Fixed Point Theory in Twisted Formal Power Series Rings over Finite Fields

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DOI: https://doi.org/10.28924/2291-8639-24-2026-347
Published: Oct 15, 2026
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Haitham Qawaqneh, Mahmood Akhtar Khan, Abdullah Shoaib, Thabet Abdeljawad, Iman Ben Salah, Fixed Point Theory in Twisted Formal Power Series Rings over Finite Fields, Int. J. Anal. Appl., 24 (2026), 347.

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Haitham Qawaqneh, Mahmood Akhtar Khan, Abdullah Shoaib, Thabet Abdeljawad, Iman Ben Salah

Abstract

In this paper we develop a complete non-Archimedean metric framework on the ring of twisted formal power series \({\widehat{R}}={\mathbb{F}_q}[[t;\sigma]]\), where \(\sigma\) denotes the Frobenius automorphism of the finite field \({\mathbb{F}_q}\). Using the natural \(t\)-adic valuation, we prove that the resulting metric is an ultrametric and that the ring is complete. We establish Banach-type fixed point theorems for contractive affine operators and obtain existence and uniqueness results for one-sided, two-sided, and mixed twisted affine equations, illustrating how the noncommutative structure naturally leads to contractions involving both left and right multiplications. Applications to iterative Frobenius systems and discrete-time dynamical models over finite fields are also given.

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