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Periodic BVPs in ODEs with time singularities. (English) Zbl 1231.34072

Summary: We show the existence of solutions to a nonlinear singular second order ordinary differential equation,

${u}^{\text{'}\text{'}}\left(t\right)=\frac{a}{t}{u}^{\text{'}}\left(t\right)+\lambda f\left(t,u\left(t\right),{u}^{\text{'}}\left(t\right)\right)$

subject to periodic boundary conditions, where $a>0$ is a given constant, $\lambda >0$ is a parameter, and the nonlinearity $f\left(t,x,y\right)$ satisfies the local Carathéodory conditions on $\left[0,T\right]×ℝ×ℝ$. Here, we study the case that a well-ordered pair of lower and upper functions does not exist and therefore the underlying problem cannot be treated by well-known standard techniques. Instead, we assume the existence of constant lower and upper functions having opposite order. Analytical results are illustrated by means of numerical experiments.

##### MSC:
 34C25 Periodic solutions of ODE 34B16 Singular nonlinear boundary value problems for ODE