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. Agarwal1 , Donal O'Regan2 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

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.