Boundary Value Problems
Volume 2009 (2009), Article ID 905769, 28 pages
doi:10.1155/2009/905769

Limit properties of solutions of singular second-order differential equations

Irena Rachůnková1 , Svatoslav Staněk1 , Ewa Weinmüller3 and Michael Zenz3

1Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, 779 00 Olomouc, Czech Republic
 3Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8-10, 1040 Wien, Austria

Abstract

We discuss the properties of the differential equation u′′(t)=(a/t)u′(t)+f(t,u(t),u′(t)), a.e. on (0,T], where a∈ℝ\{0}, and f satisfies the Lp-Carathéodory conditions on [0,T]×ℝ2 for some p>1. A full description of the asymptotic behavior for t→0+ of functions u satisfying the equation a.e. on (0,T] is given. We also describe the structure of boundary conditions which are necessary and sufficient for u to be at least in C1[0,T]. As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.