| Consider on the existence of a unique periodic solution of the following second order Hamiltonian systems:where T>0, F:R x RNâ†'R is T-periodic (T>0) in t and satisfies the following assumption:(A) F(t,x) is measurable in t for every x∈RN and continuously differ-entiable in x for a.e. t∈[0, T], and there exist a∈C(R+,R+) and b∈L1(0,T; R+) such that|F(t,x)|≤a(|x|)b(t),|â–½F(t, x)|≤a(|x|)b(t) for all x∈RN and a.e. t∈[0,T].It has been proved the existence of the periodic solutions of system (P1), however, as we all know, litter was done studying the uniqueness periodic so-lution of system(P1), only [23] had considered this situation. More precisely, he had established the following theorem:Theorem A (see Theorem1.1in [23])Suppose that (A) and there are exist f, g∈L1(0,T; R+),0<α≤1and β>0satisfies the following conditions:(F1)(â–½F(t, x)-â–½F(t,y), x-y)>0, x≠y for all t∈R, x, y∈RN,(F2)|â–½F(t,x)|≤f(t)|x|β+g(t),(F3)1/2-(2α-1β)/12-C0>0,(F4) lim inf|x|â†'∞[|x|-2α∫0T(t,x)dt-C1]>0, for all x∈RN, where Then, system (P1) has a unique T-periodic solution in HT1.It seems that Theorem A’s conditions is complicate, and we will obtain the existence of a unique periodic solution of systems (P1) under much weaker assumption by the least action principle. Our main result is the following theorem1:Theorem1Suppose (A),(F1) and the following condition is hold:(F5)∫0T F(t,x)dtâ†'+∞for all|x|â†'∞, Then, systems (P1) has a unique T-periodic solution in HT1.Consider the following second order Hamiltonian systems-u(t)=â–½F(t,u(t)),(?)t∈[0,T],(P2)where T>0, F:R×RNâ†'R is T-periodic (T>0) in t and satisfies the following assumption:(A) F(t, x) is measurable in t for every x∈RN and continuously differ-entiable in x for a.e. t∈[0,T], and there exist a∈C(R+,R+) and b∈L1(0,T;R+) such that|F(t,x)|≤a(|x|)b(t),|â–½F(t,x)|≤a(|x|)b(t) for all x∈RN and a.e. t∈[0,T].It has been proved that the existence of the periodic solutions of system-s (P2)(see [3,5-7,12,15,17-21]). Many solvability conditions are given, such as, the subquadratic condition (see [12]); the convex potential condition (see [6]); the not uniformly coercive potential condition (see [18]); the even potential condition (see [7]); the changing sign condition (see [3,19]); the γ-quasisubadditive potential condition (see [17]); the superquadratic condition (see [5,20]), the local superquadratic condition (see [15,21]). Specially, in2006, Schetcher [14] had obtained some results of the existence of solutions. More precisely, he had established the following theorems:Theorem B (see Theorem1.2in [14]). Suppose that F satisfies the following conditions: (H1) F(t,x)≥0, t∈[0, T], x∈RN.(H2) There is a constant q>2such that F(t, x)<C(|x|q+1), t∈[0, T], x∈RN,(3)and there are constants m>0,α<2Ï€2/T2such that F(t, x)≤α|x|2,|x|≤m, t∈[0, T], x∈RN.(4)(H3) There exists a constan μ>2such thatwhere Fμ(t,x)=μF(t,x)-(â–½F(t,x),x).(H4) There are constants β>(2Ï€2)/(T2)and C>0such that F(t,x)≥β|x|2,|x|>C,t∈[0,T],x∈RN Then systems (P2) has a non-constant solution.Theorem C (see Theorem1.1in [14]). The conclusion in Theorem B is the same if hypothesis (H2) is replaced by (H’2) There are constants m>0, α≤6m2/T2such that F(t,x)≤α,|x|≤m, t∈[0,T], x∈RN.In2004, Tao and Tang in [20] had obtained the following theorem:However, as far as we know, litter was done concerning non-constant T-periodic solutions of systems (P2) with local superquadratic condition. Only [21] had considered this situation. In this paper, we will consider the existence of non-constant T-periodic solutions of systems (P2) with local superquadratic condition. Our main results are the following theorems:Theorem2. Suppose that F satisfies (A),(H1),(3.2),(H3) and the following condi-tion: (H’4) There is a subset E∈[0,T] of positive measure such that lim inf|x|â†'∞>(F(t,x))/(|x|2)>0for a.e.t∈E. Then systems (P2) has at least one non-constant T-periodic solution.Theorem3. Suppose that F satisfies (A),(H1),{H’2),(H3) and (H\4). Then systems (P2) has at least one non-constant T-periodic solution. |
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