STABILITY OF tredecic FUNCTIONAL EQUATION IN MATRIX NORMED SPACES

In 1940, an interesting topic was presented by S. M. Ulam [18] triggered the study of stability problems for various functional equations. He addressed a question concerning the stability of homomorphism. In the following year, 1941, D. H. Hyers [5] was able to give a partial solution to Ulam’s question. The result of Hyers was then generalized by Aoki [1] for additive mappings. In 1978, Th. M. Rassias [14] succeeded in extending the result of Hyers theorem by weakening the condition for the Cauchy difference. The stability phenomenon that was presented by Th. M. Rassias is called the generalized Hyers-Ulam stability. In 1994, a generalization of the Rassias theorem was obtained by Gavruta [4] by replacing the unbounded Cauchy difference by a general control function. A further generalization of the Hyers-Ulam stability for a large class of mapping was obtained by Isac and Th. M. Rassias [6]. They also presented some applications in non-linear analysis, especially in fixed point theory. This terminology may also be applied to the cases of other functional equations [2, 3, 13, 15, 17, 20]. Also, the generalized Hyers-Ulam stability of functional equations and inequalities in matrix normed spaces has been studied by number of authors [7, 8, 9, 10, 12, 19]. K. Ravi and B. V. Senthil Kumar [16] discussed the general solution of undecic functional equation and proved the stability of this functional equation in quasi  normed spaces by applying the fixed point method.


ABSTRACT
In this current work, we define and find the general solution of the following tredecic functional equation

Introduction
In 1940, an interesting topic was presented by S. M. Ulam [18] triggered the study of stability problems for various functional equations. He addressed a question concerning the stability of homomorphism. In the following year, 1941, D. H. Hyers [5] was able to give a partial solution to Ulam's question. The result of Hyers was then generalized by Aoki [1] for additive mappings. In 1978, Th. M. Rassias [14] succeeded in extending the result of Hyers theorem by weakening the condition for the Cauchy difference.
The stability phenomenon that was presented by Th. M. Rassias is called the generalized Hyers-Ulam stability. In 1994, a generalization of the Rassias theorem was obtained by Gavruta [4] by replacing the unbounded Cauchy difference by a general control function. A further generalization of the Hyers-Ulam stability for a large class of mapping was obtained by Isac and Th. M. Rassias [6]. They also presented some applications in non-linear analysis, especially in fixed point theory. This terminology may also be applied to the cases of other functional equations [2,3,13,15,17,20]. Also, the generalized Hyers-Ulam stability of functional equations and inequalities in matrix normed spaces has been studied by number of authors [7,8,9,10,12,19].
K. Ravi and B. V. Senthil Kumar [16] discussed the general solution of undecic functional equation and proved the stability of this functional equation in quasi  -normed spaces by applying the fixed point method.
In this paper, we introduce the following new functional equation ( where 6227020800 = 13! is said to be tredecic functional equation since the function In Section 2, we study the tredecic functional equation (1).
In Section 3, using the fixed point technique, we prove the Hyers-Ulam stability of the functional equation (1) in matrix normed spaces.

Tredecic Functional Equation (1)
In this section, we study the tredecic functional equation (1). For this, let us consider A and B be real vector spaces.

Stability of Tredecic Functional Equation in Matrix Normed Spaces
In this section, we will investigate the Ulam-Hyers stability for the functional equation (1) Then there exists a unique tredecic mapping It follows from (22) and (23), we arrive at      and introduce the generalized metric  on M as follows: is a complete generalized metric (see also [11]). Define the mapping M M P  : by and  be an arbitrary constant with   Hence it holds that It follows from (41) that Therefore according to Theorem 2.2 in [3], there exists a mapping Y X  : T which satisfying: 1. T is a unique fixed point of P in the set , which is satisfied In other words, there exists a  satisfying , It follows from (18) . Therefore, the mapping Y X  : T is tredecic mapping. By Lemma 2.1 in [9] and (43), is a unique tredecic mapping satisfying (20 . From the definition of f and (47), we obtain that   which is a contradiction to (48). Thus the tredecic functional equation (1) is not stable for 13 = t .

Conclusion
In this investigation, we identified a general solution of tredecic functional equation and establised the generalized Ulam -Hyers stability of this functional equation in matrix normed spaces by using the fixed point method and also provided an example for non-stability.