R E S E A R C H
Open Access
A general theorem on the stability of a
class of functional equations including
monomial equations
Yang-Hi Lee
1and Soon-Mo Jung
2**Correspondence:
2Mathematics Section, College of
Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea
Full list of author information is available at the end of the article
Abstract
In this paper, we prove a general theorem for the stability of a large class of n-dimensional functional equations of the formDf(x1,x2,. . .,xn) = 0 by applying the
direct method.
MSC: Primary 39B82; secondary 39B52
Keywords: generalized Hyers-Ulam stability;n-dimensional functional equation; monomial functional equation; direct method
1 Introduction
LetV andW be real vector spaces and letnbe a positive integer. For a given mapping
f:V→W, we define a mappingDnf:V×V→W by
Dnf(x,y) := n
i=
(–)n–i
nCif(ix+y) –n!f(x)
for allx,y∈V, wherenCi=i!(nn–!i)!. A mappingf:V→Wis called a monomial mapping of degreeniff satisfies the monomial functional equationDnf(x,y) = for allx,y∈V. For example, a mappingf :V→W is called an additive mapping (or a quadratic mapping) if
f satisfies the functional equationDf(x,y) = (orDf(x,y) = ) for allx,y∈V. We notice
that if a mappingf :V →W is a monomial mapping of degreen, thenf(rx) =rnf(x) for allx∈Vand all rational numbersr(see [, ]).
In the study of Hyers-Ulam stability problems of monomial functional equations, we have followed out a routine and monotonous procedure for proving the stability of the monomial functional equations under various conditions. We can find in the books [–] a lot of references concerning the Hyers-Ulam stability of functional equations (see also [–]).
In this paper, we prove a general stability theorem that can be easily applied to the (generalized) Hyers-Ulam stability of a large class of n-dimensional functional equa-tions of the formDf(x,x, . . . ,xn) = , which includes the monomial functional equa-tions.
Throughout this paper, for a given mappingf :V→W, we define a mappingDf:Vn→
W by
Df(x,x, . . . ,xn) :=
m
i=
cif(aix+aix+· · ·+ainxn), ()
wheremis a positive integer andci,aijare real constants.
Indeed, this stability theorem can save us much trouble of proving the stability of rele-vant solutions repeatedly appearing in the stability problems for various functional equa-tions (see [–]).
It should be remarked that Bahyrycz and Olko [] applied the fixed point method to investigate the generalized Hyers-Ulam stability of the general linear equation
m
i=
Ai n
i=
aijxj
+A= .
Moreover, there are numerous recent results concerning the Hyers-Ulam stability of some particular cases of equation (). Some of them were described in the survey paper [].
In this paper, letV be a real vector space and letXandYbe a real normed space resp. a real Banach space if there is no specification.
2 Preliminaries
We introduce a lemma from the paper [], Corollary ..
Lemma . Let k= be a positive real constant,let b be a real constant,and let Y be a real normed space.Assume,moreover,thatφ:V\{} →[,∞)is a function satisfying
(x) := ∞
i=
φ(kix)
kbi <∞ ()
for all x∈V\{}and f :V→Y is an arbitrarily given mapping.If there exists a mapping F:V→Y satisfying
f(x) –F(x)≤(x) ()
for all x∈V\{}and
F(kx) =kbF(x) ()
for all x∈V,then F is a unique mapping satisfying()and().
We denote by Nthe set of all nonnegative integers,i.e., N= N∪ {}.
Lemma . Let k= be a positive real constant and let b be a real constant.Assume that a functionφ:V\{} →[,∞)satisfies condition()for all x∈V\{}.If a mapping f :V→Y satisfies f() = and
f(x) –f(kx)
kb
for all x∈V\{},then there exists a unique mapping F:V→Y satisfying()and()for all x∈V\{}resp.for all x∈V.
Proof It follows from () that
f(kkbmmx)–
f(km+lx)
kb(m+l)
≤
m+l–
i=m f(kkbiix)–
f(ki+x)
kb(i+)
= m+l–
i=m
kbi
fkix–
kbf
k·kix
≤ m+l–
i=m φ(kix)
kbi ()
for allx∈V\{}andm,l∈N. Thus, in view of (), it is easy to show that the sequence
{f(kmx)
kbm }is a Cauchy sequence for allx∈V\{}. SinceY is complete andf() = , the
se-quence{f(kkbmmx)}converges for allx∈V. Hence, we can define a mappingF:V→Yby
F(x) := lim
m→∞
f(kmx)
kbm
for allx∈V.
We easily see that equality () is true by the definition ofF. Moreover, we putm= and letl→ ∞in () to get inequality (). Obviously, in view of Lemma ., the mapping
F:V→Ysatisfying () and () is uniquely determined.
Lemma . Let k= ,b,V,Y be as in Lemma.,let f :V→Y and F:V→Y be arbitrary mappings,and let n be a fixed integer greater than.Assume that a functionϕ: (V\{})n→ [,∞)satisfies the condition
∞
i=
ϕ(kix,kix, . . . ,kixn)
kbi <∞ ()
for all x,x, . . . ,xn∈V\{}.If f satisfies the condition
Df(x,x, . . . ,xn)≤ϕ(x,x, . . . ,xn) ()
for all x,x, . . . ,xn∈V\{}and if F satisfies the equality
DF(x,x, . . . ,xn) =m→∞lim
Df(kmx,kmx, . . . ,kmxn)
kbm ()
for all x,x, . . . ,xn∈V\{},then F satisfies the equation
DF(x,x, . . . ,xn) = ()
Proof It follows from (), (), and () that
DF(x,x, . . . ,xn)=m→∞lim
Df(kmx,kmx, . . . ,kmxn)
kbm
≤ lim
m→∞ ϕ(kmx
,kmx, . . . ,kmxn)
kbm =
for allx,x, . . . ,xn∈V\{}. In other words,DF(x,x, . . . ,xn) = holds for allx,x, . . . ,xn∈
V\{}.
Lemma . Let a be a real constant satisfying a∈ {/ –, , },let N be an integer,and let n
be a fixed integer greater than.Assume thatμ:V\{} →[,∞)is a function satisfying the condition
∞
i=
μ(aix)
|a|Ni <∞ ()
for all x∈V\{}.If a mapping f :V→Y satisfies f() = and
f(ax) –aNf(x)≤μ(x) ()
for all x∈V\{},then there exists a unique mapping F:V→Y satisfying
F(ax) =aNF(x) ()
for all x∈V and
f(x) –F(x)≤ ∞
i=
μ(aix)
|a|Ni+N ()
for all x∈V\{}.
Proof It follows from () that
f(aaNmmx)–
f(am+lx)
aN(m+l)
≤
m+l–
i=m f(aaNiix)–
f(ai+x)
aN(i+)
= m+l–
i=m
f(a·aix) –aNf(aix) |a|N(i+)
≤ m+l–
i=m μ(aix)
|a|Ni+N ()
for allx∈V\{} and allm,l∈N. Thus, in view of (), it is easy to show that the
the sequence{f(aaNmmx)}converges for allx∈V. Hence, we can define a mappingF:V→Y
by
F(x) := lim
m→∞
f(amx)
aNm
for allx∈V, from which it follows that ().
Moreover, if we putm= and letl→ ∞in (), then we obtain inequality (). Finally, using Lemma ., we conclude that the mapping F satisfying () and () is uniquely determined, since we have
f(x) –F(x)≤∞ i=
μ(aix) |a|Ni+N =
∞
i=
|a|Nμ(aix) +μ(a·aix) |a|Ni+N =
∞
i=
φ(kix)
kNi
for allx∈V, where we setk:=aandφ(x) :=μ(x)
|a|N +
μ(ax)
|a|N.
Lemma . Let a,N, V,Y be defined as in Lemma.,let f :V→Y and F:V→Y be arbitrary mappings,and let n be a fixed integer greater than.Assume that a function
ϕ: (V\{})n→[,∞)satisfies the condition
∞
i=
ϕ(aix
,aix, . . . ,aixn)
|a|Ni <∞ ()
for all x,x, . . . ,xn∈V\{}.If f satisfies the inequality
Df(x,x, . . . ,xn)≤ϕ(x,x, . . . ,xn) ()
for all x,x, . . . ,xn∈V\{},and if F satisfies the condition
DF(x,x, . . . ,xn) =m→∞lim
Df(amx,amx, . . . ,amxn)
aNm ()
for all x,x, . . . ,xn∈V\{},then F satisfies the equation
DF(x,x, . . . ,xn) =
for all x,x, . . . ,xn∈V\{}.
Proof It easily follows from (), (), and () that
DF(x,x, . . . ,xn)=m→∞lim
Df(amx,amx, . . . ,amxn)
aNm
≤ lim
m→∞ ϕ(amx
,amx, . . . ,amxn)
|a|Nm =
for allx,x, . . . ,xn∈V\{}, which implies thatDF(x,x, . . . ,xn) = holds for allx,x,
3 Main results
In this section, we denote byV,X, andYa real vector space, a real normed space, and a real Banach space, respectively, if there are no specifications.
We combine Lemmas . and . in the following theorem to formulate the main theo-rem which is easily applicable. We here notice that conditions () and () are satisfied in usual cases provided we apply the direct method for proving the (generalized) Hyers-Ulam stability of various functional equations of the formDf(x,x, . . . ,xn) = .
Theorem . Let k= ,b,and n> be a positive real constant,a real constant,and an integer,respectively.Assume thatφ:V\{} →[,∞)is a function satisfying condition()
for all x∈V\{}.If a mapping f :V→Y satisfies f() = and
f(x) –f(kx)
kb ≤φ(x)
for all x∈V\{},then there exists a unique mapping F:V→Y satisfying
f(x) –F(x)≤(x)
for all x∈V\{}and
F(kx) =kbF(x)
for all x∈V.In particular,if there exists a functionϕ: (V\{})n→[,∞)satisfying
con-dition()for all x,x, . . . ,xn∈V\{}and if f satisfies
Df(x,x, . . . ,xn)≤ϕ(x,x, . . . ,xn)
and if F satisfies
DF(x,x, . . . ,xn) =m→∞lim
Df(kmx
,kmx, . . . ,kmxn)
kbm
for all x,x, . . . ,xn∈V\{},then F satisfies the equation
DF(x,x, . . . ,xn) =
for all x,x, . . . ,xn∈V\{}.
In the following theorem, letabe a real number not in{–, , }. We can easily prove this theorem by using Lemmas . and .. We also note that condition () is satisfied in usual cases when we use the direct method for proving the stability of functional equations of the formDf(x,x, . . . ,xn) = .
Theorem . Let a∈ {/ –, , },N,and n> be a real constant and integers,respectively.
Assume thatμ:V\{} →[,∞)is a function satisfying condition()for all x∈V\{}.If a mapping f :V→Y satisfies f() = and
for all x∈V\{},then there exists a unique mapping F:V→Y such that
F(ax) =aNF(x)
for all x∈V and
f(x) –F(x)≤ ∞
i=
μ(aix) |a|Ni+N
for all x∈V\{}.In particular,if there exists a functionϕ: (V\{})n→[,∞)satisfying condition()for all x,x, . . . ,xn∈V\{}and if f satisfies
Df(x,x, . . . ,xn)≤ϕ(x,x, . . . ,xn)
and if F satisfies
DF(x,x, . . . ,xn) =m→∞lim
Df(amx
,amx, . . . ,amxn)
aNm
for all x,x, . . . ,xn∈V\{},then F satisfies the equation
DF(x,x, . . . ,xn) =
for all x,x, . . . ,xn∈V\{}.
We now introduce a corollary to Lemma ..
Corollary . Assume that k,b,p,θ,δ are real constants such that k> ,k= ,p=b, θ> ,andδ> .If a mapping f :X→Y satisfies f() = and
f(kx) –kbf(x)≤δxp
for all x∈X\{},then there exists a unique mapping F:X→Y satisfying()for all x∈X and
f(x) –F(x)≤ δx p
|kb–kp| ()
for all x∈X\{}.
Proof If (k,p)∈(,∞)×(–∞,b) or (k,p)∈(, )×(b,∞), thenφ(x) := δ
kbxp satisfies
condition () with
(x) = δx p
kb–kp.
For the case of (k,p)∈(, )×(–∞,b) or (k,p)∈(,∞)×(b,∞), we setk := k to get (k,p)∈(,∞)×(–∞,b) or (k,p)∈(, )×(b,∞), and
fkx–k bf(x)=k bkbf
x k –f(x)
for allx∈X\{}. By applying the same argument as in the first case, we can see thatφ(x) :=
k pδxpsatisfies condition () with
(x) = ∞
i=
φ(k ix)
k bi =
δxp kp–kb
for allx∈X\{}.
According to Lemma ., there exists a unique mappingF:X→Ysatisfying () for all
x∈Xand () for allx∈X\{}.
The following corollary can be easily proved by using Lemma ..
Corollary . Assume that N is an integer and a, p,θ, δ are real constants such that a∈ {/ –, , },p=N,θ> ,andδ> .If a mapping f :X→Y satisfies f() = and
f(ax) –aNf(x)≤δxp ()
for all x∈X\{},then there exists a unique mapping F:X→Y satisfying()for all x∈X and
f(x) –F(x)≤ δx p
||a|N–|a|p| ()
for all x∈X\{}.
Proof If (|a|,p)∈(,∞)×(–∞,N) or (|a|,p)∈(, )×(N,∞), thenμ(x) :=δxpsatisfies condition () with
∞
i=
μ(aix) |a|Ni =
|a|Nδxp |a|N–|a|p.
When (|a|,p)∈(, )×(–∞,N) or (|a|,p)∈(,∞)×(N,∞), we set a := a to get (|a|,p)∈(,∞)×(–∞,N) or (|a|,p)∈(, )×(N,∞), and
fax–a Nf(x)=f
x a –
aNf(x) ≤
|a|Nδ
axp=ap+Nδxp
for allx∈X\{}. By arguing similarly as in the first case, we see thatμ(x) :=|a|p+Nδxp satisfies condition () with
∞
i=
μ(a ix) |a|Ni =
|a|p+Nδxp |a|N–|a|p =
|a|N
δxp |a|p–|a|N
for allx∈X\{}.
Therefore, by Lemma ., there exists a unique mappingF:X→Y such that () and () hold for allx∈Xresp. for allx∈X\{}.
4 Applications
For a given mappingf :V→Y, we use the abbreviations
Df(x,y) :=f(ax+by) –af(x) –bf(y),
Df(x,y) :=f(ax+by) +abf(x–y) –a(a+b)f(x) –b(a+b)f(y),
whereaandbare fixed nonzero real numbers witha+b= andab= . (The definitions ofDf andDf just above have nothing to do with theDnf given in the Introduction.)
Theorem . Let a and b be real constants such that a+b ∈ {/ –, , } and let ϕ : (V\{})→[,∞)be a function satisfying the condition
∞
i=
ϕ((a+b)ix, (a+b)iy) |a+b|i <∞
for all x,y∈V\{}.If a mapping f :V→Y satisfies f() = and
Df(x,y)≤ϕ(x,y) ()
for all x,y∈V\{},then there exists a unique mapping F:V→Y such that
DF(x,y) = ()
for all x,y∈V\{}and
f(x) –F(x)≤ ∞
i=
ϕ((a+b)ix, (a+b)ix) |a+b|i+
for all x∈V\{}.
Proof It follows from () that
(a+b)f(x) –f(a+b)x≤ϕ(x,x)
for allx∈V\{}. If we seta:=a+b,μ(x) :=ϕ(x,x),n:= ,N:= , andDf:=Df, thena,
N,f, andμsatisfy conditions () and (). According to Lemma ., there exists a unique mappingF:V→Ysatisfying () and (),i.e.,Fsatisfies
F(ax) =aNF(x)
for allx∈Vand
f(x) –F(x)≤ ∞
i=
ϕ(aix,aix) |a|Ni+N
for allx∈V\{}.
Moreover,ϕ,f, andFsatisfy () and (). Condition () is also satisfied due to the def-inition ofFgiven in the proof of Lemma .. Hence, in view of Theorem ., the mapping
Theorem . Let a and b be real constants such that a+b ∈ {/ –, , } and let ϕ : (V\{})→[,∞)be a function satisfying the condition
∞
i=
|a+b|iϕ
x
(a+b)i,
y
(a+b)i <∞ ()
for all x,y,z∈V\{}.If a mapping f :V→Y satisfies f() = and()for all x,y∈V\{},
then there exists a unique mapping F:V→Y satisfying()for all x∈V and
f(x) –F(x)≤ ∞
i=
|a+b|iϕ
x
(a+b)i+,
x
(a+b)i+
for all x∈V\{}.
Proof It follows from () that
a+bf(x) –f
x a+b ≤
|a+b|ϕ
x a+b,
x a+b
for allx∈V\{}. If we seta:=a+b,μ(x) :=|a|ϕ(ax,ax),n:= ,N:= , andDf :=Df,
thena,N,f, andμsatisfy conditions () and (). In view of Lemma ., there exists a
unique mappingF:V→Ysatisfying () and (),i.e.,Fsatisfies
F(ax) =aNF(x)
for allx∈Vand
f(x) –F(x)≤ ∞
i=
ϕ(ai+x,ai+x) |a|Ni+N–
for allx∈V\{}.
Furthermore,ϕ,f, andFsatisfy () and (). The truth of () follows from the definition ofFgiven in the proof of Lemma .. By Theorem ., the mappingFalso satisfies ()
for allx,y∈V\{}.
Theorem . Let a and b be real constants satisfying a+b∈ {/ –, , } and let ϕ : (V\{})→[,∞)be a function satisfying the condition
∞
i=
ϕ((a+b)ix, (a+b)iy) (a+b)i <∞
for all x,y∈V\{}.If a mapping f :V→Y satisfies f() = and
Df(x,y)≤ϕ(x,y) ()
for all x,y∈V\{},then there exists a unique mapping F:V→Y such that
for all x,y∈V\{}and
f(x) –F(x)≤ ∞
i=
ϕ((a+b)ix, (a+b)ix) (a+b)i+
for all x∈V\{}.
Proof It follows from () that
(a+b)f(x) –f(a+b)x≤ϕ(x,x)
for allx∈V\{}. If we seta:=a+b,μ(x) :=ϕ(x,x),n:= ,N:= , andDf:=Df, thena,
N,f, andμsatisfy conditions () and (). We then see by Lemma . that there exists a unique mappingF:V→Ysatisfying () and (),i.e.,Fsatisfies
F(ax) =aNF(x)
for allx∈Vand
f(x) –F(x)≤ ∞
i=
ϕ(aix,aix) |a|Ni+N
for allx∈V\{}.
Furthermore,ϕ,f, andFsatisfy () and (). Due to the definition ofFgiven in the proof of Lemma ., we obtain the validity of (). According to Theorem ., the mappingF
satisfies () for allx,y∈V\{}.
Theorem . Let a and b be real constants such that a +b ∈ {/ –, , } and let ϕ : (V\{})→[,∞)be a function satisfying the condition
∞
i=
(a+b)iϕ
x
(a+b)i,
y
(a+b)i <∞ ()
for all x,y∈V\{}.If a mapping f :V→Y satisfies f() = and()for all x,y∈V\{},
then there exists a unique mapping F:V→Y satisfying()for all x,y∈V\{}and
f(x) –F(x)≤ ∞
i=
(a+b)iϕ
x
(a+b)i+,
x
(a+b)i+ ()
for all x∈V\{}.
Proof It follows from () that
(a+b)f(x) –f
x a+b ≤
(a+b)ϕ
x a+b,
x a+b
for allx∈V\{}. If we puta:=a+b,μ(x) :=aϕ(ax,ax),n:= ,N:= , andDf :=Df,
there exists a unique mappingF:V→Y satisfying () and (),i.e.,Fsatisfies
F(ax) =aNF(x)
for allx∈Vand
f(x) –F(x)≤ ∞
i=
ϕ(ai+ x,ai+x)
|a|Ni+N–
for allx∈V\{}.
Moreover,ϕ,f, andFsatisfy () and (). From the definition ofFgiven in the proof of Lemma ., it follows that () is true. In view of Theorem ., the mappingFsatisfies
() for allx,y∈V\{}.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
Author details
1Department of Mathematics Education, Gongju National University of Education, Gongju, 314-711, Republic of Korea.
2Mathematics Section, College of Science and Technology, Hongik University, Sejong, 339-701, Republic of Korea.
Acknowledgements
The authors are grateful to anonymous referees for their kind suggestions and comments. Soon-Mo Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2013R1A1A2005557).
Received: 14 April 2015 Accepted: 23 August 2015 References
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