# Stability of Multi-Quadratic Functions in Lipschitz Spaces

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## Abstract

The algebra of Lipschitz functions on a complete metric space plays a role in non-commutative metric theory similar to that played by the algebra of continuous functions on a compact space in non-commutative topology. The notion of stability of functional equations was posed by Ulam, and then, Hyers gave the first significant partial solution in 1941. The subject has been established and developed by an increasing number of mathematicians in various spaces, particularly during the last two decades. In Lipschitz spaces, the notion of stability-type problems was introduced by Tabor and Czerwik. It should be mentioned that this subject area has been considered less attention over the recent years. In this paper, we prove stability of multi-quadratic functional equations in Lipschitz spaces by introducing the notion of a multi-symmetric left invariant mean. Indeed, we prove under certain Lipschitz conditions a family of Lipschitz functions can be approximated by multi-quadratic functions.

## Keywords

Multi-quadratic functional equation Lipschitz space stability Approximation## Mathematics Subject Classification

39B82 39B52## References

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