A Constructive C¹,α Realisation of an Oscillatory Nonlinear Difference Equation

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Original abstract

This independent preprint gives a constructive existence theorem for a class of oscillatory nonlinear difference equations. For parameters alpha between zero and one, q of unit modulus with q not equal to one, nonzero mu, and complex c, it constructs a complete orbit converging to a prescribed limiting point and a compactly supported map of class C1-alpha that realises the orbit exactly. The orbit has an explicit two-term oscillatory asymptotic expansion. The proof combines a weighted contraction argument for the asymptotic tail, a Brouwer-degree argument for finite backward extension, collision-compatible interpolation by smooth bump functions, and an exact calculation of the local derivative modulus. When c is nonzero, the constructed map is not locally of class C1-beta at the limiting point for any beta greater than alpha and at most one. This manuscript is an independent preprint and has not undergone formal peer review. It makes no claim of novelty, priority, resolution of an open problem, or a universal sharp regularity threshold. OpenAI's ChatGPT was used extensively as a research and manuscript-development tool, including assistance with symbolic derivations, proof organisation, adversarial checking, literature-search support, and preparation of the LaTeX source. It is not an author or contributor. David Anthony Perkins is the sole author and accepts sole responsibility for the theorem statements, calculations, proofs, references, and conclusions.

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