outcomes in [11 20] and some classical stability resultsconcerning the Cauchy equation (see eg [9 page 3] [15page 15 16] and [16 page 2])

Below we provide two further simple and natural exam-ples of functions 119871 and 119888 satisfying the conditions (13) and(14) The first one clearly includes the case just described

(a) 119871(119909 119910 119911) = (12057211199091199041 +12057221199101199042 +12057231199111199043)119908 for 119909 119910 119911 isin

1198830with some 119908 120572

119894 119904119894

isin R such that 120572119894

gt 0 and119908119904119894lt 0 for 119894 = 1 2 3 and 119888(119898) equiv |119898|

minusV|119908| whereV = min |119904

1| |1199042| |1199043|

(b) 119871(119909 119910 119911) = 120572119909119904119910119905119911119906 for 119909 119910 119911 isin 119883

0with some

reals 120572 gt 0 and 119904 119905 119906 such that 119904 + 119905 + 119906 lt 0Clearly if two functions 119871 satisfy the condition (14)

then so do their sum and product with suitable functions 119888Therefore we can easily produce numerous examples of suchfunctions Of course there are some other such examples thatare a bit more artificial for instance 119871(119909 119910 119911) = 119861(119860(119909)

minus119899)

for 119909 isin 1198830 where 119899 isin N 119860 119883 rarr 119883 and 119861 R rarr R

are functions with 119860(119909119898) = 119898119860(119909) and 119861(119905119898) = 119898119861(119905) for119909 isin 119883 119905 isin R and 119898 isin Z

We end the paper with a simple example of applicationsof our main result

Corollary 5 Let119883 be a normed space and1198830= 1198830Write

119863(119909 119910 119911) =100381610038161003816100381610038161003817100381710038171003817119909 + 119910 + 119911

10038171003817100381710038172

+ 1199092+

100381710038171003817100381711991010038171003817100381710038172

+ 1199112

minus1003817100381710038171003817119909 + 119910

10038171003817100381710038172

minus 119909 + 1199112minus

1003817100381710038171003817119910 + 1199111003817100381710038171003817210038161003816100381610038161003816

(45)

for119909 119910 119911 isin 119883 Assume that one of the following two hypothesesis valid

(i) There exist 1199080 120572119894 119904119894

isin R such that 120572119894 1199080119904119894

gt 0 for119894 = 1 2 3 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

(12057211199091199041 + 1205722

100381710038171003817100381711991010038171003817100381710038171199042 + 12057231199111199043)1199080

lt infin (46)

(ii) There exist reals 119904 119905 119906 such that 119904 + 119905 + 119906 gt 0 and

sup119909119910119911isin1198830

119863(119909 119910 119911)

1199091199041003817100381710038171003817119910

1003817100381710038171003817119905

119911119906

lt infin (47)

Then 119883 is an inner product space

Proof Write 119891(119909) = 1199092 for 119909 isin 119883 Then with 119871 and 119888

of the forms described in Remark 4 (with 119908 = minus1199080) from

Corollary 3 we easily derive that 119891 is a solution to (7) whichyields the statement

6 Journal of Function Spaces and Applications

References

[1] M Frechet ldquoSur la definition axiomatique drsquoune classe drsquoespacesvectoriels distancies applicables vectoriellement sur lrsquoespace deHilbertrdquoAnnals of Mathematics Second Series vol 36 no 3 pp705ndash718 1935

[2] P Jordan and J von Neumann ldquoOn inner products in linearmetric spacesrdquoAnnals ofMathematics Second Series vol 36 no3 pp 719ndash723 1935

[3] D Amir Characterizations of Inner Product Spaces OperatorTheory Advances and Applications vol 20 Birkhauser BaselSwitzerland 1986

[4] C Alsina J Sikorska and M S Tomas Norm Derivatives andCharacterizations of Inner Product Spaces World ScientificPublishing 2010

[5] S S Dragomir ldquoSome characterizations of inner product spacesand applicationsrdquo Universitatis Babes-Bolyai vol 34 no 1 pp50ndash55 1989

[6] M S Moslehian and J M Rassias ldquoA characterization of innerproduct spaces concerning an Euler-Lagrange identityrdquo Com-munications in Mathematical Analysis vol 8 no 2 pp 16ndash212010

[7] K Nikodem and Z Pales ldquoCharacterizations of inner productspaces by strongly convex functionsrdquo Banach Journal of Mathe-matical Analysis vol 5 no 1 pp 83ndash87 2011

[8] T M Rassias ldquoNew characterizations of inner product spacesrdquoBulletin des Sciences Mathematiques 2nd Serie vol 108 no 1pp 95ndash99 1984

[9] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[10] J Brzdęk ldquoRemarks on hyperstability of the Cauchy functionalequationrdquo Aequationes Mathematicae vol 86 no 3 pp 255ndash267 2013

[11] J Brzdęk ldquoHyperstability of the Cauchy equation on restricteddomainsrdquoActaMathematica Hungarica vol 141 no 1-2 pp 58ndash67 2013

[12] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[13] G Maksa ldquoThe stability of the entropy of degree alphardquo Journalof Mathematical Analysis and Applications vol 346 no 1 pp17ndash21 2008

[14] G Maksa and Z Pales ldquoHyperstability of a class of linearfunctional equationsrdquo Acta Mathematica vol 17 no 2 pp 107ndash112 2001

[15] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[16] S-M Jung Hvers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[17] J Brzdęk J Chudziak and Z Pales ldquoA fixed point approachto stability of functional equationsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 74 no 17 pp 6728ndash6732 2011

[18] J Brzdęk and K Cieplinski ldquoA fixed point approach to thestability of functional equations in non-Archimedean metricspacesrdquo Nonlinear Analysis Theory Methods amp Applications Avol 74 no 18 pp 6861ndash6867 2011

[19] L Cadariu L Gavruta and P Gavruta ldquoFixed points and gener-alized Hyers-Ulam stabilityrdquo Abstract and Applied Analysis vol2012 Article ID 712743 10 pages 2012

[20] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[21] Pl Kannappan Functional Equations and Inequalities withApplications Springer Monographs in Mathematics SpringerNew York NY USA 2009

[22] M Frechet ldquoUne definition fonctionnelles des polvnomesrdquoNouvelles Annales de Mathematique 4th serie vol 9 pp 145ndash162 1909

[23] M Frechet ldquoLes polvnomes abstraitsrdquo Journal de Mathema-tiques Pures et Appliquees 9th serie vol 8 pp 71ndash92 1929

[24] M Kuczma An Introduction to the Theory of Functional Equa-tions and Inequalities Cauchyrsquos Equation and Jensenrsquos InequalityBirkhauser Basel Switzerland 2nd edition 2009

[25] H Stetkaer Functional Equations on Groups World ScientificPublishing Singapore 2013

[26] Y H Lee ldquoOn the Hvers-Ulam-Rassias stability of the gen-eralized polynomial function of degree 2rdquo Journal of theChungcheong Mathematical Society vol 22 no 2 pp 201ndash2092009

[27] M Albert and J A Baker ldquoFunctions with bounded m-thdifferencesrdquoAnnales Polonici Mathematici vol 43 no 1 pp 93ndash103 1983

[28] D H Hyers ldquoTransformations with boundedm-th differencesrdquoPacific Journal of Mathematics vol 11 pp 591ndash602 1961

[29] Y-H Lee ldquoOn the stability of the monomial functional equa-tionrdquo Bulletin of the Korean Mathematical Society vol 45 no 2pp 397ndash403 2008

[30] M Chudziak On solutions and stability of functional equationsconnected to the Popoviciu inequality [PhD thesis] PedagogicalUniversity of Cracow Cracow Poland 2012 (Polish)

[31] W Fechner ldquoOn the Hyers-Ulam stability of functional equa-tions connected with additive and quadratic mappingsrdquo Journalof Mathematical Analysis and Applications vol 322 no 2 pp774ndash786 2006

[32] S-M Jung ldquoOn the Hyers-Ulam stability of the functionalequations that have the quadratic propertyrdquo Journal of Mathe-matical Analysis and Applications vol 222 no 1 pp 126ndash1371998

[33] J Sikorska ldquoOn a direct method for proving the Hyers-Ulam stability of functional equationsrdquo Journal of MathematicalAnalysis and Applications vol 372 no 1 pp 99ndash109 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Journal of Function Spaces and Applications

References

[1] M Frechet ldquoSur la definition axiomatique drsquoune classe drsquoespacesvectoriels distancies applicables vectoriellement sur lrsquoespace deHilbertrdquoAnnals of Mathematics Second Series vol 36 no 3 pp705ndash718 1935

[2] P Jordan and J von Neumann ldquoOn inner products in linearmetric spacesrdquoAnnals ofMathematics Second Series vol 36 no3 pp 719ndash723 1935

[3] D Amir Characterizations of Inner Product Spaces OperatorTheory Advances and Applications vol 20 Birkhauser BaselSwitzerland 1986

[4] C Alsina J Sikorska and M S Tomas Norm Derivatives andCharacterizations of Inner Product Spaces World ScientificPublishing 2010

[5] S S Dragomir ldquoSome characterizations of inner product spacesand applicationsrdquo Universitatis Babes-Bolyai vol 34 no 1 pp50ndash55 1989

[6] M S Moslehian and J M Rassias ldquoA characterization of innerproduct spaces concerning an Euler-Lagrange identityrdquo Com-munications in Mathematical Analysis vol 8 no 2 pp 16ndash212010

[7] K Nikodem and Z Pales ldquoCharacterizations of inner productspaces by strongly convex functionsrdquo Banach Journal of Mathe-matical Analysis vol 5 no 1 pp 83ndash87 2011

[8] T M Rassias ldquoNew characterizations of inner product spacesrdquoBulletin des Sciences Mathematiques 2nd Serie vol 108 no 1pp 95ndash99 1984

[9] N Brillouet-Belluot J Brzdęk and K Cieplinski ldquoOn somerecent developments in Ulamrsquos type stabilityrdquo Abstract andApplied Analysis vol 2012 Article ID 716936 41 pages 2012

[10] J Brzdęk ldquoRemarks on hyperstability of the Cauchy functionalequationrdquo Aequationes Mathematicae vol 86 no 3 pp 255ndash267 2013

[11] J Brzdęk ldquoHyperstability of the Cauchy equation on restricteddomainsrdquoActaMathematica Hungarica vol 141 no 1-2 pp 58ndash67 2013

[12] E Gselmann ldquoHyperstability of a functional equationrdquo ActaMathematica Hungarica vol 124 no 1-2 pp 179ndash188 2009

[13] G Maksa ldquoThe stability of the entropy of degree alphardquo Journalof Mathematical Analysis and Applications vol 346 no 1 pp17ndash21 2008

[14] G Maksa and Z Pales ldquoHyperstability of a class of linearfunctional equationsrdquo Acta Mathematica vol 17 no 2 pp 107ndash112 2001

[15] D H Hyers G Isac and T M Rassias Stability of FunctionalEquations in Several Variables Birkhauser Boston Mass USA1998

[16] S-M Jung Hvers-Ulam-Rassias Stability of Functional Equa-tions in Nonlinear Analysis vol 48 of Springer Optimization andIts Applications Springer New York NY USA 2011

[17] J Brzdęk J Chudziak and Z Pales ldquoA fixed point approachto stability of functional equationsrdquo Nonlinear Analysis TheoryMethods amp Applications A vol 74 no 17 pp 6728ndash6732 2011

[18] J Brzdęk and K Cieplinski ldquoA fixed point approach to thestability of functional equations in non-Archimedean metricspacesrdquo Nonlinear Analysis Theory Methods amp Applications Avol 74 no 18 pp 6861ndash6867 2011

[19] L Cadariu L Gavruta and P Gavruta ldquoFixed points and gener-alized Hyers-Ulam stabilityrdquo Abstract and Applied Analysis vol2012 Article ID 712743 10 pages 2012

[20] M Piszczek ldquoRemark on hyperstability of the general linearequationrdquo Aequationes Mathematicae 2013

[21] Pl Kannappan Functional Equations and Inequalities withApplications Springer Monographs in Mathematics SpringerNew York NY USA 2009

[22] M Frechet ldquoUne definition fonctionnelles des polvnomesrdquoNouvelles Annales de Mathematique 4th serie vol 9 pp 145ndash162 1909

[23] M Frechet ldquoLes polvnomes abstraitsrdquo Journal de Mathema-tiques Pures et Appliquees 9th serie vol 8 pp 71ndash92 1929

[24] M Kuczma An Introduction to the Theory of Functional Equa-tions and Inequalities Cauchyrsquos Equation and Jensenrsquos InequalityBirkhauser Basel Switzerland 2nd edition 2009

[25] H Stetkaer Functional Equations on Groups World ScientificPublishing Singapore 2013

[26] Y H Lee ldquoOn the Hvers-Ulam-Rassias stability of the gen-eralized polynomial function of degree 2rdquo Journal of theChungcheong Mathematical Society vol 22 no 2 pp 201ndash2092009

[27] M Albert and J A Baker ldquoFunctions with bounded m-thdifferencesrdquoAnnales Polonici Mathematici vol 43 no 1 pp 93ndash103 1983

[28] D H Hyers ldquoTransformations with boundedm-th differencesrdquoPacific Journal of Mathematics vol 11 pp 591ndash602 1961

[29] Y-H Lee ldquoOn the stability of the monomial functional equa-tionrdquo Bulletin of the Korean Mathematical Society vol 45 no 2pp 397ndash403 2008

[30] M Chudziak On solutions and stability of functional equationsconnected to the Popoviciu inequality [PhD thesis] PedagogicalUniversity of Cracow Cracow Poland 2012 (Polish)

[31] W Fechner ldquoOn the Hyers-Ulam stability of functional equa-tions connected with additive and quadratic mappingsrdquo Journalof Mathematical Analysis and Applications vol 322 no 2 pp774ndash786 2006

[32] S-M Jung ldquoOn the Hyers-Ulam stability of the functionalequations that have the quadratic propertyrdquo Journal of Mathe-matical Analysis and Applications vol 222 no 1 pp 126ndash1371998

[33] J Sikorska ldquoOn a direct method for proving the Hyers-Ulam stability of functional equationsrdquo Journal of MathematicalAnalysis and Applications vol 372 no 1 pp 99ndash109 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of