Periodic solutions of Euler-Lagrange equations with sublinear potentials in an Orlicz-Sobolev space setting

Sonia Acinas, Fernando Mazzone

Abstract


In this paper, we obtain existence results of periodic solutions of hamiltonian systems in the Orlicz-Sobolev space \(W^1L^\Phi([0,T])\). We employ the direct method of calculus of variations and we consider  a potential  function \(F\) satisfying the inequality \(|\nabla F(t,x)|\leq b_1(t) \Phi_0'(|x|)+b_2(t)\), with \(b_1, b_2\in L^1\) and  certain \(N\)-functions \(\Phi_0\).

Keywords


Periodic solution; Orlicz-Sobolev spaces; Euler-Lagrange; \(N\)-function; critical points

Full Text:

PDF

References


Acinas, S., Buri, L., Giubergia, G., Mazzone, F., Schwindt, E., Some existence results on periodic solutions of Euler-Lagrange equations in an Orlicz-Sobolev space setting, Nonlinear Anal. 125 (2015), 681-698.

Adams, R., Fournier, J., Sobolev Spaces, Elsevier/Academic Press, Amsterdam, 2003.

Conway, J. B., A Course in Functional Analysis, Springer, New York, 1985.

Fiorenza, A., Krbec, M., Indices of Orlicz spaces and some applications, Comment. Math. Univ. Carolin. 38 (3) (1997), 433-452.

Gustavsson, J., Peetre, J., Interpolation of Orlicz spaces, Studia Math. 60 (1) (1977), 33-59, URL http://eudml.org/doc/218150

Hudzik, H., Maligranda, L., Amemiya norm equals Orlicz norm in general, Indag. Math. (N.S.) 11 (4) (2000), 573-585.

Krasnoselskiı, M. A., Rutickiı, J. B., Convex Functions and Orlicz Spaces, P. Noordhoff Ltd., Groningen, 1961.

Maligranda, L., Orlicz Spaces and Interpolation, Vol. 5 of Seminarios de Matematica [Seminars in Mathematics], Universidade Estadual de Campinas, Departamento de Matematica, Campinas, 1989.

Mawhin, J., Willem, M., Critical Point Theory and Hamiltonian Systems, Springer-Verlag, New York, 1989.

Rao, M. M., Ren, Z. D., Theory of Orlicz Spaces, Marcel Dekker, Inc., New York, 1991.

Tang, C.-L., Periodic solutions of non-autonomous second-order systems with (gamma)-quasisubadditive potential, J. Math. Anal. Appl. 189 (3) (1995), 71-675.

Tang, C.-L., Periodic solutions for nonautonomous second order systems with sublinear nonlinearity, Proc. Amer. Math. Soc. 126 (11) (1998), 3263-3270.

Tang, C. L.,Wu, X.-P., Periodic solutions for second order systems with not uniformly coercive potential, J. Math. Anal. Appl. 259 (2) (2001), 386-397.

Tang, X., Zhang, X., Periodic solutions for second-order Hamiltonian systems with a p-Laplacian, Ann. Univ. Mariae Curie-Skłodowska Sect. A 64 (1) (2010), 93-113.

Tian, Y., Ge, W., Periodic solutions of non-autonomous second-order systems with a p-Laplacian, Nonlinear Anal. 66 (1) (2007), 192-203.

Wu, X.-P., Tang, C.-L., Periodic solutions of a class of non-autonomous second-order systems, J. Math. Anal. Appl. 236 (2) (1999), 227-235.

Xu, B., Tang, C.-L., Some existence results on periodic solutions of ordinary p-Laplacian systems, J. Math. Anal. Appl. 333 (2) (2007), 1228-1236.

Zhao, F., Wu, X., Periodic solutions for a class of non-autonomous second order systems, J. Math. Anal. Appl. 296 (2) (2004), 422-434.

Zhao, F., Wu, X., Existence and multiplicity of periodic solution for non-autonomous second-order systems with linear nonlinearity, Nonlinear Anal. 60 (2) (2005), 325-335.

Zhu, K., Analysis on Fock Spaces, Springer, New York, 2012.




DOI: http://dx.doi.org/10.17951/a.2017.71.2.1
Date of publication: 2017-12-18 20:31:30
Date of submission: 2017-12-15 22:17:03


Statistics


Total abstract view - 1993
Downloads (from 2020-06-17) - PDF - 624

Indicators



Refbacks

  • There are currently no refbacks.


Copyright (c) 2017 Sonia Acinas, Fernando Mazzone