Novel Fixed Point Theorems and Stability Analysis for Cyclic Contractions in b-G-metric Spaces via λ-Iteration

Abstract

This paper develops fixed point results for cyclic contractive mappings in complete $b$-$G$-metric spaces via an accelerated $\lambda$-iteration scheme. Working in a convex $b$-$G$-metric setting endowed with a metric-affine convex structure, we study the iterative process
$ x_{n+1}=W\!\left(x_n,Tx_n,\frac1\lambda\right)$ for  $(\lambda>1)$, which reduces to $x_{n+1}=\frac{(\lambda-1)x_n+Tx_n}{\lambda}$ in linear spaces. We establish existence and uniqueness of fixed points for cyclic $\phi$-contractions (including linear $\phi(t)\le \mu t$) and provide explicit a priori error estimates with geometric convergence of order $O(q^n)$, where $q=\frac{\lambda-1+\mu}{\lambda}\in(0,1)$. As an application, we obtain a unique solvability result for a cyclic system of nonlinear integral equations, together with convergence of the $\lambda$-iteration to the solution, and we give a numerical illustration of the method.