Extension of Homomorphisms-Isomorphisms and Derivatives on Quasi-Banach Algebra Based on the General Additive Cauchy-Jensen Equation ()
1. Introduction
Let
and
are two linear spaces on the same field
, and
be a linear mapping. I use the notation
(
) for corresponding the norms on
and
. In this paper, I investigate the stability of generalized homomorphisms-isomorphism and derivatives when
is a quasi-normed algebras with quasi-norm
and that
is a p-Banach algebras with p-norm
.
In fact, when
is a quasi-normed algebras with quasi-norm
and that
is a p-Banach algebras with p-norm
, I solve and prove the Hyers-Ulam-Rassias type stability of generalized Homomorphisms-isomorphism and derivatives on quasi-Banach algebra, associated to the following generalized Cauchy-Jensen additive functional equations
(1)
(2)
(3)
The study the stability of generalized homomorphisms-isomorphism and derivatives in quasi-Banach algebras originated from a question of S. M. Ulam [1] , concerning the stability of group homomorphisms.
Let
be a group and let
be a metric group with metric
. Given
, does there exist a
such that if
satisfies inequality
then there is a homomorphism
with
If the answer is affirmative, I would say that the equation of homomorphism
is stable. The concept of stability for a functional equation arises when I replace the functional equation by an inequality which acts as a perturbation of the equation. Thus the stability question of functional equations is that how do the solutions of the inequality differ from those of the given functional equation? In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces.
Let X and Y be Banach space. Assume that
satisfies
(4)
for all
and some
. Then there exists a unique additive mapping
such that
(5)
Next 1978 Th. M. Rassias [3] provided a generalization of Hyers’ Theorem which allows the Cauchy difference to be unbounded.
J. M. Rassias [4] [5] [6] built no continuity conditions are required for this result, but if f(tx) is continuous in the real variable t for each fixed
, then L is linear, and if f is continuous at a single point of E then
is also continuous. J. M. Rassias assumed the following weaker inequality
(6)
; involving a product of different powers of norms, where
and real
such that
, and retained the condition of continuity
in t for fixed x. Analogous results could be investigated with additive type equations involving a product of powers of norms.
Next 1994 Găvruta [7] Generalized the Rassias’ result. There are also many mathematicians who have built many results for this topic as [1] - [24] .
Recently, the authors studied the Hyers-Ulam-Rassias type stability for the following functional equations (see [8] [9] [10] [11] [12] )
(7)
(8)
(9)
Next
(10)
and
(11)
Final
(12)
In this paper, I have built a general problem about homomorphisms-isomorphism and derivatives on quasi-Banach algebra based on the general additive Cauchy-Jensen equation. To write these problems, I follow the ideas of mathematicians around the world see [1] - [24] . In order to provide researchers in Mathematics when building problems, there is no restriction on variables for the problem. This is what I consider an open dream problem or a bright horizon for the field of functional equations in quasi-Banach algebras.
In this paper, I solve and proved the Hyers-Ulam-Rassias type stability for functional Equations (1.1), (1.2) and (1.3), i.e., the functional equations with 3k variables. Under suitable assumptions on spaces
and
, I will prove that the mappings satisfying the functional Equation (1.1), (1.2) and (1.3).
Thus, the results in this paper are generalization of those in [8] [9] [10] [11] [12] [24] for functional equations with 3k variables. The paper is organized as follows:
In section preliminarier I remind some basic notations in such as Quasi-normed space―Quasi-Banach algebras. Some theorems
-linear mapping and Solutions of the equations see ( [3] [4] [5] [6] ).
Section 3: Constructing homomorphisms on quasi-Banach algebras for (1.1).
Section 4: Constructing isomorphisms on quasi-Banach algebras for (1.2).
Section 5: Constructing derivatives on quasi-Banach algebras for (1.3).
2. Preliminaries
2.1. Quasi-Normed Space―Quasi-Banach Algebras
Let
be a real linear space. A quasi-norm is a real-valued function on
satisfying the following:
1)
for all
and
if and only if
.
2)
for all
and all
.
3) There is a constant
such that
The pair
is called a quasi-normed space if
is a quasi-norm on
.
The smallest possible K is called the modulus of concavity of
.
A quasi-Banach space is a complete quasi-normed space.
A quasi-norm
is called a p-norm (
) if
In this case, a quasi-Banach space is called a p-Banach space.
Note: Given a p-norm, the formula
gives us a translation invariant metric on X. By the Aoki-Rolewicz Theorem [13] (see also [14] ), each quasi-norm is equivalent to some p-norm. Since it is much easier to work with p-norms, henceforth I restrict my attention mainly to p-norms.
Let
be a quasi-normed space. The quasi-normed space
is called a quasi-normed algebras if
is an algebras and there is a constant
such that
A quasi-Banach algebras is a complete quasi-normed algebras.
If the quasi-norm
is a p-norm, quasi-Banach is called p-Banach algebras.
2.2. Some Theorems
-Linear Mapping
Theorem 1. Th. M. Rassias: Let
be a mapping from a normed vector space
into a Banach space
subject to the inequality
(13)
for all
, where
and p are constants with
and
. then the limit
(14)
exists for all
and that
is the unique additive mapping satisfying
(15)
If
then (2.1) holds for
and (2.2) for
. Also, if for each
the function If f(tx) is continuous in
, then T is linear.
Theorem 2. Let
be real normed linear space and
a real complete normed linear space. Assume that
is an approximately additive mapping for which there exists constants
and
such that
satisfy inequality
(16)
then there exists a unique additive mapping linear
satisfies
(17)
If, in addition
is a transformation
is continuous in
for each fixed
, then T is an
-linear mapping.
Theorem 3. Let
be real normed linear space and
a real complete normed linear space. Assume that
is an approximately additive mapping for which there exists constants
such that
satisfy inequality
(18)
and
is a non-negative real-valued function such that
(19)
is a non-negative function of x, and the condition
(20)
holds then there exists a unique additive mapping
satisfies
(21)
If, in addition
is a transformation
is continuous in
for each fixed
, then
is an
-linear mapping.
Theorem 4. Let
be real normed linear space and
a real complete normed linear space. Assume that
is an approximately additive mapping for which there exists constants
and
such that
and f satisfy inequality
(22)
then there exists a unique additive mapping linear
satisfies
(23)
If, in addition
is a transformation
is continuous in
for each fixed
, then T is an
-linear mapping.
2.3. Solutions of the Equation
(24)
is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping.
The functional equation
(25)
is called the Jensen equation. In particular, every solution of the Jensen equation is said to be a Jensen additive mapping.
The functional equation
(26)
is called the Cauchy-Jensen equation. In particular, every solution of the Cauchy-Jensen equation is said to be a Jensen-Cauchy additive mapping.
3. Constructing Homomorphisms in Quasi-Banach Algebras
Now I construct a homomorphism for (1.1) Note that: (1.2) and (1.3) are also built exactly the same.
Here I assume that,
is a quasi-normed with norm
and that
is a p-Banach algebra with norm
. Let
be the modulus of concavity of
. Under this setting, I can show that the mappings satisfying (1.1) is homomorphisms.
Theorem 5. Let
with
and
be positive real numbers, and
be a mapping such that
(27)
(28)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique homomorphism
such that
(29)
Proof. I replace
by
in (27), I have
(30)
for all
. So
(31)
for all
. From
is p-Banach algebra so I have
(32)
for all nonnegative integers m and l with
and for all
. It follows (32) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
(33)
for all
. By (28) and (27),
(34)
for all
, for all
.
So
(35)
for all
, for all
. By lemma 5 (see 24]), the mapping
is Cauchy additive. (see the theorem of [3] )
Then mapping
is
-linear. It follows from (28) that
(36)
.
So
(37)
.
Now I prove the uniqueness of H. Assume that
is a Cauchy-Jensen additive mapping satisfying (29). Then I have
(38)
which tends to zero as
for all
. So I can conclude that
for all
. This proves the uniqueness of H. Thus the mapping
is a unique homomorphism satisfying (29). □
Theorem 6. Let
with
and
be positive real numbers, and
be a mapping such that
(39)
(40)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique homomorphism
such that
(41)
Proof. I replace
by
in (39), I have
(42)
for all
. So
(43)
for all
. Since
is a p-Banach algebra,
(44)
for all nonnegative integers m and l with
and for all
. It follows (44) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
for all
.
Moreover, letting
and passing the limit
in (44) I get (41). The rest of the proof is similar to the proof of Theorem 5. □
Theorem 7. Let
, with
and
be positive real numbers, and
be a mapping such that
(45)
(46)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique homomorphism
such that
(47)
Proof. I replace
by
in (45), I have
(48)
for all
. So
(49)
for all
. Since
is a p-Banach algebras,
(50)
for all nonnegative integers m and l with
and for all
. It follows (50) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
(51)
for all
. By (46) and (45),
(52)
for all
, for all
.
So
(53)
for all
. By lemma 5 (see [24] ), the mapping
is Cauchy additive. By proving as proof of the theorem of [3] the mapping
is
-linear.
.
It follows from (46) that
(54)
.
So
(55)
.
The rest of the proof is similar to the proof of Theorem 5 □
Theorem 8. Let
, with
and
be positive real numbers, and
be a mapping such that
(56)
(57)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique homomorphism
such that
(58)
Proof. I replace
by
in (56), I have
(59)
for all
. So
for all
. Since
is a p-Banach algebras,
(60)
for all nonnegative integers m and l with
and for all
. It follows (60) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
for all
. Moreover, letting
and passing the
in (60) I have (56),
It follows from (57) that
. So
.
The rest of the proof is similar to the proof of Theorem 5. □
4. Constructing Isomorphisms Based on Quasi-Banach Algebras
Now I construct isomorphisms for (1.2). Note that: (1.1) and (1.3) are also built exactly the same.
Here I assume that
is a quasi-Banach with norm
and unit e and that
is a p-Banach algebra with norm
and unit
. Let
be the modulus of concavity of
. Under this setting, I can show that the mappings satisfying (1.2) is isomorphisms.
Theorem 9. Let
with
and
be positive real numbers, and
be a mapping such that
(61)
(62)
for all
, for all
. If
is continuous in
each fixed
and
(63)
then the mapping
is an isomorphism.
Proof. I replace
by
in (61), I have
(64)
for all
. So
(65)
for all
. Since
is a p-Banach algebra,
(66)
for all nonnegative integers m and l with
and for all
. It follows (66) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
(67)
for all
. Moreover, letting
and passing the limit
in (66), I get
(68)
It follows from (62) that
(69)
for all
, for all
.
So
(70)
for all
, for all
. By lemma 5 (see [24] ), the mapping
is Cauchy additive. See the theorem of [3] .
The mapping
is
-linear. Since
(71)
for all
, for all
.
(72)
for all
, for all
. So the mapping
is homomorphism.
It follows from (62) that
(73)
. So the mapping
is an isomorphism. □
Theorem 10. Let
with
and
be positive real numbers, and
be a mapping such that
(74)
(75)
for all
, for all
. If
is continuous in
each fixed
and
(76)
then the mapping
is an isomorphism.
The rest of the proof is similar to the proof of Theorem 9.
Theorem 11. Let
with
and
be positive real numbers, and
be a mapping such that
(77)
(78)
for all
, for all
. If
is continuous in
each fixed
and
(79)
then the mapping
is an isomorphism.
Proof. I replace
by
in (77), I have
(80)
for all
. So
(81)
for all
. Hence
(82)
for all nonnegative integers m and l with
and for all
. It follows (82) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
(83)
for all
.
Moreover, letting
and passing the limit
in (82), I get
(84)
The rest of the proof is similar to the proof of Theorems 7 and 9. □
Theorem 12. Let
with
and
be positive real numbers, and
be a mapping such that
(85)
(86)
for all
, for all
. If
is continuous in
each fixed
and
(87)
then the mapping
is an isomorphism.
The rest of the proof is similar to the proof of Theorems 8 and 9.
5. Constructing Derivatives on Quasi-Banach Algebras
Now I construct derivatives for (1.3). Note that: (1.1) and (1.2) are also built exactly the same.
Here I assume that,
is a p-Banach algebras with norm
Let
be the modulus of concavity of
. Under this setting, I can show that the mappings satisfying (1.3) is generalized derivation.
A generalized derivations
is linear and fulfills the generalized Leibniz rue
(88)
for all
.
Theorem 13. Let
with
and
be positive real numbers, and
be a mapping such that
(89)
(90)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique generalized derivation
such that
(91)
Proof. I replace
by
in (89), I have
(92)
for all
. So
for all
. Since
is a p-Banach algebra,
(93)
for all nonnegative integers m and l with
and for all
. It follows (93) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
for all
. Moreover, letting
and passing the limit
in (93), I get (91). It follows from (89) that
for all
, for all
.
So
for all
, for all
. By lemma 5 (see [24] ), the mapping
is Cauchy additive. By the theorem of [3] the mapping
is
-linear.
It follows from (90) that
(94)
for all
, for all
. So
(95)
for all
, for all
.
Now I prove the uniqueness of
. Assume that
is a Cauchy-Jensen additive mapping satisfying (91). Then I have
which tends to zero as
for all
. So I can conclude that
for all
. This proves the uniqueness of
. Thus the mapping
is a unique generalized derivation satisfying (91). □
Theorem 14. Let
with
and
be positive real numbers, and
be a mapping such that
(96)
(97)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique generalized derivation
such that
(98)
The rest of the proof is similar to the proof of Theorem 13.
Theorem 15. Let
with
and
be positive real numbers, and
be a mapping such that
(99)
(100)
for all
, for all
. If
is continuous in
each fixed
, then there exists a unique generalized derivation
such that
(101)
Proof. I replace
by
in (99), I have
(102)
for all
. So
for all
. Since
is a p-Banach algebra,
(103)
for all nonnegative integers m and l with
and for all
. It follows (103) that the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges. So one can define the mapping
by
for all
. Moreover, letting
and passing the limit
in (103), I get (101). It follows from (99) that
(104)
for all
.
So
(105)
for all
. By lemma 5 (see [24] ), the mapping
is Cauchy additive. By the theorem of [3] the mapping
is
-linear.
It follows from (100) that
(106)
for all
. So
(107)
for all
.
Now I prove the uniqueness of
. Assume that
is a Cauchy-Jensen additive mapping satisfying (101). Then I have
(108)
which tends to zero as
for all
. So I can conclude that
for all
. This proves the uniqueness of
. Thus the mapping
is a unique generalized derivation satisfying (101). □
6. Conclusion
In this paper, I construct extensions of homomorphisms, isomorphisms, and derivatives based on Banach algebra. The fundamental contribution here is the development of a general Cauchy-Jensen equation, which serves as the cornerstone for establishing mathematical links across various research areas in mathematics, without any restrictions on generality.