Abstract
We use the degree introduced by P. Benevieri and M. Furi [3] to obtain the generalized minimal cardinal of the number of solutions of the λ- sections of those components of the set of nontrivial solutions of an abstract equation of the form \(\mathfrak{F}\)(λ, x) = 0 that are compact in one direction of the parameter; here \(\mathfrak{F}\) is a C1 Fredholm map of index 1 such that \(\mathfrak{F}\)(λ, 0) = 0 for all λ ∈ ℝ. These bounds are given in terms of the parity of the linearized Fredholm family D2\(\mathfrak{F}\)(·, 0). The parity is a local invariant measuring the change of the orientation of D2\(\mathfrak{F}\)(λ, 0) as λ crosses an interval. The set of eigenvalues of D2\(\mathfrak{F}\)(·, 0) is not assumed to be discrete. Therefore, as regards applications, the theory developed in this paper should be extremely versatile.
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López-Gómez, J., Mora-Corral, C. (2005). Generalized Minimal Cardinal of the λ-slices of the Semi-bounded Components Arising in Global Bifurcation Theory. In: Brezis, H., Chipot, M., Escher, J. (eds) Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 64. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7385-7_18
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