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

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.

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