Subharmonic solutions of first-order Hamiltonian systems

Abstract The aim of this article is to study subharmonic solutions of superquadratic and asymptotically (constant) linear nonautonomous Hamiltonian systems in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi mathvariant="double-struck">R</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:mrow> </m:msup> </m:math> {{\mathbb{R}}}^{2n} respectively, and to improve the results in Professor Liu’s [ Subharmonic solutions of Hamiltonian systems , Nonlinear Anal. 42 (2000), no. 2, 185–198, Ser. A: Theory Methods]. For the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>T</m:mi> </m:math> T -periodic Hamiltonian system, we show that for each integer <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>j</m:mi> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:math> j\ge 1 , there exists a nonconstant <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>j</m:mi> <m:mi>T</m:mi> </m:math> jT -periodic solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {x}_{j} such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {x}_{j} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {x}_{pj} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:math> p\in {\mathbb{N}} ) are geometrically distinct provided <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>&gt;</m:mo> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:math> p\gt 2n ; for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> n=1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {x}_{j} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>x</m:mi> </m:mrow> <m:mrow> <m:mi>p</m:mi> <m:mi>j</m:mi> </m:mrow> </m:msub> </m:math> {x}_{pj} ( <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="double-struck">N</m:mi> </m:math>