Stability of Nonic Functional Equations in Multi-Banach Spaces

Introduction In 1940, Ulam posed a problem concerning the stability of functional equations: Give conditions in order for a linear function near an approximately linear function to exist. An earlier work was done by Hyers [6] in order to answer Ulam’s equation [14] on approximately additive mappings. During last decades various stability problems for large variety of functional equations have been investigated by several mathematicians. A large list of references concerning in the stability of functional equations can be found. e.g.( [1], [2], [6], [7], [9]). In 2010, Liguang Wang, Bo Liu and ran Bai [10] proved the stability of a mixed type functional equations on Multi Banach Spaces. In 2010, Tian Zhou Xu, John Michael Rassias, Wan Xin Xu [13] investigated the generalized Ulam-Hyers stability of the general mixed additive-quadratic-cubic-quartic functional equation


Introduction
In 1940, Ulam posed a problem concerning the stability of functional equations: Give conditions in order for a linear function near an approximately linear function to exist.An earlier work was done by Hyers [6] in order to answer Ulam's equation [14] on approximately additive mappings.
During last decades various stability problems for large variety of functional equations have been investigated by several mathematicians.A large list of references concerning in the stability of functional equations can be found.e.g.( [1], [2], [6], [7], [9]).
In 2010, Liguang Wang, Bo Liu and ran Bai [10] proved the stability of a mixed type functional equations on Multi -Banach Spaces.In 2010, Tian Zhou Xu, John Michael Rassias, Wan Xin Xu [13] investigated the generalized Ulam-Hyers stability of the general mixed additive-quadratic-cubic-quartic functional equation In 2011, Zhihua Wang, Xiaopei Li and Th.M. Rassias [16] proved the Hyers -Ulam stability of the additive -cubic -quartic functional equations in Multi -Banach Spaces by using fixed point method.
In 2016, John M. Rassias, M. Arunkumar, E. Sathya and T. Namachivayam [8] established the (??) general solution and also proved the Felbin's type fuzzy normed space and intuitionistic fuzzy normed space using direct and fixed point method.
In this paper, we carry out the following Stability of Nonic Functional Equations where 362880 = 9! in Multi-Banach Spaces by using fixed point technique.
It is easily verified that that the function 9 = ) ( s s  satisfies the above functional equations.In other words, every solution of the nonic functional equation is called a nonic mapping.e d i t o r g j m @ g m a i l .c o m Theorem 1.1 [3], [12] and the following axioms are satisfied for each In this case, we say that   , ( is a multi -normed spaces, and take .

N  k
We need the following two properties of multi -norms.They can be found in [4].
We define an operator We assert that J is a strictly contractive operator.Given Hence,it holds that e d i t o r g j m @ g m a i l .c o m This Means that J is strictly contractive operator on  with the Lipschitz constant .According to Theorem 1.1, we deduce the existence of a fixed point of J that is the existence of mapping A In this paper, we carry out the following Stability of Nonic Functional Equations