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Zbl 1231.34072
Feichtinger, Anna; Rachuunková, Irena; Staněk, Svatoslav; Weinmüller, Ewa
Periodic BVPs in ODEs with time singularities. (English)
[J] Comput. Math. Appl. 62, No. 4, 2058-2070 (2011). ISSN 0898-1221

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 $[0,T]\times \bbfR\times \bbfR$. 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 2000:: *34C25 Periodic solutions of ODE
34B16 Singular nonlinear boundary value problems

Keywords: singular boundary value problems; periodic boundary conditions; time singularity of the first kind; lower and upper functions; opposite order; collocation methods

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