A Criterion for the Existence of the Unique Periodic Solution of One-Dimensional Periodic Differential Equation

In this paper, we discuss one-dimensional differential equation with ω -period. By using the fixed point theory, the existence of a periodic solution is obtained; by using the second Lyapunov method, the uniqueness and stability of the periodic solution are obtained.

Center Conditions for a Class of Rigid Quintic Systems

The problem of determining necessary and sufficient conditions on P and Q for system. x = - y+P(x+y),y = x+Q(x+y) to have a center at the origin is known as the Poincaré center-focus problem. So far, people has tried many ways to solve the problem of central focus. However, it is difficult to solve the center focus problem of higher order polynomial system. In this paper, we use the Poincaré and Alwash-Lloyd methods to study the center focus problem and derive the center conditions of the five periodic differential equation.

Periodic solutions for indefinite singular perturbations of the relativistic acceleration

Using the Leray–Schauder degree, we study the existence of solutions for the following periodic differential equation with relativistic acceleration and singular nonlinearity:where μ > 1 and the weight h: [0, T] → ℝ is a continuous sign-changing function. There are no a priori estimates on the set of positive solutions (a condition used in general to apply the Leray–Schauder degree), and we prove that no solution of the equation appears on the boundary of an unbounded open set during the deformation to an autonomous problem.