The authors study, by means of upper and lower solutions and by topological degree arguments, existence of solutions of some boundary value problem (BVP) of the type \[ -x''=f(t,x,x')-r\phi(t), \] subject to the three-point boundary conditions \[ x(0)=0,\;x(1)=\delta x(\eta), \] where \(r\) is a parameter, \(\eta\in (0,1)\), \(0<\delta\eta<1\) and both \(f\) and \(\phi\) are continuous functions.
The authors investigate how the number of solutions changes as the parameter \(r\) varies. They prove that there exist no solutions and there exist two solutions of this BVP for certain values of the parameter \(r\). The authors also provide a class of examples to better illustrate their theory.