Abstract and Applied Analysis
Volume 2003 (2003), Issue 14, Pages 813-821

Focal decompositions for linear differential equations of the second order

L. Birbrair,1,2 M. Sobolevsky,3 and P. Sobolevskii4

1Departamento de Algebra, Geometria y Topologia, Universidad de Valladolid, Valladolid 47005, Spain
2Departamento de Matematica, Universidade Federal do Ceará, Fortaleza Cep. 60155-760, Brazil
3Departamento de Matematica, Universidade Estadual do Ceará, Av. Paranjana, 1700, Fortaleza Cep. 60740-000, Brazil
4Institute of Mathematics, The Hebrew University, Givat Ram, Jerusalem 91904, Israel

Received 10 November 2002

Copyright © 2003 L. Birbrair et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Focal decomposition associated to an ordinary differential equation of the second order is a partition of the set of all two-points boundary value problems according to the number of their solutions. Two equations are called focally equivalent if there exists a homomorphism of the set of two-points problems to itself such that the image of the focal decomposition associated to the first equation is a focal decomposition associated to the second one. In this paper, we present a complete classification for linear second-order equations with respect to this equivalence relation.