Eigenvalue Problems for Fredholm Operators with Set-Valued Perturbations

By means of a suitable degree theory, we prove persistence of eigenvalues and eigenvectors for set-valued perturbations of a Fredholm linear operator. As a consequence, we prove existence of a bifurcation point for a non-linear inclusion problem in abstract Banach spaces. Finally, we provide applications to differential inclusions.

Characterisation and In Vitro Evaluation of Fenugreek (Trigonella foenum-graecum) Seed Gum as a Potential Prebiotic in Growing Rabbit Nutrition

Some components of soluble fibre appear to have prebiotic effects that can contribute to improving digestive health in post-weaning rabbits. In this work, a fenugreek seed gum (FGS), extracted from Trigonella foenum-graecum seeds and rich in galactomannan, was characterised. Both the pure FSG and ten substrates obtained by the inclusion of 0, 5, 10, 15 and 20 g/kg of FSG in diets rich in soluble (SF) and insoluble (IF) fibre were evaluated in vitro to determine FSG prebiotic potential for rabbit diets. FSG was rich in total sugars (630 g/kg dry matter), consisting entirely of galactose and mannose in a 1:1 ratio, and a moderate protein content (223 g/kg dry matter). Pure FSG was affected very little by in vitro digestion, as only 145 g/kg of the FSG was dissolved during the enzymatic phase. However, the linear inclusion of FSG up to 20 g/kg in growing rabbit feeds has led to a reduction in the solubility of some nutrients during in vitro enzymatic phase, especially in SF diets. Pure FSG not digested during the enzymatic phase almost completely disappeared during the in vitro fermentation phase, 984 g/kg of this indigestible fraction. However, although linear inclusion of FSG up to 20 g/kg in SF diets increased the fermented fraction, no relevant changes in the fermentation profile were observed. In conclusion, FSG satisfies two essential conditions of the prebiotic effect, showing resistance to in vitro enzymatic digestion and being totally fermented in vitro by caecal bacteria, although in vivo studies will be necessary to determine its prebiotic potential.