Mathematical hints in nonlinear problems

We investigate the possible application of an abstract mathematical method, the topological index theory and Krasnosel'ski theorem (a brief account of this theory is given at the end of the paper) for solving nonlinear physical problems, particularly when bifurcation phenomena-with one or more parameters-are expected. Various mathematical results are presented, especially in finite-dimensional cases, in which simple conditions for the existence of nontrivial branching solutions are given, together with a discussion concerning «perturbative» expansions. The case of the presence of some symmetry property is also briefly consieered.