Electron. J. Diff. Eqns., Vol. 2008(2008), No. 47, pp. 1-15.

Another understanding of fourth-order four-point boundary-value problems

Petio Kelevedjiev, Panos K. Palamides, Nedyu Popivanov

In this article we investigate the existence of positive and/or negative solutions of a classes of four-point boundary-value problems for fourth-order ordinary differential equations. The assumptions in this article are more relaxed than the known assumptions. Our technique relies on the continuum property (connectedness and compactness) of the solutions funnel (Knesser's Theorem), combined with the corresponding vector field's ones. This approach permits the extension of results (getting positive solutions) to nonlinear boundary conditions, whenever the corresponding Green's kernel is not of definite sign or there does not exist (see the last Corollary).

Submitted February 5, 2008. Published March 30, 2008.
Math Subject Classifications: 34B15, 34B25.
Key Words: Multipoint boundary value problem; positive solution; vector field; third order differential equation; Green function; Krasnoselskii's fixed point theorem.

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Petio S. Kelevedjiev
Department of Mathematics
Technical University of Sliven
8800 Sliven, Bulgaria
e-mail: keleved@mailcity.com
Panos K. Palamides
Naval Academy of Greece
Piraeus, 451 10, Greece
email: ppalam@otenet.gr   ppalam@snd.edu.gr
Nedyu I. Popivanov
Faculty of Mathematics and Informatics
"St. Kl. Ohridski" University of Sofia
1164 Sofia, Bulgaria
email: nedyu@fmi.uni-sofia.bg

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