Premium
The existence of positive solutions to a non‐local singular boundary value problem
Author(s) -
O'Regan Donal,
Staněk Svatoslav
Publication year - 2005
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.676
Subject(s) - mathematics , regularization (linguistics) , boundary value problem , singular solution , mathematical analysis , singular value , pure mathematics , boundary values , value (mathematics) , combinatorics , eigenvalues and eigenvectors , physics , quantum mechanics , artificial intelligence , computer science , statistics
Abstract We consider the non‐local singular boundary value problem$$x\prime\prime=f(x)-\mu {{q(t)x\prime(0)} \over {\int_{0}^{1}q(s) h(x(s)){\rm d}s}} h(x), \quad x(0)= x\prime(1)=0$$where q ∈ C 0 ([0,1]) and f , h ∈ C 0 ((0,∞)), lim x →0 +f ( x )=−∞, lim x →0 +h ( x )=∞. We present conditions guaranteeing the existence of a solution x ∈ C 1 ([0,1]) ∩ C 2 ((0,1]) which is positive on (0,1]. The proof of the existence result is based on regularization and sequential techniques and on a non‐linear alternative of Leray–Schauder type. Copyright © 2005 John Wiley & Sons, Ltd.