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This study has developed a unified framework for modeling economic growth through Caputo fractional differential equations. The framework has established the existence and uniqueness of solutions by employing a generalized fixed-point approach. In particular, the analysis has introduced and utilized new classes of symmetric operators, including symmetric Lipschitz-type mappings, symmetric Kannan-type contractions, and symmetric Chatterjea-type contractions. These mappings are based on a refined symmetric Lipschitz condition that enables the examination of the behavior of their iterative sequences. The study has focused on several forms of symmetric contractions defined on metric spaces endowed with a binary relation, providing a setting that generalizes and unifies various existing fixed-point theorems. This framework has extended classical results by Goebel and Sims, Goebel and Japon-Pineda, and others. Finally, to illustrate the practical significance of the theoretical findings, the developed results have been applied to demonstrate the existence of solutions for fractional models of economic growth and a related Fredholm integral equation.
Wang et al. (Mon,) studied this question.