Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 151, pp. 1-13.
Title: A singular third-order 3-point boundary-value problem with
nonpositive Green's function
Authors: Alex P. Palamides (Univ. of Peloponesse, Tripolis, Greece)
Anastasia N. Veloni (Tech. Education Inst. of Piraeus, Greece)
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.