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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 489353, 12 pages
doi:10.1155/2012/489353
Research Article

Monotonic Positive Solutions of Nonlocal Boundary Value Problems for a Second-Order Functional Differential Equation

A. M. A. El-Sayed,1 E. M. Hamdallah,1 and Kh. W. El-kadeky2

1Faculty of Science, Alexandria University, Alexandria, Egypt
2Faculty of Science, Garyounis University, Benghazi, Libya

Received 13 October 2011; Accepted 5 December 2011

Academic Editor: István Györi

Copyright © 2012 A. M. A. El-Sayed 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.

Abstract

We study the existence of at least one monotonic positive solution for the nonlocal boundary value problem of the second-order functional differential equation ๐‘ฅ ๎…ž ๎…ž ( ๐‘ก ) = ๐‘“ ( ๐‘ก , ๐‘ฅ ( ๐œ™ ( ๐‘ก ) ) ) , ๐‘ก โˆˆ ( 0 , 1 ) , with the nonlocal condition โˆ‘ ๐‘š ๐‘˜ = 1 ๐‘Ž ๐‘˜ ๐‘ฅ ( ๐œ ๐‘˜ ) = ๐‘ฅ 0 , ๐‘ฅ ๎…ž โˆ‘ ( 0 ) + ๐‘› ๐‘— = 1 ๐‘ ๐‘— ๐‘ฅ ๎…ž ( ๐œ‚ ๐‘— ) = ๐‘ฅ 1 , where ๐œ ๐‘˜ โˆˆ ( ๐‘Ž , ๐‘‘ ) โŠ‚ ( 0 , 1 ) , ๐œ‚ ๐‘— โˆˆ ( ๐‘ , ๐‘’ ) โŠ‚ ( 0 , 1 ) , and ๐‘ฅ 0 , ๐‘ฅ 1 > 0 . As an application the integral and the nonlocal conditions โˆซ ๐‘‘ ๐‘Ž ๐‘ฅ ( ๐‘ก ) ๐‘‘ ๐‘ก = ๐‘ฅ 0 , ๐‘ฅ ๎…ž ( 0 ) + ๐‘ฅ ( ๐‘’ ) โˆ’ ๐‘ฅ ( ๐‘ ) = ๐‘ฅ 1 will be considered.