Existence and Uniqueness of Fixed Point for a Higher-Order \'Ciri\'c-Type Contraction

Abstract

Let $(X,d)$ be a complete metric space and $T: X \to X$ a continuous mapping. Suppose there exist an integer $\rho > 1$ and a constant $k \in [0,\frac{1}{2})$ such that for all $x,y \in X$, $$ d(Tx,Ty) \le k \max \big\{ d(x,y),\, d(T^{\rho}x,y),\, d(T^{\rho}y,x),\, d(T^{\rho}x,x),\, d(T^{\rho}y,y) \big\}.$$ We prove that $T$ admits a unique fixed point in $X$. The argument combines asymptotic regularity, Cauchy convergence of the Picard iteration, and a standard uniqueness estimate. This result extends classical \'Ciri\'c-type contraction theorems to delayed higher-order contractions.