Existence Of Periodic Solutions For Ordinary P-Laplacian Systems

Consider the non-autonomous second-order Hamiltonian systems where T＞0, and F:[0, T]×R^N→R satisfies the following assumption:(A) F(t, x) is measurable in t for each x∈R^N and continuously differentiable in x for a.e. t∈[0, T], and there exist a∈C(R⁺, R⁺), b∈L¹ (0, T; R⁺) such that for all x∈R^N and a.e. t∈[0, T].In this paper, the minimax method in critical point theory is employed to research the existence of periodic solutions for second order Hamiltonian systems. The main results are the following two theorems ([17]).Theorem 1 Suppose that F satisfies assumptions (A). Assume further thatThen the system (HS) has at least one solution in H_T¹.Theorem 2 Suppose that F satisfies assumptions (A), (F₁) and (F₂). Assume further that Then the system (HS) has at least one solution in H_T¹.Consider the ordinary p-Laplacian systems where p＞1, T＞0 and F:[0, T]×R^N→R satisfies the previous assumption (A).Using the uniformly convex Banach space property, we prove that the compact condition also in position under the assumptions of subquadratic potential condition. The main results are the following two theorems([18]).Theorem 3 Suppose that F satisfies assumptions (A). Assume that(F₃) there exist constants 0＜μ＜p, M＞0, such that (â–½F(t, x), x)≤μF(t, x) for |x|≥M and a.e. t∈[0, T],(F₄) there exists g∈L¹(0,T) such that F(t,x)≥g(t) for all x∈R^N and a.e. t∈[0,T],(F₅) there exists a subset E of [0, T] with meas(E)＞0, such that F(t, x)→+∞as |x|→∞for a.e. t∈E.Then the system (OPS) has at least one solution in W_T^1,p.Theorem 4 Suppose that F satisfies assumptions (A), (F₃) and (F₅). Assume that(F₆) F(t,·) is (β,γ)-subconvex withγ＞0 for a.e. t∈[0, T], that is, F(t,β(x+y))≤γ(F(t, x)+F(t, y)) for all x,y∈R^N.Then the system (OPS) has at least one solution in W_T^1,p.