Periodic BVPs in ODEs with time singularities

Abstract In this paper, we show the existence of solutions to a nonlinear singular second order ordinary differential equation, u ″ ( t ) = a t u ′ ( t ) + λ f ( 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 Caratheodory conditions on [ 0 , T ] × 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.

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