Solutions for \(m\)-point BVP with sign changing nonlinearity (Q1040182)

Summary: We study the existence of positive solutions for the following nonlinear \(m\)-point boundary value problem for an increasing homeomorphism sign changing nonlinearity: \[ (\phi(u'(t)))'+a(t)f(t,u(t))=0,\quad 0<t<1, \] \[ u'(0)=\sum^{m-2}_{i=1}a_iu'(\xi_i),\;u(1)=\sum_{i=1}^k b_iu(\xi_i)-\sum^s_{i=k+1}b_iu(\xi_i)-\sum^{m-2}_{i=s+1}b_iu'(\xi_i), \] where \(\phi:R\to R\) is an increasing homeomorphism and \(\phi(0)=0\). The nonlinear term \(f\) may change sign. As an application, an example to demonstrate our results is given. The conclusions in this paper essentially extend and improve the known results.