Periodic Bouncing Solutions for Nonlinear Impact Oscillators

Abstract

We prove the existence of a periodic solution to a nonlinear impact oscillator, whose restoring force has an asymptotically linear behavior. To this aim, after regularizing the problem, we use phase-plane analysis, and apply the Poincaré-Bohl fixed point Theorem to the associated Poincaré map, so to find a periodic solution of the regularized problem. Passing to the limit, we eventually find the “bouncing solution” we are looking for.

Periodic Impact Motions at Resonance of a Particle Bouncing on Spheres and Cylinders

“…The reader will notice that the next result comes out free by the proof of the previous theorem (setting L = 0 everywhere). Such a corollary extends a previous result provided by Fonda and the author in [17].…”

“…Moreover, following the proof of Theorem 3.1, we will obtain Corollary 3.2 which extends to the resonant case a previous result obtained by Fonda and the author in [17] for impact oscillators.…”

“…where, here, x varies in R. Let us spend few words to motivate the introduction of the small constant δ > 0, which can appear unessential: it will be useful to simplify the proof of the validity of the fourth property, which appears in Definition 1.1, for the bouncing solution we are going to find (cf. [17]). It is well known that such a differential equation has at least one Tperiodic solution when n is large enough and L is fixed (cf.…”

Odd Periodic Solutions of Fully Second-Order Ordinary Differential Equations with Superlinear Nonlinearities

“…Many theorems and methods of nonlinear functional analysis have been applied to the periodic problems of (2), such as the upper and lower solutions method and monotone iterative technique [1][2][3][4], the continuation method of topological degree [5][6][7][8][9], variational method and critical point theory [10][11][12][13][14], method of phase-plane analysis [15][16][17][18][19], the Krasnoselskii's type fixed point theorem in cone [20][21][22][23], and the theory of fixed point index [24][25][26].…”

Double resonance for one-sided superlinear or singular nonlinearities

“…We have preferred to postpone their proof because the arguments we will use are totally independent by the rest of the proof of Proposition 2.11. This section is inspired by some recent results obtained by the second author in [22] for impact systems at resonance (see also [13]).…”