Approximation of general α-cubic functional equations in 2-Banach spaces
We introduce a new \alpha -cubic functional equation and investigate the generalized Hyers – Ulam stability of this functional equation in 2-Banach spaces.
Gespeichert in:
| Datum: | 2016 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2016
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/1930 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507822265991168 |
|---|---|
| author | Eskandani, G. Z. Rassias, J. M. Ескандані, Г. З. Расіас, Дж. М. |
| author_facet | Eskandani, G. Z. Rassias, J. M. Ескандані, Г. З. Расіас, Дж. М. |
| author_sort | Eskandani, G. Z. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:31:57Z |
| description | We introduce a new \alpha -cubic functional equation and investigate the generalized Hyers – Ulam stability of this functional
equation in 2-Banach spaces. |
| first_indexed | 2026-03-24T02:15:25Z |
| format | Article |
| fulltext |
К О Р О Т К I П О В I Д О М Л Е Н Н Я
UDC 517.5
G. Z. Eskandani (Univ. Tabriz, Iran),
J. M. Rassias (Nat. and Capodistrian Univ. Athens, Greece)
APPROXIMATION OF GENERAL \bfitalpha -CUBIC FUNCTIONAL EQUATIONS
IN \bftwo -BANACH SPACES
НАБЛИЖЕННЯ ЗАГАЛЬНИХ \bfitalpha -КУБIЧНИХ ФУНКЦIОНАЛЬНИХ РIВНЯНЬ
У \bftwo -БАНАХОВИХ ПРОСТОРАХ
We introduce a new \alpha -cubic functional equation and investigate the generalized Hyers – Ulam stability of this functional
equation in 2-Banach spaces.
Введено нове \alpha -кубiчне функцiональне рiвняння та вивчено узагальнену стiйкiсть Хайєрса – Улама цього функцiо-
нального рiвняння в 2-банахових просторах.
1. Introduction and preliminaries. Speaking of the stability of a functional equation, we follow
the question raised in 1940 by S. M. Ulam [29]:
When is it true that the solution of an equation differing slightly from a given one, must of
necessity be close to the solution of the given equation?
The first partial answer (in the case of Cauchy’s functional equation in Banach spaces) to Ulam’s
question was given by D. H. Hyers (see [11]). This result was generalized by Aoki [1] for additive
mappings and by Th. M. Rassias [24] for linear mappings by considering an unbounded Cauchy
difference. In 1994, a further generalization was obtained by P. Găvruta [10]. J. M. Rassias (see
[19 – 23]) solved the Ulam problem for different mappings. In addition, J. M. Rassias considered the
mixed product-sum of powers of norms control function [28].
During the last two decades, a number of papers and research monographs have been published
on various generalizations and applications of the generalized Hyers – Ulam stability to a number of
functional equations and mappings (see [8, 9, 15, 16, 25 – 27]). We also refer the readers to the
books: P. Czerwik [4] and D. H. Hyers, G. Isac and Th. M. Rassias [12].
Jun and Kim [13] introduced the functional equation
f(2x+ y) + f(2x - y) = 2f(x+ y) + 2f(x - y) + 12f(x), (1.1)
and they established the general solution and the generalized Hyers – Ulam stability problem for
functional equation (1.1). It is easy to see that the function f(x) = cx3 is a solution of (1.1). Thus,
it is natural that (1.1) is called a cubic functional equation and every solution of (1.1) is said to be a
cubic mapping. Jun et al. [14] introduced the Euler – Lagrange type cubic functional equation
f(ax+ y) + f(ax - y) = af(x+ y) + af(x - y) + 2a(a2 - 1)f(x) (1.2)
for a fixed integer a with a \not = 0,\pm 1, and they showed that functional equation (1.1) is equivalent to
functional equation (1.2).
In the 1960s, S. Gahler [6, 7] introduced the concept of linear 2-normed spaces.
c\bigcirc G. Z. ESKANDANI, J. M. RASSIAS, 2016
1430 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
APPROXIMATION OF GENERAL \alpha -CUBIC FUNCTIONAL EQUATIONS IN 2-BANACH SPACES 1431
Definition 1.1. Let X be a linear space over \BbbR with \mathrm{d}\mathrm{i}\mathrm{m}X > 1 and \| \cdot , \cdot \| : X \times X - \rightarrow \BbbR be
a function satisfying the following properties:
(a) \| x, y\| = 0 if and only if x and y are linearly dependent,
(b) \| x, y\| = \| y, x\| ,
(c) \| \lambda x, y\| = | \lambda | \| x, y\| ,
(d) \| x, y + z\| \leq \| x, y\| + \| x, z\|
for all x, y, z \in X and \lambda \in \BbbR .
Then the function \| \cdot , \cdot \| is called a 2-norm on X and the pair (X, \| \cdot , \cdot \| ) is called a linear
2-normed space. A standard example of a 2-normed space is \BbbR 2 equipped with the 2-norm defined
as \| x, y\| = the area of the triangle having vertices 0, x and y.
It follows from (d), that \| x+ y, z\| \leq \| x, z\| + \| y, z\| and
\bigm| \bigm| \| x, z\| - \| y, z\|
\bigm| \bigm| \leq \| x - y, z\| . Hence
the functions x - \rightarrow \| x, y\| are continuous functions of X into \BbbR for each fixed y \in X.
Definition 1.2. A sequence \{ xn\} in a linear 2-normed space X is called a Cauchy sequence if
there are two points y, z \in X such that y and z are linearly independent,
\mathrm{l}\mathrm{i}\mathrm{m}
m,n\rightarrow \infty
\| xm - xn, y\| = 0,
and
\mathrm{l}\mathrm{i}\mathrm{m}
m,n\rightarrow \infty
\| xm - xn, z\| = 0.
Definition 1.3. A sequence \{ xn\} in a linear 2-normed space X is called a convergent sequence
if there is an x \in X such that
\mathrm{l}\mathrm{i}\mathrm{m}
m,n\rightarrow \infty
\| xn - x, y\| = 0
for all y \in X. If \{ xn\} converges to x, write xn - \rightarrow x as n - \rightarrow \infty and call x the limit of \{ xn\} . In
this case, we also write \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty xn = x.
Definition 1.4. A linear 2-normed space in which every Cauchy sequence is a convergent se-
quence is called a 2-Banach space.
Lemma 1.1 [18]. Let (X, \| \cdot , \cdot \| ) be a linear 2-normed space. If x \in X and \| x, y\| = 0 for all
y \in X, then x = 0.
Lemma 1.2 [18]. For a convergent sequence \{ xn\} in a linear 2-normed space X,
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| xn, y\| =
\bigm\| \bigm\| \bigm\| \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
xn, y
\bigm\| \bigm\| \bigm\|
for all y \in X.
In [18] W. G Park investigated approximate additive mappings, approximate Jensen mappings
and approximate quadratic mappings in 2-Banach spaces. The superstability of the Cauchy functional
inequality and the Cauchy – Jensen functional inequality in 2-Banach spaces under some conditions
were investigated by C. Park in [17].
In this paper, we deal with the next general cubic functional equation
f(\alpha x+ y) + f(\alpha x - y) + f(x+ \alpha y) - f(x - \alpha y) =
= 2\alpha f(x+ y) + 2\alpha (\alpha 2 - 1)[f(x) + f(y)] (1.3)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1432 G. Z. ESKANDANI, J. M. RASSIAS
with \alpha \in \BbbN , \alpha \not = 1.
It is easy to see that the function f(x) = ax3 is a solution of functional equation (1.3). We will
prove the generalized Hyers – Ulam stability of equation (1.3) in 2-Banach spaces.
Let X and Y be two linear spaces. For convenience, we use the following abbreviation for a
given function f : X \rightarrow Y :
D\alpha f(x, y) := f(\alpha x+ y) + f(\alpha x - y) + f(x+ \alpha y) -
- f(x - \alpha y) - 2\alpha f(x+ y) - 2\alpha (\alpha 2 - 1)[f(x) + f(y)]
for all x, y \in X. We need the following two lemmas.
Lemma 1.3 [5]. Let X and Y be two linear spaces. If a mapping f : X \rightarrow Y satisfies the
functional equation
f(x+ \alpha y) - f(x - \alpha y) = \alpha [f(x+ y) - f(x - y)] + 2\alpha (\alpha 2 - 1)f(y)
for all x, y \in X, then f is cubic.
Lemma 1.4. Let X and Y be two linear spaces. If a mapping f : X \rightarrow Y satisfies (1.3) for all
x, y \in X, then f is cubic.
Proof. Replacing (x, y) with (0, 0) in (1.3), we get f(0) = 0. Replacing (x, y) with (x, 0) in
(1.3), we have
f(\alpha x) = \alpha 3f(x) (1.4)
for all x \in X. By setting x = 0 and using (1.4), we obtain f( - y) = - f(y) for all y \in X, that is f
is odd. Replacing (x, y) with (x, - y) in (1.3) and using oddness of f, we get
f(\alpha x - y) + f(\alpha x+ y) + f(x - \alpha y) - f(x+ \alpha y) = 2\alpha f(x - y) + 2\alpha (\alpha 2 - 1)[f(x) - f(y)]
(1.5)
for all x, y \in X. It follows from (1.3) and (1.5) that
f(x+ \alpha y) - f(x - \alpha y) = \alpha [f(x+ y) - f(x - y)] + 2\alpha (\alpha 2 - 1)f(y)
for all x, y \in X. It follows from Lemma 1.3 that f is cubic.
2. Approximate cubic mappings. Throughout this section, let X be a normed linear space, Y
be a 2-Banach space and \alpha \in \BbbN , \alpha \not = 1.
Theorem 2.1. Let \varphi : X \times X \times X - \rightarrow [0,+\infty ) be a function such that
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\alpha 3n
\varphi (\alpha nx, \alpha ny, z) = 0 (2.1)
for all x, y, z \in X. Suppose that f : X - \rightarrow Y is mapping with f(0) = 0,
\| D\alpha f(x, y), z\| \leq \varphi (x, y, z), (2.2)
and
\widetilde \varphi (x, z) =:
\infty \sum
n=0
1
\alpha 3i
\varphi (\alpha ix, 0, z) <\infty (2.3)
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
APPROXIMATION OF GENERAL \alpha -CUBIC FUNCTIONAL EQUATIONS IN 2-BANACH SPACES 1433
exists for all x, y, z \in X. Then there exists a unique cubic mapping C : X - \rightarrow Y such that
\| f(x) - C(x), z\| \leq 1
2\alpha 3
\widetilde \varphi (x, z) (2.4)
for all x, z \in X.
Proof. Setting y = 0 in (2.2), we have
\| f(\alpha x) - \alpha 3f(x), z\| \leq 1
2
\varphi (x, 0, z) (2.5)
for all x, z \in X. Replacing x with \alpha nx in (2.5) and dividing both sides of (2.5) by \alpha 3n+3, we obtain\bigm\| \bigm\| \bigm\| 1
\alpha 3n+3
f(\alpha n+1x) - 1
\alpha 3n
f(\alpha nx), z
\bigm\| \bigm\| \bigm\| \leq 1
2\alpha 3n+3
\varphi (\alpha nx, 0, z) (2.6)
for all x, z \in X and all nonnegative integers n. Hence,\bigm\| \bigm\| \bigm\| 1
\alpha 3n+3
f(\alpha n+1x) - 1
\alpha 3m
f(\alpha mx), z
\bigm\| \bigm\| \bigm\| \leq
n\sum
i=m
\bigm\| \bigm\| \bigm\| 1
\alpha 3i+3
f(\alpha i+1x) - 1
\alpha 3i
f(\alpha ix), z
\bigm\| \bigm\| \bigm\| \leq
\leq 1
2\alpha 3
n\sum
i=m
1
\alpha 3i
\varphi (\alpha ix, 0, z) (2.7)
for all x, z \in X and all nonnegative integers m and n with n \geq m. Therefore, we conclude from
(2.3) and (2.7) that the sequence
\biggl\{
1
\alpha 3n
f(\alpha nx)
\biggr\}
is a Cauchy sequence in Y for all x \in X. Since
Y is complete the sequence
\biggl\{
1
\alpha 3n
f(\alpha nx)
\biggr\}
converges in Y for all x \in X. So one can define the
mapping C : X \rightarrow Y by
C(x) := \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\alpha 3n
f(\alpha nx) (2.8)
for all x \in X. That is
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\bigm\| \bigm\| \bigm\| \bigm\| 1
\alpha 3n
f(\alpha nx) - C(x), y
\bigm\| \bigm\| \bigm\| \bigm\| = 0
for all x, y \in X. Letting m = 0 and passing to the limit n \rightarrow \infty in (2.7), we get (2.4). Now, we
show that C : X \rightarrow Y is a cubic mapping. It follows from (2.1), (2.2), (2.8) and Lemma 1.2 that\bigm\| \bigm\| D\alpha C(x, y), z
\bigm\| \bigm\| = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\alpha 3n
\| D\alpha f(\alpha
3nx, \alpha 3ny), z\| \leq
\leq \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\alpha 3n
\varphi (\alpha nx, \alpha ny, z) = 0
for all x, y, z \in X. By Lemma 1.1, D\alpha C(x, y) = 0 for all x, y \in X. So by Lemma 1.4 the mapping
C : X \rightarrow Y is cubic.
To prove the uniqueness of C, let C : X \rightarrow Y be another cubic mapping satisfying (2.4). Then
\| C(x) - C \prime (x), z\| = \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\alpha 3n
\bigm\| \bigm\| f(\alpha nx) - C \prime (\alpha nx), z
\bigm\| \bigm\| \leq
\leq 1
2\alpha 3
\mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
1
\alpha 3n
\widetilde \varphi (\alpha nx, z) = 0
for all x, z \in X. By Lemma 1.1, C(x) - C \prime (x) = 0 for all x \in X. So C = C \prime .
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1434 G. Z. ESKANDANI, J. M. RASSIAS
Remark 2.1. We can formulate a similar theorem to Theorem 2.1 in which we can define the
sequence C(x) := \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \alpha 3nf
\Bigl( x
\alpha n
\Bigr)
under suitable assumption on the function \varphi .
Corollary 2.1. Let \psi : [0,\infty ) \rightarrow [0,\infty ) be a function such that \psi (0) = 0 and
(i) \psi (ts) \leq \psi (t)\psi (s),
(ii) \psi (t) < t for all t > 1.
Suppose that f : X - \rightarrow Y is a mapping with f(0) = 0 and
\| D\alpha f(x, y), z\| \leq \psi (\| x\| ) + \psi (\| y\| ) + \psi (\| z\| ) (2.9)
for all x, y, z \in X. Then there exists a unique cubic mapping C : X \rightarrow Y satisfying
\| f(x) - C(x), z\| \leq 1
2
\psi (\| x\| )
\alpha 3 - \psi (\alpha )
+
1
2
\psi (\| z\| )
\alpha 3 - 1
(2.10)
for all x, z \in X.
Proof. Let
\varphi (x, y, z) = \psi (\| x\| ) + \psi (\| y\| ) + \psi (\| z\| )
for all x, y, z \in X. It follows from (i) that \psi (\alpha n) \leq
\bigl(
\psi (\alpha )
\bigr) n
and
\varphi (\alpha nx, \alpha ny, z) \leq
\bigl(
\psi (\alpha )
\bigr) n\bigl(
\psi (\| x\| ) + \psi (\| y\| )
\bigr)
+ \psi (\| z\| ).
By using Theorem 2.1, we obtain (2.10).
Corollary 2.2. Let q be a nonnegative real number such that q < 3 and H : [0,\infty )\times [0,\infty ) \rightarrow
\rightarrow [0,\infty ) be a homogeneous function of degree q. Suppose that f : X - \rightarrow Y is a mapping with
f(0) = 0 and
\| D\alpha f(x, y), z\| \leq H(\| x\| , \| y\| ) + \| z\|
for all x, y, z \in X. Then there exists a unique cubic mapping C : X \rightarrow Y such that
\| f(x) - C(x), z\| \leq 1
2
H(\| x\| , 0) + \| z\|
\alpha 3 - q3
(2.11)
for all x \in X.
Proof. Let
\varphi (x, y, z) = H(\| x\| , \| y\| ) + \| z\|
for all x, y, z \in X. By using Theorem 2.1, we obtain (2.11).
Corollary 2.3. Let q be a nonnegative real number such that q < 3 and H : [0,\infty )\times [0,\infty ) \rightarrow
\rightarrow [0,\infty ) be a homogeneous function of degree q. Suppose that f : X - \rightarrow Y is a mapping with
f(0) = 0 and
\| D\alpha f(x, y), z\| \leq H(\| x\| , \| y\| )\| z\|
for all x, y, z \in X. Then there exists a unique cubic mapping C : X \rightarrow Y such that
\| f(x) - C(x), z\| \leq 1
2
H(\| x\| , 0)\| z\|
\alpha 3 - q3
(2.12)
for all x, z \in X.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
APPROXIMATION OF GENERAL \alpha -CUBIC FUNCTIONAL EQUATIONS IN 2-BANACH SPACES 1435
Proof. Let
\varphi (x, y, z) = H(\| x\| , \| y\| )\| z\|
for all x, y, z \in X. By using Theorem 2.1, we obtain (2.12).
Corollary 2.4. Let p be a nonnegative real number such that p < 3. Suppose that f : X - \rightarrow Y
is a mapping with f(0) = 0 and
\| D\alpha f(x, y), z\| \leq \| x\| p + \| y\| p + \| z\|
for all x, y, z \in X. Then there exists a unique cubic mapping C : X \rightarrow Y such that
\| f(x) - C(x), z\| \leq 1
2
\| x\| p + \| z\|
\alpha 3 - q3
for all x, z \in X.
Corollary 2.5. Let r, s be nonnegative real numbers such that r + s < 3. Suppose that f :
X - \rightarrow Y is a mapping with f(0) = 0 and
\| D\alpha f(x, y), z\| \leq \| x\| r\| y\| r\| z\| p
for all x, y, z \in X. Then f is cubic.
References
1. Aoki T. On the stability of the linear transformation in Banach spaces // J. Math. Soc. Jap. – 1950. – 2. – P. 64 – 66.
2. Aczél J., Dhombres J. Functional equations in several variables. – Cambridge Univ. Press, 1989.
3. Benyamini Y., Lindenstrauss J. Geometric nonlinear functional analysis, vol. 1 // Colloq. Publ. – Providence, RI:
Amer. Math. Soc., 2000. – 48.
4. Czerwik S. Functional equations and inequalities in several variables. – New Jersey etc.: World Sci., 2002.
5. Eskandani G. Z., Rassias J. M., Gavruta P. Generalized Hyers – Ulam stability for a general cubic functional equation
in quasinormed spaces // Asian-Eur. J. Math. – 2003. – 4. – P. 413 – 425.
6. Ghler S. 2-metrische Rume und ihre topologische Struktur // Math. Nachr. – 1963. – 26. – S. 115 – 148.
7. Ghler S. Lineare 2-normierte Rumen // Math. Nachr. – 1964. – 28. – P. 1 – 43.
8. Găvruta P. On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings // J. Math. Anal. and
Appl. – 2001. – 261. – P. 543 – 553.
9. Găvruta P. An answer to question of John M. Rassias concerning the stability of Cauchy equation // Adv. Equat. and
Inequality. – 1999. – P. 67 – 71.
10. Găvruta P. A generalization of the Hyers – Ulam – Rassias stability of approximately additive mappings // J. Math.
Anal. and Appl. – 1994. – 184. – P. 431 – 436.
11. Hyers D. H. On the stability of the linear functional equation // Proc. Nat. Acad. Sci. – 1941. – 27. – P. 222 – 224.
12. Hyers D. H., Isac G., Rassias Th. M. Stability of functional equations in several variables. – Basel: Birkhäuser, 1998.
13. Jun K., Kim H. The generalized Hyers – Ulam – Rassias stability of a cubic functional equation // J. Math. Anal. and
Appl. – 2002. – 274. – P. 867 – 878.
14. Jun K., Kim H., Chang I. On the Hyers – Ulam stability of an Euler – Lagrange type cubic functional equation // J.
Comput. Anal. and Appl. – 2005. – 7. – P. 21 – 33.
15. Jung S. M. Hyers – Ulam – Rassias stability of functional equations in mathimatical analysis. – Palm Harbor: Hadronic
Press, 2001.
16. Park C. Homomorphisms between Poisson JC\ast -algebras // Bull. Braz. Math. Soc. – 2005. – 36. – P. 79 – 97.
17. Park C. Additive functional inequalities in 2-Banach spaces // J. Inequal. and Appl. – 2013. – 447. – P. 1 – 10.
18. Park W. G. Approximate additive mappings in 2-Banach spaces and related topics // J. Math. Anal. and Appl. –
2011. – 376. – P. 193 – 202.
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
1436 G. Z. ESKANDANI, J. M. RASSIAS
19. Rassias J. M. On approximation of approximately linear mappings by linear mappings // J. Funct. Anal. – 1982. –
46, № 1. – P. 126 – 130.
20. Rassias J. M. On approximation of approximately linear mappings by linear mappings // Bull. Sci. Math. – 1984. –
108, № 4. – P. 445 – 446.
21. Rassias J. M. Solution of a problem of Ulam // J. Approxim. Theory. – 1989. – 57, № 3. – P. 268 – 273.
22. Rassias J. M. Solution of a stability problem of Ulam // Discuss. Math. – 1992. – 12. – P. 95 – 103.
23. Rassias J. M. On the stability of the non-linear Euler – Lagrange functional equation // Chinese J. Math. – 1992. –
20, № 2. – P. 185 – 190.
24. Rassias Th. M. On the stability of the linear mapping in Banach spaces // Proc. Amer. Math. Soc. – 1978. – 72. –
P. 297 – 300.
25. Rassias Th. M. On a modified Hyers – Ulam sequence // J. Math. Anal. and Appl. – 1991. – 158. – P. 106 – 113.
26. Rassias Th. M., S̆emrl P. On the behaviour of mappings which do not satisfy Hyers – Ulam stability // Proc. Amer.
Math. Soc. – 1992. – 114. – P. 989 – 993.
27. Rassias Th. M., Tabor J. What is left of Hyers – Ulam stability? // J. Natur. Geom. – 1992. – 1. – P. 65 – 69.
28. Ravi K., Arunkumar M., Rassias J. M. Ulam stability for the orthogonally general Euler – Lagrange type functional
equation // Int. J. Math. Statist. – 2008. – P. 36 – 46.
29. Ulam S. M. A collection of the mathematical problems. – New York: Intersci. Publ., 1960. – P. 431 – 436.
Received 11.11.14,
after revision — 31.03.16
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 10
|
| id | umjimathkievua-article-1930 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:15:25Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/74/120d7589e74aeddb2fe874a99f229474.pdf |
| spelling | umjimathkievua-article-19302019-12-05T09:31:57Z Approximation of general α-cubic functional equations in 2-Banach spaces Наближення загальних α -кубiчних функцiональних рiвнянь у 2 -банахових просторах Eskandani, G. Z. Rassias, J. M. Ескандані, Г. З. Расіас, Дж. М. We introduce a new \alpha -cubic functional equation and investigate the generalized Hyers – Ulam stability of this functional equation in 2-Banach spaces. Введено нове \alpha -кубiчне функцiональне рiвняння та вивчено узагальнену стiйкiсть Хайєрса –Улама цього функцiонального рiвняння в 2-банахових просторах. Institute of Mathematics, NAS of Ukraine 2016-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1930 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 10 (2016); 1430-1436 Український математичний журнал; Том 68 № 10 (2016); 1430-1436 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1930/912 Copyright (c) 2016 Eskandani G. Z.; Rassias J. M. |
| spellingShingle | Eskandani, G. Z. Rassias, J. M. Ескандані, Г. З. Расіас, Дж. М. Approximation of general α-cubic functional equations in 2-Banach spaces |
| title | Approximation of general α-cubic functional equations
in 2-Banach spaces |
| title_alt | Наближення загальних α -кубiчних функцiональних рiвнянь у 2 -банахових просторах |
| title_full | Approximation of general α-cubic functional equations
in 2-Banach spaces |
| title_fullStr | Approximation of general α-cubic functional equations
in 2-Banach spaces |
| title_full_unstemmed | Approximation of general α-cubic functional equations
in 2-Banach spaces |
| title_short | Approximation of general α-cubic functional equations
in 2-Banach spaces |
| title_sort | approximation of general α-cubic functional equations
in 2-banach spaces |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1930 |
| work_keys_str_mv | AT eskandanigz approximationofgeneralacubicfunctionalequationsin2banachspaces AT rassiasjm approximationofgeneralacubicfunctionalequationsin2banachspaces AT eskandanígz approximationofgeneralacubicfunctionalequationsin2banachspaces AT rasíasdžm approximationofgeneralacubicfunctionalequationsin2banachspaces AT eskandanigz nabližennâzagalʹnihakubičnihfunkcionalʹnihrivnânʹu2banahovihprostorah AT rassiasjm nabližennâzagalʹnihakubičnihfunkcionalʹnihrivnânʹu2banahovihprostorah AT eskandanígz nabližennâzagalʹnihakubičnihfunkcionalʹnihrivnânʹu2banahovihprostorah AT rasíasdžm nabližennâzagalʹnihakubičnihfunkcionalʹnihrivnânʹu2banahovihprostorah |