Stability of general multi-Euler-Lagrange quadratic functional equations in non-Archimedean fuzzy normed spaces

Xu Tian, Rassias, Rassias John

Description

In this paper we prove the generalized Hyers-Ulam stability of the system defining general Euler-Lagrange quadratic mappings in non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods. MSC: 39B82, 39B52, 46H25. In this paper we prove the generalized Hyers-Ulam stability of the system defining general Euler-Lagrange quadratic mappings in non-Archimedean fuzzy normed spaces over a field with valuation using the direct and the fixed point methods. MSC: 39B82, 39B52, 46H25.

Sujets

• Direct method
• Fixed point iteration

Informations

• Publié par: biomed
• Publié le: 01 janvier 2012
• Nombre de lectures: 26

Extrait

XuandRassiasAdvancesinDifferenceEquations2012,2012:119
http://www.advancesindifferenceequations.com/content/2012/1/119
RESEARCH OpenAccess
Stabilityofgeneralmulti-Euler-Lagrange
quadraticfunctionalequationsin
non-Archimedeanfuzzynormedspaces
1* 2TianZhouXu andJohnMichaelRassias
*Correspondence:
[e-mail] Abstract
1SchoolofMathematics,Beijing
InthispaperweprovethegeneralizedHyers-UlamstabilityofthesystemdefiningInstituteofTechnology,Beijing,
100081,P.R.China generalEuler-Lagrangequadraticmappingsinnon-Archimedeanfuzzynormed
Fulllistofauthorinformationis spacesoverafieldwithvaluationusingthedirectandthefixedpointmethods.
availableattheendofthearticle
MSC: 39B82;39B52;46H25
Keywords: stabilityofgeneralmulti-Euler-Lagrangequadraticfunctionalequation;
directmethod;fixedpointmethod;non-Archimedeanfuzzynormedspace
1 Introduction
Let K be a field. A valuation mapping on K is a function |·| :K →R such that for any
r,s ∈K the following conditions are satisfied: (i) |r|≥ and equality holds if and only if
r=;(ii) |rs|= |r|·|s|;(iii) |r+s|≤|r|+ |s|.
Afieldendowedwithavaluationmappingwillbecalledavaluedfield.Theusualabso-
lutevaluesofRandCareexamplesofvaluations.Atrivialexampleofanon-Archimedean
valuation is the function |·| taking everything except for  into  and ||=.Inthefol-
lowingwewillassumethat|·|isnon-trivial,i.e.,thereisanr ∈Ksuchthat |r | =,. 
If the condition (iii) in the definition of a valuation mapping is replaced with a strong
triangle inequality (ultrametric): |r+s|≤ max{|r|,|s|}, then the valuation|·| is said to be
non-Archimedean. In any non-Archimedean field we have ||= |–|=and |n|≤for
alln ∈N.
Throughout this paper, we assume thatK is a valued field, X and Y are vector spaces
 over K, a,b ∈K are fixed with λ:=a +b = ,(λ :=a= ,if a=b)and n is a posi-
tiveinteger.Moreover,N standsforthesetofallpositiveintegersandR (respectively,Q)
denotesthesetofallreals(respectively,rationals).
nAmappingf :X →Y iscalledageneralmulti-Euler-Lagrangequadraticmappingifit
satisfiesthegeneralEuler-Lagrangequadraticequationsineachoftheirnarguments:

f x ,...,x ,ax +bx ,x ,...,x +f x ,...,x ,bx –ax ,x ,...,x i– i i+ n  i– i i+ ni i

  = a +b f(x ,...,x )+f x ,...,x ,x ,x ,...,x (.) n  i– i+ ni
© 2012 Xu and Rassias; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction
inanymedium,providedtheoriginalworkisproperlycited.
XuandRassiasAdvancesinDifferenceEquations2012,2012:119 Page2of19
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foralli=,...,n andallx ,...,x ,x ,x ,x ,...,x ∈X.Letting x =x =in(.),weget i– i i+ n ii i
f(x ,...,x ,,x ,...,x )=.Puttingx =in(.),wehave i– i+ n i
f(x ,...,x ,ax ,x ,...,x )+f(x ,...,x ,bx ,x ,...,x )=λf(x ,...,x ). (.) i– i i+ n  i– i i+ n  n
Replacingx byax andx bybx in(.),respectively,weobtaini i ii
f(x ,...,x ,λx ,x ,...,x ) i– i i+ n

=λ f(x ,...,x ,ax ,x ,...,x )+f(x ,...,x ,bx ,x ,...,x ).(.) i– i i+ n  i– i i+ n
From(.)and(.),onegets
f(x ,...,x ,λx ,x ,...,x )=λ f(x ,...,x)(.) i– i i+ n  n
foralli=,...,nandallx ,...,x ∈X.Ifa=bin(.),thenwehave n

f x ,...,x ,a x +x ,x ,...,x +f x ,...,x ,a x –x ,x ,...,x i– i i+ n  i– i i+ ni i

 =a f(x ,...,x )+f x ,...,x ,x ,x ,...,x.(.) n  i– i+ ni
Lettingx =x in(.),weobtainii
f(x ,...,x ,λ x ,x ,...,x )=λ f(x ,...,x)(.) i–  i i+ n  n
foralli=,...,nandallx ,...,x ∈X. n
ThestudyofstabilityproblemsforfunctionalequationsisrelatedtoaquestionofUlam
[] concerning the stability of group homomorphisms and affirmatively answered for
BanachspacesbyHyers[].TheresultofHyerswasgeneralizedbyAoki[]forapprox-
imate additive mappings and by Rassias [] for approximate linear mappings by allow-
ing the Cauchy difference operator CDf(x,y)=f(x+y)–[f(x)+f(y)] to be controlled by
p p(x +y ).In,afurthergeneralizationwasobtainedbyGăvruţa[],whoreplaced
p p(x +y )byageneralcontrolfunctionϕ(x,y).Wereferthereadertosee,forinstance,
[,–,–,,,,,,,,–]formoreinformationondifferentaspects
ofstabilityoffunctionalequations.Ontheotherhand,forsomeoutcomesonthestability
ofmulti-quadraticandEuler-Lagrange-typequadraticmappingswereferthereaderto[,
,].
ThemainpurposeofthispaperistoprovethegeneralizedHyers-Ulamstabilityofmulti-
Euler-Lagrange quadratic functional equation (.) in complete non-Archimedean fuzzy
normedspacesoverafieldwithvaluationusingthedirectandthefixedpointmethods.
2Preliminaries
Werecallthenotionofnon-Archimedeanfuzzynormedspacesoverafieldwithvaluation
and some preliminary results (see for instance [, , , , ]). For more details the
readerisreferredto[,].
Definition. LetX bealinearspaceoverafieldK withanon-Archimedeanvaluation
|·|.Afunction · :X →[,∞) issaidtobeanon-Archimedeannormifitsatisfiesthe
followingconditions:XuandRassiasAdvancesinDifferenceEquations2012,2012:119 Page3of19
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(i) x=ifandonlyifx=;
(ii) rx= |r| x,r ∈K,x ∈X;
(iii) thestrongtriangleinequality
x+y ≤ max{ x,y } , x,y ∈X.
Then (X, · )iscalledanon-Archimedeannormedspace.Byacompletenon-Archime-
deannormedspace,wemeanoneinwhicheveryCauchysequenceisconvergent.
In , Hensel discovered the p-adic numbers as a number-theoretical analogue of
powerseriesincomplexanalysis.Letpbeaprimenumber.Foranynonzerorationalnum-
rbera,thereexistsauniqueintegerr suchthata=p m/n,wheremandnareintegersnot
–rdivisiblebyp.Then |a| :=p definesanon-ArchimedeannormonQ.Thecompletionofp
Qwithrespecttothemetric d(a,b)= |a–b| isdenotedbyQ whichiscalledthe p-adicp p
nnumberfield.Notethatifp>,then | | =foreachintegernbut || <.p 
Duringthelastthreedecades,p-adicnumbershavegainedtheinterestofphysicistsfor
theirresearch,inparticular,intoproblemsderivingfromquantumphysics,p-adicstrings,
andsuperstrings(seeforinstance[]).
A triangular norm (shorter t-norm, []) is a binary operation T:[,] × [,] →
[,] which satisfies the following conditions: (a) T is commutative and associative;
(b) T(a,)= a for all a ∈ [,]; (c) T(a,b) ≤ T(c,d)whenever a ≤ c and b ≤ d for all
a,b,c,d ∈ [,]. Basic examples of continuous t-norms are the Łukasiewicz t-norm T ,L
T (a,b)=max{a+b–,},theproduct t-norm T , T (a,b)=ab andthestrongesttrian-L P P
gularnormT ,T (a,b)=min{a,b}.At-normiscalledcontinuousifitiscontinuouswithM M
respecttotheproducttopologyontheset[,] ×[,].
At-normT canbeextended(byassociativity)inauniquewaytoanm-arrayoperation
m takingfor(x ,...,x ) ∈[,] ,thevalueT(x ,...,x )definedrecurrentlybyT x =and m  m ii=
m m–T x =T(T x ,x )form ∈N.Tcanalsobeextendedtoacountableoperation,takingi i mi= i=
∞ mfor any sequence {x } in [,], the value T x is defined as lim T x.Thelimiti i∈N i m→∞ ii= i=
m ∞exists since the sequence {T x } is non-increasing and bounded from below. T xi m∈N ii= i=m
∞isdefinedasT x .m+ii=
Definition. At-normT issaidtobeofHadžić-type(H-type,wedenotebyT ∈H)ifa
mfamilyoffunctions {T (t)}foral