Common Fixed Points of Cq-Commuting Maps via Generalized Gregus-Type Inequalities

We establish the existence of common fixed points for Cq-commuting self-mappings satisfying a generalized Gregus-type inequality with quadratic terms in q-starshaped subsets of normed linear spaces. Our framework extends classical fixed point theory through: (i) Set-distance constraints δ(·, [q, ·]) generalizing norm conditions (ii) Compatibility via Cq-commutativity without full affinity requirements (iii) Reciprocal continuity replacing full map continuity. Explicit examples (e.g., Example 2.6) demonstrate the non-triviality of these extensions. As applications, we derive invariant approximation theorems for best approximation sets. Our results generalize Nashine's work {Nashine2007} and unify several known fixed point theorems.