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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.

34C25 Periodic solutions to ordinary differential equations
34B16 Singular nonlinear boundary value problems for ordinary differential equations
Full Text: DOI
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