Building Extended Homomorphism on Fuzzy Banach Algebra Based on Jensen Equation with 2k-Variables by Fixed Point Methods and Direct Methods ()
1. Introduction
Let
and
are two fuzzy normed vector spaces on the same field
, and map
be continuously on
. We use the notation
, N for corresponding the norms on
and
. In this paper, we investigate the stability of generalized Jensen-type additive function equation with 2k-variables when
is a fuzzy normed-algebras with norm
and
is a fuzzy Banach algebras with norm N.
In fact, when
is a fuzzy normed algebras with norm
and
is a fuzzy Banach algebras with norm N, we solve and prove the Hyers-Ulam-Rassias type stability of generalized Jensen-type additive function equation in fuzzy Banach algebras, associated to the Jensen type additive functional equation
(1)
The study of the stability of generalized Jensen-type additive function equation in fuzzy Banach algebras is 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
, there exists a
such that if
satisfies
then there is a homomorphism
with
Since Hyers’ answer to Ulam’s question [2] , many ideas have arisen from mathematicians who have built theories about space such as the Theory of fuzzy space. It has much progressed in developing the theory of randomness. Some mathematicians have defined fuzzy norms on a vector space from various points of view. Following Bag and Samanta [3] and Cheng and Mordeson [4] gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric was of Kramosil and Michalek type [5] and investigated some properties of fuzzy normed spaces. We use the definition of fuzzy normed spaces given in [3] [6] [7] [8] to investigate a fuzzy version of the Hyers-Ulam stability for the Jensen functional equation in the fuzzy normed algebra setting.
The functional equation
is called a quadratic functional equation. The Hyers-Ulam stability of the quadratic functional equation was proved by Skof [9] for mapping
, where
is a normed space and
is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. Czerwik [11] proved the Hyers-Ulam stability of the quadratic functional equation.
The stability problems for several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem. Such as in 2008, Choonkil Park [12] have established and investigated the Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras the following Jensen functional equation
And next in 2009, M. Éhaghi Gordji and M. Bavand Savadkouhi [13] have established and investigated the approximation of generalized stability of homomorphisms in quasi-Banach algebras the following Jensen functional equation
Next, in 2022 Ly Van An [14] have established and investigated the approximation of generalized stability of homomorphisms in quasi-Banach algebras the following Jensen type functional equation
(2)
Recently, Ly Van An continued to conduct extensive research (1.2) in the Hyers-Ulam-Rassias type on fuzzy Banach algebras for the following equation
i.e., the functional equation with 2k-variables. Under suitable assumptions on spaces
and
, we will prove that the mappings satisfying the functional (1). Thus, the results in this paper are generalization of those in [12] [13] [14] for functional equation with 2k-variables.
In this paper, I build a general homomorphism based on Jensen equation with 2k-variables on fuzzy Banach algebra. This is an extended problem for the field of homotopy research, exploiting unlimited problems of variables to build this problem based on the ideas of mathematicians around the world. See [1] - [30] . Allow me to express my deep gratitude to the mathematicians.
The paper is organized as follows:
In Section 2, we remind some basic notations in [3] [6] [7] [8] [16] [25] [30] such as Fuzzy normed spaces, extended metric space theorem and solutions of the Jensen function equation.
Section 3: Using the fixed point method, establish extended homomorphisms on fuzzy Banach algebra.
Section 4: Using the direct method, establish extended homomorphisms on fuzzy Banach algebra.
2. Preliminaries
2.1. Fuzzy Normed Spaces
Let X be a real vector space. A function
is called a fuzzy norm on X if it stabilities the following conditions: for all
and
,
1) (N1)
for
;
2) (N2)
if and only if
for all
;
3) (N3)
if
4) (N4)
;
5) (N5)
is a non-decreasing function of
and
;
6) (N6) for
,
is continuous on
.
The pair
is called a fuzzy normed vector space
1) Let
be a fuzzy normed vector space. A sequence
in X is said to be convergent or converge if there exists an
such that
for all
. In this case, x is called the limit of the sequence
and we denote it by
.
2) Let
be a fuzzy normed vector space. A sequence
in X is called Cauchy if for each
and each
there exists an
such that for all
and all
, we have
.
It is well-known that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space. We say that a mapping
between fuzzy normed vector spaces X and Y is continuous at a point
if for each sequence
converging to
in X, then the sequence
converges to
. If
is continuous at each
, then
is said to be continuous on X. Let X be an algebra and
a fuzzy normed space.
1) The fuzzy normed space
is called a fuzzy normed algebra if
for all
and all positive real numbers s and t.
2) A complete fuzzy normed algebra is called a fuzzy Banach algebra.
EXAMPLE
Let
be a normed algebra. Let
Then
is a fuzzy norm on X and
is a fuzzy normed algebra. Let
and
be fuzzy normed algebras. Then a multiplicative
-linear mapping
is called a fuzzy algebra homomorphism.
2.2. Extended Metric Space Theorem
Theorem 1. Let
be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. Then for each given element
, either
for all nonnegative integers n or there exists a positive integer
such that
1)
,
;
2) The sequence
converges to a fixed point
of J;
3)
is the unique fixed point of J in the set
;
4)
2.3. Solutions of the Equation
The functional equation
is called the Jensen equation. In particular, every solution of the Jensen equation is said to be a Jensen-additive mapping.
2.4. Complete Generalized Metric Space and Solutions of the Inequalities
Theorem 2. Let
be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. Then for each given element
, either
for all nonnegative integers n or there exists a positive integer
such that
1)
,
;
2) The sequence
converges to a fixed point
of J;
3)
is the unique fixed point of J in the set
;
4)
.
2.5. Solutions of the Inequalities
The functional equation
is called the Cauchy equation. In particular, every solution of the Cauchy equation is said to be an additive mapping.
3. Using the Fixed Point Method, Establish Extended Homomorphisms on Fuzzy Banach Algebra
Now we study extended homomorphism by fixed point method.
When
is a fuzzy normed algebra with quasi-norm
,
is a fuzzy Banach algebras with norm N. Under this setting, we need to show that the mapping must satisfy (1). These results are given in the following.
Here we assume that
is a positive integer and
.
Theorem 3. Suppose
be a function such that there exists an
(3)
for all
for
. If
be a mapping satisfying
and
(4)
(5)
for all
for
, for all
and all
. Then
exists for each
and defines a fuzzy algebras generalized homomorphism
such that
(6)
Proof. Putting
.
Replacing
by
in hypothesis (4), we have
(7)
for all
.
Now we consider the set
and introduce the generalized metric on
as follows:
where, as usual,
. That has been proven by mathematicians
is complete [18] Now we consider the linear mapping
such that
for all
. Let
be given such that
then
Hence
(8)
So
implies
. This means that
for all
. On ther hand, (6) implies that
.
By Theorem 2.5, there exists a mapping
satisfying the following:
(1) A is a fixed point of T, i.e.,
(9)
for all
. The mapping A is a unique fixed point T in the set
This implies that A is a unique mapping satisfying (9) such that there exists a
satisfying.
(2)
as
. This implies equality
for all
(3)
,
which implies the inequality
This implies that the inequality (6)
By (4), I have
(10)
for all
,
,
. So
(11)
for all
,
,
. So
Since
for all
,
,
. So
(12)
for all
,
,
. So we have
for all
,
,
. So the mapping
is additive and
-linear. From (5)
(13)
for all
, and all
.So
(14)
for all
, and all
. Since
for all
, and all
,
(15)
Thus
for all
, and all
. So that mapping
is a fuzzy algebra generalized homomorphism, as desired.
□
Theorem 4. Suppose
be a function such that there exists an
(16)
for all
, if
be a mapping satisfying
and
(17)
(18)
for all
, all
and all
. Then
exists for each
and defines a fuzzy algebras generalized homomorphism
such that
(19)
for all
and all
.
Proof. Let
be the generalized metric space defined on the proof of Theorem 3. Now we consider the linear mapping
such that
for all
.
Next putting
.
Replacing
by
in hypothesis (17), we have
(20)
for all
, all
. So
(21)
for all
, all
.
Thus
Hence
which implies that the inequality (19) holds. The rest of the proof is similar to the proof of Theorem 3.
□
4. Using the Direct Method, Establish Extended Homomorphisms on Fuzzy Banach Algebra
Now we study extended homomorphism by direct method.
Where
is a fuzzy normed algebra with quasi-norm
,
is a fuzzy Banach algebras with norm N. Under this setting, we need to show that the mapping must satisfy (1). These results are given in the following.
Here we assume that
is a positive integer and
.
Theorem 5. Suppose
be a function such that
(22)
for all
, if
be a mapping satisfying
and
(23)
uniformly on
for each
, and
(24)
uniformly on
, where
(25)
for all
, then
exists for each
and defines a fuzzy algebras generalized homomorphism
such that if for each
(26)
for all
, then
(27)
for all
.
Furthermore, the fuzzy algebra generalized homomorphism
is a unique mapping such that
(28)
uniformly on
.
Proof. We put
in (23).With
, by (23), we can exist some
such that
(29)
for all
. Next we replace
by
in hypothesis (23), and we have
(30)
for all
. By induction on n, we will show that
(31)
for all
, for all
, all
. It follows from (30) and (31) holds for
We now assume that (31) satisfies all
. Then
(32)
This completes the induction argument. Letting
and we replace n and x by q and
in (31), respectively, we get
(33)
for all
. It follows from (22) and the equality
(34)
That for a given
there is an
such that
(35)
for all
and
. Now we deuce since (31) that
(36)
for all
, all
and all
. It follows from (36) that the sequence
is a Cauchy sequence for all
. Since
is a fuzzy complete (fuzzy Banach space), the sequence
converges. So one can define the mapping
by
(37)
In other words, for each
and
(38)
Now we are fixed
and
. Since
there is an
such that
Hence for each
, we get
(39)
for all
, for all
and for all
. It follows from (30) and (31) holds for
We now assume that (31) satisfies all
. Then
(40)
for all
and all
.
Thus
for all
, by
,
Hence the mapping
is additive.
Next we replace
by
in hypothesis (23).
, by (23), then exists
such that
(41)
It follows from (41), we have
for all
and all
.
Similarly, it follows from (24) that
for all
. We now assume that
and
satisfied (26). Put
(42)
for all
.
Suppose by the same reasoning as in the beginning of the proof, one can deuce from (26) that
(43)
for all
, then for all positive integer n. Suppose
we have
(44)
Combining (43) and (44). If
be a mapping satisfying
and the fact that
, we observe that
For large enough
. Thanks to the continuity of the function
we see that
Now I give
, we conclude that
In the end I still have to prove the uniqueness. Suppose
be another additive mapping satisfying (27) and (28). Fix
. Given
, follow (28) for A, and
, then exist
such that
for all
and
. With fixed
then exists
such that
for all
. From
(45)
(46)
It follows that
for all
. So
,
. □
Theorem 6. Suppose
be a function such that
(47)
for all
. If
be a mapping satisfying
and
(48)
uniformly on
for each
, and
(49)
uniformly on
, where
(50)
for all
. Then
exists for each
and defines a fuzzy algebras generalized homomorphism
such that if for each
(51)
for all
, then
(52)
for all
.
Furthermore, the fuzzy algebra generalized homomorphism
is a unique mapping such that
(53)
uniformly on
.
5. Conclusion
In this paper, I built the existence of extended homomorphism on fuzzy Banach algebra based on Jensen equation 2k variables by two methods such as fixed point and direct method to check.