Coupled mapping Structure under Zulfia Invexity via Decomposition

Invexity generalizes convexity by replacing linear displacement with a directionmapping. In this paper, we introduce a structural framework in which admissibledirections are not prescribed but generated through a decompositionZ(x, u) = η(x, u) + ∆(x, u).Within this framework, Zulfia invexity is formulated through a coupled admis-sibility system. Specifically, a pair of differentiable functions satisfiesf (x) − f (u) ≥ ∇f (u)T Z(x, u), A(x) − A(u) ≥ ∇A(u)T Z(x, u), ∀x, u ∈ X.A sufficiency result is established under this coupled structure. Classical invex-ity is recovered as a limiting case when the structural component ∆(x, u) vanishes,while convexity arises when, in addition, η(x, u) = x − u. This provides a mini-mal structural extension of invexity in which admissible directions are generatedintrinsically and governed by a coupled mapping mechanism.