Abstract
We investigate the existence and multiplicity of solutions for second-order Hamiltonian systems satisfying generalized periodic boundary value conditions at resonance by means of the index theory, the critical point theory without compactness assumptions, the least action principle, the saddle point reduction theorem, and the minimax method. Applying the results to second-order HS satisfying periodic boundary value conditions, we obtain some new results.
Highlights
Introduction and main resultsSolutions of Hamiltonian systems are very important in applications
The existence and multiplicity of solutions for Hamiltonian systems via critical point theory have been studied by many authors
By means of critical point theory, the least action principle, and the minimax method, the existence and multiplicity of periodic solutions for second-order Hamiltonian systems with periodic boundary conditions were extensively studied in the cases where the gradient of the nonlinearity is bounded sublinearly and linearly, and many interesting results are given in [5, 9, 10, 13,14,15,16,17,18,19, 22]
Summary
Solutions of Hamiltonian systems are very important in applications. In recent years, the existence and multiplicity of solutions for Hamiltonian systems via critical point theory have been studied by many authors (see [2, 5,6,7,8,9,10, 12,13,14,15,16,17,18,19,20,21,22]). Theorem 1.2 Assume that V (t, x) satisfies (A), (A1), and (A4) there exist f , g ∈ L1([0, 1], R+) with νMs (B1 + f (t)In) = 0 and isM(B1 + f (t)In) = isM(B1) + νMs (B1) such that. Corollary 1.9 Assume that V (t, x) satisfies (A), (A4), and (H5) there exists c0 > 0 large enough such that lim inf |x|–2α. Remark 1.10 As T = 1, in Theorems 1–3 of [14] assume that V (t, x) satisfies (A), (H4), and (A4) there exist α ∈ [0, 1) and f , g ∈ L1([0, 1], R+) such that for all x ∈ Rn and a.e. t ∈ [0, 1]; (H5,1). Corollary 1.12 Assume that V (t, x) satisfies (A), (A4), and (H5) there exists c0 > 0 large enough such that (1.3) or (1.4) hold for x ∈ ker(Λ – (2kπ)2)
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