Multiple sign-changing solutions for fourth order \(m\)-point boundary value problems (Q866525)

The existence and multiplicity of nontrivial solutions for fourth order \(m\)-point boundary value problems \[ x^{(4)}(t)=f(x(t),-x''(t)) , \qquad t \in (0,1)\,, \] \[ x(0)= 0\,, \quad x(1)=\sum^{m-2}_{i=1}a_{i} x(\eta_i)\,, \] \[ \; \quad x''(0)= 0\,, \quad x''(1)=\sum^{m-2}_{i=1}a_{i} x''(\eta_i)\,, \] is studied, where \(0< a_{i}<1,\, \, \sum^{m-2}_{i=1}a_{i}<1\,, \) \(\,0<\eta_{1} <\eta_{2}< \dots <\eta_{m-2}<1\,,\,m\geq 3\) and \(\,f \,\) is a continuous function. By means of the theory of fixed point index in a cone and Leray-Schauder degree and under general conditions on the nonlinearity, it is proved that there exist at least six different nontrivial solutions. (If \(f\) is odd, then there exist at least eight different nontrivial solutions.)