International Journal of Differential Equations
Volume 2011 (2011), Article ID 261963, 16 pages
http://dx.doi.org/10.1155/2011/261963
Research Article

On a Constructive Approach for Derivative-Dependent Singular Boundary Value Problems

R. K. Pandey1 and Amit K. Verma2

1Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, India
2Department of Mathematics, BITS Pilani, Rajasthan, Pilani 333031, India

Received 20 May 2011; Revised 24 August 2011; Accepted 13 September 2011

Academic Editor: Alberto Cabada

Copyright © 2011 R. K. Pandey and Amit K. 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 present a constructive approach to establish existence and uniqueness of solution of singular boundary value problem − ( 𝑝 ( 𝑥 ) 𝑦 ( 𝑥 ) ) = 𝑞 ( 𝑥 ) 𝑓 ( 𝑥 , 𝑦 , 𝑝 𝑦 ) for 0 < 𝑥 ≤ 𝑏 , 𝑦 ( 0 ) = 𝑎 , 𝛼 1 𝑦 ( 𝑏 ) + 𝛽 1 𝑝 ( 𝑏 ) 𝑦 ( 𝑏 ) = 𝛾 1 . Here 𝑝 ( 𝑥 ) > 0 on ( 0 , 𝑏 ) allowing 𝑝 ( 0 ) = 0 . Further 𝑞 ( 𝑥 ) may be allowed to have integrable discontinuity at 𝑥 = 0 , so the problem may be doubly singular.