Electron. J. Diff. Equ., Vol. 2014 (2014), No. 175, pp. 1-10.

Three-point third-order problems with a sign-changing nonlinear term

Johnny Henderson, Nickolai Kosmatov

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.

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Johnny Henderson
Department of Mathematics
Baylor University
Waco, TX 76798-7328, USA
email: Johnny_Henderson@baylor.edu
Nickolai Kosmatov
Department of Mathematics and Statistics
University of Arkansas at Little Rock
Little Rock, AR 72204-1099, USA
email: nxkosmatov@ualr.edu

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