Periodic And Subharmonic Solutions Of Second-order Hamiltonian Systems

Consider the second-order Hamiltonian systemswhere A(t) is a real symmetric n x n matrix made of continuous functions, T > 0 and F : R × RN → 4 R is T-periodic in its first variable and satisfies the following assumption:(A) F(t,x) is measurable in t for all x ∈ RN, continuously differentiable in x for a.e. t ∈ [0,T], and there exist a ∈ C(R+,R+) and b ∈ Ll(0,T;R+) such that\F(t,x)\ ≤ a(|x|)b(t), for all x∈ RN and a.e. t∈ [0,T].In this paper, we studied the second order Hamiltonian systems (HS1) under the condition "superquadratic" and "subquadratic" by mini- max method, and obtained some theorems about periodic solutions of (HS1). Moreover, we have studied the subharmonic solutions of (HS2) under the "superquadratic" and "subquadratic" condition. In fact, the subharmonic solutions is the XT-period solutions, where K is a positive integer. The main results are the following theorems.Theorem 1 Suppose F satisfies conditions (A) andwhere μ > 0 is a constant, (·,·) is the usual inner product of RN and are consecutive eigenvalues of the operator -d2/dt2+ A(t). Then there exists a T- period solution of system (HS1).'Supported by National Natural Science Foundation of China, by Major Project of Science and Technology of MOE, P.R.C and by the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of MOE, P.R.CTheorem 2 Suppose F satisfies conditions (A) andwhere μ > 0 is a constant and Ak-1 < k are consecutive eigenvalues of the operator -d2/dt2+A(t). Then there exists a T- period solution of system (HS1).Corollary 1 Suppose F satisfies conditions (.4), (1) (or (3)) and exists a constant A > A0 such that= A unif. for a.e. t [0,T],where AO is the minimal eigenvalue of the operator -d2/dt2 + A(t). Then there exists a T- period solution of system (HS1).Theorem 3 Suppose F satisfies conditions (A), (1) and unif. for a.e. t ∈ [0, T], (5) for all x ∈ RN and for a.e. t 6 [0,T], (6)where m > 0, 2 < r < n + 2, g ∈ L1(Q,T;R) and k-1< k are consecutive eigenvalues of the operator -d2/dt2+A(t). Then there exists a non-constant T- period solution of system (HS1).Theorem 4 Suppose F satisfies conditions (A), (3), (5) and (6). Then there exists a non-constant T-period solution of system (HS1).Theorem 5 Suppose F satisfies conditions (A), (1) andThen there exist infinite distinct subharmonic solutions of system (HS2).Theorem 6 Suppose F satisfies conditions (A), (3) and (7)- (10). Then there exist infinite distinct subharmonic solutions of system...