Generalized mean $(\psi ,\phi )$-cyclic contractions: unified fixed point theorems and Proinov-type cyclic attractors

In this article, we introduce generalized mean $(\psi ,\phi )$-cyclic contractions by integrating the concepts of Reich-type contractions, $(\psi ,\phi )$-contractions and cyclic contractions on complete metric spaces. We establish fixed point theorems for these generalized cyclic contractions under suitable conditions on the control functions ψ and ϕ. By replacing the Reich-type contraction with a Ćirić-type contraction, we further define the generalized Ćirić-type $(\psi ,\phi )$-cyclic contraction and prove corresponding fixed point results. As a special case, we present the Proinov-type cyclic contraction and develop the Hutchinson-Barnsley theory for this setting, leading to the existence of a Proinov-type cyclic attractor. An explicit example is provided to illustrate the construction of such an attractor, demonstrating applicability beyond Banach-type cases. These results unify and extend several well-known fixed point theorems, offering a broader framework for cyclic mappings and their applications in fractal theory.