Singular mixed boundary value problem

Abstract

We study singular boundary value problems with mixed boundary conditions of the form u ″ + f ( t , u , u ′ ) = 0 , u ′ ( 0 ) = 0 , u ( T ) = 0 , where [ 0 , T ] ⊂ R , D = ( 0 , ∞ ) × ( − ∞ , 0 ) , f is a nonnegative function and satisfies the Carathéodory conditions on ( 0 , T ) × D . Here, f can have a time singularity at t = 0 and/or t = T and a space singularity at x = 0 and/or y = 0 . We present conditions for the existence of solutions positive on [ 0 , T ) and having continuous first derivatives on [ 0 , T ] .