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This paper aims to provide a careful and self-contained introduction to the theory of topological degree in Euclidean spaces. It is intended for people mostly interested in analysis and, in general, a heavy background in algebraic or differential topology is not required., Comment: 25 pages

We study, by means of a topological approach, the forced oscillations of second order functional retarded differential equations subject to periodic perturbations. We consider a delay-type functional dependence involving a gamma probability distribution. By a linear chain trick we obtain a first order system of ODE's whose $T$-periodic solutions correspond to those of the functional equation., Comment: 15 pages, 4 figures

We consider the propagation of surface shear waves in a half-plane, whose shear modulus $\mu(y)$ and density $\rho(y)$ depend continuously on the depth coordinate $y$. The problem amounts to studying the parametric Sturm-Liouville equation on a half-line with frequency $\omega$ and wave number $k$ as the parameters. The Neumann (traction-free) boundary condition and the requirement of decay at infinity are imposed. The condition of solvability of the boundary value problem determines the dispersion spectrum $\omega(k)$ for the corresponding surface wave. We establish the criteria for non-existence of surface waves and for the existence of $N(k)$ surface wave solutions, with $N(k) \to \infty$ as $k \to \infty$. The most intriguing result is a possibility of the existence of infinite number of solutions, $N(k)=\infty$, for any given $k$. These three options are conditioned by the properties of $\mu(y)$ and $\rho(y)$., Comment: 19 pages, 2 figures

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed coupled and nonautonomous differential equations on manifolds. By using degree-theoretic methods we obtain a global continuation result for the $T$-periodic solutions of the considered equations.

We investigate the structure of the set of $T$-periodic solutions to periodically perturbed coupled delay differential equations on differentiable manifolds. By using fixed point index and degree-theoretic methods we prove the existence of branches of $T$-periodic solutions to the considered equations. As main application of our methods, we study a generalized version of the sunflower equation.

By using topological methods, mainly the degree of a tangent vector field, we establish multiplicity results for $T$-periodic solutions of parametrized $T$-periodic perturbations of autonomous ODEs on a differentiable manifold $M$. In order to provide insights into the key notion of $T$-resonance, we consider the elementary situations $M = \mathbb{R}$ and $M = \mathbb{R}^2$. So doing, we provide more comprehensive analysis of those cases and find improved conditions., Comment: 10 figures

In this paper we investigate the properties of the set of T-periodic solutions of semi-explicit parametrized Differential-Algebraic Equations with non-autonomous constraints of a particular type. We provide simple, degree theoretic conditions for the existence of branches of T-periodic solutions of the considered equations. Our approach is based on topological arguments about differential equations on implicitly defined manifolds, combined with elementary facts of matrix analysis.

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., Comment: 15 pages, 2 figures, to appear in Communications in Contemporary Mathematics. arXiv admin note: substantial text overlap with arXiv:1102.1562

We study the structure of the set of harmonic solutions to perturbed nonautonomous, T-periodic, separated variables ODEs on manifolds. The perturbing term is allowed to contain a finite delay and to be T-periodic in time.

When the plane is pie-sliced in $n\leq 4$ parts (with nonempty interior and common vertex at the origin) our main result provides a sufficient condition for any map $L$, that is continuous and piecewise linear relatively to this slicing, to be invertible. Some examples show that the assumptions of the theorem cannot be relaxed too much. In particular, convexity of the slices cannot be dropped altogether when $n=4$. This result cannot be plainly extended to a greater number of slices. Our result is proved by a combination of linear algebra and topological arguments., Comment: 14 pages

We study the set of $T$-periodic solutions of a class of $T$-periodically perturbed Differential-Algebraic Equations, allowing the perturbation to contain a distributed and possibly infinite delay. Under suitable assumptions, the perturbed equations are equivalent to Retarded Functional (Ordinary) Differential Equations on a manifold. Our study is based on known results about the latter class of equations., Comment: 13 pages. Revision: Incorporate changes suggested by readers. Corrected a few typos across the paper, definition of BU added, revised the (previously incorrect) definition of solution of RFDAE, made slight changes in the Introduction. Replacement of Dec. 6, 2012: introduced further changes suggested by referee, bundled addendum/erratum containing a corrected version of Lemma 5.5 and Corollary 5.7

We apply topological methods to obtain global continuation results for harmonic solutions of some periodically perturbed ordinary differential equations on a $k$-dimensional differentiable manifold $M \subseteq \mathbb{R}^m$. We assume that $M$ is globally defined as the zero set of a smooth map and, as a first step, we determine a formula which reduces the computation of the degree of a tangent vector field on $M$ to the Brouwer degree of a suitable map in $\mathbb{R}^m$. As further applications, we study the set of harmonic solutions to periodic semi-esplicit differential-algebraic equations., Comment: 14 pages, final version, to appear in Nonlinear Differential Equations and Applications

This paper gives sufficient conditions for a class of bang-bang extremals with multiple switches to be locally optimal in the strong topology. The conditions are the natural generalizations of the ones considered in previous papers for more specific cases. We require both the strict bang-bang Legendre condition, and the second order conditions for the finite dimensional problem obtained by moving the switching times of the reference trajectory., Comment: 43 pages

We study a particular class of autonomous Differential-Algebraic Equations that are equivalent to Ordinary Differential Equations on manifolds. Under appropriate assumptions we determine an easy-to-use straightforward formula for the computation of the degree of the associated tangent vector field that does not require any explicit knowledge of the manifold. We use this formula to study the set of harmonic solutions to periodic perturbations of our equations. Two different classes of applications are provided., Comment: 16 pages Changes in new version: A few typos across the paper; Wrong cross references at page one; Statement and proof of Theorem 5.1 revised; $\deg(F,U)=1$ in Example 5.2; $p_n\to p_0$ in the proof of Lemma 5.5; inequality instead of equality in line 18, page 15; Assumption $A$ bounded added in Corollary 5.3

We apply topological methods to the study of the set of harmonic solutions of periodically perturbed autonomous ordinary differential equations on differentiable manifolds, allowing the perturbing term to contain a fixed delay. In the crucial step, in order to cope with the delay, we define a suitable (infinite dimensional) notion of Poincar\'e $T$-translation operator and prove a formula that, in the unperturbed case, allows the computation of its fixed point index., Comment: 14 pages