A solution set analysis of a nonlinear operator equation using a Leray-Schauder type fixed point approach
In a Banach linear subspace \(\mathcal L^n\) of the space \(C(X)\) of all continuous real valued functions on an infinite compact Hausdorff space \(X\), the solution set of a nonlinear operator equation is investigated. The subspace \(\mathcal L^n\) of a finite defect \( n \) is defined by a set of linear Radon-Stieltjes integral equations with respect to some linearly independent Radon measures. The considered nonlinear operator equation has a special form, namely, it has a linear closed surjective operator in the left hand side, and a nonlinear continuous mapping in the right hand side. To be able to use topological methods of Leray-Schauder fixed point type, the considered operator equation is transformed into another new equation by introducing a new extended operator in the left hand side and a new extended nonlinear mapping in the right hand side. Based on the Leray-Schauder theorem (described in the appendix of the paper), the authors prove a result describing the solution set of the considered nonlinear operator equation.
scientific article; zbMATH DE number 5706260