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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 $ℒ\left(\lambda \right)u+ℛ\left(\lambda ,u\right)=0$ for $\left(\lambda ,u\right)\in ℝ×U$, where $ℒ\left(\lambda \right)$ is a Fredholm operator of index 0 and $ℛ\in C\left(ℝ×U;U\right)$ is compact on bounded sets and ${lim}_{u\to 0}ℛ\left(\lambda ,u\right)/\parallel u\parallel =0$.

The main result of the present paper concerns the bounded components of positive solutions emanating from $\left(\lambda ,u\right)=\left(\lambda ,0\right)$. 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)