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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 (λ)u+(λ,u)=0 for (λ,u)×U, where (λ) is a Fredholm operator of index 0 and C(×U;U) is compact on bounded sets and lim u0 (λ,u)/u=0.

The main result of the present paper concerns the bounded components of positive solutions emanating from (λ,u)=(λ,0). The proofs are based on refined techniques from modern bifurcation theory.

47J05Equations involving nonlinear operators (general)
47J15Abstract bifurcation theory
35B30Dependence of solutions of PDE on initial and boundary data, parameters
35B32Bifurcation (PDE)
35B50Maximum principles (PDE)