In this paper, the authors investigate the general solution and generalized Ulam -Hyers stability of a new type of n-dimensional additive functional equation ∑ ∑ ∑ ∑ ∑ with 3 n > in Banach space and Banach Algebra using direct and fixed point methods.

The object of the present paper is to assess speculation of the Hyers-Ulam stability theorem for the complex additive functional equation on abelian groups and stability results have been gotten by a fixed point technique. This technique demonstrates that the stability is identified with some fixed point of an appropriate operator.

In this paper, authors verify the generalized Ulam -Hyers stability of the following Euler -Lagrange additive functional equationin Intuitionistic Fuzzy Banach Spaces using direct and fixed point methods.

In this paper, the authors established the solution of the additive functional equation and inequality$$f(x)+f(y+z)-f(x+y)=f(z)$$and$$\|f(x)+f(y+z)-f(x+y)\| \leq\|f(z)\| .$$We also prove that the above functional equation and inequality are stable in Banach space in the sense of Ulam, Hyers, Rassias. An application of this functional equation is also studied.

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