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$(K, L, a)$ homeomorphism; $a$-homogeneous operator; $a$-stably solvable map
Surjectivity results of Fredholm alternative type are obtained for nonlinear operator equations of the form ${\lambda} T(x)-S(x)=f$, where $T$ is invertible, and $T,S$ satisfy various types of homogeneity conditions. We are able to answer some questions left open by Fu\v{c}'{\i}k, Ne\v{c}as, Sou\v{c}ek, and Sou\v{c}ek. We employ the concept of an $a$-{stably-solvable} operator, related to nonlinear spectral theory methodology. Applications are given to a nonlinear Sturm-Liouville problem and a three point boundary value problem recently studied by Gupta, Ntouyas and Tsamatos.
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