General Solution and Stability of Quattuordecic Functional Equation in Quasi β-Normed Spaces ()
1. Introduction
The first stability problem concerning group homomorphisms was raised by Ulam [1] in 1940. He stated that if
is a group and let
be a metric group with metric
: Given
, does there exist a δ > 0 such that if a mapping
satisfies the inequality
for all
, then there exists a homomorphism
with
for all
?
The case of approximately additive functions was solved by D. H. Hyers [2] under the assumption that both E1 and E2 are Banach spaces. He stated that for
and
such that
for all
, then there exists a unique additive mapping
such that
for all
. This result is called Hyers-Ulam stability.
Hyers Theorem was generalized by Th. M. Rassias [3] for linear mappings by considering an unbounded Cauchy difference. The stability problem of several functional equations has been extensively investigated by a number of authors, and there are many interesting results concerning this problem [4] - [17] .
Very recently the general solution and the stability of the quintic and sextic functional equation in quasi-b-normed spaces via fixed point method were discussed by [18] . The general solution, the stability of the septic and Octic functional equations, viz.
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and
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in quasi-b -normed spaces were investigated by T. Z. Xu et al. [18] .
J. M. Rassias and Mohamed Eslamian discussed the general solution of a Nonic functional equation
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and proved the stability of nonic functional equation [19] in quasi-b-normed spaces by applying the fixed point method.
A fixed point approach for the stability of Decic functional equation
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in quasi-b-normed spaces was investigated by K. Ravi et al. [20] .
Very recently, K. Ravi and Senthil Kumar discussed the undecic and duodecic functional equation and its stability in quasi-b-normed spaces.
In this paper, the authors are interested in finding the general solution and stability of Quattuordecic functional equation
(1)
where
in quasi-b-normed spaces by using fixed point method.
The functional Equation (1) is called Quattourdecic functional equation because the function
satisfies the Equation (1).
In Section 2, we have given necessary definitions. In Section 3, we discuss the general solution of the functional Equation (1). In Section 4, we investigate the stability of Quattuordecic functional Equation (1) in quasi-b-normed spaces and we provide a counter example to show that the functional Equation (1) is not stable.
2. Preliminaries
We recall some basic concepts concerning quasi-b-normed spaces introduced by J. M. Rassias and H. M. Kim [14] in 2009. Let b be a fixed real number with
, and let K denote either R or C. Let X be linear space over K. A quasi-b-norm
is a real valued function on X satisfying the following three conditions:
1)
, for all
; and
iff
,
2)
for all
, and all
,
3) there is a constant
such that
.
For all
. A quasi-b -normed space is a pair
, where
is a quasi-b on X. The smallest possible K is called the modules of concavity of
. A quasi-b-Ba- nach space is a complete quasi-b-normed space. A quasi-b-norm
is called a
-norm
if
.
In this space a quasi-b-Banach space is called a
-Banach space. We can refer to [18] for the concept of quasi-normed spaces and p-Banach spaces. Given a p -norm, the formula
gives us a translation invariant metric on X. By the Aoki-Rolewicz theorem, each quasi-norm is equal to some p-norm. Since it is much easier to work with p-norms then quasi-norms, we restrict our attention mainly to p-norms.
Using fixed point theorem, Xu et al. [18] proved the following impotent lemma.
Lemma 1. Let
be fixed,
with
, and
be a function such that there exists an
with
for all
. Let
be a mapping satisfying
(2)
Then there exists a uniquely determined mapping
such that
(3)
3. General Solution of Functional Equation
In this section, let X and Y be vector spaces. In the following Theorem, we investigate the general solution of the functional Equation (1).
Theorem 1. A function
is a solution of the Quattuordecic functional Equation (1) if and only if f is of the form
for all
, where
is the diagonal of the 14-additive symmetric mapping
.
Proof. Assume that f satisfies the functional Equation (1). Replacing
by
in (1), we have
. Replacing
by
in (1), we get
(4)
Substituting
by
in (1), we obtain
(5)
Subtracting Equations (5) and (4), we get
(6)
Replacing
with
in (1), one gets
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and
(7)
Replacing
with
in (1), one gets
(8)
Subtracting the Equations (7) and (8), we obtain
(9)
Replacing
with
in (1) and multiplying by 14, we have
(10)
Subtracting Equations (9) and (10), we obtain
(11)
Replacing
with
in (1) and multiplying by 91, we have
(12)
Subtracting Equations (11) and (12), we have
(13)
Replacing
with
in (1) and multiplying by 364, we have
(14)
Subtracting Equations (13) and (14), we obtain
(15)
Replacing
with
in (1) and multiplying by 1001, we obtain
(16)
Subtracting Equations (15) and (16), one gets
(17)
Replacing
with
in (1) and multiplying by 2002, we have
(18)
Subtracting Equations (17) and (18), we obtain
(19)
Replacing
with
in (1) and multiply by 3003, we have
(20)
Subtracting Equations (19) and (20), one gets
(21)
Replacing
with
in (1) and multiplying by 1716, we have
(22)
Subtracting Equations (20) and (21), we have
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or
(23)
On the other hand, one can rewrite the functional Equation (1) in the form
(24)
for all
. By ( [17] , Theorems 3.5 and 3.6), f is a generalized polynomial function of degree at most 14, that is, f is of the form
(25)
for all
.
Here,
is an arbitrary element of y and
is the diagonal of the i-addi- tive symmetric map
(
) by
and
, for all
, we get
and the function f is even. Thus, we have
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it follows that
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Using Equations (25) and
, we obtain
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for all
and
. It follows that
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for all
. Hence
.
Conversely, assume that
for all
, where
is the diagonal of the 14-additive symmetric map
from
(26)
and
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for all
and
. We see that f satisfies the Equation (1). This completes the proof of the Theorem.
4. Stability of Quattuordecic Functional Equation
Throughout this section, we assume that X is a linear space, Y is a
Banach space with
-norm
. Let K be the modulus of concavity of
. We establish the following stability for the Quarttuordecic functional equation in quasi b-normed spaces. For a given mapping
, we define the difference operator
(27)
Theorem 2. Let
be fixed and
be a function such that there exists an
with
for all
. Let
be a mapping satisfying
(28)
for all
. Then there exists a unique Quattuordecic mapping
such that
(29)
for all
, where
(30)
Proof. Replacing
in (28), we get
(31)
Replacing
by
in (28), we arrive that
(32)
Replacing
by
in (28), we have
(33)
From Equations (32) and (33), we obtain
(34)
Replacing
with
in (28), we arrive that
(35)
for all
. By (31), (34) and (35), we have
(36)
Replacing
with
in (28), we have
(37)
From (36) and (37), we arrive that
(38)
Replacing
with
in (28), one finds that
(39)
Utilizing (38) and (39), we find that
(40)
Replacing
with
in (28), we obtain
(41)
From (40) and (41), we arrive at
(42)
Replacing
with
in (28), we obtain
(43)
Using Equations (42) and (43), we get
(44)
Replacing
with
in (28), one finds that
(45)
From (44) and (45), we arrive at
(46)
Replacing
with
in (28), we obtain
(47)
Using Equations (46) and (47), one gets that
(48)
Replacing
with
in (28), we have
(49)
Using Equation (48) and (49), we obtain
(50)
Replacing
with
in (28), we obtain
(51)
From (50) and (51), we arrive at
(52)
Therefore,
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for all
. By Lemma 2.1, there exists a unique mapping
such that
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and
(53)
for all
. It remains to show that Q is a Quattuordecic mapping. From (28), we have
(54)
for all
and
. Here
, for all
.
Therefore, the mapping
is a Quattuordecic mapping. The following corollary is an immediate consequence of Theorem 4.1 concerning the stability of Quattuordecic functional Equation (1).
Corollary 1. Let X be a quasi a-normed space with quasi a-norm
, and let Y be a
Banach Space with
-norm
. Let
be a positive number
with
and let
be a mapping satisfying
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for all
. Then there exists a unique quattuordecic mapping
such that
(55)
where
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The following example shows that the assumption
cannot be omitted in
Corollary 4.2. This example is a modification of well known example of Gajda [6] for the additive functional inequality.
Example 1. Let
be defined by
(56)
consider the function
to be defined by
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Then f satisfies the following functional inequality
(57)
Proof. It is easy to see that f is bounded by
on
. If
or
, then
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for all
. Now, suppose that
Then there exists a non-negative integer k such that
(58)
Hence
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and
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Hence
for all
. From the definition of f and the inequality (58), we obtain that
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Therefore, f satisfies (57) for all
. Now, we claim that functional Equation (1) is not stable for
in above Corollary (4.2)
.
Suppose on the contrary that there exists a Quattuordecic mapping
and constant
such that
Then there exists a constant
such that
for all rational numbers x (see (25)). So we obtain the following inequality
(59)
Let
with
. If x is a rational number in
, then
for all
and in this case we get
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which contradicts the inequality (59).