Existence and multiplicity of periodic solutions for some second order Hamiltonian systems

Abstract

In this paper, we study the existence of nontrivial periodic solutions for the second order Hamiltonian systemsü(t) + ∇F(t, u(t)) = 0, where F(t, x) is either nonquadratic or superquadratic as |u| → ∞. Furthermore, if F(t, x) is even in x, we prove the existence of infinitely many periodic solutions for the general Hamiltonian systemsü(t)where A(•) is a continuous T-periodic symmetric matrix. Our theorems mainly improve the recent result of Tang and Jiang [X.H. Tang, J. Jiang, Existence and multiplicity of periodic solutions for a class of second-order Hamiltonian systems, Comput. Math. Appl. 59 (2010) 3646-3655].

Infinitely Many Solutions for a Generalized Periodic Boundary Value Problem without the Evenness Assumption

“…In his pioneer paper [1] of 1978, Rabinowitz studied for the existence of periodic solutions for Hamiltonian systems via the critical point theory. From then on, with the aid of the critical point theory, the existence of infinitely many periodic solutions for Hamiltonian systems has been extensively investigated in some papers (see [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]) and the excellent books (see [18][19][20]).…”

Existence of Periodic Solutions for Second-Order Ordinary p-Laplacian Systems

“…Wu, Chen and Teng [7] provided the variational principle of system (3). At the same time, some existence results of system (3) were obtained by using critical point theorem.…”

“…The existence and multiplicity of periodic solutions to problem (2) were obtained on various hypotheses on the potential function F(t, x) or nonlinearity ∇F(t, x) (see, Refs. [1][2][3][4][5][6]).…”