A nonlinear singular perturbation problem

TL;DR

This paper investigates a nonlinear singular perturbation problem involving an operator equation in a Hilbert space, providing conditions for solution existence and convergence as the perturbation parameter approaches zero, with an example involving a nonlinear integral operator.

Contribution

It offers new sufficient conditions for the existence and convergence of solutions to a nonlinear singular perturbation problem where the linearized operator is not invertible.

Findings

Established conditions for solution existence.

Proved convergence of solutions as perturbation vanishes.

Presented an application with a nonlinear integral operator.

Abstract

Let F(u_\ve)+\ve(u_\ve-w)=0 \eqno{(1)} where $F$ is a nonlinear operator in a Hilbert space $H$, $w\in H$ is an element, and $\ve>0$ is a parameter. Assume that $F(y)=0$, and $F'(y)$ is not a boundedly invertible operator. Sufficient conditions are given for the existence of the solution to \eqref{e1.1} and for the convergence $\lim_{\ve\to 0}\|u_\ve-y\|=0$. An example of applications is considered. In this example $F$ is a nonlinear integral operator.