Limit Properties of Solutions of Singular Second-Order Differential Equations. (29th June 2009) - Rachůnková, Irena; Staněk, Svatoslav; Weinmüller, Ewa; Zenz, Michael
Abstract : We discuss the properties of the differential equationu ′′ ( t ) = ( a / t ) u ′ ( t ) + f ( t, u ( t ), u ′ ( t ) ), a.e. on( 0, T ], wherea ∈ ℝ \ { 0 }, andf satisfies theL p -Carathéodory conditions on[ 0, T ] × ℝ 2 for somep > 1 . A full description of the asymptotic behavior fort → 0 + of functionsu satisfying the equation a.e. on( 0, T ] is given. We also describe the structure of boundary conditions which are necessary and sufficient foru to be at least inC 1 [ 0, T ] . As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.