Generalized Fixed-Point Theorems in Complete Metric Spaces and Their Applications

Abstract

Fixed-point theory provides a fundamental framework for establishing existence, uniqueness, and con-structive approximation of solutions to nonlinear equations. This paper presents a unified treatment of generalized contractive mappings in complete metric spaces by considering a symmetric Hardy–Rogers-type condition incorporating Banach-, Kannan-, and Chatterjea-type distance terms. For a self-mapping T on a complete metric space (X, d), we study the inequality

d(Tx, Ty) ≤ ad(x, y) + b[d(x, Tx) + d(y, Ty)] + c[d(x, Ty) + d(y, Tx)],

where a, b, c ≥ 0 and a + 2b + 2c < 1. An explicit contraction factor for successive Picard iterates is ob-tained, and it is proved that every mapping satisfying the proposed condition possesses a unique fixed point. The convergence of the Picard sequence is established without imposing continuity of the map-ping as a separate assumption. Quantitative a priori and a posteriori error bounds are derived, together with an Ulam–Hyers-type stability estimate for approximate fixed points. Classical contraction principles of Banach, Kannan, and Chatterjea are recovered as immediate special cases. Applications are presented to Fredholm-type nonlinear integral equations and finite-dimensional nonlinear iterative systems. A nu-merical illustration demonstrates geometric convergence and confirms the theoretical error behavior. The framework emphasizes the usefulness of generalized metric contractions as a common language for exis-tence theory, iterative approximation, and stability analysis in nonlinear problems. To connect threshold behavior with intervention-relevant parameters, we calculate normalized forward sensitivity indices of R0, which reveal the dominant drivers of transmission and persistence. We then propose the LRPS method in a numerically robust form in terms of fractional power series coefficients and a convolution formula for the nonlinear incidence term. Extensive simulations are performed over long horizons to show the influence of the transmission rate and the fractional order on TB dynamics, with smaller values of γ resulting in more persistent transients in accordance with the memory effect. As a final step, we validate the LRPS method using an alternative Adams-Bashforth-Moulton predictor-corrector scheme for Caputo FDEs and compare runtime and accuracy with the standard fractional predictor-corrector approaches. The results show the LRPS method is a robust tool for long horizon approximations with low computational costs and is suitable for an integrated analytical-numerical approach in the context of fractional TB modeling and control.