Electronic Journal of Differential Equations,
Vol. 2014 (2014), No. 175, pp. 1-10.
Title: Three-point third-order problems with a sign-changing nonlinear term
Authors: Johnny Henderson (Baylor Univ., Waco, TX, USA)
Nickolai Kosmatov (Univ. of Arkansas, Little Rock, AR, USA)
Abstract:
In this article we study a well-known boundary value problem
$$\displaylines{
u'''(t) = f(t, u(t)), \quad 0 < t < 1, \cr
u(0) = u'(1/2) = u''(1)=0.
}$$
With $u'(\eta)=0$ in place of $u'(1/2)=0$, many authors studied the existence
of positive solutions of both the positone problems with $\eta \geq 1/2$ and
the semi-positone problems for $\eta > 1/2$. It is well-known that the standard
method successfully applied to the semi-positone problem with $\eta > 1/2$
does not work for $\eta =1/2$ in the same setting. We treat the latter as a
problem with a sign-changing term rather than a semi-positone problem.
We apply Krasnosel'skii's fixed point theorem [4] to obtain positive solutions.
Submitted June 10, 2014. Published August 14, 2014.
Math Subject Classifications: 34B15, 34B16, 34B18.
Key Words: Green's function; fixed point theorem; positive solutions;
third-order boundary-value problem.