Abstract

In this paper, we propose a shrinking double inertial framework for approximating common fixed points of a countable family of quasi-nonexpansive mappings in Hilbert spaces. The framework combines a shrinking projection technique with a double inertial extrapolation mechanism, and strong convergence of the generated sequence is established under suitable assumptions. The shrinking strategy ensures strong convergence to a common fixed point, whereas the double inertial terms introduce additional extrapolation flexibility into the iterative process. To illustrate the theoretical results, a numerical example in an infinite-dimensional function space is presented using quasi-nonexpansive mappings. Motivated by variational inequality problems involving monotone and Lipschitz continuous operators, two further shrinking double inertial algorithms are developed using projected gradient and extragradient-type constructions. The proposed algorithms are then incorporated into Extreme Learning Machine models for air quality prediction. Experiments are conducted for five target pollutants, namely, NOx(GT), NMHC(GT), C6H6(GT), CO(GT), and NO2(GT), using target-specific MRMR-selected feature sets. Predictive performance is evaluated using RMSE, MAE, and R2. The numerical results show that the proposed algorithms achieve competitive predictive performance across the five targets. The projected gradient and extragradient frameworks exhibit complementary behavior depending on the target variable and problem structure, suggesting that different variational mapping constructions may be suitable for different prediction tasks. These findings demonstrate the theoretical flexibility and practical applicability of the proposed frameworks to fixed-point problems, variational inequalities, and constrained machine learning models.

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