Abstract and Applied Analysis
Volume 2011 (2011), Article ID 826451, 12 pages
http://dx.doi.org/10.1155/2011/826451
Research Article

Existence of Positive Solutions for a Fourth-Order Periodic Boundary Value Problem

Yongxiang Li

Department of Mathematics, Northwest Normal University, Lanzhou 730070, China

Received 10 April 2011; Revised 7 July 2011; Accepted 14 July 2011

Academic Editor: Ferhan M. Atici

Copyright © 2011 Yongxiang Li. 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

The existence results of positive solutions are obtained for the fourth-order periodic boundary value problem 𝑢 ( 4 ) − 𝛽 𝑢 + 𝛼 𝑢 = 𝑓 ( 𝑡 , 𝑢 , 𝑢 ) , 0 ≤ 𝑡 ≤ 1 , 𝑢 ( 𝑖 ) ( 0 ) = 𝑢 ( 𝑖 ) ( 1 ) ,   𝑖 = 0 , 1 , 2 , 3 , where 𝑓 ∶ [ 0 , 1 ] × ℝ + × ℝ → ℝ + is continuous, 𝛼 , 𝛽 ∈ ℝ , and satisfy 0 < 𝛼 < ( ( 𝛽 / 2 ) + 2 𝜋 2 ) 2 , 𝛽 > − 2 𝜋 2 , ( 𝛼 / 𝜋 4 ) + ( 𝛽 / 𝜋 2 ) + 1 > 0 . The discussion is based on the fixed point index theory in cones.