Abstract and Applied Analysis 
Volume 2006 (2006), Article ID 96826, 30 pages
doi:10.1155/AAA/2006/96826

General existence principles for nonlocal boundary value problems with φ-Laplacian and their applications

Ravi P. Agarwal,1 Donal O'Regan,2 and Svatoslav Staněk3

 1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne 32901-6975, Florida, USA
2Department of Mathematics, National University of Ireland, Galway, Ireland
3Department of Mathematical Analysis, Faculty of Science, Palacký University, Tomkova 40, Olomouc 779 00, Czech Republic
Received 1 April 2005; Accepted 12 May 2005

Abstract

The paper presents general existence principles which can be used for a large class of nonlocal boundary value problems of the form (φ(x′))′=f1(t,x,x′)+f2(t,x,x′)F1x+f3(t,x,x′)F2x,α(x)=0, β(x)=0, where fj satisfy local Carathéodory conditions on some [0,T]×𝒟j⊂ℝ2, fj are either regular or have singularities in their phase variables (j=1,2,3), Fi:C1[0,T]→C0[0,T](i=1,2), and α,β:C1[0,T]→ℝ are continuous. The proofs are based on the Leray-Schauder degree theory and use regularization and sequential techniques. Applications of general existence principles to singular BVPs are given.