Multiplicity of periodic bouncing solutions for generalized impact Hamiltonian systems

Delong Huang,Fei Guo

doi:10.1186/s13661-019-1169-1

Abstract

Applying the Generalized Nonsmooth Saddle Point Theorem, we obtain multiple nontrivial periodic bouncing solutions for systems ddot{x}=f(t,x) with new conditions. In particular, we generalize the collision axis from x=0 to the axis x=a, where a is an arbitrary constant.

Highlights

Consider the following Hamiltonian system with an obstacle, that is, x = f (t, x), t ∈ R \ W, (1.1)associated with the conditions⎧ ⎪⎪⎨x(t–) = –x(t+), t ∈ W, ⎪⎪⎩xx((tt)) ≥ a, = x(t + T ), ∀t ∈ R, ∀t ∈ R, (1.2)
Different from the papers [3, 7] and [9], we focus on the nontrivial kT-periodic bouncing solutions for system (1.1) with a new condition separating whether or not a is equal to 0
Definition 2.1 Function φ satisfies the nonsmooth (PS) condition if every sequence {xn} ⊂ E, such that {φ(xn)} is bounded and λ(xn) → 0 for n → ∞, has a strongly convergent subsequence, where λ(x) = infx∗∈∂φ(x) x∗ E∗, E∗ is the dual space of E, and ∂φ(x) denotes the Clarke generalized gradient of φ

Summary

Introduction

In 2017, Nie (see [9]) first proved a Generalized Nonsmooth Saddle Point Theorem, which is applied to impact Hamiltonian systems, obtained nontrivial kT-periodic bouncing solutions for system (1.1) with another sublinear condition. Definition 2.1 (see [5]) Function φ satisfies the nonsmooth (PS) condition if every sequence {xn} ⊂ E, such that {φ(xn)} is bounded and λ(xn) → 0 for n → ∞, has a strongly convergent subsequence, where λ(x) = infx∗∈∂φ(x) x∗ E∗ , E∗ is the dual space of E, and ∂φ(x) denotes the Clarke generalized gradient of φ. Suppose that functional φ satisfies the nonsmooth (PS) condition, and for some x0 ∈ X, there exists a constant r > 0 such that maxv∈V∩∂Br φ(v + x0) < infx∈X φ(x).

Set x

Choose ε

For every k