Bounded components of positive solutions of abstract fixed point equations: Mushrooms, loops and isolas (Q1763211)

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 \({\mathcal L}(\lambda)u+{\mathcal R}(\lambda,u)=0\) for \((\lambda ,u)\in \mathbb R\times U\), where \({\mathcal L}(\lambda)\) is a Fredholm operator of index \(0\) and \({\mathcal R}\in C(\mathbb R\times U;U)\) is compact on bounded sets and \(\lim_{u\rightarrow 0}{\mathcal R}(\lambda,u)/\| u\| =0\). 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.