## On a third-order three-point boundary value problem at resonance.(English)Zbl 0722.34014

The problems $$\mp (u'''+\pi^ 2u')+g(x,u,u',u'')=e(x)$$, $$u'(0)=u'(1)=u(\eta)=0$$, $$0\leq \eta \leq 1$$ are reduced to $$\mp (y''+\pi^ 2y)+g(x,\int^{x}_{\eta}y(t)dt,y(x),y'(x))=e(x)$$, $$y(0)=y(1)=0$$, $$u(x)=\int^{x}_{\eta}y(t)dt$$ and then, under some assumptions, solved by the Mawhin continuation theorem. The existence, uniqueness theorems and theorems on the difference of two solutions are proved.

### MSC:

 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34K10 Boundary value problems for functional-differential equations 45J05 Integro-ordinary differential equations