| Consider the second order Hamiltonian systemswhere T > 0, A(t) is an N×N symmetric matrix, continuous and T-periodic in t, F : R×RN→R is T-periodic in t and satisfies the following condition(A) F(t, x) is measurable in t for every x∈RN and continuously differentiable in x for a.e. t∈[0,T], and there exist a∈C(R+,R+),b∈L1(0,T;R+) such thatfor all x∈RN and a.e. t∈[0, T].In this paper, the minirnax method in critical point theory is employed to search the existence of periodic and subharmonic solutions for a class of second order Hamiltonian systemswith not uniformly coercive potential, and with superquadratic potential.First, consider the case A = 0, then systems (HS1) becomesOur main results are the following theorems. Theorem 1 Suppose that F(t,x) = G(x) + H(t,x) satisfies assumption (A). Assume that there exist r < 4π2/T2 and g∈L1(0,T;R+) such thatfor all x, y∈RN andfor all x∈RN and a.e. t∈[0,T]. Assume also that there existsγ∈L1(0,T) such thatfor all x∈RN and a.e. t∈[0,T], and that there exists a subset E of [0,T] with meas(E) > 0 such thatfor a.e. t∈E. Then problem (HS2) has at least one solution inTheorem 2 Suppose that F(t, x) = G(x) + H(t,x) satisfies assumption (A) and (1), and there exist f, g∈L1 (0, T; R+) such thatfor all x∈RN and a.e. t∈[0,T]. Assume thatas |x|→+∞uniformly for a.e. t∈[0,T]. Then problem (HS2) has at least one soluiton inHT1Theorem 3 Suppose that F(t, x) = G(x) + H(t, x) satisfies assumption (A), (1), (2) and (3). Assume that there exists a subset E of [0, T] with meas E > 0 such thatfor a.e. t∈E. Then problem (HS2) has at least one soluiton in HT1.Theorem 4 Suppose that F(t, x) = G(x) + H(t,x) satisfies assumption (A), (1), (2), (3) and (4). Assume that there existsδ> 0,ε> 0 and an integer k > 0 such that for all x∈RN and a.e. t∈[0,T], andfor all |x|≤δand a.e. t∈[0,T], whereω= 2π/T. Then problem (HS2) has at least one soluiton in HT1.Then we consider the general case where A(t) is an N×N symmetric matrix, continuous and T-periodic in t.Theorem 5 Suppose F satisfies (A) and the following conditions :Assume that there existλ> 2 andβ>λ- 2 such thatIf 0 is an eigenvalue of -d2/dt2 - A(t) (with periodic boundary condition), assume alsofor some r > 0. Then problem (HS1) has at least one nontrivial T-periodic solution.Theorem 6 Suppose that A(t) = m2ω2I, where m is a nonnegative integer,ω= 2π/T and I is the unit matrix of order N, and F satisfies (A), (5), (6), (7) and the following conditions :Then there exists a sequence {kj} (?) N, kj→∞, and corresponding distinct kjT periodic solutions of problem (HS1).Consider the second-order discrete Hamiltonian system where△u(t) = u(t+1)-u(t),△2u(t) =△(△u(t)), b∈C(R,R) and there exists a positive integer T such that b(t + T) = b(t) for all t∈Z, Z is the set of all integers, V∈C1(RN, R), (?)V(x) denotes the gradient of V(x) in x.We obtain some existence results of periodic solutions for the second order discrete Hamiltonian systems with a change of sign in potential by the minimax methods in critical point theory. Our main results are the following theorems.Theorem 7. Suppose that b(t) and V(x) = a|x|μ+ W(x), where a > 0,μ> 2, W∈C1(RN,R) satisfies the following assumptions(b1) b∈C(R, R), there exists a positive integer T, such that for any t∈Z, b(t+T) = b(t),∑t=1Tb(t)=0 and b (?) 0.(b2) There exists a T-periodic functional e : Z→RN such that e|Z[1,T]\N1 =0,∑t∈N1= 0, e (?) 0 andwhere N1 = {t∈Z[1,T] : b(t) > 0}.(W1) There existα0∈(0,2B-1sin2(?)))and r0 > 0 such thatwhere B = max{b(t) : t∈Z[1,T]}, Z[n1,n2] = Z∩[n1,n2] for every n1,n2∈Z with n1≤n2(W2) There exists a constant G0> 0 such thatThen system (DHS) possesses at least one nontrivial solutions with period T.Theorem 8 Suppose thatμ> 2, d∈C(R, R) satisfying(d1) There exists a positive integer T, such that for any t∈Z, d(t+T) = d(t),∑t=1Td(t) = 0 and d (?) 0.(d2) There exists a T-periodic functional e : Z→RN such that e|Z[1,T]\N1 = 0,∑t∈N1e(t)=0,e(?)0 andwhere N1 = {t∈Z[1,T] : d{t) > 0}.Assume that H : Z×RN→R, H (t, x) is continuously differentiable in x for every t∈Z and T-periodic in t for all x∈RN, such that (H1)∑t=1TH(t,x) H(t, x)≥0 for all x∈RN.(H2) There existα0∈(0,1-cos(2π/T)) and r0 > 0 such that(H3) There exists M0 > 0 such thatThen problempossesses at least one nonzero T-periodic solution.
|
PDF Full Text Request