On Stability of the Cauchy Equation on Solvable Groups
The notion of $(ψ, γ)$-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, Trans. Amer. Math. Soc., 354, 4455 (2002)]. It was shown that the Cauchy equation $f (xy) = f (x) + f (y)$ is $(ψ, γ)$-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev...
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| author | Faiziev, V. A. Sahoo, P. K. Файзієв, В. А. Сахо, П. К. |
| author_facet | Faiziev, V. A. Sahoo, P. K. Файзієв, В. А. Сахо, П. К. |
| author_sort | Faiziev, V. A. |
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| datestamp_date | 2019-12-05T09:49:13Z |
| description | The notion of $(ψ, γ)$-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, Trans. Amer. Math. Soc., 354, 4455 (2002)]. It was shown that the Cauchy equation $f (xy) = f (x) + f (y)$ is $(ψ, γ)$-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev and P. K. Sahoo, Publ. Math. Debrecen, 75, 6 (2009)], it was proved that the Cauchy equation is $(ψ, γ)$-stable on step-two solvable groups and step-three nilpotent groups. In the present paper, we prove a more general result and show that the Cauchy equation is $(ψ, γ)$-stable on solvable groups. |
| first_indexed | 2026-03-24T02:17:33Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 512.5
V. A. Faĭziev (Tver State Agricultural Academy, Russia),
P. K. Sahoo (Univ. Louisville, USA)
ON STABILITY OF CAUCHY EQUATION ON SOLVABLE GROUPS*
ПРО СТIЙКIСТЬ РIВНЯННЯ КОШI НА РОЗВ’ЯЗНИХ ГРУПАХ
The notion of (ψ, γ)-stability was introduced in [Faĭziev V. A., Rassias Th. M., Sahoo P. K. The space of (ψ, γ)-additive
mappings on semigroups // Trans. Amer. Math. Soc. – 2002. – 354. – P. 4455 – 4472]. It was shown that the Cauchy
equation f(xy) = f(x) + f(y) is (ψ, γ)-stable on any Аbelian group as well as any meta-Abelian group. In [Faĭziev
V. A., Sahoo P. K. On (ψ, γ)-stability of Cauchy equation on some noncommutative groups // Publ. Math. Debrecen. –
2009. – 75. – P. 67 – 83], it was proved that the Cauchy equation is (ψ, γ)-stable on step-two solvable groups and step-three
nilpotent groups. In our paper, we prove a more general result and show that the Cauchy equation is (ψ, γ)-stable on
solvable groups.
Поняття (ψ, γ)-стiйкостi введено в роботi [Faĭziev V. A., Rassias Th. M., Sahoo P. K. The space of (ψ, γ)-additive
mappings on semigroups // Trans. Amer. Math. Soc. – 2002. – 354. – P. 4455 – 4472]. Було показано, що рiвняння
Кошi f(xy) = f(x) + f(y) є (ψ, γ)-стiйким як на довiльнiй абелевiй групi, так i на довiльнiй метабелевiй групi. В
роботi [Faĭziev V. A., Sahoo P. K. On (ψ, γ)-stability of Cauchy equation on some noncommutative groups // Publ. Math.
Debrecen. – 2009. – 75. – P. 67 – 83] доведено, що рiвняння Кошi є (ψ, γ)-стiйким як на двоступеневих розв’язних
групах, так i на триступеневих нiльпотентних групах. В нашiй роботi доведено бiльш загальний результат i
показано, що рiвняння Кошi є (ψ, γ)-стiйким на розв’язних групах.
1. Introduction. In 1940, S.M. Ulam [11] posed the following fundamental problem. Given a
group G1, a metric group (G2, d) and a positive number ε, does there exist a δ > 0 such that if f :
G1 → G2 satisfies d(f(xy), f(x)f(y)) < δ for all x, y ∈ G1, then a homomorphism T : G1 → G2
exists with d(f(x), T (x)) < ε for all x, y ∈ G1? Interested reader should see S. M. Ulam [11] for a
discussion of such problems, as well as D. H. Hyers [8], Th. M. Rassias [9], J. Aczél and J. Dhombres
[1], G. L. Forti [7], and P. K. Sahoo and Pl. Kannappan [10]. The first affirmative answer to this
problem was given by D. H. Hyers [8] in 1941.
On a group G, the Cauchy functional equation f(x+ y) = f(x) + f(y) takes the form f(xy) =
= f(x) + f(y) for all x, y ∈ G. In connection with Hyers’ result the following question arises.
Let G be an arbitrary group and let a mapping f : G → R (the set of reals) be such that the set,
D, defined by {f(xy) − f(x) − f(y) |x, y ∈ G} is bounded. Is it true that there is a mapping T :
G→ R that satisfies T (xy)− T (x)− T (y) = 0 for all x, y ∈ G, and the set {T (x)− f(x) |x ∈ G}
is bounded. A negative answer was given in 1987 by G.L. Forti [6]. He constructed a real-valued
function f on the free group F2 of rank two (and also on a free semigroup S2 of rank two) such that
the set D is bounded but for any additive function T, the function f(x)− T (x) is not bounded. It is
worth pointing out that in 1987, Făiziev in [2] gave a description of all such functions f on the free
product of semigroups A ∗ B. In [3], it was established that the Cauchy functional equation is not
stable on the free product A ∗B of groups A and B unless A = B = Z2.
* This paper was partially supported by an IRIG grant from the Office of the VP for Research, and an A&S grant from
the College Arts and Sciences, University of Louisville.
c© V. A. FAĬZIEV, P. K. SAHOO, 2015
1000 ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON STABILITY OF CAUCHY EQUATION ON SOLVABLE GROUPS 1001
The first paper to extend Hyers’ result to a class nonabelian groups and semigroups was [2]. The
notion of (ψ, γ)-stability of the Cauchy functional equation was introduced in [5]. In [5] among
other results, it was proven that the Cauchy functional equation f(xy) = f(x)+f(y) is (ψ, γ)-stable
on any abelian group as well as any meta-Abelian (step-two nilpotent) group. It was also shown
that any arbitrary group A can be embedded into a group G, where the Cauchy functional equation
is (ψ, γ)-stable. In [4], the (ψ, γ)-stability of the Cauchy functional equation on step-two solvable
groups and step-three nilpotent groups were treated. In this paper, we show that the Cauchy functional
equation is (ψ, γ)-stable on solvable groups.
2. Definitions and notations. We introduce some relevant definitions and notations that will be
useful for the proof of the main result.
Definition 1. A quasicharacter of a semigroup G is a real-valued function f on G such that the
set {f(xy)− f(x)− f(y)|x, y ∈ G} is bounded.
Definition 2. By a pseudocharacter of a group G we mean its quasicharacter f that satisfies
f(xn) = nf(x) for all x ∈ G and all n ∈ Z.
The set of quasicharacters of a group G is a vector space (with respect to the usual operations of
addition of functions and their multiplication by numbers), which will be denoted by KX(G). The
subspace of KX(G) consisting of pseudocharacters will be denoted by PX(G) and the subspace
consisting of real additive characters of the group G, will be denoted by X(G). We say that a
pseudocharacter ϕ of the group G is nontrivial if ϕ /∈ X(G).
Next we recall some notions from [5] that we need for this paper. Let R+
0 = [0,∞) be the set
of nonnegative numbers and R+ = (0,∞) be the set of positive numbers. Let G be an arbitrary
group. Throughout this paper, the function ψ : R+
0 → R+ is considered to be an increasing function
satisfying the following three additional conditions:
(1) ψ(t1 t2) ≤ ψ(t1)ψ(t2) for all t1, t2 ∈ R+
0 ,
(2) ψ(t1 + t2) ≤ ψ(t1) + ψ(t2) for all t1, t2 ∈ R+
0 ,
(3) limn→∞
ψ(n)
n
= 0, n ∈ N.
Throughout this paper, by γ we will mean a function γ : G→ R+
0 satisfying the inequality γ(xy) ≤
≤ γ(x) + γ(y) + d for all x, y ∈ G and some real nonnegative constant d. It is obvious that for any
x ∈ G and for any m ∈ N, the function γ satisfies the inequality
γ(xm) ≤ mγ(x) + (m− 1) d. (1)
Definition 3. Let G be an arbitrary group and E a Banach space. Further, let ψ : R+
0 → R+
and γ : G → R+
0 be the functions as described above. The mapping f : G → E is said to be a
(ψ, γ)- quasiadditive mapping if there exists a θ ∈ R+ such that
‖f(xy)− f(x)− f(y)‖ ≤ θ [ψ(γ(x)) + ψ(γ(y))] ∀x, y ∈ G (2)
holds.
It is clear that the set of all (ψ, γ)-quasiadditive mappings from G to E is a real linear space
relative to the usual operations. Let us denote it by KAMψ,γ(G;E).
Definition 4. Let ϕ : G→ E be a mapping from the groupG to a Banach space E. The mapping
ϕ is said to be a (ψ, γ)-pseudoadditive mapping if it is a (ψ, γ)-quasiadditive mapping satisfying
ϕ(xn) = nϕ(x) for all x ∈ G and for each n ∈ Z.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
1002 V. A. FAĬZIEV, P. K. SAHOO
We denote the space of all (ψ, γ)-pseudoadditive mappings from a group G to a Banach space
E by PAMψ,γ(G;E). By HOM(G;E) we mean the set of all homomorphisms from G to E. By
Bψ,γ(G;E) we denote the linear space of functions from G to E over reals satisfying the relation:
‖f(x)‖ ≤ c ψ(γ(x)) for some c > 0 and for all x ∈ G.
Definition 5. The Cauchy functional equation
f(xy) = f(x) + f(y) ∀x, y ∈ G (3)
is said to be (ψ, γ)-stable for the pair (G;E) if for any f : G→ E satisfying the functional inequality
‖ f(xy)− f(x)− f(y) ‖ ≤ θ [ψ(γ(x)) + ψ(γ(y))] ∀x, y ∈ G, (4)
there is a solution g : G→ E of functional equation (3) such that the function f(x)− g(x) belongs
to the space Bψ,γ(G;E).
It was shown in [5] that the equation (3) is (ψ, γ)-stable for the pair (G;E) if and only if
PAMψ,γ(G;E) = HOM(G;E). The following result is one of the main results in [5]. Let E1 and
E2 be Banach spaces over reals. Then the equation (3) is (ψ, γ)-stable for the pair (G,E1) if and only
if it is (ψ, γ)-stable for the pair (G,E2). In view of this result it is not important which Banach space
is used on the range. Thus one may consider the (ψ, γ)-stability of the functional equation (3) on the
pair (G,R). Let us simplify the following notations: In the case E = R the spaces KAMψ,γ(G;R),
PAMψ,γ(G;R), and HOM(G;R) will be denoted by KXψ,γ(G), PXψ,γ(G), X(G), respectively.
Further, we will call a (ψ, γ)-additive map a (ψ, γ)-quasicharacter, and a (ψ, γ)-pseudoadditive map
a (ψ, γ)-pseudocharacter. It is clear that if γ is a constant function then PXψ,γ(G) = PX(G). If
f ∈ PXψ,γ(G), then it known (see [5]) that the (ψ, γ)-pseudocharacter satisfies (i) f(xy) = f(yx)
for any x, y ∈ G, and (ii) f(xy) = f(x) + f(y) if xy = yx.
3. Some preliminary results.
Theorem 1. Suppose G is a group, H a normal subgroup of G and A a subgroup of G. Let G
be the semidirect product of A and H, that is G = AnH. If f belongs to PXψ,γ(G), f
∣∣
A
∈ X(A),
and f
∣∣
H
∈ X(H), then f ∈ X(G).
Proof. Let a ∈ A and v ∈ H, then for any n ∈ N the following relation:
(av)n = anva
n−1
va
n−2
. . . vav, (5)
holds, where va
i
denotes the element a−ivai. The element va
n−1
va
n−1
. . . vav belongs to H and
va
n−1
va
n−2
. . . vav = a−(n−1)va(n−1)a−(n−2)va(n−2) . . . a−1vav =
= a−(n−1)vava . . . vav. (6)
Therefore
γ(va
n−1
va
n−2
. . . vav) = γ(a−n(av)n) ≤
≤ γ(a−n) + γ((av)n) + d ≤
≤ nγ(a−1) + (n− 1) d+ nγ(av) + (n− 1)d+ d ≤
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON STABILITY OF CAUCHY EQUATION ON SOLVABLE GROUPS 1003
≤ nγ(a−1) + (n− 1) d+ n [ γ(a) + γ(v) + d ] + (n− 1) d+ d ≤
≤ n
[
γ(a−1) + γ(a) + γ(v)
]
+ (3n− 1) d
and
ψ
(
γ(va
n−1
va
n−2
. . . vav)
)
≤
≤ ψ
(
n [γ(a−1) + γ(a) + γ(v)] + (3n− 1) d
)
≤
≤ ψ(n)ψ( γ(a−1) + γ(a) + γ(v) ) + ψ(n)ψ(3d).
Let g = av. Then for any n ∈ N, since f ∈ PXψ,γ(G), the following relation:
|f(gn)− f(an)− f(van−1
va
n−1
. . . vav)| ≤
≤ θ
[
ψ(γ(an)) + ψ(γ(va
n−1
va
n−1
. . . vav))
]
≤
≤ θ
[
ψ(n)ψ(γ(a) + d) + ψ(γ(va
n−1
va
n−1
. . . vav))
]
≤
≤ θ ψ(n)
[
ψ(γ(a) + d) + ψ(γ(a−1) + γ(a) + γ(v)) + ψ(3d)
]
(7)
holds. Moreover, since, f ∈ PXψ,γ(G), f(xy) = f(yx) for any x, y ∈ G. Hence it follows that,
for any x, y ∈ G, the relation f(y−1xy) = f(x) holds. This implies that f is invariant under inner
automorphisms of group G. Therefore, since f
∣∣
H
∈ X(H) and function f is invariant with respect
to inner automorphisms of the group G, we get
f(va
n−1
va
n−1
. . . vav) =
= f
(
va
n−1
)
+ f
(
va
n−1
)
+ . . .+ f (va) + f(v) = n f(v).
Therefore from (7) it follows
n |f(g)− f(a)− f(v)| ≤
≤ θ ψ(n)
[
ψ(γ(a) + d) + ψ(γ(a−1) + γ(a) + γ(v)) + ψ(3d)
]
.
The latter relation implies
|f(g)− f(a)− f(v)| ≤
≤ θ ψ(n)
n
[
ψ(γ(a) + d) + ψ(γ(a−1) + γ(a) + γ(v)) + ψ(3d)
]
(8)
for any n ∈ N. This implies that
f(av) = f(a) + f(v) ∀a ∈ A ∀v ∈ H. (9)
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
1004 V. A. FAĬZIEV, P. K. SAHOO
Now if a, b ∈ A and u, v ∈ H, then from (9) it follows that
f(avbu) = f(abvbu) = f(ab) + f(vbu) =
= f(a) + f(b) + f(vb) + f(u) =
= f(a) + f(b) + f(v) + f(u) =
= f(a) + f(v) + f(b) + f(u) = f(av) + f(bu).
Hence f ∈ X(G) and the theorem is proved.
The next theorem is a corollary of Theorem 1.
Theorem 2. Let G be a group. Suppose that G is a semidirect product G = A n H of its
subgroups A and H, where H is normal. Suppose that the Cauchy functional equation is (ψ, γ)-
stable on A and H. Then it is (ψ, γ)-stable on G.
Proof. To prove this theorem it is enough to show that PXψ,γ(G) = X(G). Since X(G) ⊆
⊆ PXψ,γ(G), we only need to show that PXψ,γ(G) ⊆ X(G). Hence, let f ∈ PXψ,γ(G). Since by
hypothesis the Cauchy functional equation is (ψ, γ)-stable on A and H, therefore f
∣∣
A
∈ X(A), and
f
∣∣
H
∈ X(H). Hence by Theorem 1 we have f ∈ X(G). This proves that PXψ,γ(G) ⊆ X(G) and
the Cauchy functional equation is (ψ, γ)-stable on G.
Theorem 2 is proved.
4. Stability on solvable groups. In this section, we prove the main result of this paper using
Theorem 2.
Theorem 3. The Cauchy equation is (ψ, γ)-stable on any solvable group.
Proof. We prove this theorem by induction on the degree of solvability. The solvable groups
of degree one are abelian groups. We know that for abelian groups the theorem is true (see [5]).
Suppose that the theorem is true for all solvable groups having degree of solvability no more then k
and let us prove it for solvable groups with degree k + 1.
Let G2 be free solvable group of degree k + 1 with free generators a, b. Denote by A and B
subgroups of G, generated by a and b, respectively and let G′2 denote the commutator subgroup of
G2. It is well known that G2 is a semidirect product G2 = An (BnG′2). The commutator subgroup
G′2 is a solvable group of degree k and B is a cyclic group. By Theorem 2, the Cauchy functional
equation is (ψ, γ)-stable on subgroup H = B n G′2. Now the group G2 is a semidirect product
G2 = AnH again by Theorem 2, the Cauchy equation is (ψ, γ)-stable on G2 = AnH.
Now let K be an arbitrary solvable group of degree k + 1. Suppose that the Cauchy equation is
not (ψ, γ)-stable on the solvable group K. It means that the set PXψ,γ(K) r X(K) is not empty.
Let ϕ ∈ PXψ,γ(K)rX(K). Hence, there exist x, y ∈ K such that ϕ(xy) 6= ϕ(x) +ϕ(y). Let L be
a subgroup of K generated by elements x and y. Since G2 is free solvable group of degree k+1, the
mapping π(a) = x, π(b) = y can be uniquely extended to epimorphism π : G2 → L. Let γ∗ = γ ◦π.
Then function f = ϕ ◦ π belongs to the space PXψ,γ∗(G2). Since, PXψ,γ∗(G2) = X(G2) we see
that f ∈ X(G2). However, f(ab) = ϕ(π(ab)) = ϕ(xy) 6= ϕ(x) + ϕ(y) = f(a) + f(b) which is
a contradiction to the fact f ∈ X(G2). Hence PXψ,γ(K) r X(K) is not nonempty. Therefore
PXψ,γ(K) = X(K) and the Cauchy equation is (ψ, γ)-stable on G.
Theorem 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
ON STABILITY OF CAUCHY EQUATION ON SOLVABLE GROUPS 1005
1. Aczél J., Dhombres J. Functional equations in several variables // Encycl. Math. and its Appl. – Cambridge: Cambridge
Univ. Press, 1989.
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3. Faĭziev V. A. On the stability of a functional equation on groups // Uspechi Mat. Nauk. – 1993. – 48. – S. 193 – 194.
4. Faĭziev V. A., Sahoo P. K. On (ψ, γ)-stability of Cauchy equation on some noncommutative groups // Publ. Math.
Debrecen. – 2009. – 75. – P. 67 – 83.
5. Faĭziev V. A., Rassias Th. M., Sahoo P. K. The space of (ψ, γ)-additive mappings on semigroups // Trans. Amer.
Math. Soc. – 2002. – 354. – P. 4455 – 4472.
6. Forti G. L. Remark 11 // Rept of 22nd Int. Symp. Funct. Equat.: Aequat. Math. – 1985. – 29. – P. 90 – 91.
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P. 143 – 190.
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9. Rassias Th. M. On the stability of the linear mapping in Banach spaces // Proc. Amer. Math. Soc. – 1978. – 72. –
P. 297 – 300.
10. Sahoo P. K., Kannappan Pl. Introduction to functional equations. – Boca Raton: CRC Press, 2011.
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Received 21.09.12,
after revision — 07.01.14
ISSN 1027-3190. Укр. мат. журн., 2015, т. 67, № 7
|
| id | umjimathkievua-article-2039 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:17:33Z |
| publishDate | 2015 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/12/911a6db5685a92ec6d31912d482f5312.pdf |
| spelling | umjimathkievua-article-20392019-12-05T09:49:13Z On Stability of the Cauchy Equation on Solvable Groups Про стійкість рівняння Коші на розв'язних групах Faiziev, V. A. Sahoo, P. K. Файзієв, В. А. Сахо, П. К. The notion of $(ψ, γ)$-stability was introduced in [V. A. Faiziev, Th. M. Rassias, and P. K. Sahoo, Trans. Amer. Math. Soc., 354, 4455 (2002)]. It was shown that the Cauchy equation $f (xy) = f (x) + f (y)$ is $(ψ, γ)$-stable both on any Abelian group and on any meta-Abelian group. In [V. A. Faiziev and P. K. Sahoo, Publ. Math. Debrecen, 75, 6 (2009)], it was proved that the Cauchy equation is $(ψ, γ)$-stable on step-two solvable groups and step-three nilpotent groups. In the present paper, we prove a more general result and show that the Cauchy equation is $(ψ, γ)$-stable on solvable groups. Поняття $(ψ, γ)$-стійкості введено в роботі [Fahiev V. A., Rassias Th. M., Sahoo P. K. The space of $(ψ, γ)$-additive mappings on semigroups//Trans. Amer. Math. Soc. - 2002. - 354. - P. 4455-4472]. Було показано, що рівняння Коші $f (xy) = f (x) + f (y)$ є $(ψ, γ)$-стійким як на довільній абелевій групі, так i на довільній метабелевій групі. В роботі [Farnev V. A., Sahoo P. K. On $(ψ, γ)$-stability of Cauchy equation on some noncommutative groups // Publ. Math. Debrecen. - 2009. - 75. - P. 67-83] доведено, що рівняння Коші є $(ψ, γ)$-стійким як на двоступеневих розв'язних групах, так i на триступеневих нільпотентних групах. В нашій роботі доведено більш загальний результат i показано, що рівняння Коші є $(ψ, γ)$-стійким на розв'язних групах. Institute of Mathematics, NAS of Ukraine 2015-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2039 Ukrains’kyi Matematychnyi Zhurnal; Vol. 67 No. 7 (2015); 1000-1005 Український математичний журнал; Том 67 № 7 (2015); 1000-1005 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/2039/1095 https://umj.imath.kiev.ua/index.php/umj/article/view/2039/1096 Copyright (c) 2015 Faiziev V. A.; Sahoo P. K. |
| spellingShingle | Faiziev, V. A. Sahoo, P. K. Файзієв, В. А. Сахо, П. К. On Stability of the Cauchy Equation on Solvable Groups |
| title | On Stability of the Cauchy Equation on Solvable Groups |
| title_alt | Про стійкість рівняння Коші на розв'язних групах |
| title_full | On Stability of the Cauchy Equation on Solvable Groups |
| title_fullStr | On Stability of the Cauchy Equation on Solvable Groups |
| title_full_unstemmed | On Stability of the Cauchy Equation on Solvable Groups |
| title_short | On Stability of the Cauchy Equation on Solvable Groups |
| title_sort | on stability of the cauchy equation on solvable groups |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2039 |
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