General existence principles for nonlocal boundary value problems with <mml:math alttext="$\PHI$" id="E1" xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>φ</mml:mi></mml:math>-Laplacian and their applications

Agarwal, Ravi P.; O'Regan, Donal; Stanek, Svatoslav

doi:https://doi.org/10.1155/AAA/2006/96826

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Volume 2006 | Article ID 096826 | https://doi.org/10.1155/AAA/2006/96826

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

Ravi P. Agarwal,¹Donal O'Regan,²and Svatoslav Stanek³

Received01 Apr 2005

Accepted12 May 2005

Published20 Feb 2006

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.

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Copyright

Copyright © 2006 Ravi P. Agarwal 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.

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