General Solution and Stability of Quattuordecic Functional Equation in Quasi β-Normed Spaces

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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 E_{1} and E_{2} 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.

and

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

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

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

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

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

it follows that

Using Equations (25) and, we obtain

for all and. It follows that

for all. Hence.

Conversely, assume that for all, where is the diagonal of the 14-additive symmetric map from

(26)

and

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,

for all. By Lemma 2.1, there exists a unique mapping such that

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

for all. Then there exists a unique quattuordecic mapping such that

(55)

where

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

Then f satisfies the following functional inequality

(57)

Proof. It is easy to see that f is bounded by on. If or, then

for all. Now, suppose that Then there exists a non-negative integer k such that

(58)

Hence

and

Hence for all. From the definition of f and the inequality (58), we obtain that

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

which contradicts the inequality (59).

References

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