Abstract
In this paper, we investigate the general solution of a new quadratic functional equation of the form ∑ 1 ≤ i < j < k ≤ r ϕ l i + l j + l k = r − 2 ∑ i = 1 , i ≠ j r ϕ l i + l j + − r 2 + 3 r − 2 / 2 ∑ i = 1 r ϕ l i . We prove that a function admits, in appropriate conditions, a unique quadratic mapping satisfying the corresponding functional equation. Finally, we discuss the Ulam stability of that functional equation by using the directed method and fixed-point method, respectively.
Highlights
The stability problem of functional equations originated from a question of Ulam [1] concerning about the stability
Functional equations have substantially grown to become an important branch of this field
In [10], the authors deal with a comprehensive illustration of the stability of functional equations, and in [11], the authors studied functional equations and inequalities in several variables
Summary
The stability problem of functional equations originated from a question of Ulam [1] concerning about the stability. Consider the functional equation as follows: φðl + mÞ + φðl − mÞ = 2φðlÞ + φðmÞ, ð1Þ Every solution of the quadratic functional equation is a quadratic mapping. We investigate the general solution of a new quadratic functional equation of the form
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