MATHEMATIC.q.
_
REVUD D'ANAI]YSE NUMÉRIQUE ET DE THÉORIE DD I]'APPROXIMATIONL'ANALYSE
NUMÉRIQUEET
I,A.T'IIÉORIE ÐE
L'APPROXIMATION Tome 12, No2,
1983, PP. 113-123GENERAI, SOI,UTION OF THE ARCTANGËNT
FUNCTIONAL EQUATION
by
BORISL,{V CRSTICI, IOAN MUNTEAN a¡d NECUI,AE VORNICESCU
(Timiçoara)
(Cluj-NaPoca)Abstraet
We
shall
showthat
evet'y soiution/: R* R of the functional
equa-tion
f(*) + fU) : Í
frY*r) for all
n,! eR with'ry <
1,which is bounded or measurable on an
interval
of positive length, possessesa finite derivative at the point x:0 and has the
f.ormf(x):"f'(O)
..
arctanx, x e R.
Nonmeasurablesolutions of this
equatioriare
exhi-bited.
The
general solutionsfor functional
equations consid.ered.by I.
Sta-mate
and.N.
Ghircoiagiuand by H.
Kiesewetterare
derived.from
our results und.er weaker hypotheseson
unknown functions.1.
IntroductionThere are many methods
to
define whatis
commonly callecl arctangentfunction. The
methodsof
Euclid.ean geometryfirst
introducethe
direct trigonometric functions sine, cosineand tangent by
quotientsof
lengthsof
some adequate straightline
segments, and, thenthe
arctangent functionf, by
inversionof the restriction to the - interval l-:, J 2' + Tlof
2Lthetan-
gent function. The
methodof definite integrals
generatesthe
values of 2 - L'a¡alyse nuroérlque et la théo¡le de I'approxlmation Tome 12, nr,2, 1983I74 B CRSTICI, I MUNTEAN and N VollNIctìScU
the
arctangentf,
bYa,
ARCTÀNGENT FUNCTION.AL EQUATION 115
3
further, when the condition
%y<7 in (1.2) is
replacedH. Kiesewetter t8l
showedthat the null function f(x) :
the
single continuoussolution of the functional
equationby
xy # _7,0, ø eR
isr
(%t -9--
1e1ø eR
J 0
4+Q
l+t'
1+
(see
[6], p.389, and [4]).'l'he
methodof
recuirent-sequences- proposedü-A -rr"'i*ià'[7]
arrd'subseqrrct-ttlv dcvcloped
in [9'],.pp' 33-36'
an¿IiO], pp. 20-'27,
introdtrcesthc
arctangcnt functronlt
DYÍ"(*) :lim
2"'2,,(x)lot x eR"
(1.4) Í(*) +l(y):f ii:";) for all x, ! = R with *! +.\.'
2, would
allowthe
reductionof
the onal ecluationfor the
functionng the difficulties
concerningt
equation, : we - shall ; directlY/: Il--* R of the functional
èqqa.l. / is ,;tt;*tt
regularitv
conditions:2.f is R,
'3./is :
I4.
"f
ís
5.
/ is stlr,
possesses
u -the'foim
(1.5) Í(*):f'(0)'arctan x, x
e.R.As an application of these results,
the
general solutiorrs for, functional equations córisidereilby I. Stamate and-N. Ghircoiaçiu ll2l'
a_n_d tryH'.
Kiesewetter[B]
are -d.erived under general hypothesesof
boundednessor measurability of
unknown functions'2. Dilfercntiable
solutionsand
continuous solutionsLet
usfirst
remarkthat for a function/:R-"R sptisfying
(1'2) we have(2,1) , Í.(0):0
:'(Pal
x: Y:0 into (l'2)),
and(2.2) Í(-*) : -Í(x) for x eR
(ptrty: -xinto
(1.2) anduse(2.1));from
(1.2)withy -
% we also obtain(2.3) 2f(x):f(."n i tot J\ ' " lt-xr1 lxl<7. r: ,. '
,;under the differentiability
condition of/
on R,the
following theoremis
essentiallydue to G. H. Hard.y [6], p. 360:
:THEoRDM
2,1. IÍ /: R* R is
þointx :
he functioshow
th e
R:For
Rhaving btain
where
zo@): x
anð' 2,1t:
',(x) f.otn2O i
þ,@))other
methodsto
definethe
arctangentfunction are
reviewedin
[10],pp. 2-5.
The
numbern,
definedin thc
Euciidean geonr.etry.asthe
quotient,r, oiirr"
-iàngthof any
circleof
positive radiusby its
cliameter, appearsinio
later metîodsth;;;gh ih" i*tio1""
Tz:
4f "(I)
and' Ts: 2'¡mf
t(x) ' respectiveiy.' It
iutä
be expected.,of
corlfse,that the
enumerated methods afe no-thire
elsebut diff;;;;-;t*eedingå to
introduce orle and'the
same func- tion,-that is the following
equlities(1.1)
.f'Lx): Írlx) : f"i!x) for all ø e
R'are
true,
which then imlly
ær:
7!2 :.æ3. Bach oJthe
functionsf ,' f ,
and/, rr1irfi". the
same conditionalfunctional
equation(L2) Í(*)
-rf(y) : f (:+r) for all
x,! e R with xv < f
lscc 16l.
pp' 390-391,
forf,,
anð'[7] for/'),
the strorrg conc]'itiorrof
diffc-ientia¡iiiiv on R and the
equalitics(1.3) /i(o) : f|p):,fi(o) :
1'We
intend.to
derive(1.1
and some weaker regularityconditions on the
funcDeno'te
by arctan l,' l.'3\d /'
andby
æ thc"orrJupäáingä,t-t"r' rn i" it'zl
haé been considcredby w, Alt
12) evenin a
more generalsetting, but"without taking
into accountof its condîtioo"l
"turncîer.
Whenthã condition xy <
1fu (i'2)
is
replacedby xy>ï;th"-r"""tiãit f@f J
arctanx' x e:R'
satisfics ano-ther functional
equation :Í(*) +Í(y):f(i"r) f æ'sign x for
alrx, i e Il with xv >
7"Using (2.2), (2.1) and
the continuity
of.f al 0 we
gettim
f
(k): Í(x) -
timf(h) : f(*).
h+x è+0
To
provethe differentiability
off at x :
0, choose a positive number ø suchthat
ø( l,
integrate (1.2)with
respectto y in [0, 1]
and obtain{2.4) Í(*):ø+lr(}fi)ø,
Ixet-ø,øt
where å
is a
constant.ett"l tn"
changeof
variablez:!-lY, ye[0, l],
we
have L-try
,=f,, f,lc[- o,f,] ""a | { xz , l¡, o
md
(2.4) becomesARCTANGENT FUNCTIONAL EQUATION 777
3. Boundetl
solutionsand
monotone solutionsTHEoREM 3.1.
If /: R--*R is a
bound,ed'function on
am interaal' of þositiue Lengtk and. satisfies tkefunctional
equation (1.2), tkenf is
d'ifferen-tiabl,e
at x :0
and, (1.5) hold,s.Proof.
I.et xo eF,
a¡ 0 and
supposethat
thereis an Z > 0
suchthat lf(fl < ¿ for
all! =
f-xo- a, %o*
ø1. Choose ó¡
0with blxol 1 |
and.
å(1I
x'r)<
ø(1- blxoù. If x e l-b, bl, then xxs
<,blxol < |
and. for
J-
xiro
l-t*o
we have
l!-xol:l*l
and,
so
(1.2) yields(3.1) tf@)t:l-tø,1+r(itu_,."11* trt,.l
I+ tÍu)t < M, xe l-b,bl,
where
U:lf(xo)l+L.
Now,
we-assertthat/is
continuousat x':0.
Supposing the contrary,there exists
an
E¡ 0
suchthat for
each integern )- I
we canindicate
anumber ,,=l- +, +1 with lÍ@,)l > ", rt follows from (3.1) that
there is a subsequence (denoted
in
the same manner) ofthe
seilueflce (x,),>r and. thereis a
number c suchthat
Icl > e
and,(3.2) fi*,: o and tlaf@,) :
c.By induction we associate
with
eachinteger k ) 0 a
sequênce(x!),>,, having the
properties(3.3) li* *I:0 and ¡rnf(xï) :2b'c'
Namely,
lork:0
weput %o: I
and'xf;: fi,for all n 21,
and we seethat the equalities (3.3) revert to
(3.2).Admitting that the
sequence(x!),r,r with the
properties (3.3)is
constructed-,we first
determine anindex np¡1 so large
as l*l,l <
1for all n 2
np+t andthen
weput
*I+':-2!=
'fornÞtp+t.
I -(xÍ\
Then
lim *!,*t :0
antlÍ(*1*\ :
2f (x!,) (see (2.3)), hencelimf(x!+r) :2
.2'
c:2þ+r
. c.116 B. CRSTtrCI, ¡1. MUNTEAN and N. VoRNICESCU 5
l*z
4
-*l* +
h)< l,
whenceby
(2.2), (1.2),(2,1) and the differentiability
of.
f at 0 we
derivelim :lim!.¡( u
-.1
: t .tim/0-10):l'Q)
h+o h+o It -\l+*(x!h)) t+*, Þ+o h llr2
Thus,
/
is differentiableat
eachpoint r e R
and.f'(x) :
{':',,,
hencethereis ac eRsuch that f(x):Í'(0).arctan xlc, ø eR;The
equalities
(2.1) and
arctan0:0
yielð.f(ù):,f'(0).arctan x, x eF..
fn the
caseof
continuous solutions we have:TTTEoREM 2.2.
IÍf
:R-* Ris
ø continuousfunction attheþoint x:0
ønd
søtisfigtjlrt- functional
equation (1.2), thenf is
diJferentløbl,eøt
tkisþoint
ønd, (1.5) hold.s.Proof
.
We sha1lfirst
transferthe continuity
of.f
fuomx :
Oto
anypoint x eR. For all h e R sufficiently
closeto ø
wehave -xh 1l
and,
by
(1.2) we obtainÍþc) -f Í(-
h): t
f(* - hI.
lt+
rn)r*rã a
l-xxo ó
t+nâ l-blxol <a.
lþ):bl(tlx,) __JþL
(l I xz)z ¿,1-*
a
I
,xel-a,ø1.
The
theoremon differentiation of
integralswith
re6pectto a
parameter ensuresthe
differentiation off at x :
0 and. Theorem2.1 applies. \
118 B. CRSTICI, L MUNTEAN and N. VORNICESCU
l*tr,
ø ll2
6 I ARCTANGENT FUNCTIONAL EQUATION 119
Now
choosean integer h
>-0
suchthat
2h. e
)-M + l. It
followsfrom
(3.3)that
there existsaî
tx>
nr so largethat
|*:l < b
anð, lÍ@X) I>
> 2r.ltl - | >
2re- I
whence,in virtue of
(3.1),we arrive at the
con-tradiction ,M )- lf(x:)
l>
2"e- | >
1l1. Consequently,/ must be
conti-nuous
át x:0 andlTh"ot"- 2.2
applies.., #,
#,if i
íi,"v,
o
{;ir'.::,: ;# *T ; :' i;f:
isfies
the functionø|, equøtion (7.2) length,0 e I,
suclt.tkøt xf(x) >
0for_al,l-2.=I (or xf(x) 40for
ø1,1,x eI), thenf is
d,ifferentiableat x:0
ønd (1.5) lrclds.
.
Proof. ^U:_t-rS-(2.2) we may admit that I
hasthe form I :
10,al,
where ø>
0.we
shall provethat if xf(x) )
0for
allx e r, then/is i"ót""- singcn L Let x, z e /with x 12. Theny:ftr>0, y 42, xy <l
an! f(ù ).0,
henceÍ(,) : r (i=r) :.r(x) + Í(v)
>Í(*).
" Simila,ily,if^xf(x) (.0 for
a77x e
1,then/is
decreasing on _LIn
both cases Corollary3.2
applies.3.\.If /:R-" R is ø
d.ifferentiøbte(or
continuous) func-,in
R^and_søiisfj.es the functionøI equøtion (1.2), tken
f
isx :0
and. (1.5) holds.4.
Measurable solutions and nonmeasurablo solutions' I,etþ, {, x eRwithþ*q, Iþl <qandqlxl<
1. Thehomographicruncrion k: h,: lþ, ql-*130, #;1,
d.efinedby
(4.r)
h(v)::Yt!elþ,q1,
r-fiy
is
strictly
increasing-and continuous togetherwith
íts inverse. Consequently,/ø maps^open (closed) sets_into o_pe4 (clõsed, ¡espectively) sets and. thä ima[é
by h of an
openinterval lø, blCLþ, gl,
ø<
ó,is ihe
openinterval
e(4.2) h(la, b[) : ]ø l[,where 6-!J-" and,d.- rtb.r
Moreover, '-:' l-bx
(4.3) (b
-
o)'T,EMMA
4.1.
Tkeimøge by h' of any nxe(Ìs1,t/øbl,e setÐClþ' ql is
measu'vøble ønd. satisfies
(4.4)
mesh(E) , ,+--'
(1-l
qlxl)" rnes -Ð'Proof.
we
usethe following well-known
characLetizationof
rneasu-rable sets
in R (cf. [11], pp. 73-76):
a setM ínP. is
measurableif
andonlyiffor each; > 0therôèxist
all openset Gf
[È ancla closedset]i f fl
such
that F CluI f G and nes
(G\F) <
".I,et
e¡ 0.
SinceE is
measurable, there exist an open set G6 lþ' ql
and a
closedset Ff IlsuchthatFCECGand (4.5) mes(G\P) < \-qVl)'-'e.
l*x'The
open detG
..F
can be represented as unionof a
countable farnily ofmutually
disjoirrt open intervals : G\ F :
LJ {1' :h > l}
(cf.
[11],p.
54),:and so
the
measureof
G\.. F is
given b1'(4.6)
mes (G\ F) : 'f) -". r,.
The
irnages G,: h(G)
anó,¡;, - h(F).Ïton"n
and closed, respectively, andverify Frçh(E) f
G..Irrorn
(4.3), (4.6) and (4.5) we derive(4.7)
mes (G1'.. F,) <
111e,j /z(G ... -{i)<
mes(U {k(Ir):Ä > 1})
<*f *", h(Ir) <,;itr-".)),,'"0 1*: (,'+#4y 'mcs(G .F) <
u,i.e. the
measurabilityof h(E) is
proved.Ihe inequality $.\
canbe
dedrlcedfroin a known result (cf'
[11],pp.228-229) .
Hcwever',for the
sakeof
completenesswe
present here ã-direct proofof
(4.4). The open set G, can also bewritten
asunion of
a countablõfamil¡' of rnutually disjoint
openintervals:
G,: U {"f,:
h,> l},
where
f r: fc,
d*1. Denoteby ar: ø
and.br:
bthe
numbers obtainedfrom
tñËlasi
irvo'èqualitiesií
(4.2)lor
c: cr
and d,: d,
Cleally,,f, :
: k(Lr), where Ln: lør, b*[.
Moreover,(4.8) GCU{Lr:h>t}.
Ind.eed, supposingthe contrary, there exists
an y e Gwitlny ø l) {Lr: lt
>-)
1). For the numb"rz: h(y) e
k(G): Grit
must exist an inte.ge¡ k'->
7ru"h thut , = Jr:
h(Lr), heîce thereis
any' e Lr wltl¡
z: h(y')'
Now,the injectivity of h'leads to the contradiction y : !' =
Lr.Using
(4.3)and
(4.8)we
obtain-9- I å
mes.Emes G,
:
Jt F9s./¡ >
(1 +qwry þi
mesL, >'
11¡
qln1y'(r-q
720
which together
with
G,:
h(E)U
(G'\
¿(E))Ch(E) U (G'\F')
anð' (4.7)yield
rrres
h(E) >
mes G1-
rnes (G,\FJ > ---:! i- -
eSince o
is an artritrary
positive number, (4.4)is
proved'TrrEoRÞM 4.2.
IÍ /:R-'R is a'
measurablefønction
otuøn
interual'of
þositiae lcngth and' søtisfies tloefunctional
equati,on(l'2),
tkenf is
d'iffe-ientiøble
at x :0 and
(1.5) hold's.Proof. Our
argurnentis
basedon the
Banach'smethod [3] for
theintegration of
Cauchyfunctional
equation.the
interval
lxo- a,
1(o* ø],
where'!ir'1,""î'"åTî"jo:'#"0il:'i:it1
is a continuo's runction s: lx,-., îT*þ*tljtår:,iJ; 118-11e)
there(4.9) mesl/.i,
whereH:{x =lxo- d, no*øl:Í(x)+
SØ)\.Let
Ebe a positive
number. Sinceg is uniformly
continuous,there is
a8>0with
(4.10) le@)-
S(?r)l< efor tt,a elx¡-ttr, %otø1, lu-al <
Ð.Pttt þ : xo- |,
ø: xo* !,
and,(4.11) 4 : min
^'-^'ìz("0{=-f- ¡
a7' z¡t, * -L.
+,1@o*o)l' --t
Z¡t+
ø')I'
ILet x eR with lxl<r¡. Clearly, lþl <q, þ+q
anð.qlxl<1.
Accor-ding to
(4.11)the function h: k*
definedin (a.l) fulfils
k(þ)>
no-
aanð, k:(q)
1
xoI
ø, hence(4.12) h(lþ,
qDC lro -
&, xo-f
ø1.By
(4.9), the measurable setE:lþ, qt\H
satisfies mesE> a'- i :1o' 44 In virtue of Lemma 4.1
we have(4.r3)
mesh(E) ' > -Ïelq-=+:+¡l-". (ltqlx\'' Ílsn)' 3 B2
4!
since qr¡
< ls -
1.The
inequalitiesin
(4.9)and
(4.13) showthat the
ser k'(E)\ I/
isnonvoid, hence
by $J2)
there is a numberz e
h(E)Ck(lþ,
ql)C lx, - a,
xoI al
g ARCTANGENT FUNCTIONÄL ESUATION 727
with
z 4 H,
anð.so there exists an! eE such ttat lÏy: h(y):
z'Therefore,
! ê H ""a !_ - 4 H.
Tlnelast relations can be written in the
form(4.r4) f(t): eu) and rl:=,): t(i:, -,,ì :
e(,)By
(a.11) we obtainlz - yl :
Ixl lLt: < n'!f-t: <2nl + q') (
8'l-xY L-qn
hence
from
(1.2), (4.14) and (4.10)we
derivetÍ@)-/(0) t: tÍ@)t:l¡í-", -Í(v) l: ttt') -su)t<u
and.
the continuity
o1.f at x : 0 is
proved'RÞMARr( 4.3.- G.
"Hamel t5]
clonstructed.a
d.iscontinuous solutiong: R-" R of the
Cauchyfunctional
eqnation(4.15)
g(u+
u):
g(u)+
g(ü)for all
ot',a eH"
We shall prove
that the function/:R-'R
definedþv Í@): g(arctanx)
is an
unbäunded. and. nonmeasutuËl" solutionof
(1.2)on any interval
ofpositivc
length.----in"fun"ction/satisfies
(1.2) since,if x, y e
Rand.xy <7,from
(4.15)we
derive(4,16) Í(*) + Í(y) :
g(atctanx) |
g(arctaî!) :
:
g(arctanx I
arctan!) :
B(arctani+r\ : f (U, å)
Now,
/
is neither bounded noï rueasufable on any ínhetvalof positive.]9ngth,i"ã"ää, -*pposing
the contrary,
(4.16) together_with .Theorems3.1
and¿.2
*"ittlitãptvihe continuity
off
airx:0. B'lrt g(y):f(tany), y e
=l-;,å[,.ogwouldbecontinuousaty:0andtheobtainedcon-
tradiction
achievesthe proof of our
assertion'5.
Some applieationsUnd.er
the differentiability
condition of/
onR the
following corollaryhas been proved
by I.
Stamate andN.
Ghircoiaçiu ^[12].:-- ðó"oir,,lnv 5.i.
Letf, g, h:I\-,R
be giuen functi,ons ult'iclt, satisfythe
cond,itional,functionøl
eqøat'ion(5.1) Í(*) + eU) : r(=*) for
øtt' %,! eR' uitk xv 1r'
B. CRSTICI, ,I. MUNTEAN and N. VORNICESCU 8
722 B. CRSTJCI, '¡. MUNTEAN and N. VORNICESCU 11 ARCTANGENT FUNCTTONAL EQUATION 723 10
,
then theerl
by
the tod.h(x):
d then
withy: 0,
we find(s.2)
h(z): f(0) + gþ)
and, h(z): Í(z) l- s(0) for alt z e
R,whence
(5.3)
e(z):f(z) +
s(0)- f(0) for alt z =
R.Define the functior.q.jl*R by ç(*)
=f(x) _f(0)
and.remark that,
according
to
(s.2) and (5.8), the eqiratiò'ils.ij tãr."í ìrí" iåì,,
ç(x) l- ç(y): r(=+rl for all
x,! eR with xy <
t.Now, from Theorems 3.1 and 4.2
it
forlowsthat
g is differentiableat x :
o13d.it is given bv qV): p'(O).arcta' *, * ='F., ì;-flr"
conclusion of Corollary5.I is
immediáte.' '
's on
R the
followingresult
has been bound.-ed.(or
meøsotrøbte) function sfies the cond.itional functionat equø-rems
3.1
and.4.2itfollows
rhatf,, u,hilit"f,"nit':i ;':àÏati",'l
"ii::.
ssed.
by
^(1.5)..
From
(1.4)and
(2.2)we get 2f(JD _
1): flt), zf(^lî ¡
*
1): Í(-r): -/(1)
and/(1) +f(J,_\:fdr*1) : _f,f:J,
[6] If arily, G. 11., A course of þure watkemøti.cs, Fourth Eclition, Cambriclge University [7] I{urw
l8l I(iese
Math.-Natur. Reihe ; 12t' 417'421, (1965).
[9] Maak,w., Di.fferential - und, Integtalrechnung,4.Ãtflage,vanclenhoeck uncl Ruprecht, Göttingen, 1969.
t10] Muntean i., Elementøry trør¿scendental functions (Romanian), Universitatea ,,Babeç-Bolyai", Cluj-NaPoca, 1982.
t11l Natansoa,'I. P., Theory of fiutctions of a real aariabl'¿, Seconcl Etlition (Russian),
Gos. Izdat. Tehn.-'I'eoret. Lit., Moscow, 1957.
t12l St am.ate, f. ancl Ghircoiagir, ñ., Fuøctional' equalions defining trigonotn-etri.c ftr,ncl'iotcs (Romanian), Bul. $tiin!, fnst. Politehn. Cluj, 10' 57-60, (1967)'
Rece¡ved 1s.VI.1983
Føaultatea d,¿ malemaliad.
Uniuersitøteø B abeç- B olyøi
Str. Kogãl,niceønu
I
3400Ctruj-Naþocø
whence ,f(1)
:0. Using
(1.5)with
ø: 0,
weobtain
0:/(1) : Í'(0)
.arctan l:f'(0).; ,
henceÍ(x) :0 for all z e
R.RÞFERENCES
tl I A c z éI, !., V nagleichungen ultd, ihre Anuaod.tmgen,, Birkhäuser
Verlag, iget.
L2l Alt, W., Üter ciner yeellen Veyänd,erlichcn, uelche cin røtio¡tal¿s Ad,d,itionstheorew besi,tzen, Deutsche Math., 5, l_12. llé40)
[3] Banach, s., t2 Søz t'équation fonc!,ionneue ¡ø fil:iø l'¡üí,'äo¿"-. Math., 1, [4] Eberlei eletnenlary trønscendental, functions, Amer. Math. Monthly
81 ). '
[5] Ilamel, ie Zahlen unil die unstetige l-ösungen.d,ey Funhüionalgl,eichung
f@