Existence for Singular Periodic Problems: A Survey of Recent Results

Abstract

We present a survey on the existence of periodic solutions of singular differential equations. In particular, we pay our attention to singular scalar differential equations, singular damped differential equations, singular impulsive differential equations, and singular differential systems.

Periodic solutions of a class of indefinite singular differential equations

“…During the past few decades, scholars mainly focused on the periodic problem of the equations with definite singularities, either attractive type or repulsive type, and many excellent results have been obtained. We just refer the reader to the literature [5] and the monographs [20,22] for the surveys on this problem. When nonlinearities with indefinite singularities are taken into account, the periodic problem becomes slightly more tricky and still remains insufficiently investigated.…”

A periodic boundary value problem with switchingsunder nonlinear perturbations

“…Perturbation theory is proved to be a powerful tool in the qualitative and quantitative study of periodic solutions to the Hill, Mathieu, van der Pol equations, and many other important mathematical models in nonlinear mechanics and physics (see, e.g., [3]). Without pretending to be complete, we also mention the existence results concerning periodic boundary value problems for nonlinear differential equations with singularities [4,5], second-order non-autonomous nonlinear equations on the positive cone [6], superlinear second-order equations with positive solutions [7], first-order problem with resonance and nonlinear impulses [8], Hamiltonian systems with nonsmooth potentials [9], and second-order equations with a convection term [10].…”

Affine-Periodic Solutions for Dissipative Systems

“…The related topics had ever captured the main field in periodic solutions theory from the 1960s to the 1990s. For some litratures, see, for example, [1][2][3][4][5][6][7][8][9][10][11][12].…”