Zulfia Fixed Point Theorem: A Contraction Principle on Relational Spaces

AbstractThe Banach Contraction Principle (1922) [1] guarantees a unique fixed pointfor a contraction on a complete metric space. Its framework requires a metric, andits classical applications rely on differentiability or convexity.After more than 100 years, this principle is recovered as the zero-deviation caseof the Zulfia parent structure introduced in this paper. We here introduce the ZulfiaFixed Point Theorem on a general relational space. The imposed metric is replacedby the Zulfia Decomposition Mapping [3].Z(x, u) = η(x, u) + ∆(x, u), Z(x, x) = 0, (1)without assuming any metric, norm, or topology. The contraction condition isreplaced by the Zulfia constantκZ := ∥∆∥∥η∥ ∈ [0, 1), (2)defined as the ratio of ∆ to η, without requiring differentiability or convexity of theobjective f .Under κZ < 1, the Zulfia gradient flow converges to a unique fixed point x∗ of theZulfia operator, and x∗ is a global minimizer of f . The Banach principle is recoveredas the special case η(x, u) = x−u and ∆(x, u) ≡ 0 on a complete metric space. Fourexamples demonstrate applicability where Banach fails: non-metric divergences,nonsmooth functions, nonconvex objectives, and failed constraint qualifications