Alessandro Calamai

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

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 C1 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.

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.

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.

Nontrivial solutions of boundary value problems for second-order functional differential equations

In this paper, we present a theory for the existence of multiple nontrivial solutions for a class of perturbed Hammerstein integral equations. Our methodology, rather than to work directly in cones, is to utilize the theory of fixed point index on affine cones. This approach is fairly general and covers a class of nonlocal boundary value problems for functional differential equations. Some examples are given in order to illustrate our theoretical results.

On the degree for oriented quasi-Fredholm maps: its uniqueness and its effective extension of the Leray-Schauder degree

In a previous paper, the first and third author developed a~degree theory for oriented locally compact perturbations of

Periodic perturbations of constrained motion problems on a class of implicitly defined manifolds

C\sp{1}

Bifurcation Results for a Class of Perturbed Fredholm Maps

Fredholm maps of index zero between real Banach spaces. In the spirit of a celebrated Amann--Weiss paper, we prove that this degree is unique if it is assumed to satisfy three axioms: Normalization, Additivity and Homotopy invariance. Taking into account that any compact vector field has a canonical orientation, from our uniqueness result we shall deduce that the above degree provides an effective extension of the Leray--Schauder degree.

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We study forced oscillations on differentiable manifolds which are globally defined as the zero set of appropriate smooth maps in some Euclidean spaces. Given a T-periodic perturbative forcing field, we consider the two different scenarios of a nontrivial unperturbed force field and of perturbation of the zero field. We provide simple, degree-theoretic conditions for the existence of branches of T-periodic solutions. We apply our construction to a class of second-order differential-algebraic equations.