Algorithm for equations of Hammerstein type and applications

TL;DR

This paper develops a new technique to prove strong convergence for nonlinear Hammerstein integral equations without relying on uncertain constants, and applies it to physical problems like pendulum oscillations.

Contribution

A novel approach for strong convergence in Hammerstein equations that avoids the need for certain constants, demonstrated through physical system applications.

Findings

Established strong convergence results for nonlinear Hammerstein equations.

Applied the method to analyze forced oscillations of a pendulum.

Provided numerical examples illustrating convergence of iterative sequences.

Abstract

Equations of Hammerstein type cover large variety of areas and are of much interest to a wide audience due to the fact that they have applications in numerous areas. Suitable conditions are imposed to obtain a strong convergence result for nonlinear integral equations of Hammerstein type with monotone type mappings. A technique which does not involve the assumption of existence of a real constant whose calculation is unclear has been used in this study to obtain the strong convergence result. Moreover, our technique is applied to show the forced oscillations of finite amplitude of a pendulum as a specific example of nonlinear integral equations of Hammerstein type. Numerical example is given for the illustration of the convergence of the sequences of iteration. These are done to demonstrate to our readers that this approach can be applied to problems arising in physical systems.