A necessary and sufficient condition for positive solutions for fourth-order multi-point boundary value problems with p-Laplacian

AbstractInthispaper,weconsidertheexistenceofpositivesolutionsforaclassofsingularfourth-order p-Laplacianequationwithmulti-point boundary conditions. By means of a monotone iterative technique, a necessary and sufficient condition for the existence of apseudo-C 3 [0,1] positive solution is established. Also, the existence and uniqueness of the pseudo-C 3 [0,1] positive solution, theiterative sequence of the solution and an error estimation for the pseudo-C 3 [0,1] positive solution are obtained.c 2007 Elsevier Ltd. All rights reserved. MSC: 34B15; 34B25Keywords: Necessary and sufficient condition; p-Laplacian; Pseudo-C 3 [0,1] positive solutions; Cone 1. IntroductionIn this paper, we consider the existence of positive solutions for the singular fourth-order multi-point boundaryvalue problem with p-Laplacian[ϕ p (u 00 t))] 00 = f ,)), 0 < 1u(0) = m X −2i=1 a i u(ξ ), u(1) = 0, u 00 (0) = m X −2i=1 b u 00 (η ), u 00 (1) = 0,(1.1)whereϕ p (t) = |t| p−2 t, p ≥ 2,0 < a i ,b i ,ξ i ,η

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