A continuation lemma and the existence of periodic solutions of perturbed planar Hamiltonian systems with sub-quadratic potentials

Zaihong Wang, Tiantian Ma

DOI: http://dx.doi.org/10.12775/TMNA.2018.037


n this paper, we study the existence of periodic solutions of perturbed planar Hamiltonian systems of the form $$ \begin{cases} x'=f(y)+p_1(t,x,y), \\ y'=-g(x)+p_2(t,x,y). \end{cases} $$% We prove a continuation lemma for a given planar system and further use it to prove that this system has at least one $T$-periodic solution provided that $g$ has some sub-quadratic potentials.


Continuation lemma; sub-quadratic potential; periodic solution

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A. Boscaggin, Subharmonic solutions of planar Hamiltonian systems: a rotation number approach, Adv. Nonlinear Stud. 11 (2011), 77–103.

A. Boscaggin and M. Garrione, Resonance and rotation numbers for planar Hamiltonian systems: Multiplicity results via the Poincaré–Birkhoff theorem, Nonlinear Anal. 74 (2011), 4166–4185.

T. Ding, R. Iannacci and F. Zanolin, On periodic solutions of sublinear Duffing equations, J. Math. Anal. Appl. 158 (1991), 316–332.

T. Ding, R. Iannacci and F. Zanolin, Existence and multiplicity results for periodic solutions of semilinear Duffing equations, J. Differential Equations 105 (1993), 364–409.

M.L. Fernandes and F. Zanolin, Periodic solutions of a second order differential equation with one-sided growth restrictions on the restoring term, Arch. Math. 51 (1988), 151–163.

A. Fonda and L. Ghirardelli, Multiple periodic solutions of Hamiltonian systems in the plane, Topol. Methods Nonlinear Anal. 36 (2010), 27–38.

A. Fonda and A. Sfecci, A general method for the existence of periodic solutions of differential systems in the plane, J. Differential Equations 252 (2012), 1369–1391.

M. Garrione, Resonance at the first eigenvalue for fist order systems in the plane: vanishing Hamiltonians and the Landesman–Lazer conditions, Differential Integral Equations 25 (2012), 505–526.

A.C. Lazer, On Schauder’s fixed point theorem and forced second order non-linear oscillations, J. Math. Anal. Appl. 21 (1968), 421–425.

T. Ma and Z. Wang, A continuation lemma and its applications to periodic solutions of Rayleigh differential equations with subquadratic potential conditions, J. Math. Anal. Appl. 385 (2012), 1107–1118.

J. Mawhin, Equivalence theorems for nonlinear operator equations and coincidence degree theory for some mappings in locally convex topological vector spaces J. Differential Equations 12 (1972), 610–636.

J. Mawhin and J. Ward, Periodic solutions of second order forced Liénard differential equations at resonance, Arch. Math. 41 (1983), 337–351.

P. Omari and F. Zanolin, Nonresonance conditions on the potential for a second-order periodic boundary value problem, Proc. Amer. Math. Soc. 117 (1993), 125–135.

R. Reissig, Periodic solutions of a second order differential equation including a one-sided restoring term, Arch. Math. 33 (1979), 85–90.

K. Schmitt, Periodic solutions of a forced nonlinear oscillator involving a onesided restoring force, Arch. Math. 31 (1978), 70–73.

Z. Wang and T. Ma, Periodic solutions of planar Hamiltonian systems with asymmetric nonlinearities, Boundary Value Problems, 2017, No. 46.


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