We define a notion of degree for a class of perturbations of nonlinear Fredholm maps of index zero between infinite-dimensional real Banach spaces. Our notion extends the degree introduced by Nussbaum for locally α-contractive perturbations of the identity, as well as the recent degree for locally compact perturbations of Fredholm maps of index zero defined by the first and third authors

We discuss the solvability of the following strongly nonlinear BVP:, a is a positive, continuous function and f is a Carathéodory nonlinear function. We give conditions for the existence and non-existence of heteroclinic solutions in terms of the behavior of y → f (t, x, y) and y → Φ(y) as y → 0, and of t → f (t, x, y) as |t| → +∞. Our approach is based on fixed point techniques suitably combined to the method of upper and lower solutions.

We prove a global bifurcation result for T -periodic solutions of the T -periodic delay differential equation x (t) = λf (t, x(t), x(t − 1)) depending on a real parameter λ 0. The approach is based on the fixed point index theory for maps on ANRs.

We consider a discontinuous system exhibiting a, possibly non-smooth, homoclinic trajectory. We assume that the critical point lies on the discontinuity level. We study the persistence of such a trajectory when the system is subject to a smooth non-autonomous perturbation. We use a Mel'nikov type approach and we introduce conditions which enable us to reformulate the problem in the setting of smooth systems so that we can follow the outline of the classical theory.

We prove a global bifurcation result for T -periodic solutions of the delay T -periodic differential equation x (t) = λf (t, x(t), x(t − 1)) on a smooth manifold (λ is a nonnegative parameter). The approach is based on the asymptotic fixed point index theory for C 1 maps due to Eells-Fournier and Nussbaum. As an application, we prove the existence of forced oscillations of motion problems on topologically nontrivial compact constraints. The result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.