Abstract
LetE be a Banach lattice which consists of some functionsu:Ω→ℂ over a fixed domain Ω. This article is concerned with the local stability of a non-zero positive solutionu *∈E to the Hammerstein equationu(x)=\( = \int\limits_\Omega {k(x,y)f(y,u(y))d\mu } (y),\) x∈Ω, wherek≥0, andf:Ω×[0, ∞)→[0, ∞) is not necessarily increasing in the second variable. It is assumed thatf(x, 0)=0 and\(\left| {\frac{{\partial f(x,u)}}{{\partial u}}} \right|< \frac{{f(x,u)}}{u}\) forx∈Ω,u>0. Under some mild additional hypotheses onE, k, μ andf it is proved that the spectral radius of the Fréchet derivative atu * of the Hammerstein operator is less than one. Also the impact of local stability on secondary bifurcations is investigated. The proof of the main result is based on the spectral theory for completely continuous and irreducible positive operators on Banach lattices.
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Takáč, P. The local stability of positive solutions to the Hammerstein equation with a nonmonotonic nemytskii operator. Monatshefte für Mathematik 106, 313–335 (1988). https://doi.org/10.1007/BF01295289
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DOI: https://doi.org/10.1007/BF01295289