This paper investigates the existence of solutions to subquadratic operator equations with convex nonlinearities and resonance by means of the index theory for self-adjoint linear operators developed by Dong and dual least action principle developed by Clarke and Ekeland. Applying the results to subquadratic convex Hamiltonian systems satisfying several boundary value conditions including Bolza boundary value conditions, generalized periodic boundary value conditions and Sturm–Liouville boundary value conditions yield some new theorems concerning the existence of solutions or nontrivial solutions. In particular, some famous results about solutions to subquadratic convex Hamiltonian systems by Mawhin and Willem and Ekeland are special cases of the theorems.

本文利用Dong提出的自伴线性算子的指数理论以及Clarke和Ekeland提出的对偶最小作用原理，研究了具有凸非线性和共振的次二次算子方程解的存在性。将结果应用到满足几个边界值条件（包括Bolza边界值条件，广义周期边界值条件和Sturm-Liouville边界值条件）的次二次凸哈密顿系统，得出了有关解或非平凡解存在性的一些新定理。特别是，有关Mawhin和Willem和Ekeland的次二次凸哈密顿系统的解的一些著名结果是这些定理的特例。