International Journal of Differential Equations
Volume 2011 (2011), Article ID 986948, 13 pages
http://dx.doi.org/10.1155/2011/986948
Research Article

Nonlinear Singular BVP of Limit Circle Type and the Presence of Reverse-Ordered Upper and Lower Solutions

Amit K. Verma1 and Lajja Verma2

1Department of Mathematics, BITS Pilani, Pilani, Rajasthan 333031, India
2Department of Mathematics and Astronomy, University of Lucknow, Lucknow 226007, India

Received 27 May 2011; Accepted 3 July 2011

Academic Editor: Mohamed A. El-Gebeily

Copyright © 2011 Amit K. Verma and Lajja Verma. 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.

Abstract

We consider the following class of nonlinear singular differential equation − ( 𝑝 ( 𝑥 ) 𝑦 ′ ( 𝑥 ) ) ′ + 𝑞 ( 𝑥 ) 𝑓 ( 𝑥 , 𝑦 ( 𝑥 ) , 𝑝 ( 𝑥 ) 𝑦 ′ ( 𝑥 ) ) = 0 , 0 < 𝑥 < 1 subject to the Neumann boundary condition 𝑦 ′ ( 0 ) = 𝑦 ′ ( 1 ) = 0 . Conditions on 𝑝 ( 𝑥 ) and 𝑞 ( 𝑥 ) ensure that 𝑥 = 0 is a singular point of limit circle type. A simple approximation scheme which is iterative in nature is considered. The initial iterates are upper and lower solutions which can be ordered in one way ( 𝑣 0 ≤ 𝑢 0 ) or the other ( 𝑢 0 ≤ 𝑣 0 ) .