### Existence of non-collision periodic solutions for second order singular dynamical systems

#### Abstract

In this paper, we study the existence of non-collision

periodic solutions for the second order singular dynamical systems.

We consider the systems where the potential have a repulsive or

attractive type behavior near the singularity. The proof is based on

Schauder's fixed point theorem involving a new type of cone. The

so-called strong force condition is not needed and the nonlinearity

could have sign changing behavior. We allow that the Green function

is non-negative, so the critical case for the repulsive case is

covered. Recent results in the literature are generalized and

improved.

periodic solutions for the second order singular dynamical systems.

We consider the systems where the potential have a repulsive or

attractive type behavior near the singularity. The proof is based on

Schauder's fixed point theorem involving a new type of cone. The

so-called strong force condition is not needed and the nonlinearity

could have sign changing behavior. We allow that the Green function

is non-negative, so the critical case for the repulsive case is

covered. Recent results in the literature are generalized and

improved.

#### Keywords

Second order dynamical system; non-collision periodic solution; strong singularity; weak singularity; Schauder's fixed point theorem

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