A rigorous fixed point framework is developed for linear and nonlinear second-order mechanical vibration systems governed by ordinary differential equations with damping, stiffness, external forcing, and nonlinear restoring forces. The governing equation is reformulated as a Volterra operator equation in an appropriate Banach space, providing a convenient integral formulation for the construction of a Picard fixed point iteration. Global existence, uniqueness, and convergence are established under suitable contraction conditions, with explicit bounds for the contraction constant and corresponding geometric convergence of the iterative sequence. Energy-based stability estimates are derived for both forced and unforced systems, providing bounds on the mechanical energy and clarifying the dissipative behavior of the underlying dynamics. A discrete quadrature-based implementation is developed for the Volterra formulation and is analyzed in terms of its convergence order and stability properties under appropriate conditions. Numerical experiments for representative linear and nonlinear vibration problems are presented to validate the theoretical findings and to demonstrate the accuracy, convergence, and effectiveness of the proposed fixed point approach.
Torabi, P. (2026). A Globally Convergent Fixed Point Framework for Linear and Nonlinear Mechanical Vibration Systems. Analytical and Numerical Solutions for Nonlinear Equations, (), -. doi: 10.22128/ansne.2026.3360.1225
MLA
Parvin Torabi. "A Globally Convergent Fixed Point Framework for Linear and Nonlinear Mechanical Vibration Systems", Analytical and Numerical Solutions for Nonlinear Equations, , , 2026, -. doi: 10.22128/ansne.2026.3360.1225
HARVARD
Torabi, P. (2026). 'A Globally Convergent Fixed Point Framework for Linear and Nonlinear Mechanical Vibration Systems', Analytical and Numerical Solutions for Nonlinear Equations, (), pp. -. doi: 10.22128/ansne.2026.3360.1225
VANCOUVER
Torabi, P. A Globally Convergent Fixed Point Framework for Linear and Nonlinear Mechanical Vibration Systems. Analytical and Numerical Solutions for Nonlinear Equations, 2026; (): -. doi: 10.22128/ansne.2026.3360.1225
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