A Novel Inertial-Type Iteration Algorithm: Convergence, Data Dependence, and Applications in Image Deblurring and Fractal Generation

This study introduces a novel inertial-type iteration algorithm based on the Normal S iteration for the class of almost contraction mappings in Banach spaces. Traditional fixed point iterations often suffer from slow convergence and high computational cost; to address these limitations, the proposed framework incorporates an adaptive inertial-type parameter. We establish strong convergence of the algorithm and derive explicit a posteriori error estimates under weak contractive conditions. In addition, we demonstrate the asymptotic equivalence of the NS inertial-type trajectories with the classical Normal S iteration, provide a comprehensive weak w 2 —stability analysis, and obtain sharp upper bounds for the data dependence problem. The practical performance of the algorithm is evaluated in two distinct computational domains: image deblurring via wavelet-based ℓ 1 regularization and the generation of complex fractal patterns, including Julia and Mandelbrot sets. Numerical results show that the proposed inertial-type iteration algorithm significantly outperforms existing methods—such as Picard, Mann, Ishikawa, and standard Normal S iterations—achieving faster convergence, higher PSNR values in image restoration, and more stable basins of attraction in fractal visualizations. These findings highlight the effectiveness and versatility of the NS inertial-type iteration algorithm approach for both theoretical analysis and real-world applications.