A solution set analysis of a nonlinear operator equation using a Leray–Schauder type fixed point approach

Here we study the solution set of a nonlinear operator equation in a Banach subspace $L^n\subset C\left(X\right)$ by reducing it to a Leray–Schauder type fixed point problem. The subspace $L^n$ is of finite codimension $n\in Z_{+}$ in $C\left(X\right)$, with $X$ an infinite compact Hausdorff space, and is defined by conditions ${\alpha }_i^{\ast }\left(f\right)≔{\int }_{X}f\left(x\right)d{\mu }_i\left(x\right)=0,f\in C\left(X\right)$, with norms $\Vert {\mu }_i\Vert =1,i=1,\dots ,n$.