Fabio Zanolin

A Continuation Approach To Superlinear Periodic Boundary-value-problems

This paper deals with the problem of the existence of T-periodic solutions for the first order differential system x’ = I;( t, x), (1.1) where F: [0, T] x R” + R” is a Caratheodory function. In what follows, we prove some results for the solvability of the periodic BVP in the case when the dimension ,of the space is even. Such a limitation is motivated by our interest in applications to the second order equation d’+ g(t, u, u’)=O, (1.2) which takes the form of (1.1) when it is written as the equivalent system u’ = v 0’ = -g(t, u, v). (1.3)

Permanence and Positive Periodic Solutions for Kolmogorov Competing Species Systems

We prove the existence of periodic solutions in a compact attractor of (R+)n for the Kolmogorov system x′i= xifi(t, x1, …, xn), i = l, …, n in the competitive case. Extension to differential delay equations are con- sidered too. Applications are given to Lotka-Volterra systems with periodic coefficients.

Periodic solutions of the Liénard equation with one-sided growth restrictions

In this article we are concerned with the problem of the existence of periodic solutions to the periodically forced scalar Lienard equation x” + f(x) x’ + g(x) = e( t ) (’ = d/dt), (1) where ,f, g, e: R + R are continuous functions and e(.) is periodic. We are looking for solutions of (1) having the same period of the forcing term e(.). Throughout the paper, we assume, without loss of generality (see e.g. [22; 36, p. 693]), that

Periodic solutions of Duffing's equations with superquadratic potential

Abstract This paper is devoted to the study of harmonic and subharmonic solutions for the second order scalar nonlinear Duffings equation x ″ + g ( x ) = p ( t , x , x ′), where g and p are continuous functions with p bounded and periodic in the first variable and g satisfying the assumption g(x) sign (x) → + ∞, as ¦x¦ → + ∞ . Among other results, we prove the existence of infinitely many harmonic and subharmonic solutions (of any order) p = p ( t ) and if the potential G ( x ) of g ( x ) satisfies certain conditions of superquadratic growth at ∞. The new existence results can be applied to situations in which the more classical superlinear growth condition g(x) x → + ∞ , as ¦x¦ → + ∞ , is not satisfied. In this manner, various preceding theorems are improved and sharpened (see the “Introduction” for more details). Proofs are based on a generalized version of the Poincare-Birkhoff “twist” theorem due to W. Ding.

Periodic oscillations for a nonlinear suspension bridge model

Abstract We look for time-periodic solutions of the suspension bridge equation. Lazer and McKenna showed that for a certain configuration of the parameters, one may expect the existence of large-amplitude periodic solutions having the same period as the forcing term. We prove the existence of large-scale subharmonic solutions.

Nonlinear perturbations of differential operators with nontrivial kernel and applications to third-order periodic boundary value problems

This paper deals with the solvability of the nonlinear operator equations in normed spaces Yx = EGx + J where dp is a linear map with possible nontrivial kernel. Applications are given to the existence of periodic solutions for the thirdorder scalar differential equation x”’ + ax” + bx’ + cx + g(t, x) = p(t) under various conditions on the interaction of g(t, x)/x with spectral contigurations of a, b, and c.

On the use of time-maps for the solvability of nonlinear boundary value problems

(T > 0 is a fixed positive constant). The study of the periodic problem for equation (1.1) (or for some of its generalizations) represents a central subject in the qualitative theory of ordinary differential equations and it has been widely developed by the introduction of powerful tools from nonlinear functional analysis. See e.g. [22, 11, 9, 8, 16] and the references therein, for a source of various different techniques which can be used for this purpose. A classical method to deal with problem (1.1)-(1.2) consists into the search of fixed points of the translation operator (Poincar~-Andronov map) ~ : (x o, Yo) ~ (x (T; Xo, Yo), y (T; xo, Yo)) associated to the equivalent planar system

On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill's equations

Abstract We present some results which show the rich and complicated structure of the solutions of the second order differential equation ẍ + w(t)g(x) = 0 when the weight w(t) changes sign and g is sufficiently far from the linear case. New applications, motivated by recent studies on the superlinear Hill’s equation in [57, 58, 59], are then proposed for some asymptotically linear equations and for some sublinear equations with a sign-indefinite weight. Our results are based on a fixed point theorem for maps which satisfy a stretching condition along the paths on two-dimensional cells.

A topological approach to superlinear indefinite boundary value problems

We obtain the existence of infinitely many solutions with prescribed nodal properties for some boundary value problems associated to the second order scalar equation

Fixed points, periodic points, and coin-tossing sequences for mappings defined on two-dimensional cells

\ddot{x} + q(t) g(x) = 0