Fixed Point Theorem for Path-Averaged Contractions in Complete Suprametric Spaces

We introduce and study Path-Averaged Contractions (PA-contractions) in complete suprametric spaces, where the distance function satisfies the relaxed triangle inequality \[d(x,y) \le d(x,z) + d(z,y) + \rho\, d(x,z)d(z,y), \quad \rho \ge 0.\]We prove that if a PA-contraction $T$ admits an orbit $\{T^n x_0\}$ with uniformly bounded successive distances, i.e., $C_{x_0} = \sup_{k \ge 0} d(T^k x_0, T^{k+1}x_0) < +\infty$, and if the parameters satisfy\[\left(1 + \rho C_{x_0}\right) \alpha^{1/N} < 1,\]where $\alpha \in (0,1)$ and $N \in \mathbb{N}$ are the PA-contraction constants, then the Picard iterates $T^n x_0$ converge to a fixed point of $T$. Moreover, any fixed point of $T$ is unique. We establish global Lipschitz continuity of PA-contractions and compare them with Banach and Kannan contractions. In particular, every Banach contraction is a PA-contraction, but the converse fails; similarly, the classes of PA- and Kannan contractions are incomparable. This extends recent fixed point results to a broader nonlinear metric framework motivated by multiplicative error models and perturbed geometry.