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Bounded components of positive solutions of abstract fixed point equations: Mushrooms, loops and isolas. (English) Zbl 1066.47063
Let $U$ be an ordered Banach space whose positive cone is normal and has nonempty interior. This paper is devoted to the study of the nonlinear abstract equation ${\cal L}(\lambda)u+{\cal R}(\lambda,u)=0$ for $(\lambda ,u)\in \Bbb R\times U$, where ${\cal L}(\lambda)$ is a Fredholm operator of index $0$ and ${\cal R}\in C(\Bbb R\times U;U)$ is compact on bounded sets and $\lim_{u\rightarrow 0}{\cal R}(\lambda,u)/\Vert u\Vert =0$. The main result of the present paper concerns the bounded components of positive solutions emanating from $(\lambda,u)=(\lambda,0)$. The proofs are based on refined techniques from modern bifurcation theory.

MSC:
 47J05 Equations involving nonlinear operators (general) 47J15 Abstract bifurcation theory 35B30 Dependence of solutions of PDE on initial and boundary data, parameters 35B32 Bifurcation (PDE) 35B50 Maximum principles (PDE)
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References:
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