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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)=a tu ' (t)+λf(t,u(t),u ' (t))

subject to periodic boundary conditions, where a>0 is a given constant, λ>0 is a parameter, and the nonlinearity f(t,x,y) satisfies the local Carathéodory conditions on [0,T]××. 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:
34C25Periodic solutions of ODE
34B16Singular nonlinear boundary value problems for ODE
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