R E S E A R C H
Open Access
Stability of a functional equation connected
with Reynolds operator
Jaeyoung Chung
**Correspondence:
[email protected] Department of Mathematics, Kunsan National University, Kunsan, 573-701, Korea
Abstract
LetGbe a commutative semigroup,K=RorCandF:G→Kn. Generalizing the
stability of the functional equationF(x◦g(y)) –F(x)F(y) = 0 with bounded difference (Najdecki in J. Inequal. Appl. 2007:79816, 2007), we prove the stability of the above functional equation with unbounded differences. We also give a more precise description for bounded components ofF= (f1,f2,. . .,fn).
MSC: 39B82
Keywords: bounded solution; exponential function; involution; Reynolds operator; stability
1 Main results
Throughout this paper,G,◦is a commutative semigroup with an identitye,Rthe set of real numbers, Cthe set of complex numbers, K=R or C, ≥, and g :G→G
andφ:G→[,∞) are given functions. For (a,a, . . . ,an), (b,b, . . . ,bn)∈Kn, we define (a,a, . . . ,an)(b,b, . . . ,bn) = (ab,ab, . . . ,anbn). A functionσ:G→Gis said to bean
involutionifσ(x◦y) =σ(x)◦σ(y) andσ(σ(x)) =xfor allx,y∈G. A functionm:G→Kn is calledan exponential functionprovided thatm(x◦y) =m(x)m(y) for allx,y∈G.
Generalizing the result of Ger and Šemrl [], Najdecki [] proved the stability of the functional equation
Fx◦g(y)–F(x)F(y) = (.)
in the class of functionsF:G→Kn. The particular cases of (.) are the exponential equa-tionf(xy) =f(x)f(y) (see Aczél and Dhombres [] and Baker []) and the equation
fxf(y)=f(x)f(y) (.)
for allx,y∈K\ {}, wheref :K\ {} →K\ {}(see Brzde¸k [], Brzde¸k, Najdecki and Xu [] and Chudziak and Tabor [] for related equations). As mentioned in [, ], (.) arises in averaging theory applied to the turbulent fluid motion and is connected with the Reynolds operator (see Marias []), the averaging operator and the multiplicatively symmetric operator (see []). Moreover, the equation (.) is connected with a description of some associative operations,i.e., the binary operation◦: (K\ {})×(K\ {})→K\ {} defined byx◦f(y) =xf(y) is associative if and only iff satisfies (.) (see [] for more
details). We also refer the reader to Belluot, Brzde¸k and Ciepliński [] and Brzde¸k and Ciepliński [] for some recent developments on the issues of stability and superstability for functional equations.
The main result of Najdecki [] is the following.
Theorem . Let F:G→Kn,F= (f
,f, . . . ,fn)satisfy
Fx◦g(y)–F(x)F(y)≤ (.)
for all x,y∈G with any norm · inKn.Then there exist ideals I,J⊂Knsuch thatKn=
I⊕J,PF is bounded and QF satisfies(.),where P:Kn→I,Q:Kn→J are the natural
projections.
In this paper, generalizing the above result we consider the functional inequalities
Fx◦g(y)–F(x)F(y)≤φ(y), (.)
F
x◦g(y)–F(x)F(y)≤φ(x) (.)
for allx,y∈Gwith any norm · inKn(see [] for related results). Throughout this paper we denote
L={j:fjis bounded,j= , , . . . ,n},
K={j:fjis unbounded,j= , , . . . ,n},
whereF= (f,f, . . . ,fn).
Theorem . Let F:G→Kn,F= (f,f, . . . ,fn)satisfy(.)for all x,y∈G with any norm
· inKn.Assume that one of the following two conditions is fulfilled.
(i) gis an involution,
(ii) for eachj∈K,there exists a sequencexn,n= , , , . . .(possibly depending onj)such
that
|fj(xn)| +φ(xn)→ ∞
asn→ ∞. (.)
Then there exist ideals I,J⊂Knsuch thatKn=I⊕J,PF is bounded and QF satisfies (.),where P:Kn→I, Q:Kn→J are the natural projections.Moreover, Q(F◦g–)is
exponential provided g is bijective.
Remark The case (ii) of Theorem . includes Theorem ..
Theorem . Let F:G→Kn,F= (f
,f, . . . ,fn)satisfy(.)for all x,y∈G with any norm
· in Kn.Assume that g is an involution. Then there exist ideals I,J⊂Kn such that
Kn=I⊕J,PF is bounded,QF satisfies(.),where P:Kn→I,Q:Kn→J are the natural
If we replace · by the usual norm · uonKndefined by
(a,a, . . . ,an)u=
|a|+|a|+· · ·+|an|,
we can estimatePF(in Theorem . and Theorem .) as follows.
Theorem . The following two statements are valid. (a)If F:G→Kn,F= (f
,f, . . . ,fn)satisfies(.),then PF satisfies
PF(y)u≤
√ |L|
+ + φ(y) (.)
for all y∈G,where|L|denotes the number of the elements of L.In particular,if|L|= and G is a group,then PF satisfies either
+ – φ(y)≤PF(y)u≤
+ + φ(y) (.)
for all y∈B:={y∈G:φ(y) <},or
PF(y)u≤
– – φ(y) (.)
for all y∈B.
(b)If F:G→Kn,F= (f
,f, . . . ,fn)satisfies(.),then PF satisfies(.).In particular if G
is a group,g is surjective and|L|= ,then PF satisfies(.)or(.).
2 Proofs
Letg:G→Gandφ:G→[,∞) be given. We first consider the stability of the functional equation
fx◦g(y)–f(x)f(y) = (.)
in the class of functionsf :G→K,i.e., we investigate both bounded and unbounded functionsf:G→Ksatisfying the functional inequalities
fx◦g(y)–f(x)f(y)≤φ(y), (.)
fx◦g(y)–f(x)f(y)≤φ(x) (.)
for allx,y∈G.
Lemma . Assume that g=σis an involution and f :G→Kis an unbounded function
satisfying the inequality(.).Then f is exponential and satisfies(.).In particular,if G is-divisible,then f has the form
f(x) =m
x◦σ(x)
(.)
Proof Choose a sequencexn∈G,n= , , , . . . , such that|f(xn)| → ∞asn→ ∞. Putting particular, ifGis -divisible, then we can write
f(x) =f
Lemma . Let f :G→Kbe an unbounded function satisfying(.).Assume that there
for allx∈G,n= , , , . . . . Lettingn→ ∞in (.) we have
Lemma . Assume that g is bijective and f :G→Kis an unbounded function satisfying
the inequality(.).Then f ◦g–is an exponential function.
Proof Choose a sequencexn∈G,n= , , , . . . , such that|f(xn)| → ∞asn→ ∞. Putting
Proof of Theorem. Since every two norms inKnare equivalent, from (.) there exists
α> such that bounded andQFsatisfies (.). Ifgis bijective, then by Lemma .,fj◦g–are exponential function for allj∈K, which impliesQ(F◦g–) is an exponential function. This completes
the proof.
Lemma . Assume that g=σis an involution and f :G→Kis an unbounded function
has the form
f(x) =m
x◦σ(x)
(.)
for all x∈G,where m:G→Kis an exponential function.
Proof Choose a sequenceyn∈G,n= , , , . . . , such that|f(yn)| → ∞asn→ ∞. Putting
y=yn,n= , , , . . . , in (.), dividing the result by|f(yn)|and lettingn→ ∞we have
f(x) = lim
n→∞
f(x◦σ(yn))
f(yn)
. (.)
Puttingx=ein (.) and replacingybyσ(y) in the result we have
f(y) –f(e)f
σ(y)≤φ(e) (.)
for allx,y∈G. Multiplying both sides of (.) byf(y) and using (.), (.), and (.) we have
f(y)f(x) = lim
n→∞
f(y)f(x◦σ(yn))
f(yn)
= lim
n→∞
f(y◦σ(x◦σ(yn)))
f(yn)
= lim
n→∞
f(e)f(σ(y)◦x◦σ(yn))
f(yn)
=f(e)fσ(y)◦x (.)
for allx,y∈G. Puttingx=ein (.) we have
f(y) =fσ(y) (.)
for ally∈G. From (.) and (.) we have
f(y) –f(e)≤φ(e) (.)
for ally∈G. Sincef is unbounded, from (.) we havef(e) = . Thus,f satisfies (.). This
completes the proof.
Proof of Theorem. From (.), as in (.) there existsα> such that
fj
x◦g(y)–fj(x)fj(y)≤αφ(x) (.)
for allx,y∈G,j∈ {, , . . . ,n}. Applying Lemma . to (.) for eachj∈Kwe find thatfj satisfies (.) for allj∈K, which implies thatQFsatisfies (.). This completes the proof.
Lemma . Let f :G→Kbe a bounded function satisfying(.).Then f satisfies
f(y)≤
+ + φ(y) (.)
for all y∈G.In particular,G is a group and let B={y∈G:φ(y) <},then f satisfies either
+ – φ(y)≤f(y)≤
+ + φ(y) (.)
for all y∈B,or
f(y)≤
– – φ(y) (.)
for all y∈B.
Proof LetMf=supx∈G|f(x)|. Using the triangle inequality with (.) we have
f(x)f(y)≤f
x◦g(y)+φ(y)≤Mf +φ(y) (.)
for allx,y∈G. Taking the supremum of the left hand side of (.) with respect tox∈G
we getMf|f(y)| ≤Mf+φ(y) for ally∈G. Thus, we have
Mff(y)–
≤φ(y) (.)
for ally∈G. From (.) we have
f(y)f(y)– ≤φ(y) (.)
for ally∈G. Solving the inequality (.) we get (.). Now, we assume thatGis a group. Replacingxbyx◦g(y)–in (.) and using the triangle inequality we have
f(x)≤fx◦g(y)–f(y)+φ(y)≤Mff(y)+φ(y) (.)
for allx,y∈G. Taking the supremum of the left hand side of (.) with respect tox∈G
we getMf≤Mf|f(y)|+φ(y) for ally∈G. Thus, we have
Mf
–f(y)≤φ(y) (.)
for ally∈G. From (.) and (.) we have
f(y) –f(y)≤Mf –f(y)≤φ(y) (.)
for ally∈G. For each fixedy∈B, solving the inequality (.) we get
+ – φ(y)≤f(y)≤
or
On the other hand, from (.) we have
f(y) –f(y)≥
Proof Using the triangle inequality with (.) we have
f(x)f(y)≤fx◦g(y)+φ(x)≤Mf+φ(x) (.) and using the triangle inequality we have
Proof of Theorem. From Lemma . and Lemma ., for eachj∈Lwe have
fj(y)≤
+ + φ(y) (.)
for ally∈G. Thus, from (.) we have
PF(y)u= j∈L
fj(y)
≤ √
|L|
+ + φ(y)
for ally∈G, which gives (.). Now, if|L|= , sayL={j}we have
PF(y)
u=fj(y)
for all y∈G. Thus, the inequalities (.) and (.) follow immediately from (.) and
(.). This completes the proof.
Competing interests
The author declares that he has no competing interests.
Author’s contributions
The author is the only person who is responsible to this work.
Acknowledgements
The author is very thankful to the referees for valuable suggestions that improved the presentation of the paper. This work was supported by Basic Science Research Program through the National Foundation of Korea (NRF) funded by the Korea Government (no. 2012R1A1A008507).
Received: 6 March 2014 Accepted: 23 May 2014 Published:30 May 2014
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