Generalized Hyers–Ulam Stability of an \(n\)-Dimensional Cubic Functional Equation in Banach Spaces with an Application via Fixed Point Method

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DOI: https://doi.org/10.28924/2291-8639-24-2026-301
Published: Sep 21, 2026
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Balaanandhan Radhakrishnan, Poluboina Snikitha, Siriluk Donganont, Poornima R, Kandhasamy Tamilvanan, Generalized Hyers–Ulam Stability of an \(n\)-Dimensional Cubic Functional Equation in Banach Spaces with an Application via Fixed Point Method, Int. J. Anal. Appl., 24 (2026), 301.

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Balaanandhan Radhakrishnan, Poluboina Snikitha, Siriluk Donganont, Poornima R, Kandhasamy Tamilvanan

Abstract

In this work, we investigate the generalized Hyers–Ulam stability of an \(n\)-dimensional cubic functional equation \[\begin{aligned} f\!\left( \sum_{i=1}^{n} ia_{i} \right) &= (3-n) \sum_{1 \leq i <j \leq n} f(ia_{i} + ja_{j}) +\sum_{1 \leq i < j < k \leq n} f(ia_i + ja_j + ka_{k}) \\ &\quad\ + \left( \frac{n^2 - 5n + 6}{2} \right) \sum_{i=0}^{n-1} (i+1)^3\, f(a_{i+1}) \end{aligned}\] in Banach spaces. The problem is studied by employing both direct analytical technique and fixed point method, particularly Banach’s Contraction Principle and the alternative fixed point theorem. We establish the existence and uniqueness of exact cubic mappings that approximate the given functional equation under suitable control functions. Explicit stability estimates are obtained, which ensure that the deviation of approximate solutions remains bounded. To demonstrate the applicability of the theoretical results, an illustrative example is provided. A perturbed cubic mapping is constructed and shown to satisfy the stability conditions. Furthermore, a numerical comparison supported by tabular data is included to verify the theoretical bounds and to highlight the effectiveness of the proposed approach.

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