Pierluigi Benevieri

Periodic solutions for nonlinear equations with mean curvature-like operators

Abstract We give an existence result for a periodic boundary value problem involving mean curvature-like operators in the scalar case. Following [R. Manasevich, J. Mawhin, Periodic solutions for nonlinear systems with p -Laplacian-like operators, J. Differential Equations 145 (1998), 367–393], we use an approach based on the Leray–Schauder degree.

On the concept of orientability for Fredholm maps between real Banach manifolds

In [ A simple notion of orientability for Fredholm maps of index zero between Banach manifolds and degree , Ann. Sci. Math. Quebec 22 (1998), 131–148] we introduced a concept of orientation and topological degree for nonlinear Fredholm maps between real Banach manifolds. In this paper we study properties of this notion of orientation and we compare it with related results due to Elworthy-Tromba and Fitzpatrick-Pejsachowicz-Rabier.

A degree theory for locally compact perturbations of Fredholm maps in Banach spaces

We present an integer valued degree theory for locally compact perturbations of Fredholm maps of index zero between (open sets in) Banach spaces (quasi-Fredholm maps, for short). The construction is based on the Brouwer degree theory and on the notion of orientation for nonlinear Fredholm maps given by the authors in some previous papers. The theory includes in a natural way the celebrated Leray-Schauder degree.

Global continuation of periodic solutions for retarded functional differential equations on manifolds

We consider T-periodic parametrized retarded functional differential equations, with infinite delay, on (possibly) noncompact manifolds. Using a topological approach, based on the notions of degree of a tangent vector field and of the fixed point index, we prove a global continuation result for T-periodic solutions of such equations.Our main theorem is a generalization to the case of retarded equations of a global continuation result obtained by the last two authors for ordinary differential equations on manifolds. As corollaries we obtain a Rabinowitz-type global bifurcation result and a continuation principle of Mawhin type.MSC:34K13, 34C40, 37C25, 70K42.

A Continuation Result for Forced Oscillations of Constrained Motion Problems with Infinite Delay

Abstract We prove a global continuation result for T-periodic solutions of a T-periodic parametrized second order retarded functional differential equation on a boundaryless compact manifold with nonzero Euler-Poincaré characteristic. The approach is based on the fixed point index theory for locally compact maps on ANRs. As an application, we prove the existence of forced oscillations of retarded functional motion equations defined on topologically nontrivial compact constraints. This existence result is obtained under the assumption that the frictional coefficient is nonzero, and we conjecture that it is still true in the frictionless case.

A degree theory for a class of perturbed Fredholm maps

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 -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.

About the Sign of Oriented Fredholm Operators between Banach Spaces

We give conditions for an oriented family of bounded Fredholm operators of index zero between Banach spaces to have a sign jump. In particular, we discuss criteria for detecting the sign jump in some special situations. For instance, when a sort of Crandall-Rabinowitz condition for bifurcation is assumed or in the case of a family of Leray-Schauder type. Finally, some examples of ordinary differential operators are presented to illustrate the meaning of the abstract results.

Retarded Functional Differential Equations on Manifolds and Applications to Motion Problems for Forced Constrained Systems

Abstract We prove an existence result for T-periodic retarded functional differential equations of the type xʹ(t) = f(t, xt), where f is a T-periodic functional tangent vector field on a smooth manifold. As an application we show that any constrained system acted on by a periodic force, possibly with delay, admits a forced oscillation provided that the constraint is a topologically nontrivial compact manifold and the frictional coefficient is nonzero. We conjecture that the same assertion holds true even in the frictionless case.

Atypical Bifurcation Without Compactness

We prove a global bifurcation result for an abstract equation of the type Lx + λh(λ, x) = 0, where L : E → F is a linear Fredholm operator of index zero between Banach spaces and h : R × E → F is a C1 (not necessarily compact) map. We assume that L is not invertible and, under suitable conditions, we prove the existence of an unbounded connected set Σ of nontrivial solutions of the above equation (i.e. solutions (λ, x) with λ 6= 0) such that the closure of Σ contains a trivial solution (0, x). This result extends previous ones in which the compactness of h was required. The proof is based on a degree theory for Fredholm maps of index zero developed by the first two authors.

On the product formula for the oriented degree for Fredholm maps of index zero between Banach manifolds

One of the most important and deep properties of the Leray-Schauder degree is the well-known Leray Product Formula for the computation of the degree of a composite map (see, e.g., [3], [14], [15], [17], [19]). In this paper, using the concept of boundary set of a map introduced in [1], among other results we give an extension of the Leray formula (Theorem 3.5) and we provide, as a consequence, a simple proof of the generalized Jordan-Brouwer Separation Theorem due to Leray (see [14]). As it is well-known, the integer valued degree has been extended by several authors to the framework of Fredholm maps between real Banach manifolds. A pioneering work in this direction is due to Elworthy and Tromba (see [8], [9]). In [1], still in the context of nonlinear Fredholm maps, the first two authors introduce an elementary notion of oriented map (see below) which differs from the one given in [10] in some aspects which are pointed out in [2]. By means of this notion they define an integer valued degree which coincides, for a large variety of maps, with the degree introduced in [10] and can be considered an evolution of the oriented degree of Elworthy-Tromba. This work contains two versions of the Product Formula for the oriented degree of [1], namely Theorem 3.1 and Theorem 3.7. The first one is the analogue of Theorem 3.5. The second one is a more general formula containing, as a particular case, an extended additivity property for the degree of oriented maps.