### Multiple periodic solutions of Hamiltonian systems in the plane

#### Abstract

Our aim is to prove a multiplicity result for periodic solutions

of Hamiltonian systems in the plane, by the use of the Poincaré-Birkhoff

Fixed Point Theorem. Our main theorem generalizes previous results obtained

for scalar second order equations by Lazer and McKenna [< i> Large scale oscillatory behaviour

in loaded asymmetric systems< /i> , Ann. Inst. H. Poincaré Anal. Non Linéaire < b> 4< /b> (1987), 243–274] and

Del Pino, Manasevich and Murua [< i> On the number of $2\pi$-periodic

solutions for $u''+g(u) =s(1+h(t))$ using the Poincaré–Birkhoff Theorem< /i> ,

J. Differential Equations < b> 95< /b> (1992), 240–258].

of Hamiltonian systems in the plane, by the use of the Poincaré-Birkhoff

Fixed Point Theorem. Our main theorem generalizes previous results obtained

for scalar second order equations by Lazer and McKenna [< i> Large scale oscillatory behaviour

in loaded asymmetric systems< /i> , Ann. Inst. H. Poincaré Anal. Non Linéaire < b> 4< /b> (1987), 243–274] and

Del Pino, Manasevich and Murua [< i> On the number of $2\pi$-periodic

solutions for $u''+g(u) =s(1+h(t))$ using the Poincaré–Birkhoff Theorem< /i> ,

J. Differential Equations < b> 95< /b> (1992), 240–258].

#### Keywords

Multiplicity of periodic solutions; nonlinear boundary value problems; Poincaré-Birkhoff Theorem

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