Electron. J. Diff. Eqns., Vol. 2007(2007), No. 151, pp. 1-13.

A singular third-order 3-point boundary-value problem with nonpositive Green's function

Alex P. Palamides, Anastasia N. Veloni

Abstract:
We find a Green's function for the singular third-order three-point BVP
$$ u'''(t)=-a(t)f(t,u(t)),\quad u(0)=u'(1)= u''(\eta )=0 $$
where $0\leq \eta <1/2$. Then we apply the classical Krasnosel'skii's fixed point theorem for finding solutions in a cone. Although this problem Green's function is not positive, the obtained solution is still positive and increasing. Our techniques rely on a combination of a fixed point theorem and the properties of the corresponding vector field.

Submitted October 11, 2007. Published November 13, 2007.
Math Subject Classifications: 34B15, 34B18, 34B10, 34B16.
Key Words: Three-point singular boundary-value problem; fixed point in cones; third-order differential equation; positive solution; Green's function; vector field.

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Alex P. Palamides
Department of Telecommunications Science and Technology
University of Peloponnese
Karaiskaki Str., 22100 Tripolis, Greece
email: palamid@uop.gr
Anastasia N. Veloni
Technological Education Institute of Piraeus
Department of Electronic Computer Systems
P. Ralli Ave. & Thivon Ave. 250
Aigaleo 12244, Athens, Greece
email: aveloni@teipir.gr

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