Theory of Approximate Functional Equations. in Banach algebras, inner product spaces and amenable groups

139 Pages·2016·1.38 MB·English

by  Madjid Eshaghi Gordji

Theory of Approximate Functional
Equations
Theory of Approximate
Functional Equations
in Banach algebras, inner product spaces
and amenable groups
Madjid Eshaghi Gordji
Institute for Cognitive Science Studies,
Shahid Beheshti University, Tehran, Iran
Sadegh Abbaszadeh
Intelligent systems and perception recognition
laboratory, CS group of Mathematics
department, Shahid Beheshti University,
Tehran, Iran
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Introduction 1
In 1940, an interesting talk presented by Stanislaw M. Ulam triggered the study of
stability problems for various functional equations. In his talk, Ulam discussed the
notion of the stability of mathematical theorems considered from a rather general
point of view: When is it true that by changing a little the hypothesesof a theorem,
onecanstillassertthatthethesisofthetheoremremainstrueorapproximatelytrue?
Particularly,foreverygeneralfunctionalequationonecanaskthefollowingquestion:
whenisittruethatthesolutionofanequationdifferingslightlyfromagivenonemust
ofnecessitybeclosetothesolutionofthegivenequation?Indeed,Ulamproposedthe
followingproblem:
GivenagroupG ,ametricgroupG withthemetricd(·,·)andapositivenumberε,
1 2
doesthereexistδ >0suchthat,ifamappingf :G →G satisfies
1 2
d(f(xy),f(x)f(y))≤δ
forallx,y∈G ,thenahomomorphismh:G →G existswith
1 1 2
d(f(x),h(x))≤ε
forallx ∈ G ?Iftheproblemacceptsasolution,theequationissaidtobestable(see
1
[90]).
In tryingto solve problemsof this kind,most authorshave consideredhomomor-
phisms between different groups or vector spaces, rather than automorphisms, for
perturbations.
Inthefollowingyear,1941,DonaldH.Hyers[6]wasabletogiveapartialsolution
to Ulam’s question that was the first significant breakthroughand step toward more
solutions in this area. He considered the case of approximately additive mappings
betweenBanachspacesandprovedthefollowingresults.
Theorem1.0.1. LetE andE beBanachspacesandletf :E −→E satisfiesthe
1 2 1 2
followingcondition:thereisaconstant(cid:4) ≥0suchthat
(cid:6)f(x+y)−f(x)−f(y)(cid:6)≤(cid:4)
f(2nx)
forallx,y∈E .Thenthelimith(x)= lim existsforallx∈E andh:E →
1 n→∞ 2n 1 1
E isauniquelineartransformationsuchthat
2
(cid:6)f(x)−h(x)(cid:6)≤(cid:4).
TheoryofApproximateFunctionalEquations.http://dx.doi.org/10.1016/B978-0-12-803920-5.00001-8
©2016ElsevierLtd.Allrightsreserved.
2 TheoryofApproximateFunctionalEquations
Theorem 1.0.2. If under the hypotheses of Theorem 1.0.1 f(tx) is continuous in
t∈Rforeachx∈E ,thenthemappinghisR-linear.
1
The term Hyers-Ulam stability originates from this historical background. The
method that was provided by Hyers is called the direct method, and produces the
additive mapping h. The direct method is the most important and powerful tool for
studyingthestabilityofvariousfunctionalequations.
The theorem of Hyers was generalized by Aoki [1] for additive mappings by
consideringanunboundedCauchydifference.
Theorem1.0.3(Aoki[1]). Iff(x) isan approximatelylineartransformationfrom
(cid:8)
EintoE,ie,
(cid:6)f(x+y)−f(x)−f(y)(cid:6)≤K ((cid:6)x(cid:6)p+(cid:6)y(cid:6)p), forallx,y∈E ,
0 1
whereK ≥ 0and0 ≤ p < 1,thenthereisauniquelineartransformationQ(x)near
0
f(x),ie,thereexistsK ≥0suchthat
(cid:6)f(x)−Q(x)(cid:6)≤K(cid:6)x(cid:6)p.
In1978,ThemistoclesM.Rassias[8]succeededinextendingtheresultofHyers’s
theorembyweakeningtheconditionfortheCauchydifference.
Theorem 1.0.4. Let E and E be two Banach spaces and let f : E −→ E be
1 2 1 2
a mappingsuch thatf(tx) is continuousin t ∈ R for eachfixedx. Assume thatthere
existsε >0and0≤p<1suchthat
(cid:6)f(x+y)−f(x)−f(y)(cid:6)≤ε((cid:6)x(cid:6)p+(cid:6)y(cid:6)p), forallx,y∈E .
1
ThenthereexistsauniquelinearmappingT :E →E suchthat
1 2
2ε
(cid:6)f(x)−T(x)(cid:6)≤ (cid:6)x(cid:6)p, forallx∈E .
2−2p 1
The stability phenomenon that was presented by Th.M. Rassias is called the
generalizedHyers-Ulamstability. This terminologymay also be applied to the cases
of other functional equations. Since then, a large number of papers have been
publishedin connectionwithvariousgeneralizationsofUlam’sproblemandHyers’s
theorem.
There are cases in which each approximate mapping is actually a true mapping.
Insuchcases,wesaythatthefunctionalequationishyperstable.Indeed,afunctional
equation is hyperstable if every solution satisfying the equation approximately is an
exact solution of it. For the history and various aspects of this theory, we refer the
readerto[2–5].
The concept of Hyers-Ulam stability is quite significant in realistic problems in
numericalanalysis, biology,and economics.This conceptactually means that if one
is studying a Hyers-Ulam stable system, then one does not have to reach the exact
solution (which usually is quite difficult or time consuming). Hyers-Ulam stability
Introduction 3
guaranteesthatthereisacloseexactsolution.Thisisquiteusefulinmanyapplications,
eg,numericalanalysis,optimization,biology,andeconomics,etc., where findingthe
exactsolutionisquitedifficult.Italsohelps,ifthestochasticeffectsaresmall,tousea
deterministicmodeltoapproximateastochasticone.
Thenotionofstabilityarosenaturallyinproblemsofmechanics.Thereitinvolves,
mathematicallyspeaking,thecontinuityofthesolutionofaprobleminitsdependence
on initials parameters. This continuity may be defined in various ways. Often it is
sufficientto provetheboundednessofthesolutionsforarbitrarilylongtimes, eg,the
boundednessofthedistancebetweenthepointrepresentingthesystematanytimefor
theinitialpoint,etc.Needlesstosay,problemsofstabilityoccurinotherbranchesof
physicsand,inaway,alsoeveninpuremathematics.
Unfortunately,therearenobookscompletelydealingwiththestabilityproblemof
functionalequationsinBanachalgebras,innerproductspaces,andamenablegroups.
Moreover,inmoststabilitytheoremsforfunctionalequations,thecompletenessofthe
targetspaceoftheunknownfunctionscontainedintheequationisassumed.Recently,
thequestionofwhetherthestabilityofafunctionalequationimpliesthiscompleteness
has been investigatedby several authors.In this book, we are goingto deal with the
above-mentioneddevelopmentsinthetheoryofapproximatefunctionalequations.
Thisbookintroducesthelatestandnewresultsonthefollowingtopics:
(1) approximateCauchyfunctionalequationsandcompleteness;
(2) stabilityofbinary,ternary,andfoursomemixturesoffunctionalequations;
(3) approximationsofhomomorphismsandderivationsindifferentBanachalgebras;
(4) stabilityoffunctionalequationsininnerproductspaces;and
(5) amenabilityofgroups(semigroups)andthestabilityoffunctionalequations.
Approximate Cauchy functional 2
equations and completeness
2.1 Theorem of Hyers
Thefirstresultonthestabilityoffunctionalequationswasgivenin1941byHyers[6].
HewasabletogiveapartialsolutiontoUlam’squestionthatwasthefirstsignificant
breakthroughandsteptowardmoresolutionsinthisarea.
ThesolutionsoftheCauchy(additive)functionalequationf(x+y) = f(x)+f(y)
have been investigated for many spaces. Any solution f(x) of this equation will be
calledalineartransformation.Tomakethestatementoftheproblemprecise,letEand
E(cid:2) beBanachspacesandletδ beapositivenumber.Alineartransformationf(x)ofE
intoE(cid:2)willbecalledaδ-lineartransformationif
(cid:3)f(x+y)−f(x)−f(y)(cid:3)≤δ
forallx and y in E.Thenthe problemmaybe stated asfollows.Doesthereexistfor
eachε > 0a δ > 0suchthat,to eachδ-lineartransformationf(x)ofE intoE(cid:2) there
correspondsalineartransformationl(x)ofEintoE(cid:2)satisfyingtheinequality
(cid:3)f(x)−l(x)(cid:3)≤ε
for all x in E? Hyers answered this question in the affirmative, and showed that δ
maybetakenequaltoε.Thisisclearlyabestresult,sincethetransformationf(x) =
L(x)+cwhereL(x)islinearandc<εisevidentlyanε-lineartransformationforwhich
l(x)=L(x).
Theorem2.1.1(Hyers[6]). LetEandE(cid:2) beBanachspacesandletf : E → E(cid:2) be
f(2nx)
aδ-lineartransformation.Thenthelimitl(x) = lim existsforallx ∈ E and
n→∞ 2n
l:E →E(cid:2)istheonlylineartransformationsuchthat
(cid:3)f(x)−l(x)(cid:3)≤ε
forallx∈E
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©2016ElsevierLtd.Allrightsreserved.
6 TheoryofApproximateFunctionalEquations
Proof. For any x in E the inequality (cid:3)f(2x) − 2f(x)(cid:3) < δ is obvious from the
definition of δ-linear transformations. On replacing x by x/2 in this inequality and
dividingby2weseethat(cid:3)(1/2)f(x)−f(x/2)(cid:3)<δ/2forallxinE.Maketheinduction
assumption
(cid:3)2−nf(x)−f(2−nx)(cid:3)<δ(1−2−n). (2.1)
Onthebasisofthelasttwoinequalities,wefindthat
(cid:3)2−1f(2−nx)−f(2−n−1x)(cid:3)<δ/2
and
(cid:3)2−n−1f(x)−2−1f(2−nx)(cid:3)<δ(1/2−2−n−1).
Hence,
(cid:3)2−n−1f(x)−f(2−n−1x)(cid:3)<δ(1−2−n−1).
Therefore,sincetheinductionassumption[2.1]isknowntobetrueforn=1,itistrue
forallpositiveintegersnandallx∈E.
Putq (x)=f(2nx)/2n,wherenisapositiveintegerandxisinE.Then
n
f(2mx) f(2nx) f(2m−n·2nx)−2m−nf(2nx)
q (x)−q (x)= − = .
m n 2m 2n 2m
Therefore,ifm<nwecanapplyinequality[2.1]andwrite
(cid:3)q (x)−q (x)(cid:3)<δ(1−2m−n)/2m
m n
for all x ∈ E. Hence, {q (x)} is a Cauchy sequence for each x ∈ E, and since E is
n
complete,there exists a limit functionl(x) = limn→∞qn(x). Let x and y be anytwo
pointsofE.Sincef(x)isδ-linear,
(cid:3)f(2nx+2ny)−f(2nx)−f(2ny)(cid:3)≤δ
forallpositiveintegersn.Ondividingby2nandlettingnapproachinfinity,weseethat
l(x+y)=l(x)+l(y).
Ifwereplacexby2nxintheinequality[2.1],wefindthat
(cid:3)f(2nx)/2n−f(x)(cid:3)≤δ(1−2−n).
Hence,inthelimit,(cid:3)l(x)−f(x)(cid:3)≤δ.
ApproximateCauchyfunctionalequationsandcompleteness 7
SupposethattherewasanotherlineartransformationL(x)satisfyingtheinequality
(cid:3)L(x)−f(x)(cid:3) ≤ δ for all x ∈ E, and such that l(y) (cid:8)= L(y) for some point y of E.
Foranyintegern > 2δ/(cid:3)l(y)−L(y)(cid:3),itisobviousthat(cid:3)l(ny)−L(ny)(cid:3) > 2δ,which
contradictstheinequalities(cid:3)L(ny)−f(ny)(cid:3)≤δand(cid:3)l(ny)−f(ny)(cid:3)≤δ.Hence,l(x)
istheuniquelinearsolutionoftheinequality(cid:3)l(x)−f(x)(cid:3)≤δ.
Nextthecharacterofthelineartransformationl(x)whencontinuityrestrictionsare
placedonthetransformationf(x)isinvestigated.
Theorem 2.1.2 (Hyers [6]). If under the hypotheses of Theorem 2.1.1 we sup-
pose that f(x) is continuous at a single point y of E, then l(x) is continuous
everywhereinE.
Proof. Assume, contrary to the theorem, that the linear transformation l(x) is not
continuous.Thenthereexistsanintegerkandasequenceofpointsx ofEconverging
n
tozerosuchthat(cid:3)I(x )(cid:3) > l/k forallpositiveintegersn.Letmbeanintegergreater
n
than3kδ.Then
(cid:3)l(mx +y)−l(y)(cid:3)=(cid:3)l(mx )(cid:3)>3δ.
n n
Ontheotherhand,
(cid:3)l(mx +y)−l(y)(cid:3)≤(cid:3)l(mx +y)−f(mx +y)(cid:3)
n n n
+(cid:3)f(mx +y)−f(y)(cid:3)+(cid:3)f(y)−l(y)(cid:3)<3δ
n
forsufficientlylargen,sincelimn→∞f(mxn+y)=f(y).Thiscontradictionestablishes
thetheorem.
Corollary2.1.3(Hyers[6]). IfunderthehypothesesofTheorem2.1.1wesuppose
thatforeachx∈Ethefunctionf(tx)isacontinuousfunctionoftherealvariabletfor
t∈R,thenl(x)ishomogeneousofdegreeone.
Proof. Forfixedx,f(tx)isacontinuousδ-lineartransformationoftherealaxisinto
E(cid:2).ByTheorems2.1.1and2.1.2,l(tx)iscontinuousint andhence,beinglinear,l(x)
ishomogeneousofdegreeone.
In 1991,Baker [7] used the Banachfixed pointtheoremto provethe Hyers-Ulam
stabilityforsinglevariablefunctionalequations.
Proposition 2.1.4 (Baker [7]). Suppose S is a nonempty set, φ : S → S, E is a
Banachspace,λisascalar,|λ|<1,δ >0,andf :S→Esuchthat
(cid:3)λf(φ(x))−f(x)(cid:3)≤δ
forallx ∈ S.Thenthereexistsauniqueh : S → E suchthatλh(φ(x)) = h(x)forall
x∈Sand
(cid:3)f(x)−h(x)(cid:3)≤δ(1−|λ|)
forallx∈S.
Theorem2.1.5(Baker[7]). Suppose(Y,ρ)isacompletemetricspaceandT :Y →
Y isacontraction(forsomeλ∈[0,1),ρ(T(x),T(y))<λρ(x,y)forallx,y∈Y).Also
8 TheoryofApproximateFunctionalEquations
supposethatu ∈ Y,δ > 0andρ(u,T(u)) < δ.Thenthereexistsauniquep ∈Y such
thatp=T(p).Moreover,ρ(u,p)<δ/(1−λ).
Theorem2.1.6(Baker[7]). SupposeSisanonemptyset;(X,d)isacompletemetric
space;φ :S→S,F :S×X →X,0≤λ<1;and
d(F(t,u),F(t,v))≤λd(u,v)
forallt∈Sandallu,v∈S.Alsosupposethatg:S→X,δ >0and
d(g(t),F(t,g(φ(t))))≤δ (2.2)
forallt∈S.Thenthereisauniquefunctionf :S→Xsuchthat
f(t)=F(t,f(φ(t))) (2.3)
forallt∈Sand
d(f(t),g(t))≤δ/(1−λ) (2.4)
forallt∈S.
Proof. Let Y = {a : S → X|sup{d(a(t),g(t))|t ∈ S} < ∞}. For a,b ∈ Y
define ρ(a,b) = sup{d(a(t),b(t))|t ∈ S}. Then g ∈ Y, ρ is a metric on Y, and
convergence with respect to ρ means uniform convergence on S with respect to d.
Moreover,thecompletenessofXwithrespecttodimpliesthecompletenessofY with
respecttoρ.
Fora∈Y,defineT(a):S→Xby
(T(a))(t)=F(t,a(φ(t)))
forallt∈S.ThenT mapsY intoY.Ifa,b∈Y thenforallt∈S,
d((T(a))(t),(T(b))(t))=d(F(t,a(φ(t))),F(t,b(φ(t))))
≤λd(a(φ(t)),b(φ(t)))
≤λρ(a◦φ,b◦φ)
≤λρ(a,b).
Thus,
ρ(T(a),T(b))≤λρ(a,b)
for all a,b ∈ Y. But Eq. [2.2] means that ρ(g,T(g)) ≤ δ. Hence, according to
Theorem2.1.5,thereisauniquef inY suchthatf = T(f)andρ(g,f) ≤ δ/(1−λ).
Thatis,Eqs.[2.3]and[2.4]hold.