Periodic solutions of second-order systems with subquadratic convex potential

Yiwei Ye

doi:10.14232/ejqtde.2015.1.43

Abstract

In this paper, we investigate the existence of periodic solutions for the second order systems at resonance: ( ¨ u(t) + m 2 w 2 u(t) +rF(t, u(t)) = 0 a.e. t2 (0, T), u(0) u(T) = ˙ u(0) ˙ u(T) = 0, where m > 0, the potential F(t, x) is convex in x and satisfies some general subquadratic conditions. The main results generalize and improve Theorem 3.7 in J. Mawhin and M. Willem (Critical point theory and Hamiltonian systems, Springer-Verlag, New York, 1989).

Highlights

Introduction and main resultsConsider the second order Hamiltonian systems u(t) + m2ω2u(t) + ∇F(t, u(t)) = 0 a.e. t ∈ [0, T], (1.1)u(0) − u(T) = u (0) − u (T) = 0, where T > 0, ω = 2π/T and m > 0 is an integer
We investigate the existence of periodic solutions for the second order systems at resonance: u(t) + m2ω2u(t) + ∇F(t, u(t)) = 0 u(0) − u(T) = u (0) − u (T) = 0, a.e. t ∈ [0, T], where m > 0, the potential F(t, x) is convex in x and satisfies some general subquadratic conditions
Brézis [2], Mawhin and Willem [6] proved an existence theorem for semilinear equations of the form Lu = ∇F(x, u), where L is a noninvertible linear selfadjoint operator and F is convex with respect to u and satisfies a suitable asymptotic quadratic growth condition

Summary

Introduction

We investigate the existence of periodic solutions for the second order systems at resonance: u(t) + m2ω2u(t) + ∇F(t, u(t)) = 0 u(0) − u(T) = u (0) − u (T) = 0, a.e. t ∈ [0, T], where m > 0, the potential F(t, x) is convex in x and satisfies some general subquadratic conditions. The potential F : [0, T] × RN → R satisfies the following assumption: (A) F(t, x) is measurable in t for every x ∈ RN and continuously differentiable in x for a.e. t ∈ [0, T], and there exist a ∈ C(R+, R+), b ∈ L1(0, T; R+) such that

Results

Conclusion