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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, $u''(t)=\frac{a}{t}u'(t)+\lambda f(t,u(t),u'(t))$ subject to periodic boundary conditions, where $$a>0$$ is a given constant, $$\lambda >0$$ is a parameter, and the nonlinearity $$f(t,x,y)$$ satisfies the local Carathéodory conditions on $$[0,T]\times \mathbb{R}\times \mathbb{R}$$. 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 to ordinary differential equations 34B16 Singular nonlinear boundary value problems for ordinary differential equations
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