Electron. J. Diff. Equ., Vol. 2011 (2011), No. 32, pp. 1-29.

Multiple positive solutions for singular multi-point boundary-value problem with general growth on the positive half line

Smail Djebali, Karima Mebarki

This work is devoted to the existence of nontrivial positive solutions for a class of second-order nonlinear multi-point boundary-value problems on the positive half-line. The novelty of this work is that the nonlinearity may exhibit a singularity at the origin simultaneously with respect to the solution and its derivative; moreover it satisfies quite general growth conditions far from the origin, including polynomial growth. New existence results of single, twin and triple solutions are proved using the fixed point index theory on appropriate cones in weighted Banach spaces together with two-functional and three-functional fixed point theorems. The singularity is treated by means of approximation and compactness arguments. The proofs of the existence results rely heavily on several sharp estimates and useful properties of the corresponding Green's function.

Submitted August 26, 2010. Published February 23, 2011.
Math Subject Classifications: 34B15, 34B16, 34B18, 34B40.
Key Words: Fixed point; multiple solutions; multi-point; singularity; infinite interval; cone.

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Smaïl Djebali
Department of Mathematics, E.K.S.,
PO Box 92, 16050 Kouba. Algiers, Algeria
email: djebali@ens-kouba.dz, djebali@hotmail.com
Karima Mebarki
Department of Mathematics
A.E. Mira University, 06000. Bejaia, Algeria
email: mebarqi@hotmail.fr

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